b_1$ where $b_1$ is given by Proposition 4.1, and let $v \\\\in C([0, \\\\frac{1}{b}), \\\\mathcal{X})$ be the solution of $(\\\\mathrm{NLS}_b)$ given by Proposition 4.1. It follows that there exist $C, \\\\delta > 0$ , $\\\\eta > \\\\frac{N}{2}$ and $f_0, \\\\omega_0 \\\\in L^{\\\\infty}(\\\\mathbb{R}^N) \\\\cap C(\\\\mathbb{R}^N)$ with $\\\\| f_0 \\\\|_{L^{\\\\infty}} \\\\leq \\\\frac{1}{2}$ and $\\\\langle \\\\cdot \\\\rangle^n \\\\omega_0 \\\\in L^{\\\\infty}(\\\\mathbb{R}^N)$ such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[257,838,928,861]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left\\\\| \\\\langle \\\\cdot \\\\rangle^ {\\\\eta} [ v (t, \\\\cdot) - \\\\omega_ {0} (\\\\cdot) \\\\psi (t, \\\\cdot) e ^ {- i \\\\theta (t, \\\\cdot)} ] \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\leq C (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2 \\\\alpha} - \\\\delta}, \\\\tag {5.1}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[64,871,262,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for all $0 \\\\leq t < \\\\frac{1}{b}$ , where \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[246,898,928,944]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\psi (t, x) = \\\\left(\\\\frac {1 + f _ {0} (x)}{1 + f _ {0} (x) + \\\\frac {2 \\\\alpha | \\\\Im \\\\lambda |}{b (2 - N \\\\alpha)} | v _ {0} (x) | ^ {\\\\alpha} [ (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2}} - 1 ]}\\\\right) ^ {\\\\frac {1}{\\\\alpha}} \\\\tag {5.2}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[68,44,334,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"T. Cazenave, Z. Han and I. Naumkin \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[661,44,928,57]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Nonlinear Analysis 205 (2021) 112243 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[488,947,507,958]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[66,88,104,102]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[400,98,928,130]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\theta (t, x) = \\\\frac {\\\\Re \\\\lambda}{\\\\Im \\\\lambda} \\\\log \\\\psi (t, x). \\\\tag {5.3}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,133,152,148]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moreover, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[433,144,928,180]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left| \\\\omega_ {0} \\\\right| ^ {\\\\alpha} = \\\\frac {\\\\left| v _ {0} \\\\right| ^ {\\\\alpha}}{1 + f _ {0}}. \\\\tag {5.4}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,184,167,199]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In addition, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[352,195,928,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n(1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2}} \\\\| v \\\\| _ {L ^ {\\\\infty}} ^ {\\\\alpha} \\\\xrightarrow [ t \\\\uparrow \\\\frac {1}{b} ]{b (2 - N \\\\alpha)} \\\\frac {b (2 - N \\\\alpha)}{2 \\\\alpha | \\\\Im \\\\lambda |} \\\\tag {5.5}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,238,514,253]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and there exist two constants $0 < a \\\\leq A < \\\\infty$ such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[257,262,928,285]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\na (1 - b t) ^ {\\\\left(\\\\frac {1}{\\\\alpha} - \\\\frac {N}{2}\\\\right) \\\\left(1 - \\\\frac {N}{2 n}\\\\right)} \\\\leq \\\\| v (t) \\\\| _ {L ^ {2}} \\\\leq A (1 - b t) ^ {\\\\left(\\\\frac {1}{\\\\alpha} - \\\\frac {N}{2}\\\\right) \\\\left(1 - \\\\frac {N}{2 n}\\\\right)}, \\\\tag {5.6}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[64,297,209,316]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for all $0 \\\\leq t < \\\\frac{1}{b}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[64,334,863,350]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof. Using the a priori estimates of Proposition 4.1, we follow the proof of [2, Proposition 4.1]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[64,351,928,385]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We recall that (4.2) and (4.3) hold. Also, since $b_{1} \\\\geq b_{0}$ of Proposition 3.2, $v$ satisfies (3.32) and (3.22). In particular \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[336,382,928,405]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n| v (t, x) | \\\\leq C \\\\min \\\\left\\\\{\\\\langle x \\\\rangle^ {- n}, (1 - b t) ^ {\\\\frac {2 - N \\\\alpha}{2 \\\\alpha}} \\\\right\\\\}, \\\\tag {5.7}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[64,411,930,448]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for all $x \\\\in \\\\mathbb{R}^N$ and $\\\\frac{1}{2b} \\\\leq t < \\\\frac{1}{b}$ . We let $f \\\\in C((0, \\\\frac{1}{b}) \\\\times \\\\mathbb{R}^N)$ be defined by (3.34). It follows from (4.2), (4.3), (3.37) with $K_1$ replaced by $5\\\\widetilde{K}$ , and (3.38) that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[183,456,812,491]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\| f (t, \\\\cdot) - f (s, \\\\cdot) \\\\| _ {L ^ {\\\\infty} (\\\\mathbb {R} ^ {N})} \\\\leq C \\\\int_ {t} ^ {s} (1 - b \\\\tau) ^ {- \\\\frac {2 - N \\\\alpha}{2} - 2 \\\\sigma} d \\\\tau \\\\leq C (1 - b t) ^ {1 - \\\\frac {2 - N \\\\alpha}{2} - 2 \\\\sigma},\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[64,500,928,536]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for all $0 \\\\leq t < s < \\\\frac{1}{b}$ . Thus we see that $f(t, \\\\cdot)$ is convergent in $L^{\\\\infty}(\\\\mathbb{R}^{N})$ as $t \\\\uparrow \\\\frac{1}{b}$ . Then $f$ can be extended to a continuous function $[0, \\\\frac{1}{b}] \\\\to L^{\\\\infty}(\\\\mathbb{R}^{N})$ . We set \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[262,545,928,586]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nf _ {0} (x) = f \\\\left(\\\\frac {1}{b}, x\\\\right) = - \\\\alpha \\\\int_ {0} ^ {\\\\frac {1}{b}} | v _ {0} (x) | ^ {\\\\alpha} | v (s, x) | ^ {- \\\\alpha - 1} L (s, x) d s, \\\\tag {5.8}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,592,356,611]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"so that $f_0 \\\\in L^\\\\infty(\\\\mathbb{R}^N) \\\\cap C(\\\\mathbb{R}^N)$ and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[340,619,928,642]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left\\\\| f (t) - f _ {0} \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\leq C (1 - b t) ^ {1 - \\\\frac {2 - N \\\\alpha}{2} - 2 \\\\sigma}. \\\\tag {5.9}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,654,248,672]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moreover, using (4.3), \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[435,668,928,699]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left\\\\| f (t) \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\leq \\\\frac {1}{4}, \\\\tag {5.10}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,702,266,722]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for all $0 \\\\leq t \\\\leq \\\\frac{1}{b}$ , and so \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[444,718,928,749]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left\\\\| f _ {0} \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\leq \\\\frac {1}{4}. \\\\tag {5.11}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,752,409,769]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In particular, $1 + f_0 > 0$ , so that by (5.2), \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[433,781,928,800]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n0 < \\\\psi (t, x) \\\\leq 1 \\\\tag {5.12}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[64,811,696,830]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for all $0 \\\\leq t < \\\\frac{1}{b}$ and $x \\\\in \\\\mathbb{R}^N$ . Moreover, it follows from (5.10) and (5.11) that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[420,838,928,871]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\max \\\\left\\\\{\\\\frac {1}{H}, \\\\frac {1}{\\\\widetilde {H}} \\\\right\\\\} \\\\leq 2, \\\\tag {5.13}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,880,479,898]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for all $0 \\\\leq t < \\\\frac{1}{b}$ , where $H$ is defined by (3.33) and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[250,908,744,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\widetilde {H} (t, x) = 1 + f _ {0} (x) + \\\\frac {2 \\\\alpha | \\\\Im \\\\lambda |}{b (2 - N \\\\alpha)} | v _ {0} (x) | ^ {\\\\alpha} [ (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2}} - 1 ].\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[68,44,334,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"T. Cazenave, Z. Han and I. Naumkin \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[661,44,928,57]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Nonlinear Analysis 205 (2021) 112243 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[488,947,507,958]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"32 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[66,88,415,105]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note also that by (5.10), (5.11) and (3.18), \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[235,113,759,185]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\langle x \\\\rangle^ {n \\\\alpha} \\\\min \\\\{\\\\widetilde {H}, H \\\\} \\\\geq \\\\frac {2 \\\\alpha | \\\\Im \\\\lambda |}{b (2 - N \\\\alpha)} (\\\\langle x \\\\rangle^ {n} | v _ {0} (x) |) ^ {\\\\alpha} [ (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2}} - 1 ] \\\\\\\\ \\\\geq \\\\frac {2 \\\\alpha | \\\\Im \\\\lambda |}{b (2 - N \\\\alpha) K ^ {\\\\alpha}} [ (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2}} - 1 ], \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,192,128,204]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"so that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[345,200,928,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left. \\\\max \\\\left\\\\{\\\\frac {1}{H}, \\\\frac {1}{\\\\widetilde {H}} \\\\right\\\\} \\\\leq C (1 - b t) ^ {\\\\frac {2 - N \\\\alpha}{2}} \\\\langle x \\\\rangle^ {n \\\\alpha}, \\\\right. \\\\tag {5.14}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[64,237,483,258]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for $\\\\frac{1}{2b} \\\\leq t < \\\\frac{1}{b}$ . Moreover, it follows from (5.9) that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[351,264,928,301]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left\\\\| \\\\frac {\\\\widetilde {H}}{H} - 1 \\\\right\\\\| _ {L ^ {\\\\infty}} = \\\\left\\\\| \\\\frac {f (t) - f _ {0}}{H} \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\leq C. \\\\tag {5.15}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,307,165,320]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We now set \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[402,316,928,355]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\widetilde {v} (t, x) = \\\\left(\\\\frac {\\\\left| v _ {0} (x) \\\\right| ^ {\\\\alpha}}{\\\\widetilde {H} (t , x)}\\\\right) ^ {\\\\frac {1}{\\\\alpha}}. \\\\tag {5.16}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,359,536,374]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It follows from (3.32), (5.16), (5.9), (5.13) and (5.14) that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[216,384,779,421]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left\\\\| | v (t, \\\\cdot) | ^ {\\\\alpha} - \\\\widetilde {v} (t, \\\\cdot) ^ {\\\\alpha} \\\\right\\\\| _ {L ^ {\\\\infty}} = \\\\left\\\\| (\\\\langle \\\\cdot \\\\rangle^ {n} | v _ {0} |) ^ {\\\\alpha} \\\\frac {f (t) - f _ {0}}{\\\\langle \\\\cdot \\\\rangle^ {n \\\\alpha} H \\\\widetilde {H}} \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\leq C (1 - b t) ^ {1 - 2 \\\\sigma},\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,429,102,442]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[281,448,711,510]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\left\\\\| \\\\frac {| v (t , \\\\cdot) | ^ {\\\\alpha} - \\\\widetilde {v} (t , \\\\cdot) ^ {\\\\alpha}}{\\\\langle \\\\cdot \\\\rangle^ {n \\\\alpha}} \\\\right\\\\| _ {L ^ {\\\\infty}} = \\\\left\\\\| (\\\\langle \\\\cdot \\\\rangle^ {n} | v _ {0} |) ^ {\\\\alpha} \\\\frac {f (t) - f _ {0}}{\\\\langle \\\\cdot \\\\rangle^ {2 n \\\\alpha} H \\\\widetilde {H}} \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\\\\\ \\\\leq C (1 - b t) ^ {1 - 2 \\\\sigma + \\\\frac {2 - N \\\\alpha}{2}}, \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,520,538,539]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for all $\\\\frac{2}{b} \\\\leq t < \\\\frac{1}{b}$ . Therefore, given any $0 \\\\leq \\\\rho \\\\leq 1$ , we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[297,545,928,582]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left\\\\| \\\\frac {\\\\left| v (t , \\\\cdot) \\\\right| ^ {\\\\alpha} - \\\\widetilde {v} (t , \\\\cdot) ^ {\\\\alpha}}{\\\\langle \\\\cdot \\\\rangle^ {\\\\rho n \\\\alpha}} \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\leq C (1 - b t) ^ {1 - 2 \\\\sigma + \\\\rho \\\\frac {2 - N \\\\alpha}{2}}, \\\\tag {5.17}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,591,520,610]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for all $\\\\frac{2}{b} \\\\leq t < \\\\frac{1}{b}$ . Next, we introduce the decomposition \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[371,618,928,638]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nv (t, x) = \\\\omega (t, x) \\\\psi (t, x) e ^ {- i \\\\theta (t, x)}, \\\\tag {5.18}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[64,650,928,680]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\psi (t,x)$ and $\\\\theta (t,x)$ are defined by (5.2) and (5.3) respectively. Differentiating (5.18) with respect to $t$ we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[365,679,928,712]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\ni \\\\partial_ {t} \\\\omega = i \\\\frac {e ^ {i \\\\theta}}{\\\\psi} \\\\partial_ {t} v - i \\\\omega \\\\frac {\\\\partial_ {t} \\\\psi}{\\\\psi} - \\\\omega \\\\partial_ {t} \\\\theta . \\\\tag {5.19}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,716,619,733]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On the other hand, it follows easily from (5.2), (5.3) and (5.16) that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[382,741,610,774]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\frac {\\\\partial_ {t} \\\\psi}{\\\\psi} = \\\\Im \\\\lambda (1 - b t) ^ {- \\\\frac {4 - N \\\\alpha}{2}} \\\\widetilde {v} ^ {\\\\alpha},\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[391,776,610,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\partial_ {t} \\\\theta = \\\\Re \\\\lambda (1 - b t) ^ {- \\\\frac {4 - N \\\\alpha}{2}} \\\\widetilde {v} ^ {\\\\alpha}.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,809,542,826]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Therefore, we deduce from (5.19), (5.18), and $(\\\\mathrm{NLS}_b)$ that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[299,835,928,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} i \\\\partial_ {t} \\\\omega = i \\\\frac {e ^ {i \\\\theta}}{\\\\psi} \\\\partial_ {t} v - \\\\omega (1 - b t) ^ {- \\\\frac {4 - N \\\\alpha}{2}} \\\\lambda \\\\widetilde {v} ^ {\\\\alpha} \\\\\\\\ = \\\\frac {e ^ {i \\\\theta}}{\\\\psi} (i \\\\partial_ {t} v - \\\\lambda (1 - b t) ^ {- \\\\frac {4 - N \\\\alpha}{2}} \\\\widetilde {v} ^ {\\\\alpha} v) \\\\tag {5.20} \\\\\\\\ = \\\\frac {e ^ {i \\\\theta}}{\\\\psi} (- \\\\Delta v + \\\\lambda (1 - b t) ^ {- \\\\frac {4 - N \\\\alpha}{2}} (| v | ^ {\\\\alpha} - \\\\widetilde {v} ^ {\\\\alpha}) v). \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[68,44,334,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"T. Cazenave, Z. Han and I. Naumkin \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[661,44,928,57]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Nonlinear Analysis 205 (2021) 112243 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[488,947,507,956]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"33 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[66,88,415,103]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We claim that there exists $\\\\eta > 0$ such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[450,118,926,134]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n0 \\\\leq \\\\eta \\\\leq n, \\\\tag {5.21}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[439,138,928,156]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\eta \\\\geq n (1 - \\\\alpha), \\\\tag {5.22}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[463,157,928,188]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\eta > \\\\frac {N}{2}, \\\\tag {5.23}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[382,190,928,221]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n2 \\\\sigma + \\\\frac {2 - N \\\\alpha}{2} \\\\left[ \\\\frac {\\\\eta + n \\\\alpha}{n \\\\alpha} \\\\right] < 1. \\\\tag {5.24}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[64,227,926,262]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Indeed, if $\\\\alpha \\\\geq \\\\frac{4}{5}$ , we let $\\\\eta = \\\\frac{n}{5}$ . Conditions (5.21) and (5.22) are obviously satisfied, and (5.23) follows from the first inequality in (3.1). Moreover, (5.24) is equivalent to \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[398,270,595,303]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n2 \\\\sigma < \\\\frac {N}{2} \\\\left(\\\\alpha + \\\\frac {1}{5}\\\\right) - \\\\frac {1}{5 \\\\alpha}.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[64,309,928,382]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since $\\\\alpha \\\\geq \\\\frac{4}{5}$ and $N \\\\geq 1$ , the right-hand side of the above inequality is $\\\\geq \\\\frac{1}{4}$ . Since $\\\\sigma < \\\\frac{1}{8}$ by the fourth inequality in (3.2) and (1.5) we see that (5.24) is satisfied. If $\\\\alpha < \\\\frac{4}{5}$ , we let $\\\\eta = n(1 - \\\\alpha)$ , so that (5.21) and (5.22) hold. Moreover, $\\\\eta \\\\geq \\\\frac{n}{5}$ so that (5.23) follows from the first inequality in (3.1). Furthermore, (5.24) is equivalent to $\\\\sigma < \\\\frac{(N + 2)\\\\alpha - 2}{4\\\\alpha}$ , which is a consequence of the last inequality in (3.2). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[88,382,476,398]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We fix $\\\\eta > 0$ satisfying (5.21)-(5.24) and we let \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[283,405,928,439]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\delta = 1 - \\\\frac {2 - N \\\\alpha}{2 \\\\alpha} - 2 \\\\sigma - \\\\frac {2 - N \\\\alpha}{2} \\\\left[ \\\\frac {\\\\eta - n (1 - \\\\alpha)}{n \\\\alpha} \\\\right] > 0. \\\\tag {5.25}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,446,281,460]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It follows from (5.20) that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[200,469,928,526]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\parallel \\\\langle : \\\\rangle^ {\\\\eta} \\\\partial_ {t} \\\\omega \\\\parallel_ {L ^ {\\\\infty}} \\\\\\\\ \\\\leq \\\\left\\\\| \\\\langle \\\\cdot \\\\rangle^ {n} \\\\frac {v}{\\\\psi} \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\left(\\\\left\\\\| \\\\frac {\\\\Delta v}{\\\\langle \\\\cdot \\\\rangle^ {n - \\\\eta} | v |} \\\\right\\\\| _ {L ^ {\\\\infty}} + | \\\\lambda | (1 - b t) ^ {- \\\\frac {4 - N \\\\alpha}{2}} \\\\left\\\\| \\\\frac {| v | ^ {\\\\alpha} - \\\\widetilde {v} ^ {\\\\alpha}}{\\\\langle \\\\cdot \\\\rangle^ {n - \\\\eta}} \\\\right\\\\| _ {L ^ {\\\\infty}}\\\\right). \\\\tag {5.26} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,533,531,549]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Applying (3.18) (3.32) (5.2), (5.11) and (5.15), we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[314,557,928,601]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left\\\\| \\\\langle \\\\cdot \\\\rangle^ {n} \\\\frac {v}{\\\\psi} \\\\right\\\\| _ {L ^ {\\\\infty}} = \\\\left\\\\| \\\\langle \\\\cdot \\\\rangle^ {n} v _ {0} \\\\left(\\\\frac {1}{1 + f _ {0}} \\\\frac {\\\\tilde {H}}{H}\\\\right) ^ {\\\\frac {1}{\\\\alpha}} \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\leq C. \\\\tag {5.27}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,609,314,625]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moreover, by (4.2) and (3.15), \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[314,632,928,667]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left\\\\| \\\\frac {\\\\Delta v}{\\\\langle \\\\cdot \\\\rangle^ {n - \\\\eta} | v |} \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\leq \\\\left\\\\| \\\\frac {\\\\Delta v}{| v |} \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\leq 5 \\\\widetilde {K} (1 - b t) ^ {- 2 \\\\sigma}. \\\\tag {5.28}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,676,685,695]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Furthermore, since $0 \\\\leq n - \\\\eta \\\\leq n\\\\alpha$ , it follows from (5.17) with $\\\\rho = \\\\frac{n - \\\\eta}{n\\\\alpha}$ that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[312,702,928,735]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left\\\\| \\\\frac {\\\\left| v \\\\right| ^ {\\\\alpha} - \\\\widetilde {v} ^ {\\\\alpha}}{\\\\langle \\\\cdot \\\\rangle^ {n - \\\\eta}} \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\leq C (1 - b t) ^ {1 - 2 \\\\sigma + \\\\frac {(n - \\\\eta) (2 - N \\\\alpha)}{2 n \\\\alpha}}. \\\\tag {5.29}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,745,351,761]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We deduce from (5.26)-(5.29) that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[253,769,926,818]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\left\\\\| \\\\langle \\\\cdot \\\\rangle^ {\\\\eta} \\\\partial_ {t} \\\\omega \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\leq C (1 - b t) ^ {- 2 \\\\sigma} + C (1 - b t) ^ {- 2 \\\\sigma - \\\\frac {2 - N \\\\alpha}{2} [ \\\\frac {\\\\eta - n (1 - \\\\alpha)}{n \\\\alpha} ]} \\\\tag {5.30} \\\\\\\\ \\\\leq C (1 - b t) ^ {- 2 \\\\sigma - \\\\frac {2 - N \\\\alpha}{2} [ \\\\frac {\\\\eta - n (1 - \\\\alpha)}{n \\\\alpha} ]}. \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,828,376,843]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Applying (5.24) and (5.25), we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[191,851,803,876]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\| \\\\langle \\\\cdot \\\\rangle^ {\\\\eta} (\\\\omega (t) - \\\\omega (s)) \\\\| _ {L ^ {\\\\infty}} \\\\leq C (1 - b t) ^ {1 - 2 \\\\sigma - \\\\frac {2 - N \\\\alpha}{2} [ \\\\frac {\\\\eta - n (1 - \\\\alpha)}{n \\\\alpha} ]} = C (1 - b t) ^ {\\\\frac {2 - N \\\\alpha}{2 \\\\alpha} + \\\\delta},\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[64,886,926,917]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for all $\\\\frac{1}{2b} \\\\leq t < s < \\\\frac{1}{b}$ . We conclude that there exists $\\\\omega_0 \\\\in L^{\\\\infty}(\\\\mathbb{R}^N) \\\\cap C(\\\\mathbb{R}^N)$ such that $\\\\langle x \\\\rangle^\\\\eta \\\\omega_0 \\\\in L^\\\\infty (\\\\mathbb{R}^N)$ and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[330,917,928,940]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left\\\\| \\\\langle \\\\cdot \\\\rangle^ {\\\\eta} \\\\left(\\\\omega (t) - \\\\omega_ {0}\\\\right) \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\leq C (1 - b t) ^ {\\\\frac {2 - N \\\\alpha}{2 \\\\alpha} + \\\\delta}, \\\\tag {5.31}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[68,44,334,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"T. Cazenave, Z. Han and I. Naumkin \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[661,44,928,57]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Nonlinear Analysis 205 (2021) 112243 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[488,947,507,956]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"34 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[64,87,928,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for all $\\\\frac{1}{2b} \\\\leq t < \\\\frac{1}{b}$ . Moreover, (5.18) and (5.27) imply that $\\\\| \\\\langle \\\\cdot \\\\rangle^n\\\\omega (t)\\\\|_{L^\\\\infty} \\\\leq C$ , so that $\\\\langle \\\\cdot \\\\rangle^n\\\\omega_0 \\\\in L^\\\\infty (\\\\mathbb{R}^N)$ . Applying (5.18), (5.12) and (5.31), we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[209,134,786,157]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\| \\\\langle \\\\cdot \\\\rangle^ {\\\\eta} (v (t) - \\\\omega_ {0} \\\\psi e ^ {- i \\\\theta}) \\\\| _ {L ^ {\\\\infty}} = \\\\| \\\\langle \\\\cdot \\\\rangle^ {\\\\eta} \\\\psi (\\\\omega (t) - \\\\omega_ {0}) \\\\| _ {L ^ {\\\\infty}} \\\\leq C (1 - b t) ^ {\\\\frac {2 - N \\\\alpha}{2 \\\\alpha} + \\\\delta},\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[64,169,928,206]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for all $\\\\frac{1}{2b} \\\\leq t < \\\\frac{1}{b}$ . This proves the asymptotic estimate (5.1) for $\\\\frac{1}{2b} \\\\leq t < \\\\frac{1}{b}$ . For $0 \\\\leq t \\\\leq \\\\frac{1}{2b}$ it is clearly satisfied by possibly choosing $C$ larger. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[86,208,566,223]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now, we prove (5.4). It follows from (5.17) and (5.18) that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,239,928,260]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left\\\\| \\\\left| \\\\omega (t) \\\\right| ^ {\\\\alpha} \\\\psi (t) ^ {\\\\alpha} - \\\\widetilde {v} (t) ^ {\\\\alpha} \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\leq C (1 - b t) ^ {1 - 2 \\\\sigma}. \\\\tag {5.32}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,273,382,289]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Using (5.2) and (5.16), we deduce that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[310,300,685,335]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left\\\\| \\\\frac {| \\\\omega (t) | ^ {\\\\alpha} (1 + f _ {0}) - | v _ {0} | ^ {\\\\alpha}}{\\\\widetilde {H}} \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\leq C (1 - b t) ^ {1 - 2 \\\\sigma}.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,346,279,363]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since by (3.18) and (5.11) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[406,359,590,382]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\widetilde {H} \\\\leq C (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2}},\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,394,209,408]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"we conclude that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[266,405,928,440]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left\\\\|\\\\left| \\\\omega (t) \\\\right| ^ {\\\\alpha} \\\\left(1 + f _ {0}\\\\right) - \\\\left| v _ {0} \\\\right| ^ {\\\\alpha} \\\\right\\\\| _ {L ^ {\\\\infty}} \\\\leq C \\\\left(1 - b t\\\\right) ^ {1 - \\\\frac {2 - N \\\\alpha}{2} - 2 \\\\sigma} \\\\underset {t \\\\rightarrow \\\\frac {1}{b}} {\\\\longrightarrow} 0. \\\\tag {5.33}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,447,411,463]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Applying (5.31), we obtain property (5.4). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[86,466,290,481]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Next we prove (5.5). Set \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[365,479,928,502]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nZ (t, x) = (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2}} \\\\widetilde {v} (t, x) ^ {\\\\alpha}. \\\\tag {5.34}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,512,281,528]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It follows from (5.16) that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[262,540,733,587]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nZ (t, x) = \\\\frac {\\\\left| v _ {0} (x) \\\\right| ^ {\\\\alpha} (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2}}}{1 + f _ {0} (x) + \\\\frac {2 \\\\alpha | \\\\Im \\\\lambda |}{b (2 - N \\\\alpha)} \\\\left| v _ {0} (x) \\\\right| ^ {\\\\alpha} [ (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2}} - 1 ]}.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,598,369,614]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since $1 + f_{0} \\\\geq 0$ by (5.11), we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[341,625,650,665]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nZ (t, x) \\\\leq \\\\frac {b (2 - N \\\\alpha)}{2 \\\\alpha | \\\\Im \\\\lambda |} \\\\frac {1}{1 - (1 - b t) ^ {\\\\frac {2 - N \\\\alpha}{2}}}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,676,128,688]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"so that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[367,685,628,727]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\lim _ {t \\\\uparrow \\\\frac {1}{b}} \\\\sup _ {\\\\| Z (t) \\\\| _ {L ^ {\\\\infty}}} \\\\leq \\\\frac {b (2 - N \\\\alpha)}{2 \\\\alpha | \\\\Im \\\\lambda |}.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,731,306,747]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moreover, $1 + f_0 \\\\leq 2$ , so that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[297,758,696,805]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nZ (t, 0) \\\\geq \\\\frac {| v _ {0} (0) | ^ {\\\\alpha} (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2}}}{2 + \\\\frac {2 \\\\alpha | \\\\Im \\\\lambda |}{b (2 - N \\\\alpha)} | v _ {0} (0) | ^ {\\\\alpha} [ (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2}} - 1 ]}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,816,415,832]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since $|v_0(0)| > 0$ by (3.18), we deduce that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[369,843,625,882]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\liminf_{t\\\\uparrow \\\\frac{1}{b}}\\\\| Z(t)\\\\|_{L^{\\\\infty}}\\\\geq \\\\frac{b(2 - N\\\\alpha)}{2\\\\alpha|\\\\Re\\\\lambda|}.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,893,209,908]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Thus we see that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[393,904,928,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\| Z (t) \\\\| _ {L ^ {\\\\infty}} \\\\underset {t \\\\rightarrow \\\\frac {1}{b}} {\\\\longrightarrow} \\\\frac {b (2 - N \\\\alpha)}{2 \\\\alpha | \\\\Re \\\\lambda |}. \\\\tag {5.35}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[68,44,334,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"T. Cazenave, Z. Han and I. Naumkin \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[661,44,928,57]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Nonlinear Analysis 205 (2021) 112243 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[490,947,507,956]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"35 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[66,88,422,105]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Applying (5.34) and (5.17), we deduce that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[211,115,781,173]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\left\\\\| \\\\left(1 - b t\\\\right) ^ {- \\\\frac {2 - N \\\\alpha}{2}} | v (t) | ^ {\\\\alpha} - Z (t) \\\\right\\\\| _ {L ^ {\\\\infty}} = \\\\left(1 - b t\\\\right) ^ {- \\\\frac {2 - N \\\\alpha}{2}} \\\\| | v (t) | ^ {\\\\alpha} - \\\\widetilde {v} (t) ^ {\\\\alpha} \\\\| _ {L ^ {\\\\infty}} \\\\\\\\ \\\\leq C (1 - b t) ^ {1 - \\\\frac {2 - N \\\\alpha}{2} - 2 \\\\sigma} \\\\underset {t \\\\to \\\\frac {1}{b}} {\\\\longrightarrow} 0, \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[64,185,332,200]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"so that (5.5) follows from (5.35). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[86,202,669,221]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally we prove (5.6), and we let $\\\\frac{1}{2b} \\\\leq t < \\\\frac{1}{b}$ . It follows from (5.7) that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[193,231,797,328]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\int_ {\\\\mathbb {R} ^ {N}} | v (t) | ^ {2} = \\\\int_ {\\\\langle x \\\\rangle > (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2 n \\\\alpha}}} | v (t) | ^ {2} + \\\\int_ {\\\\langle x \\\\rangle < (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2 n \\\\alpha}}} | v (t) | ^ {2} \\\\\\\\ \\\\leq C \\\\int_ {\\\\langle x \\\\rangle > (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2 n \\\\alpha}} \\\\langle x \\\\rangle^ {- 2 n} + C} \\\\int_ {\\\\langle x \\\\rangle < (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2 n \\\\alpha}} \\\\left(1 - b t\\\\right) ^ {\\\\frac {2 - N \\\\alpha}{\\\\alpha}}} \\\\\\\\ \\\\leq C (1 - b t) ^ {\\\\frac {2 - N \\\\alpha}{\\\\alpha} (1 - \\\\frac {N}{2 n})}, \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,339,654,355]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"which proves the upper estimate in (5.6). Next, using (5.11), we see that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[330,365,663,411]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\widetilde {H} ^ {\\\\frac {1}{\\\\alpha}} \\\\leq \\\\left\\\\{ \\\\begin{array}{l l} C, & \\\\langle x \\\\rangle > (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2 n \\\\alpha}}, \\\\\\\\ C \\\\langle x \\\\rangle^ {- n}, & \\\\langle x \\\\rangle < (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2 n \\\\alpha}}. \\\\end{array} \\\\right.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,421,626,438]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Applying (3.32) and $|v_0(x)| \\\\geq K^{-1} \\\\langle x \\\\rangle^{-n}$ (by (3.18)), we deduce that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[319,448,674,491]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n| v (t, x) | \\\\geq \\\\left\\\\{ \\\\begin{array}{l l} c \\\\langle x \\\\rangle^ {- n}, & \\\\langle x \\\\rangle > (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2 n \\\\alpha}}, \\\\\\\\ c, & \\\\langle x \\\\rangle < (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2 n \\\\alpha}}, \\\\end{array} \\\\right.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,502,279,518]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for some $c > 0$ . Therefore, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[196,529,799,566]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\int_ {\\\\mathbb {R} ^ {N}} | v (t) | ^ {2} \\\\geq c \\\\int_ {\\\\langle x \\\\rangle > (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2 n \\\\alpha}} \\\\langle x \\\\rangle^ {- 2 n}} + c \\\\int_ {\\\\langle x \\\\rangle < (1 - b t) ^ {- \\\\frac {2 - N \\\\alpha}{2 n \\\\alpha}}} \\\\left(1 - b t\\\\right) ^ {\\\\frac {2 - N \\\\alpha}{\\\\alpha}},\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,575,687,591]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"from which the lower estimate in (5.6) follows. This completes the proof. $\\\\square$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[88,610,477,626]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We are now in a position to prove Theorem 1.1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[66,645,553,661]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof of Theorem 1.1. Let $v_{0} \\\\in \\\\mathcal{X}$ satisfy (1.11), and set \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[351,673,643,706]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK = \\\\| v _ {0} \\\\| _ {\\\\mathcal {X}} + \\\\left(\\\\inf _ {x \\\\in \\\\mathbb {R} ^ {N}} \\\\langle x \\\\rangle^ {n} | v _ {0} (x) |\\\\right) ^ {- 1}.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[64,718,928,754]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $b \\\\geq b_{1}$ , where $b_{1}$ is given by Proposition 4.1 for this value of $K$ , and let $v \\\\in C([0, \\\\frac{1}{b}), \\\\mathcal{X})$ be the solution of $(\\\\mathrm{NLS}_b)$ given by Proposition 4.1. Let $f_{0}, \\\\omega_{0}, \\\\theta, \\\\psi$ be given by Proposition 5.1. Set \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[242,764,928,800]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nu (t, x) = (1 + b t) ^ {- \\\\frac {N}{2}} e ^ {i \\\\frac {b | x | ^ {2}}{4 (1 + b t)}} v \\\\left(\\\\frac {t}{1 + b t}, \\\\frac {x}{1 + b t}\\\\right), t \\\\geq 0, x \\\\in \\\\mathbb {R} ^ {N}. \\\\tag {5.36}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[64,811,926,855]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It follows that $u \\\\in C([0,\\\\infty), H^1(\\\\mathbb{R}^N))$ is the solution of (NLS) with the initial value $u_0(x) = e^{i\\\\frac{b|x|^2}{4}}v_0(x)$ . Since $\\\\eta > \\\\frac{N}{2}$ , the estimate (5.1) implies \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[259,865,735,888]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\| v (t, x) - \\\\omega_ {0} (x) \\\\psi (t, x) e ^ {- i \\\\theta (t, x)} \\\\| _ {L ^ {\\\\infty} \\\\cap L ^ {2}} \\\\leq C (1 - b t) ^ {\\\\frac {2 - N \\\\alpha}{2 \\\\alpha} + \\\\delta}.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[66,901,930,936]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Then, (1.12) follows from (5.1). By using (5.36), we see that (1.13) and (1.14) are consequences of (5.5) and (5.6), respectively. This completes the proof. $\\\\square$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[68,44,334,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"T. Cazenave, Z. Han and I. Naumkin \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[661,44,928,57]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Nonlinear Analysis 205 (2021) 112243 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[488,947,507,956]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"36 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[66,88,161,103]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[69,119,928,700]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"[1] T. Cazenave, S. Correia, F. Dickstein, F.B. Weissler, A Fujita-type blowup result and low energy scattering for a nonlinear Schrödinger equation, São Paulo J. Math. Sci. 9 (2) (2015) 146-161, http://dx.doi.org/10.1007/s40863-015-0020-6, MR3457455. \\\",\\\"[2] T. Cazenave, Z. Han, Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation, Discrete Contin. Dyn. Syst. 40 (8) (2020) 4801-4819, http://dx.doi.org/10.3934/dcds.2020202. \\\",\\\"[3] T. Cazenave, Z. Han, Y. Martel, Blowup on an arbitrary compact set for a Schrödinger equation with nonlinear source term, J. Dynam. Differential Equations (2020) http://dx.doi.org/10.1007/s10884-020-09841-8. \\\",\\\"[4] T. Cazenave, Y. Martel, L. Zhao, Finite-time blowup for a Schrödinger equation with nonlinear source term, Discrete Contin. Dyn. Syst. 39 (2) (2019) 1171-1183, http://dx.doi.org/10.3934/dcds.2019050, MR3918212. \\\",\\\"[5] T. Cazenave, I. Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Commun. Contemp. Math. 19 (2) (2017) 1650038, http://dx.doi.org/10.1142/S0219199716500383, 20, MR3611666. \\\",\\\"[6] T. Cazenave, I. Naumkin, Modified scattering for the critical nonlinear Schrödinger equation, J. Funct. Anal. 274 (2) (2018) 402–432, http://dx.doi.org/10.1016/j.jfa.2017.10.022, MR3724144. \\\",\\\"[7] T. Cazenave, F.B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys. 147 (1992) 75–100, http://dx.doi.org/10.1007/BF02099529, MR1171761. \\\",\\\"[8] G.M. Constantine, T.H. Savits, A multivariate Faa di Bruno formula with applications, Trans. Amer. Math. Soc. 348 (2) (1996) 503-520, http://dx.doi.org/10.1090/S0002-9947-96-01501-2, MR1325915. \\\",\\\"[9] M.C. Cross, P.C. Hohenberg, Pattern formation outside of equilibrium, Rev. Modern Phys. 65 (3) (1993) 851-1112, http://dx.doi.org/10.1103/RevModPhys.65.851. \\\",\\\"[10] J. Ginibre, T. Ozawa, G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 60 (2) (1994) 211-239, MR1270296, http://archive.numdam.org/article/AIHPA_1994_60_2_211_0.pdf. \\\",\\\"[11] J. Ginibre, G. Velo, On a class of nonlinear Schrödinger equations. III. Special theories in dimensions 1, 2 and 3, Ann. Inst. H. Poincaré 28 (1978) 287-316, MR0498408, http://archive numdam.org/article/AIHPA_1978_28_3_287_0.pdf. \\\",\\\"[12] J. Ginibre, G. Velo, On a class of nonlinear Schrödinger equations. II. Scattering theory, general case, J. Funct. Anal. 32 (1) (1979) 33-71, http://dx.doi.org/10.1016/0022-1236(79)90077-6, MR0533219. \\\",\\\"[13] N. Hayashi, C. Li, P.I. Naumkin, Time decay for nonlinear dissipative Schrödinger equations in optical fields, Adv. Math. Phys. (2016) 7, http://dx.doi.org/10.1155/2016/3702738, Art. ID 3702738, MR3465033. \\\",\\\"[14] N. Hayashi, C. Li, P.I. Naumkin, Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations, Commun. Pure Appl. Anal. 16 (6) (2017) 2089-2104, http://dx.doi.org/10.3934/cpaa.2017103, MR3693873. \\\",\\\"[15] N. Kita, A. Shimomura, Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity, J. Differential Equations 242 (1) (2007) 192-210, http://dx.doi.org/10.1016/j.jde.2007.07.003, MR2361107. \\\",\\\"[16] N. Kita, A. Shimomura, Large time behavior of solutions to Schrödinger equations with a dissipative nonlinearity for arbitrarily large initial data, J. Math. Soc. Japan 61 (1) (2009) 39-64, http://dx.doi.org/10.2969/jmsj/06110039, MR2272871. \\\",\\\"[17] A. Mielke, The Ginzburg-Landau equation in its role as a modulation equation, in: Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, 2002, pp. 759-834, http://dx.doi.org/10.1016/S1874-575X(02)80036-4, MR1901066. \\\",\\\"[18] K. Nakanishi, T. Ozawa, Remarks on scattering for nonlinear Schrödinger equations, NoDEA Nonlinear Differ. Equ. Appl. 9 (1) (2002) 45-68, http://dx.doi.org/10.1007/s00030-002-8118-9, MR1891695. \\\",\\\"[19] A. Shimomura, Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Comm. Partial Differential Equations 31 (7-9) (2006) 1407-1423, http://dx.doi.org/10.1080/03605300600910316, MR2254620. \\\",\\\"[20] K. Stewartson, J.T. Stuart, A non-linear instability theory for a wave system in plane Poiseuille flow, J. Fluid Mech. 48 (1971) 529-545, http://dx.doi.org/10.1017/S0022112071001733, MR0309420. \\\",\\\"[21] W.A. Strauss, Nonlinear scattering theory at low energy: sequel, J. Funct. Anal. 43 (1981) 281-293, http://dx.doi.org/10.1016/0022-1236(81)90063-X, MR0614228. \\\"]\",\n \"page_idx\": \"36\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[69,44,334,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"T. Cazenave, Z. Han and I. Naumkin \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[661,44,928,57]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Nonlinear Analysis 205 (2021) 112243 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[488,947,507,956]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"37 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.1016/j.na.2020.112243"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Physics/Mathematics and Computational Science"},"sha256":{"kind":"string","value":"7a0910da74dc28ac359d36130da8f0040e7a65045d9cd1cbecb3ca010639ddbc"},"title":{"kind":"string","value":"Asymptotic behavior for a dissipative nonlinear Schrödinger equation"}}},{"rowIdx":41,"cells":{"abstract":{"kind":"string","value":"Glucocorticoids inhibit angiogenesis in the femoral head, which fails to nourish the bone tissue and leads to osteonecrosis. Restoring angiogenesis is not only essential for vessel formation, but also crucial for osteogenesis. Poly (L-lactic acid) (PLLA) is commonly used in the bone tissue engineering field. Panax notoginseng saponins (PNS) and osteopractic total flavone (OTF) promote angiogenesis and osteogenesis, respectively. We designed a sequentially releasing PLLA scaffold including PLLA loaded with OTF (inner layer) and PLLA loaded with PNS (outer layer). We assessed the osteogenic effect of angiogenesis in this scaffold by comparing it with the one-layered scaffold (PLLA embedded with OTF and PNS) in vivo. Results from the micro-CT showed that the data of bone mineral density (BMD), bone volume (BV), and percent bone volume (BV/TV) in the PO-PP group were significantly higher than those in the POP group (p < 0.01). Histological analyses show that the PO-PP scaffold exhibits better angiogenic and osteogenic effects compared with the one-layered scaffold. These might result from the different structures between them, where the sequential release of a bi-layer scaffold achieves the osteogenic effect of vascularization by initially releasing PNS in the outer layer. We further explored the possible mechanism by an immunohistochemistry analysis and an immunofluorescence assay. The results showed that the protein expressions of vascular endothelial growth factor (VEGF) and platelet endothelial cell adhesion molecule-1(CD31) in the PO-PP scaffold were significantly higher than those in the POP scaffold (p < 0.01); the protein expressions of osteocalcin (OCN), osteopontin (OPN), and alkaline phosphatase (ALP) in the PO-PP scaffold were significantly higher than those in the POP scaffold (p < 0.05). Upregulating the expressions of angiogenic and osteogenic proteins might be the possible mechanism."},"author":{"kind":"string","value":"Feng, Guiyu|Zhang, Pingxin|Huang, Jian|Yu, Yao|Yang, Fenghe|Zhao, Xueqian|Wang, Wei|Li, Dongyang|Sun, Song|Niu, Xufeng|Chai, Limin|Li, Jinyu"},"content_list":{"kind":"list like","value":[{"bbox":"[58,124,107,136]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Article ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[58,142,899,251]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sequential Release of Panax Notoginseng Saponins and OsteopRACTIC Total Flavone from Poly (L-Lactic Acid) Scaffold for Treating Glucocorticoid-Associated Osteonecrosis of Femoral Head ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[58,269,840,300]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Guiyu Feng $^{1,\\dagger}$ , Pingxin Zhang $^{1,\\dagger}$ , Jian Huang $^{2,\\dagger}$ , Yao Yu $^{3,\\dagger}$ , Fenghe Yang $^{3}$ , Xueqian Zhao $^{4}$ , Wei Wang $^{1}$ , Dongyang Li $^{1}$ , Song Sun $^{1}$ , Xufeng Niu $^{3,*}$ , Limin Chai $^{1,*}$ and Jinyu Li $^{2,*}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,459,243,471]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Citation:Feng,G.; Zhang,P. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,473,245,486]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Huang, J.; Yu, Y.; Yang, F.; Zhao, X.; ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,488,231,500]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Wang, W.; Li, D.; Sun, S.; Niu, X. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,502,230,514]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"et al. Sequential Release of Panax ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,516,200,527]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Notoginseng Saponins and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,530,226,541]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Osteopractic Total Flavone from ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,543,221,555]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Poly (L-Lactic Acid) Scaffold for ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,558,242,568]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Treating Glucocorticoid-Associated ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,573,231,584]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Osteonecrosis of Femoral Head. $J.$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,587,201,598]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Funct. Biomater. 2023, 14, 31. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,602,238,612]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.3390/jfb14010031 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[58,624,233,633]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Academic Editor: Aurelio Salerno ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[58,644,208,655]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Received: 23 November 2022 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,660,200,669]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Revised: 23 December 2022 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,675,203,685]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Accepted: 23 December 2022 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,688,191,699]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Published: 4 January 2023 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,706,149,730]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0729/2673904523d27fc444cad8e3f1d0e65bcea691218b204d294e26f3a4e4531c11.jpg","list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[57,737,250,833]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[277,321,929,448]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"<sup>1</sup> Key Laboratory of Chinese Internal Medicine of Ministry of Education and Beijing, Dongzhimen Hospital, Beijing University of Chinese Medicine, Beijing 100700, China \",\"$^{2}$ Department of Orthopedic, Dongzhimen Hospital, Beijing University of Chinese Medicine, Beijing 100700, China \",\"<sup>3</sup> Key Laboratory of Biomechanics and Mechanobiology (Beihang University), Ministry of Education, Beijing Advanced Innovation Center for Biomedical Engineering, School of Biological Science and Medical Engineering, Beihang University, Beijing 100083, China \",\"4 Yuquan Hospital Affiliated to Tsinghua University, Beijing 100040, China \",\"* Correspondence: nxf@buaa.edu.cn (X.N.); liminchai@hotmail.com (L.C.); leery_5566@163.com (J.L.) \",\"These authors contributed equally to this work. \"]","page_idx":"0","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[278,466,939,771]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract: Glucocorticoids inhibit angiogenesis in the femoral head, which fails to nourish the bone tissue and leads to osteonecrosis. Restoring angiogenesis is not only essential for vessel formation, but also crucial for osteogenesis. Poly (L-lactic acid) (PLLA) is commonly used in the bone tissue engineering field. Panax notoginseng saponins (PNS) and osteopractic total flavone (OTF) promote angiogenesis and osteogenesis, respectively. We designed a sequentially releasing PLLA scaffold including PLLA loaded with OTF (inner layer) and PLLA loaded with PNS (outer layer). We assessed the osteogenic effect of angiogenesis in this scaffold by comparing it with the one-layered scaffold (PLLA embedded with OTF and PNS) in vivo. Results from the micro-CT showed that the data of bone mineral density (BMD), bone volume (BV), and percent bone volume (BV/TV) in the PO-PP group were significantly higher than those in the POP group $(p < 0.01)$ . Histological analyses show that the PO-PP scaffold exhibits better angiogenic and osteogenic effects compared with the one-layered scaffold. These might result from the different structures between them, where the sequential release of a bi-layer scaffold achieves the osteogenic effect of vascularization by initially releasing PNS in the outer layer. We further explored the possible mechanism by an immunohistochemistry analysis and an immunofluorescence assay. The results showed that the protein expressions of vascular endothelial growth factor (VEGF) and platelet endothelial cell adhesion molecule-1(CD31) in the PO-PP scaffold were significantly higher than those in the POP scaffold $(p < 0.01)$ ; the protein expressions of osteocalcin (OCN), osteopontin (OPN), and alkaline phosphatase (ALP) in the PO-PP scaffold were significantly higher than those in the POP scaffold $(p < 0.05)$ . Upregulating the expressions of angiogenic and osteogenic proteins might be the possible mechanism. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,789,939,817]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords: Poly (L-lactic acid); osteopractic total flavone; panax notoginseng saponins; osteogenesis; angiogenesis ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,59,115,100]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"# ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[119,60,221,98]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Journal of Functional Biomaterials ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[865,61,942,97]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"MDPI ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[58,967,445,978]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31. https://doi.org/10.3390/jfb14010031 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[774,967,939,978]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"www.mdpi.com/journal/jfb ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[278,108,391,122]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[278,125,939,277]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Glucocorticoid-associated osteonecrosis of the femoral head (GONFH) is known as bone death, resulting from the use of chronic glucocorticoids [1]. Glucocorticoids could directly damage endothelial cells, impair new blood vessels formation, and decrease the vascular supply to bone tissue [2]. The vascular supply to the femoral head involves lateral and medial circumflex femoral arteries, which penetrates the femoral head and supply most of the femoral epiphysis. Vascular damage easily leads to the inability to deliver nutrients to the bone tissue of the femoral head, leading to femoral head necrosis [1]. Restoring angiogenesis is not only essential for vessel formation, but also crucial for osteogenesis [3,4]. Angiogenesis and osteogenesis, closely related to each other, are both crucial for bone repair. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,279,939,460]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bone tissue engineering is a promising strategy for bone repair [5,6]. Biomaterials represent a key element in bone tissue engineering. Poly (L-lactic acid) (PLLA), a highly versatile biodegradable aliphatic polyester, is commonly used as degradable biomaterial [7]. It has been widely used in biomedical and surgical fields, such as surgical equipment and fusion substance, tissue engineering scaffolds, and drug carriers including orthopedic treatment, owing to its safety and low toxicity [8-10]. The most commonly used PLLA in tissue engineering has been approved by FDA [11]. Recent medical studies confirmed that the implant of an electrospun, silicon-doped vaterite/PLLA membrane used to treat the calvaria defect in the rabbit model could enhance the proliferation of pre-osteoblasts and promote new bone formation [12]. Implanted PLLA also could improve the attachment, proliferation, and differentiation of pre-osteoblasts, and lead to the effective osteo-conductivity contributing to new bone formation in vitro and in vivo [13,14]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,461,939,583]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In addition to biomaterials, drug delivery systems comprise another key element in bone tissue engineering. The sequential treatment of collagen scaffolds with bone morphogenetic protein-2 (BMP-2) following stromal cell derived factor-1 (SDF-1) showed superior new bone regeneration compared with their simultaneous treatment [15]. The initial release of BMP-2, followed by the sequential release of alendronate (ALN) in collagen-hydroxyapatite composite scaffold, significantly increases osteogenic activity owing to the synergistic effect of BMP-2 and ALN. In this study, we aim to obtain a sequentially releasing PLLA scaffold, which enhances the osteogenic effect by restoring angiogenesis. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,584,939,795]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Panax notoginseng saponins (PNS) is the main ingredient of Panax Notoginseng (a medicinal herb). Treatment by PNS promotes angiogenesis, as PNS activates the VEGF-KDR and PI3K-Akt-eNOS pathways, which contribute to angiogenesis [16]. Furthermore, PNS induces the production of VEGF-A, VEGFR-1, and VEGFR-2 in bone marrow mesenchymal stem cells (BMSCs) [17]. OsteopRACTic total flavone (OTF) is the main ingredient of Drynariae Rhizoma (a medicinal herb). Several studies confirmed that OTF could improve the formation of new bone. OTF in vivo experiments could significantly promote the formation of epiphyses and bone trabeculae, increase bone mineralization, accelerate the differentiation and mineralization of osteoblasts, and inhibit the bone resorption of osteoclasts [18-21]. In this study, we constructed a bi-layered PLLA scaffold including PLLA embedded with OTF (PO, inner layer) and PLLA embedded with PNS (PP, outer layer). We hypothesize that PNS in the outer layer would be released initially, followed by the OTF release of the inner layer. We further presumed that this sequential release model could enhance the osteogenic effect by promoting angiogenesis. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,796,939,903]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We designed this sequentially released PO-PP scaffold to adopt the advantage of this sequential release model: PNS being initially released, followed by the release of OTF. We also assessed the osteogenic effect of angiogenesis (a key advantage of sequential release) in the PO-PP scaffold by comparing it with a one-layered PLLA scaffold (PLLA embedded with OTF and PNS) in vivo. We hypothesized that this sequentially released scaffold (PO-PP) could enhance the osteogenic effect by restoring angiogenesis at the basis of initially released PNS-promoting angiogenesis. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,62,233,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[897,61,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 of 17 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[277,108,465,122]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. Material and Methods ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[277,125,600,142]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.1. Preparation of One- and Bi-Layer Scaffolds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[277,144,939,312]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Poly (L-lactic acid) (PLLA, molecular weight: $2.00 \\times 10^{5}$ ) was purchased from Evonik Industries AG (Essen, German). 1,4-Dioxane (molecular formula: $\\mathrm{C_4H_8O_2}$ , molecular weight: 88.10, standard quality), Tetrahydrofuran (THF, molecular formula: $\\mathrm{C_4H_8O}$ , molecular weight: 72.11, standard quality), and N, N-Dimethylformamide (DMF, molecular formula: $\\mathrm{C_3H_7NO}$ , molecular weight: 73.10, standard quality) were purchased from Macklin Biochemical Co., Ltd. (Shanghai, China). PNS was purchased from Yunnan Baiyao Group Wenshan Qihua Co., Ltd. (Kunming, China). OTF was purchased from Beijing Qihuang Pharmaceutical Co., Ltd. (Beijing, China). The purities of purchased PNS and OTF were detected by high performance liquid chromatography (HPLC) analysis, and the results showed the purities of purchased OTF and PNS reached the standards (Supplementary Figures S1 and S2). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[277,313,939,461]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The one-layer scaffold, named POP (PLLA+OTF+PNS), was created through thermally induced phase separation (TIPS). At first, $10\\%$ PLLA was dissolved in 1,4-dioxane, and water was then added into this solution, stirred at $60^{\\circ}\\mathrm{C}$ for $1\\mathrm{h}$ (1,4-dioxane used in former step:water = 87:13). Then, OTF $(8\\mathrm{mg})$ and PNS $(5\\mathrm{mg})$ were added into the solution and stirred at $60^{\\circ}\\mathrm{C}$ for $20\\mathrm{min}$ . When OTF:PNS = 8:5, the release concentration of both OTF and PNS reached $100~\\mu \\mathrm{g / mL}$ so that the proliferation and differentiation of osteoblasts and the proliferation of vascular endothelial cells were promoted in the concentration of $100~\\mu \\mathrm{g / mL}$ [22]. The solution was poured into a plastic mold (4 mm diameter) and placed at $30^{\\circ}\\mathrm{C}$ for $1\\mathrm{h}$ , and $-20^{\\circ}\\mathrm{C}$ for $1\\mathrm{h}$ , after which the solution was freeze-dried for $24\\mathrm{h}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[277,465,939,600]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The bi-layer scaffold, named PO-PP, included two layers (inner layer: PLLA+ OTF; outer layer: PLLA+ PNS). The inner layer (PO) was prepared by the same method, and the solution was poured into a plastic mold (3 mm diameter). OTF was added into this layer. The outer layer (PP) was created by an electrospun membrane. The $10\\%$ PLLA was dissolved in THF and DMF at $60^{\\circ}\\mathrm{C}$ for 1 h. PNS $(5\\mathrm{mg})$ was added into the solution. The resulting solution was electrospun into fibers (voltage: +13, -2 kV; extrusion speed: 0.0025 mm/s). The inner layer was rolled to receive the electrospun membrane by a receiving device (running at 10 rpm) at receiving distance 20 cm. All scaffolds were sterilized by $^{60}\\mathrm{Co}$ irradiation and stored at room temperature before use. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[277,616,621,631]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2. Characterization of PO-PP and POP Scaffolds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[277,635,939,846]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The microstructures of PO-PP and POP scaffolds were observed by field emission-scanning electron microscopy (FE-SEM, FEI Quanta 250 FEG, Hillsboro, OR, USA). The round surfaces of scaffolds were placed on the platform and the pressing speed was $1\\mathrm{mm / min}$ , pressing to $80\\%$ length of scaffolds. The maximum compressive strength of the PO layer ( $3\\mathrm{mm}$ diameter, $16\\mathrm{mm}$ height) and POP scaffold ( $4\\mathrm{mm}$ diameter, $16\\mathrm{mm}$ height) was assessed through a universal testing machine (EZ-LX, SHIMADZU, Kyoto, Japan). The pycnometer method was used to assess the porosity of the porous structure (PO layer and POP scaffold). The mass $m_{1}(\\mathrm{g})$ of the scaffolds was weighed. The pycnometer filled with deionized water also likewise weighed $m_{2}(\\mathrm{g})$ ; the sample was placed into the pycnometer and soaked adequately. Then, the pycnometer was weighed $m_{3}(\\mathrm{g})$ after filling up with water. The mass of the remaining water and the pycnometer weighed $m_{4}(\\mathrm{g})$ , which were assessed after the soaked sample taken out. The porosity of the scaffolds $(\\varepsilon)$ was calculated based on foregoing data, as shown in the equation $\\varepsilon = (m_{3} - m_{4} - m_{1}) / (m_{2} - m_{4}) \\times 100\\%$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[277,847,939,908]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The loaded OTF and PNS were evaluated by a Fourier transform infrared spectroscope (FTIR, FTIR-7600, Lambda Scientific, Rüsselsheim, Germany). The sheared scaffold and potassium bromide were mixed at a ratio of 1: 100, pressed, and scanned by FTIR at a scanning range of $4000 - 400\\mathrm{cm}^{-1}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,61,233,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[897,61,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 of 17 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[278,106,939,229]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The release medium was PBS $(37^{\\circ}\\mathrm{C})$ and shaken at $100\\mathrm{rpm}$ . The cumulative release rate of OTF was measured at $283\\mathrm{nm}$ in quartz plate by ultraviolet spectrophotometry equipped with a microplate reader (Thermo Multiskan FC, Waltham, MA, USA). The cumulative release rate of PNS was measured through the vanillin-perchloric acid chromogenic method. After the solvent evaporated, prepared $5\\%$ vanillin-glacial acetic acid $(0.2\\mathrm{mL})$ and perchloric acid $(0.8\\mathrm{mL})$ were added. The mixture was heated at $60^{\\circ}\\mathrm{C}$ for $20\\mathrm{min}$ in a thermostatic water bath, cooled for $5\\mathrm{min}$ in a cold water bath, and then $5\\mathrm{mL}$ glacial acetic acid was added. Finally, PNS was measured at $545\\mathrm{nm}$ by a microplate reader. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,243,519,260]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.3. Animal Grouping and Surgery ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[278,262,939,397]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Thirty rabbits were used in this experiment. All rabbits were administered $10~\\mu \\mathrm{g / kg}$ of lipopolysaccharide (LPS, Sigma-Aldrich, St. Louis, MO, USA) fluid intravenously. Each rabbit was then given an intramuscular injection of $20\\mathrm{mg / kg}$ methylprednisolone (MPS, Macklin, Shanghai, China) every $24\\mathrm{h}$ , which was repeated three times. And then, two weeks later, a femoral head necrosis model was successfully induced, which was reported in our previously published article [23]. The thirty rabbits were randomly divided into three groups 12 weeks after modeling: the PO-PP group, imbedded with PO-PP scaffold; the POP group, imbedded with POP scaffold; and the control group, imbedded with no scaffold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,399,939,581]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The surgery was conducted two weeks after the third MPS injection. Femoral head perforation was only performed on the right side. A $25\\mathrm{mm}$ skin incision was made to expose the right hip joint. A bone tunnel (4 mm diameter, 16 mm depth) from the hole to the femoral head through the mid-axis of femoral neck was created by guiding a C-arm fluoroscope. The PO-PP and POP scaffolds (PO-PP group, $4\\mathrm{mm} \\times 16\\mathrm{mm}$ PO-PP scaffold; POP group, $4\\mathrm{mm} \\times 16\\mathrm{mm}$ POP scaffold; control group, no scaffold) were inserted into the bone tunnels. These rabbits were intramuscularly injected with 20 wu of penicillin twice a day for two weeks. After 12 weeks post-surgery, the experimental rabbits were sacrificed by intravenously injecting $150\\mathrm{mg/kg}$ bodyweight sodium pentobarbital. The whole right femur was resected, and the soft tissues were removed. Finally, the whole right femurs were fixed in $10\\%$ neutral formalin for micro-CT scanning, as well as histological, immunohistochemistry, and immunofluorescence analyses. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,596,440,612]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.4. Micro-CT Analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[278,614,939,705]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The harvested femoral head samples were assessed by a micro-CT scanner (Skyscan1076, Bruker, Karlsruhe, Germany) at $40\\mathrm{kVP}$ , $250~\\mu \\mathrm{A}$ for $800~\\mathrm{ms}$ . The region of interest (ROI, $4\\mathrm{mm}$ diameter in the center of the bone tunnel) was selected and reconstructed by a CT-analyzer. The bone mineral density (BMD, g/cc), bone surface density (BS/TV, 1/mm), percent bone volume (BV/TV, %), trabecular number (Tb.N, 1/mm), and bone volume (BV, mm³) were calculated based on the data from the micro-CT scans. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,720,453,736]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.5. Histological Analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[278,739,939,830]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The femoral heads were decalcified with $14\\%$ EDTA solution (pH7.2) for 28 days. The decalcified samples were embedded in paraffin and cut into $4\\mu \\mathrm{m}$ sections. The Hematoxylin and Eosin (H&E) (Servicebio, Wuhan, China), Masson trichrome (MT) (Servicebio), Goldner's trichrome (Servicebio), and Tartrate-resistant acid phosphatase (TRAP) (Servicebio) stainings were executed on the sections. The histological sections were observed with a light microscope (Olympus, Tokyo, Japan). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,845,574,861]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.6. Immunohistochemistry (IHC) Analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[278,864,939,923]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The quantity and distribution of the vascular endothelial growth factor (VEGF), osteopontin (OPN), and osteocalcin (OCN) in the bone samples were detected through IHC analysis. The sections were incubated for heat-induced epitope retrieval in citrate buffer at $60^{\\circ}\\mathrm{C}$ , then blocked with $1\\%$ hydrogen peroxide/methanol (Sigma-Aldrich, St. Louis, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,61,233,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[897,61,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 of 17 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[278,106,939,274]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"MO, USA) for $30\\mathrm{min}$ at $25^{\\circ}\\mathrm{C}$ . After blocking, the sections were incubated with primary antibodies VEGF (1: 100; Novus NBP2-45235, Bio-Techne Corporation, Minneapolis, MN, USA), OPN (1: 100; Novus NB110-89062, Bio-Techne Corporation, Minneapolis, MN, USA) and OCN (1: 100; Novus NBP2-89037, Bio-Techne Corporation, Minneapolis, MN, USA) overnight at $4^{\\circ}\\mathrm{C}$ . Next, the samples were incubated with goat anti-mouse secondary antibodies (1: 200; Servicebio G23301, Wuhan servicebio technology CO., LTD, Wuhan, China) for $1\\mathrm{h}$ at room temperature. The positive staining sites were colored by $3,3^{\\prime}$ -diaminobenzidine (Dako Liquid DAB, Carpinteria, CA, USA) color solution. Finally, the sections were counterstained with hematoxylin. The sections were then visualized with a light microscope, and the positive staining sites were assessed by Image J software (Image J, V1.8.0, Bethesda, MD, USA). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,290,541,304]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.7. Immunofluorescence Assay (IMA) ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[278,308,939,490]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IMA was performed to detect the quantity and distribution of the platelet endothelial cell adhesion molecule-1 (PECAM-1/CD31) and alkaline phosphatase (ALP). After heat-induced epitope retrieval and blockage, the slides were incubated with primary antibodies for CD31 (1:20; Novus NB600-562, Bio-Techne Corporation, Minneapolis, MN, USA) and ALP (1: 100; Novus NB600-540, Bio-Techne Corporation, Minneapolis, MN, USA) overnight at $4^{\\circ}\\mathrm{C}$ . Subsequently, the samples were incubated with goat anti-rabbit secondary antibodies (CD31:1:400, Servicebio GB25303, Wuhan servicebio technology CO., LTD, Wuhan, China; ALP:1:300, Servicebio GB21303, Wuhan servicebio technology CO., LTD, Wuhan, China) for 1 h at room temperature. Ultimately, the samples were counterstained with DAPI (Servicebio) for 10 min at room temperature. Images were captured with a Leitz/Leica TCSSP2 microscope (Leica Lasertechnik GmbH, Heidelberg, Germany) and assessed by ImageJ software (Image J, V1.8.0, Bethesda, MD, USA). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,505,440,520]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.8. Statistical Analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[278,524,939,584]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"All data were presented as mean $\\pm$ standard deviation (SD). Differences among the groups were assessed using one-way analysis of variance (ANOVA) with LSD for multiple comparisons in this study. A $p$ value $< 0.05$ was considered as statistically significant. All data analyses were performed using the SPSS 23.0 software (IBM, Armonk, NY, USA). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,600,351,612]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3. Results ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[278,618,500,633]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.1. Characterization of Scaffolds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[278,636,939,925]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The PO-PP scaffold is a bi-layer scaffold including PLLA embedded with OTF (PO, inner layer) and PLLA embedded with PNS (PP, outer layer), which achieves sequential delivery of PNS and OTF; PNS of the outer layer is initially released, followed by the release of OTF from the inner layer. The initially released PNS could promote new vessel formation, which would further enhance new bone formation by providing sufficient nutrition to the bone tissue. The inner OTF was released later, which could promote new bone formation. In this manner, better and faster new bone formation is obtained. Enhancing the osteogenic effect of vascularization (PNS in the outer layer) is the key technique employed by our scaffold. The PO layer has a porous structure, and the PP layer has an electrospinning net. The POP scaffold has a one-layer scaffold, including PLLA embedded with OTF and PNS (POP), which did not obtain sequential release and released OTF and PNS simultaneously. It only had a porous structure (Figure 1A,B). For the PO-PP scaffold, the pore diameter was about $10 - 38\\mu \\mathrm{m}$ (PO layer), and the electrospinning diameter was about $0.7 - 1.7\\mu \\mathrm{m}$ (PP layer). For the POP scaffold, the pore diameter ranged within 14-31 $\\mu \\mathrm{m}$ . Table 1 indicates that porous materials (PO layer, POP scaffold) meet the minimum strength requirement of cancellous bone repair (1 MPa) and the minimum porosity requirement of vessel formation (50%) [24,25]. OTF was successfully loaded into the PO layer of the PO-PP scaffold with hydroxyl groups $(3470~\\mathrm{cm}^{-1})$ and benzene ring (1654 $\\mathrm{cm}^{-1}$ ), which was the characteristic absorption peak of OTF in the FTIR image (Figure 2A). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,61,233,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[897,61,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 of 17 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[275,106,939,244]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PNS with hydroxyl structures $(3470~\\mathrm{cm}^{-1})$ and a carbon-carbon double bond $(1639~\\mathrm{cm}^{-1})$ was successfully loaded into PP layer of PO-PP scaffold in the FTIR image (Figure 2A). In this manner, OTF and PNS were both successfully loaded into the POP scaffold. For cumulative drug release, no difference was found between the PO-PP and POP scaffolds in OTF (Figure 2B). The PNS cumulative release was faster in the PO-PP scaffold than in the POP scaffold, which not only achieves better vessel formation, but also enhances the osteogenic effect of vascularization of the PO-PP scaffold to some degree (Figure 2B). The weight-averaged molecular weight (Mw) of PLLA was about $2\\times 10^{5}$ and slowly degraded in vivo, which was reported in our previous published article [26]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[270,258,643,378]","code_body":null,"code_caption":null,"image_caption":"[\"A \"]","image_footnote":"[]","img_path":"images/0043/caaa87863a6120893a980d94e4d1fde7c7d0994647c0322e2e5bf896d268e47c.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[694,277,813,378]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0533/15a91e7c44b7cd59cf50e6fd563ee3de7235ca0fce07aee6d33d5277982cb2d4.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[176,391,403,520]","code_body":null,"code_caption":null,"image_caption":"[\"B \"]","image_footnote":"[]","img_path":"images/0532/0b4eab84a6bfb6923b988774b84782bf006801bb5d7f054cdc641a78e12a3e61.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[406,391,633,520]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0919/5decaad71ec4b108a5eb1a850f8c273ae64d5065086a7b05008f88ea7bf0a80a.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[636,391,863,520]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 1. Scaffold characteristics. (A) Structural diagrams of PO-PP and POP scaffolds. PO-PP scaffold is a bi-layer scaffold, including inner layer (PO layer: PLLA embedded with OTF) and outer layer (PP layer: PLLA embedded with PNS). POP scaffold is a one-layer scaffold (PLLA embedded with OTF and PNS). (B) SEM images of PO-PP and POP scaffolds. PO layer exhibits porous structure at magnifications of $4500\\\\times$ ; PP layer shows electrospinning net at magnifications of $2500\\\\times$ ; POP scaffold has a porous structure at magnifications of $4500\\\\times$ . \"]","image_footnote":"[]","img_path":"images/0138/ae5fc31579a32780a9c6e913023dbfc4d9420640ce19ac0e15a33b3f24b36a49.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[57,61,233,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[897,61,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 of 17 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[300,110,507,241]","code_body":null,"code_caption":null,"image_caption":"[\"A \"]","image_footnote":"[]","img_path":"images/0287/2e102b79367aa42884c1e7f1539a54db01a392d23a03c6c2de8b4d1f89148e27.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[512,111,722,243]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0490/de01a740d271bbf4bdc0d6a21a3b5f8cb41ef1a0916195f36a3936659ada9f5e.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[722,111,924,242]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0392/23be232e017c3df9b621bec556f966443ae76f1629a4a0791115ff3de75da1c9.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[337,289,566,418]","code_body":null,"code_caption":null,"image_caption":"[\"B \"]","image_footnote":"[]","img_path":"images/0941/b3bfe358cccb12b11289ed3be1ac6fe2406a6814036680cac277a8954c2ca59e.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[660,288,887,421]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 2. Scaffold characteristics. (A) FTIR diagram of scaffolds including PO layer, PP layer, and POP scaffold. (B) Cumulative release profiles of OTF and PNS from POP and PO-PP scaffolds at different times; data are presented as mean $\\\\pm$ SD, $n = 10$ . \"]","image_footnote":"[]","img_path":"images/0602/0198750b637bc564c474c5029cb346fb5c4b3e0ecd4e227b15d6c672fb7f6eb1.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[275,508,939,595]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0020/0a27103b75f1ffc5211e03bb1fed19896ff490be76dc4c6f5a8bd49d4e68b39d.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":"<table><tr><td>Scaffold</td><td>Porosity (%)</td><td>Maximum Compressive Strength (MPa)</td></tr><tr><td>PO layer in PO-PP scaffold</td><td>89.56 ± 2.15</td><td>1.46 ± 0.13</td></tr><tr><td>POP scaffold</td><td>91.06 ± 2.13</td><td>1.34 ± 0.12</td></tr></table>","table_caption":"[\"Table 1. Porosity and maximum compressive strength of PO(PO-PP) and POP. \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[275,609,803,625]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.2. Micro-CT Analysis for Bone Formation Treated by Scaffold Implantation ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[275,628,939,825]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The micro-CT analysis was conducted twelve weeks after surgery. As shown in Figure 3, the scaffolds were successfully implanted into the femoral head of animals. The reconstructed 3D image indicates that the new bone formation in the PO-PP and POP groups was significantly better than that in the control group; and further PO-PP scaffold exhibits more new bone formation than the POP scaffold. The BMD, BV, and BV/TV of the PO-PP and POP groups were significantly higher than those of the control group $(p < 0.05)$ ; furthermore, the data of BMD, BV, and BV/TV in the PO-PP group show significant differences compared with those in the POP group $(p < 0.01)$ . The data of Tb.N and BS/TV in the PO-PP and POP groups show significant differences compared with those in the control group $(p < 0.05)$ . These results indicate that PO-PP and POP scaffolds all have a good osteogenesis effect. The osteogenic capacity of the PO-PP scaffold is superior to that of the POP scaffold, which indicates the advantage of sequential delivery (the key technique of PO-PP scaffold). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,61,233,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[897,61,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7 of 17 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[137,109,875,470]","code_body":null,"code_caption":null,"image_caption":"[\"A \",\"Figure 3. New bone formation in bone tunnel. (A) Scanning image (black): the focus of the red and green lines depicts the scaffold, (1) Coronal plane, (2) Sagittal plane, (3) Cross-section. Representative 3D micro-CT images (blue) within a ROI of central $3\\\\mathrm{mm}$ in diameter of the bone tunnel, (4) Coronal plane, (5) Cross-section. (B) Quantitative analysis of new bone in bone tunnel: BMD, Tb.N, BV, BV/TV, BS/TV. $\\\\mathfrak{n} = 10$ ; Error bar represents SD. ANOVA test with LSD analysis is used for statistical analysis; $^* p < 0.05$ vs. control group or POP group; $^{**}p < 0.01$ vs. control group or POP group. \"]","image_footnote":"[]","img_path":"images/0185/b974dd787ddd412f625b1327f260944854cb6adf2b11ebf8c35bb7143b4e7bd6.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[273,583,559,598]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.3. Histological Analysis of Bone Repair ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[272,600,941,783]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the process of repairing GONFH, vessels delivered nutrients to the femoral head and improved the new bone formation in the necrotic area. In this experiment, newly formed vessel and bone around the tunnel were assessed by HE staining. As shown in Figure 4A, there were more new vessels and trabecular bones filled at the injury site of the femoral head of the PO-PP and POP groups as compared with that of the control group. The repair effect of the PO-PP group was superior to that of the POP group. MT staining was commonly used to identify collagen (blue) and osteoid tissue (red). A mount of collagen fibers and osteoid tissue were filled around the defect regions in the treatment groups (Figure 4B). The therapeutic effect of the PO-PP group was superior to that of the POP group. The results of HE and Masson staining indicated better angiogenic and osteogenic effects of the PO-PP scaffold. It is the contribution of the PO-PP scaffold's sequential release that enhances the osteogenic effect of vascularization. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,60,233,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[897,60,941,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"8 of 17 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[163,112,391,296]","code_body":null,"code_caption":null,"image_caption":"[\"A \",\"Control \"]","image_footnote":"[]","img_path":"images/0489/8a31bf0a65c34dd674ba0947321271de50c4274719705e8cf1e5be1a5894ab41.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[398,112,628,296]","code_body":null,"code_caption":null,"image_caption":"[\"PO-PP \"]","image_footnote":"[]","img_path":"images/0735/e4c15c7480b1e900af704d7bf08922dfb4ac8b09d8ef7dab236630c77bdf6c5f.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[638,112,865,296]","code_body":null,"code_caption":null,"image_caption":"[\"POP \"]","image_footnote":"[]","img_path":"images/0717/7d7302224936bea7903dd54c4bdc098efcca4d1d8b1d4302cd971c233e0334c2.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[163,318,391,500]","code_body":null,"code_caption":null,"image_caption":"[\"B \",\"Control \"]","image_footnote":"[]","img_path":"images/0664/21f6fb2ad594d137fc70b110463ffa7115837b593e4799bd80edde2d82a5bba3.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[400,319,626,500]","code_body":null,"code_caption":null,"image_caption":"[\"PO-PP \"]","image_footnote":"[]","img_path":"images/0502/5d361ef1b6823ee8ef26fc8003f6a2483f63af55ac989ba554e60cced7b0508c.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[636,319,863,406]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0774/a2ea538fb413f9128c76f9f5e5995000fc89b607aba3845f6ccb7ad14bb6cda0.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[636,406,863,500]","code_body":null,"code_caption":null,"image_caption":"[\"POP \",\"Figure 4. Histological analysis of bone repair after scaffold implantation. (A) H&E staining of decalcified sections (bone tunnel, black circle) at magnifications of $20 \\\\times$ and $200 \\\\times$ . The $200 \\\\times$ images represent higher magnifications of areas within black boxes. White arrow indicates trabecular bone, blue arrow indicates vessel, bars = 500 and $50 \\\\mu \\\\mathrm{m}$ for $20 \\\\times$ and $200 \\\\times$ , respectively. (B) Masson trichrome staining of decalcified sections (bone tunnel: black circle) at magnifications of $20 \\\\times$ and $200 \\\\times$ . $200 \\\\times$ images are higher magnifications of the areas within black boxes. White arrow indicates collagen (blue) and osteoid tissue (red), bars = 500 and $50 \\\\mu \\\\mathrm{m}$ for $20 \\\\times$ and $200 \\\\times$ , respectively. \"]","image_footnote":"[]","img_path":"images/0641/36401d9d832a3d9aecbc07bf0c1492996805804ea8e81c1353f3808c1c3b4da6.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[272,630,939,934]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bone, a dynamic tissue, remodels continuously throughout life, which constitutes an important process for bone repair [27]. Bone remodeling is tightly regulated by the balance between osteoblasts and osteoclasts [27]. Osteoid, a collagen-rich matrix, is produced by osteoblasts and mineralized extracellularly by nanosized calcium phosphate (CaP) [28]. The formed osteoid in the mineralized bone represents the osteogenic capacity. Goldner staining was conducted to distinguish between osteoid tissue (red) and mineralized bone tissue (green). TRAP staining, used to reveal the osteoclast activity, was also performed to assess the osteoclastic bone resorption [29]. The results of Goldner staining (Figure 5A) show a large amount of osteoid tissue around the tunnel, appearing near the implantation site of the treatment groups. The amount of osteoid tissue in the PO-PP group was more than those in the POP group, which indicates better promoting effect of osteogenic activity in the PO-PP scaffold. As shown in Figure 5B, a large quantity of TRAP-positive red cells was detected near the tunnel of the control group, while the defect regions in the PO-PP and POP groups were surrounded with a small quantity of TRAP-positive red cells after implantation. The quantity of TRAP-positive red cells was lower in the PO-PP group than in the POP group, which indicates a better inhibiting effect of osteoclastic activity in the PO-PP scaffold. In summary, the PO-PP scaffold has better effects in promoting osteogenic activity and inhibiting osteoclastic activity, owing to the sequential release of the PO-PP scaffold. Initially released PNS not only promotes vessel formation, but also enhances new bone formation by providing sufficient nutrition to bone tissue. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,60,233,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[897,60,941,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"9 of 17 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[171,117,394,292]","code_body":null,"code_caption":null,"image_caption":"[\"A \",\"Control \"]","image_footnote":"[]","img_path":"images/0866/a3bbf1e41e129d514f5ac1066bed65ba53a61c73028e0d6b89fd598dac1dc0f4.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[405,117,628,293]","code_body":null,"code_caption":null,"image_caption":"[\"PO-PP \"]","image_footnote":"[]","img_path":"images/0134/530d6a58986fb520fedba5df7bc8360b402acee2da8cfda58a8ba89f59d785fa.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[638,116,863,292]","code_body":null,"code_caption":null,"image_caption":"[\"POP \"]","image_footnote":"[]","img_path":"images/0354/7d3bf6200510c9532b0162c4913c9bcb285afe0b17f7374add46c5a96789b661.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[169,304,394,477]","code_body":null,"code_caption":null,"image_caption":"[\"B \",\"Control \"]","image_footnote":"[]","img_path":"images/0711/cfd169b210e5c81b992615feec930d716e798ae6c0cc36bcd6c29312c241f746.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[403,305,626,476]","code_body":null,"code_caption":null,"image_caption":"[\"PO-PP \"]","image_footnote":"[]","img_path":"images/0678/6c4c47c795534e725a2fb2854451fd5972676ee625e12b21f421ab6df0d7c542.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[638,305,860,477]","code_body":null,"code_caption":null,"image_caption":"[\"POP \",\"Figure 5. Histological analysis of bone repair after scaffold implantation. (A) Goldner staining of decalcified sections (bone tunnel: black circle) at magnifications of $20\\\\times$ and $200\\\\times$ ; $200\\\\times$ images are higher magnifications of areas within black boxes. White arrow indicates osteoid tissue, bars = 500 and $50~{\\\\mu\\\\mathrm{m}}$ for $20\\\\times$ and $200\\\\times$ , respectively. (B) TRAP staining of decalcified sections at magnifications of $20\\\\times$ and $200\\\\times$ , $200\\\\times$ images are higher magnifications of areas within black boxes. Red arrow indicates osteoclast, bars = 500 and $50~{\\\\mu\\\\mathrm{m}}$ for $20\\\\times$ and $200\\\\times$ , respectively. \"]","image_footnote":"[]","img_path":"images/0886/07ec784a3902441f1e58dacac59c78ba7111170f3525dcdcb7353042705d9389.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[275,602,776,616]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.4. Distribution and Quantity of VEGF and CD31 in Implantation Sites ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[275,619,939,832]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Vascularization during bone repair supports the recruitment of circulating osteoblast-lineage cells for osteogenesis [30]. This is the key to the sequential release of the PO-PP scaffold. Therefore, to investigate the effector mechanism of vascularization, VEGF and CD31 stainings were performed (Figures 6A and 7A). Positive cells were stained brown for VEGF (IHC), and green for CD31(IMA). The distribution and quantity of VEGF proteins was significantly increased in the treatment groups compared with those in the control group $(p < 0.01)$ , and furthermore, the PO-PP scaffold showed much higher VEGF protein expression when compared with the POP scaffold $(p < 0.01)$ (Figure 6B). The same observation was made on the CD31 protein in the treatment groups, compared with that of the control group around the tunnel after implantation $(p < 0.01)$ . The PO-PP scaffold also showed significantly higher CD31 protein expression compared with the POP scaffold $(p < 0.01)$ (Figure 7B). The higher protein expressions of VEGF and CD31 in the PO-PP scaffold might be attributed to the sequential release of the PO-PP scaffold, where PNS is released first and at a faster release rate. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,61,233,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[890,61,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10 of 17 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[109,106,633,410]","code_body":null,"code_caption":null,"image_caption":"[\"B \"]","image_footnote":"[]","img_path":"images/0522/3f73e41954e71340874d3ad112e0e70c61bae014efe87ab0d9ab9def8259517c.jpg","list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[663,191,878,355]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0447/deb5af8008bccd667601af3925c3bc6fac35fe0a29900ffedee8c22d4c73b1db.jpg","list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[115,529,640,806]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 6. Immunohistochemistry staining of VEGF after scaffold implantation. (A) Distribution and quantity of VEGF in decalcified sections (bone tunnel, black circle) at magnifications of $20\\\\times$ and $200\\\\times$ . $200\\\\times$ images are higher magnifications of areas within black boxes. Red arrow indicates VEGF positive cells, bars = 500 and $50~{\\\\mu\\\\mathrm{m}}$ for $20\\\\times$ and $200\\\\times$ , respectively. (B) Quantitative analysis of VEGF expression. $\\\\mathfrak{n} = 10$ in each group, error bar represents SD. ANOVA test with LSD analysis was used for statistical analysis, $^{*}p < 0.05$ vs. control group or POP group; $^{**}p < 0.01$ vs. control group or POP group. \",\"B \"]","image_footnote":"[]","img_path":"images/0644/b573f5eb729c2325e71f36da6fab1dcfe55468874fff782bcfd27ff82488807b.jpg","list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[663,612,870,771]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 7. Immunofluorescence staining of CD31 after scaffold implantation. (A) Distribution and quantity of CD31 in decalcified sections (bone tunnel: blue circle) at magnifications of $20\\\\times$ and $200\\\\times$ . $200\\\\times$ images are higher magnifications of areas within blue boxes. White arrow indicates CD31 positive cells, bars = 500 and $50~{\\\\mu\\\\mathrm{m}}$ for $20\\\\times$ and $200\\\\times$ , respectively. (B) Quantitative analysis of CD31 expression. $\\\\mathfrak{n} = 10$ in each group, error bar represents SD. ANOVA test with LSD analysis was used for statistical analysis; $*p < 0.05$ vs. control group or POP group; \\\\*\\\\* $p < 0.01$ vs. control group or POP group. \"]","image_footnote":"[]","img_path":"images/0368/d42c762ed1d468f4c4ffd3cf0715d137e027ab89b0ad5b70a83cd8d4690edf65.jpg","list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[55,60,233,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[890,60,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"11 of 17 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[277,108,811,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.5. Distribution and Quantity of OCN, OPN, and ALP in Implantation Sites ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[277,127,939,263]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To explore the osteogenesis mechanism, the distribution and quantity of OCN, OPN, and ALP around implantation sites were assessed (Figures 8-10). The positive cells were stained brown for OCN and OPN (IHC), and red for ALP (IMA). The distribution and quantity of OCN, OPN, and ALP in the treatment groups were increased significantly compared to those in the control group $(p < 0.01)$ . Furthermore, the data of OCN, OPN, and ALP expression in the PO-PP scaffold showed significant differences when compared with those in the POP scaffold $(p < 0.05)$ . This represented the advantage of the sequential release of the PO-PP scaffold, which not only provided the osteogenic effect of vascularization (PP layer), but also the osteogenic effect of OTF (PO layer). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[275,276,709,517]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0426/23eb6f47cd838a7d11a6da8ea5e7827e7d9e05fd55597a846fe0aaf2afbf2288.jpg","list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[729,274,910,480]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0814/4dd6e7a62275819fc7dde755d5e59e01b6ed44c6e9e0647c2daa96f8d26a92b3.jpg","list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[161,631,633,897]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 8. Immunohistochemistry staining of OCN after scaffold implantation. (A) Distribution and quantity of OCN in decalcified sections (bone tunnel: black circle) at magnifications of $20\\\\times$ and $200\\\\times$ . $200\\\\times$ images are higher magnifications of areas within black boxes. Red arrow indicates OCN positive cells, bars = 500 and $50~{\\\\mu\\\\mathrm{m}}$ for $20\\\\times$ and $200\\\\times$ , respectively. (B) Quantitative analysis of OCN expression. $n = 10$ in each group, error bar represents SD. ANOVA test with LSD analysis was used for statistical analysis, $^{*}p < 0.05$ vs. control group or POP group; $^{**}p < 0.01$ vs. control group or POP group. \"]","image_footnote":"[]","img_path":"images/0253/526a5a2e3b84123b44d385bac37928d80828fa0ea702f50566e3d2a3254bfde1.jpg","list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[640,630,830,847]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 9. Immunohistochemistry staining of OPN after scaffold implantation. (A) Distribution and quantity of OPN in decalcified sections (bone tunnel, black circle) at magnifications of $20\\\\times$ and $200\\\\times$ . \"]","image_footnote":"[]","img_path":"images/0907/5742e661e2cbe9406f0548c2dcc19d336a9ae6ea15e3d9bfebf138448e5b51d1.jpg","list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[57,61,233,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[890,61,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"12 of 17 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[272,106,941,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$200\\times$ images are higher magnifications of areas within black boxes. Red arrow indicates OPN positive cells, bars $= 500$ and $50~\\mu \\mathrm{m}$ for $20\\times$ and $200\\times$ , respectively. (B) Quantitative analysis of OPN expression. $n = 10$ in each group, error bar represents SD. ANOVA test with LSD analysis was used for statistical analysis, $^{*}p < 0.05$ vs. control group or POP group, $^{**}p < 0.01$ vs. control group or POP group. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[134,190,623,461]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0196/5113ce52c44d094872778a9f32f6357e23d3f92e97c753115d66d597ef4340c1.jpg","list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[638,188,860,426]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 10. Immunofluorescence staining of ALP after scaffold implantation. (A) Distribution and quantity of ALP expression in decalcified sections (bone tunnel: blue circle) at magnifications of $20\\\\times$ and $200\\\\times$ . $200\\\\times$ images are higher magnifications of areas within blue boxes. White arrow indicates ALP positive cells, bars = 500 and $50~{\\\\mu\\\\mathrm{m}}$ for $20\\\\times$ and $200\\\\times$ , respectively. (B) Quantitative analysis of ALP expression. $\\\\mathfrak{n} = 10$ in each group, error bar represents SD. ANOVA test with LSD analysis was used for statistical analysis, $^{*}p < 0.05$ vs. control group or POP group; $^{**}p < 0.01$ vs. control group or POP group. \"]","image_footnote":"[]","img_path":"images/0631/0afa736ab7e3e7436397dfc22da0cc952dc42009fbff26c08a49af1c501aa7fd.jpg","list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[273,579,381,593]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4. Discussion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[272,598,939,703]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"GONFH, characterized by progressive deterioration of the hip joint [31], is a serious complication occurring after long-term or excess administration of clinical glucocorticoids [32]. With the development of GONFH, the femoral head will collapse, ultimately requiring total hip arthroplasty (THA) [33,34]. About 30-60,000 new cases of GONFH occur in the United States every year [35], and approximately 8.12 million people suffer from GONFH in China [36]. GONFH can create substantial financial, physical, and emotional burdens for patients and their families. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,705,941,934]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Glucocorticoids have an inhibitory function on angiogenesis in the femoral head, such that synthetic glucocorticoid treatment causes vascularity disruption at the site [37]. Osteogenesis requires an adequate blood supply [38], and vascular damage easily results in an inability to deliver nutrients to the femoral head [1]. Angiogenesis and osteogenesis are closely related to femoral head repair [4]. To achieve the osteogenic effect of vascularization, we aim to design a sequentially drug-releasing scaffold. PLLA, a biodegradable and biocompatible scaffold material, has been widely used as carrier for delivering drugs or bioactive factors in orthopedic treatment [39]. PNS, the main ingredient of Panax Notoginseng, is able to promote angiogenesis. It thus prevents the loss of vascular endothelial cells [40]. OTF, the major component of Drynariae Rhizoma, promotes osteogenesis. Several studies showed that OTF promotes bone formation in large tibial defects (LTDs) [18], and thus increases the bone density and mechanical strength of tail suspension rats [41]. In this study, we designed a sequentially released bi-layer PLLA scaffold, where the inner layer contained OTF, and the outer layer contained PNS. PNS in the outer layer was initially released to promote vessel formation and, more importantly, to enhance the ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,59,233,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[890,59,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"13 of 17 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[278,108,939,198]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"osteogenic effect of vascularization (the key technique employed in our study). The OTF in the inner layer was released later to promote new bone formation. We also made a one-layer PLLA scaffold containing OTF and PNS, which were successfully loaded into scaffolds. The PO-PP and POP scaffolds released PNS and OTF steadily and durably. The PNS release was faster in the PO-PP scaffold than in the POP scaffold, which also supported the osteogenic effect of vascularization. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,200,939,471]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"After scaffold implantation, we assessed new bone formation through a micro-CT. The results of reconstructed 3D image showed that the new bone formation in the treatment groups was significantly better than that in the control group; and the PO-PP scaffold had more new bone formation than the POP scaffold. The data of BMD, BV, and BV/TV in the treatment groups were significantly higher than those in the control group with $p < 0.05$ ; and the data of BMD, BV, and BV/TV in the PO-PP group were significantly higher than those in the POP group with $p < 0.01$ . These results indicated that both scaffolds had good osteogenesis effect, while the osteogenic effect of the PO-PP scaffold was superior to that of the POP scaffold. We observed the formation of new vessels and new bone via HE staining. The results showed that both scaffolds exhibited more vessel and bone formation around the defect when compared with the control group, and more vessel and bone formation was observed in the PO-PP scaffold than in the POP scaffold. Because bone formation is tightly regulated by the balance between osteoblasts and osteoclasts [27], we observed the osteogenic activity by Masson staining, Goldner staining, and the osteoclastic activity by TRAP staining. The results of these stainings indicate that both scaffolds promote osteogenic activity and inhibit osteoclastic activity. Interestingly, superior effects (promoting osteogenic activity and inhibiting osteoclastic activity) were found in the PO-PP scaffold compared with the POP scaffold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,475,939,624]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the process of promoting bone repair through angiogenesis and osteogenesis, the PO-PP scaffold not only promotes more vessel formation via the faster-released PNS, but the PNS in the outer layer enhanced the osteogenic effect of vascularization based on the sequential release system. Therefore, sequential release, which represent the key technology in our study, not only achieved more angiogenesis, but also indirectly promoted more new bone formation. Moreover, the OTF in the inner layer also promoted new bone formation. The one-layer scaffold (POP) obtained a slow PNS release. Furthermore, the simultaneous release of PNS and OTF lost the advantage of the sequential release, thus depriving PNS of its role in indirect osteogenesis. Therefore, a sequential release is the key technology in the PO-PP scaffold to achieve better bone repair. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,625,939,928]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To explore the possible mechanisms of angiogenesis and osteogenesis, the distribution and quantity of key proteins were evaluated through IHC and IMA. VEGF (a homodimeric disulfide bound glycoprotein) promotes endothelial growth [42]. CD31 (the most abundant constitutive co-signaling receptor glycoprotein on endothelial cells) is highly expressed at endothelial cell-cell junctions [43], and forms the first-line of cellular contact with the blood [44]. The results in our study showed that the protein expressions of VEGF and CD31 in the treatment groups were significantly higher than those in the control group with $p < 0.01$ , and the protein expressions of VEGF and CD31 in the PO-PP scaffold were significantly higher than those in the POP scaffold with $p < 0.01$ . The enhancing effect of the PO-PP scaffold might be attributed to the sequential release of the PO-PP scaffold where PNS is released first and at a faster release rate. OCN synthesized by osteoblasts is known for its strong binding affinity to bone minerals [45]. OPN, which binds to extracellular calcium via acidic (aspartate and glutamate) and phosphorylated (serine and threonine) residues, plays a key role in nucleating bone minerals [46]. ALP (a homodimer anchored to the membrane of osteoblasts) is essential for biomineralization [47]. The results indicated that the protein expressions of OCN, OPN, and ALP in the treatment groups were significantly higher than those in the control group with $p < 0.01$ , and the protein expressions of OCN, OPN, and ALP in the PO-PP scaffold were significantly higher than those in the POP scaffold with $p < 0.05$ . The promotion function of the PO-PP scaffold was better than that of the POP scaffold. Based on the sequential release ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,62,233,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[890,61,939,70]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"14 of 17 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[272,106,939,156]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"advantage of the bi-layer scaffold, the PO-PP scaffold had a better angiogenic and osteogenic effect. Research into the mechanisms further found that the PO-PP scaffold achieved the above effects by promoting the expression of angiogenic and osteogenic proteins. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,168,391,182]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5. Conclusions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[272,186,944,369]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We fabricated a sequentially released bi-layer scaffold called a PO-PP scaffold. In this PO-PP scaffold, PNS was initially released, followed by the release of OTF to obtain the osteogenic effect of angiogenesis (the key advantage of our scaffold). Compared with the POP scaffold (with concomitant release of OTF and PNS), the PO-PP scaffold had better effects in promoting new bone formation and blood vessel formation, promoting osteoid tissue formation, and inhibiting osteoclastic activity. Upregulating the expression of angiogenic proteins (VEGF and CD31) and osteogenic proteins (OCN, OPN and ALP) might be the possible mechanism. Therefore, the sequentially released PO-PP scaffold proposes a new therapy for bone repair. Based on the osteogenic effect of angiogenesis of the sequentially released PO-PP scaffold, we want to apply this theory to osteoporotic bone defect in rats (different disease and different species) in future to explore the possible effects and mechanisms. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,382,941,494]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Supplementary Materials: The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/jfb14010031/s1, Figure S1: Purity identification of osteoptractic total flavone (OTF). A, HPLC ion chromatogram of ATF (Reference substance): Neoeriocitrin and Naringin. B, HPLC ion chromatogram of ATF (Sample): Neoeriocitrin and Naringin. C, the molecular formulas of Neoeriocitrin and Naringin; Figure S2: Purity identification of panax notoginseng saponins (PNS). A, HPLC ion chromatogram of PNS (Reference substance): Ginsenoside Rg 1 and Ginsenoside Re. B, HPLC ion chromatogram of PNS (Sample): Ginsenoside Rg 1 and Ginsenoside Re. C, the molecular formulas of Ginsenoside Rg 1 and Ginsenoside Re ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,505,941,562]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Author Contributions: Writing—original draft, G.F.; Conceptualization, L.C.; X.N.; F.Y.; J.L.; D.L. and X.Z.; Supervision, L.C.; Writing—review & editing, L.C.; Resources, Y.Y.; X.N.; J.H.; S.S. and W.W.; Visualization, P.Z. All authors have read and agreed to the published version of the manuscript. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,574,941,605]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Funding: This research was financially supported by the National Natural Science Foundation of China (82074463). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,610,941,654]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Institutional Review Board Statement: The study was conducted according to the guidelines of the Declaration of Helsinki, and approved by the Ethics Committee of Beijing University of Aeronautics and Astronautics (protocol code: BM20220026, date of approval: 2022.03.04). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,659,594,674]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Informed Consent Statement: Not applicable. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,679,939,709]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Data Availability Statement: The data presented in this study are available on request from the corresponding author. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,714,707,728]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conflicts of Interest: The authors declare no conflict of interest. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,743,147,756]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[55,761,941,926]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. Chang, C.; Greenspan, A.; Gershwin, M.E. The pathogenesis, diagnosis and clinical manifestations of steroid-induced osteonecrosis. J. Autoimmun. 2020, 110, 102460. \",\"2. Zalavras, C.; Shah, S.; Birnbaum, M.J.; Frenkel, B. Role of apoptosis in glucocorticoid-induced osteoporosis and osteonecrosis. Crit. Rev. Eukaryot. Gene Expr. 2003, 13, 221-235. \",\"3. Li, J.; Fan, L.; Yu, Z.; Dang, X.; Wang, K. The effect of deferoxamine on angiogenesis and bone repair in steroid-induced osteonecrosis of rabbit femoral heads. Exp. Biol. Med. 2015, 240, 273-280. \",\"4. Liao, Y.; Zhang, P.; Yuan, B.; Li, L.; Bao, S. Pravastatin Protects Against Avascular Necrosis of Femoral Head via Autophagy. Front. Physiol. 2018, 9, 307. \",\"5. Tang, D.; Tare, R.S.; Yang, L.Y.; Williams, D.F.; Ou, K.L.; Oreffo, R.O. Biofabrication of bone tissue: Approaches, challenges and translation for bone regeneration. Biomaterials 2016, 83, 363-382. \",\"6. Farokhi, M.; Mottaghitalab, F.; Shokrgozar, M.A.; Ou, K.L.; Mao, C.; Hosseinkhani, H. Importance of dual delivery systems for bone tissue engineering. J. Control Release 2016, 225, 152-169. \"]","page_idx":"14","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[55,60,233,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[890,60,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"15 of 17 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[53,106,941,916]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"7. Yao, C.H.; Lai, Y.H.; Chen, Y.W.; Cheng, C.H. Bone Morphogenetic Protein-2-Activated 3D-Printed Polylactic Acid Scaffolds to Promote Bone Regrowth and Repair. Macromol. Biosci. 2020, 20, e2000161. \",\"8. Fairag, R.; Rosenzweig, D.H.; Ramirez-Garcialuna, J.L.; Weber, M.H.; Haglund, L. Three-Dimensional Printed Polylactic Acid Scaffolds Promote Bone-like Matrix Deposition in Vitro. ACS Appl. Mater. Interfaces 2019, 11, 15306-15315. \",\"9. Ritz, U.; Gerke, R.; Götz, H.; Stein, S.; Rommens, P.M. A New Bone Substitute Developed from 3D-Prints of Polylactide (PLA) Loaded with Collagen I: An In Vitro Study. Int. J. Mol. Sci. 2017, 18, 2569. \",\"10. Hu, J.; Liu, X.; Ma, P.X. Induction of osteoblast differentiation phenotype on poly(L-lactic acid) nanofibrous matrix. Biomaterials 2008, 29, 3815-3821. \",\"11. Iqbal, N.K.; Khan, A.S.; Asif, A.; Yar, M.; Haycock, J.W.; Rehman, I.U. Recent concepts in biodegradable polymers for tissue engineering paradigms: A critical review. Int. Mater. Rev. 2018, 64, 91-126. \",\"12. Obata, A.; Hotta, T.; Wakita, T.; Ota, Y.; Kasuga, T. Electrospun microfiber meshes of silicon-doped vaterite/poly(lactic acid) hybrid for guided bone regeneration. Acta Biomater. 2010, 6, 1248-1257. \",\"13. Jang, T.S.; Lee, E.J.; Jo, J.H.; Jeon, J.M.; Kim, M.Y.; Kim, H.E.; Koh, Y.H. Fibrous membrane of nano-hybrid poly-L-lactic acid/silica xerogel for guided bone regeneration. J. Biomed. Mater. Res. Part B Appl. Biomater. 2012, 100, 321-330. \",\"14. Hwang, C.; Park, S.; Kang, I.G.; Kim, H.E.; Han, C.M. Tantalum-coated polylactic acid fibrous membranes for guided bone regeneration. Mater. Sci. Eng. C Mater. Biol. Appl. 2020, 115, 111112. \",\"15. Hwang, H.D.; Lee, J.T.; Koh, J.T.; Jung, H.M.; Lee, H.J.; Kwon, T.G. Sequential Treatment with SDF-1 and BMP-2 Potentiates Bone Formation in Calvarial Defects. Tissue Eng. Part A 2015, 21, 2125-2135. \",\"16. Hong, S.J.; Wan, J.B.; Zhang, Y.; Hu, G.; Lin, H.C.; Seto, S.W.; Kwan, Y.W.; Lin, Z.X.; Wang, Y.T.; Lee, S.M. Angiogenic effect of saponin extract from Panax notoginseng on HUVECs in vitro and zebrafish in vivo. Phytother. Res. 2009, 23, 677-686. \",\"17. Zheng, H.; Liu, C.; Ou, Y.; Zhang, Y.; Fu, X. Total saponins of Panax notoginseng enhance VEGF and relative receptors signals and promote angiogenesis derived from rat bone marrow mesenchymal stem cells. J. Ethnopharmacol. 2013, 147, 595-602. \",\"18. Sun, W.; Li, M.; Zhang, Y.; Huang, Y.; Zhan, Q.; Ren, Y.; Dong, H.; Chen, J.; Li, Z.; Fan, C.; et al. Total flavonoids of rhizoma drynariae ameliorates bone formation and mineralization in BMP-Smad signaling pathway induced large tibial defect rats. Biomed. Pharmacother. 2021, 138, 111480. \",\"19. Wong, R.W.; Rabie, B.; Bendeus, M.; Hägg, U. The effects of Rhizoma Curculiginis and Rhizoma Drynariae extracts on bones. Chin. Med. 2007, 2, 13. \",\"20. Chen, L.L.; Lei, L.H.; Ding, P.H.; Tang, Q.; Wu, Y.M. Osteogenic effect of Drynariae rhizoma extracts and Naringin on MC3T3-E1 cells and an induced rat alveolar bone resorption model. Arch. Oral Biol. 2011, 56, 1655-1662. \",\"21. Jeong, J.C.; Kang, S.K.; Youn, C.H.; Jeong, C.W.; Kim, H.M.; Lee, Y.C.; Chang, Y.C.; Kim, C.H. Inhibition of Drynariae Rhizoma extracts on bone resorption mediated by processing of cathepsin K in cultured mouse osteoclasts. Int. Immunopharmacol. 2003, 3, 1685-1697. \",\"22. Li, J.; Jiang, J.; Zhao, X.; Sun, Q.; Zheng, C.; Bai, C.; Liu, C.; Jia, Y. Effects of osteopractic total flavone on proliferation of vascular endothelial cells of osteoblasts—Vascular endothelial cells co-culture system in nano-hydroxyapatite collagen composite. Chin. Med. Zhenggu 2019, 31, 1-8. \",\"23. Yang, F.G.; Li, J.; Niu, X. Improved modeling strategy and result evaluation of steroid-induced femoral head necrosis in rabbits. J. Beijing Univ. Aeronaut. Astronaut. 2022, 1-8. \",\"24. Lai, Y.; Li, Y.; Cao, H.; Long, J.; Wang, X.; Li, L.; Li, C.; Jia, Q.; Teng, B.; Tang, T.; et al. Osteogenic magnesium incorporated into PLGA/TCP porous scaffold by 3D printing for repairing challenging bone defect. Biomaterials 2019, 197, 207-219. \",\"25. Fiume, E.; Tulyaganov, D.; Ubertalli, G.; Verné, E.; Baino, F. Dolomite-Foamed Bioactive Silicate Scaffolds for Bone Tissue Repair. Materials 2020, 13, 628. \",\"26. Yang, F.; Niu, X.; Gu, X.; Xu, C.; Wang, W.; Fan, Y. Biodegradable Magnesium-Incorporated Poly(l-lactic acid) Microspheres for Manipulation of Drug Release and Alleviation of Inflammatory Response. ACS Appl Mater. Interfaces 2019, 11, 23546-23557. \",\"27. Kim, J.M.; Lin, C.; Stavre, Z.; Greenblatt, M.B.; Shim, J.H. Osteoblast-Osteoclast Communication and Bone Homeostasis. Cells 2020, 9, 2073. \",\"28. Kalia, P.; Vizcay-Barrena, G.; Fan, J.P.; Warley, A.; Di Silvio, L.; Huang, J. Nanohydroxyapatite shape and its potential role in bone formation: An analytical study. J. R. Soc. Interface 2014, 11, 20140004. \",\"29. Nemoto, Y.; Higuchi, K.; Baba, O.; Takano, Y. Multinucleate osteoclasts in medaka as evidence of active bone remodeling. Bone 2007, 40, 399-408. \",\"30. Eghbali-Fatourechi, G.Z.; Lamsam, J.; Fraser, D.; Nagel, D.; Riggs, B.L.; Khosla, S. Circulating osteoblast-lineage cells in humans. N. Engl. J. Med. 2005, 352, 1959-1966. \",\"31. Jilka, R.L.; Weinstein, R.S.; Bellido, T.; Roberson, P.; Parfitt, A.M.; Manolagas, S.C. Increased bone formation by prevention of osteoblast apoptosis with parathyroid hormone. J. Clin. Investig. 1999, 104, 439-446. \",\"32. Chen, L.; Wang, B.Z.; Xie, J.; Zhang, R.Y.; Jin, C.; Chen, W.K.; Fang, K.H.; Hong, C.X.; Xu, T.H.; Huang, C.B.; et al. Therapeutic effect of SIRT3 on glucocorticoid-induced osteonecrosis of the femoral head via intracellular oxidative suppression. Free Radic. Biol. Med. 2021, 176, 228-240. \",\"33. Weinstein, R.S. Glucocorticoid-induced osteonecrosis. Endocrine 2012, 41, 183-190. \",\"34. Jing, X.; Du, T.; Yang, X.; Zhang, W.; Wang, G.; Liu, X.; Li, T.; Jiang, Z. Desferoxamine protects against glucocorticoid-induced osteonecrosis of the femoral head via activating HIF-1 $\\\\alpha$ expression. J. Cell. Physiol. 2020, 235, 9864–9875. \"]","page_idx":"15","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[55,60,233,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[890,60,941,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"16 of 17 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[53,106,941,491]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"35. Fukushima, W.; Fujioka, M.; Kubo, T.; Tamakoshi, A.; Nagai, M.; Hirota, Y. Nationwide epidemiologic survey of idiopathic osteonecrosis of the femoral head. Clin. Orthop. Relat. Res. 2010, 468, 2715-2724. \",\"36. Zhao, D.W.; Yu, M.; Hu, K.; Wang, W.; Yang, L.; Wang, B.J.; Gao, X.H.; Guo, Y.M.; Xu, Y.Q.; Wei, Y.S.; et al. Prevalence of Nontraumatic Osteonecrosis of the Femoral Head and its Associated Risk Factors in the Chinese Population: Results from a Nationally Representative Survey. Chin. Med. J. 2015, 128, 2843-2850. \",\"37. Weinstein, R.S.; Hogan, E.A.; Borrelli, M.J.; Liachenko, S.; O'Brien, C.A.; Manolagas, S.C. The Pathophysiological Sequence of Glucocorticoid-Induced Osteonecrosis of the Femoral Head in Male Mice. Endocrinology 2017, 158, 3817-3831. \",\"38. Liu, X.; Chai, Y.; Liu, G.; Su, W.; Guo, Q.; Lv, X.; Gao, P.; Yu, B.; Ferbeyre, G.; Cao, X.; et al. Osteoclasts protect bone blood vessels against senescence through the angiogenin/plexin-B2 axis. Nat. Commun. 2021, 12, 1832. \",\"39. Kanczler, J.M.; Ginty, P.J.; Barry, J.J.; Clarke, N.M.; Howdle, S.M.; Shakesheff, K.M.; Oreffo, R.O. The effect of mesenchymal populations and vascular endothelial growth factor delivered from biodegradable polymer scaffolds on bone formation. Biomaterials 2008, 29, 1892-1900. \",\"40. Hu, H.; Chen, Y.; Zou, Z.; Li, L.; Wei, F.; Liu, C.; Ling, Z.; Zou, X. Panax Notoginseng Saponins Prevent Bone Loss by Promoting Angiogenesis in an Osteoporotic Mouse Model. Biomed. Res. Int. 2020, 2020, 8412468. \",\"41. Song, S.; Gao, Z.; Lei, X.; Niu, Y.; Zhang, Y.; Li, C.; Lu, Y.; Wang, Z.; Shang, P. Total Flavonoids of Drynariae Rhizoma Prevent Bone Loss Induced by Hindlimb Unloading in Rats. Molecules 2017, 22, 1033. \",\"42. Theis, V.; Theiss, C. VEGF-A Stimulus for Neuronal Development and Regeneration in the CNS and PNS. Curr. Protein Pept. Sci. 2018, 19, 589-597. \",\"43. Lertkiatmongkol, P.; Liao, D.; Mei, H.; Hu, Y.; Newman, P.J. Endothelial functions of platelet/endothelial cell adhesion molecule-1 (CD31). Curr. Opin. Hematol. 2016, 23, 253-259. \",\"44. Caligiuri, G. Mechanotransduction, immunoregulation, and metabolic functions of CD31 in cardiovascular pathophysiology. Cardiovasc. Res. 2019, 115, 1425-1434. \",\"45. Dowd, T.L.; Rosen, J.F.; Li, L.; Gundberg, C.M. The three-dimensional structure of bovine calcium ion-bound osteocalcin using 1H NMR spectroscopy. Biochemistry 2003, 42, 7769-7779. \",\"46. Gericke, A.; Qin, C.; Spevak, L.; Fujimoto, Y.; Butler, W.T.; Sorensen, E.S.; Boskey, A.L. Importance of phosphorylation for osteopontin regulation of biomineralization. Calcif. Tissue Int. 2005, 77, 45-54. \",\"47. Nizet, A.; Cavalier, E.; Stenvinkel, P.; Haarhaus, M.; Magnusson, P. Bone alkaline phosphatase: An important biomarker in chronic kidney disease-mineral and bone disorder. Clin. Chim. Acta 2020, 501, 198-206. \"]","page_idx":"16","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[53,505,941,548]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Disclaimer/Publisher's Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,61,233,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2023, 14, 31 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[890,60,941,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"17 of 17 ","text_format":null,"text_level":null,"type":"page_number"}],"string":"[\n {\n \"bbox\": \"[58,124,107,136]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Article \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[58,142,899,251]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sequential Release of Panax Notoginseng Saponins and OsteopRACTIC Total Flavone from Poly (L-Lactic Acid) Scaffold for Treating Glucocorticoid-Associated Osteonecrosis of Femoral Head \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[58,269,840,300]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Guiyu Feng $^{1,\\\\dagger}$ , Pingxin Zhang $^{1,\\\\dagger}$ , Jian Huang $^{2,\\\\dagger}$ , Yao Yu $^{3,\\\\dagger}$ , Fenghe Yang $^{3}$ , Xueqian Zhao $^{4}$ , Wei Wang $^{1}$ , Dongyang Li $^{1}$ , Song Sun $^{1}$ , Xufeng Niu $^{3,*}$ , Limin Chai $^{1,*}$ and Jinyu Li $^{2,*}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,459,243,471]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Citation:Feng,G.; Zhang,P. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,473,245,486]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Huang, J.; Yu, Y.; Yang, F.; Zhao, X.; \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,488,231,500]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Wang, W.; Li, D.; Sun, S.; Niu, X. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,502,230,514]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"et al. Sequential Release of Panax \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,516,200,527]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Notoginseng Saponins and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,530,226,541]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Osteopractic Total Flavone from \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,543,221,555]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Poly (L-Lactic Acid) Scaffold for \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,558,242,568]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Treating Glucocorticoid-Associated \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,573,231,584]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Osteonecrosis of Femoral Head. $J.$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,587,201,598]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Funct. Biomater. 2023, 14, 31. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,602,238,612]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.3390/jfb14010031 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[58,624,233,633]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Academic Editor: Aurelio Salerno \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[58,644,208,655]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Received: 23 November 2022 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,660,200,669]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Revised: 23 December 2022 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,675,203,685]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Accepted: 23 December 2022 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,688,191,699]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Published: 4 January 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,706,149,730]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0729/2673904523d27fc444cad8e3f1d0e65bcea691218b204d294e26f3a4e4531c11.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[57,737,250,833]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[277,321,929,448]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"<sup>1</sup> Key Laboratory of Chinese Internal Medicine of Ministry of Education and Beijing, Dongzhimen Hospital, Beijing University of Chinese Medicine, Beijing 100700, China \\\",\\\"$^{2}$ Department of Orthopedic, Dongzhimen Hospital, Beijing University of Chinese Medicine, Beijing 100700, China \\\",\\\"<sup>3</sup> Key Laboratory of Biomechanics and Mechanobiology (Beihang University), Ministry of Education, Beijing Advanced Innovation Center for Biomedical Engineering, School of Biological Science and Medical Engineering, Beihang University, Beijing 100083, China \\\",\\\"4 Yuquan Hospital Affiliated to Tsinghua University, Beijing 100040, China \\\",\\\"* Correspondence: nxf@buaa.edu.cn (X.N.); liminchai@hotmail.com (L.C.); leery_5566@163.com (J.L.) \\\",\\\"These authors contributed equally to this work. \\\"]\",\n \"page_idx\": \"0\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[278,466,939,771]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract: Glucocorticoids inhibit angiogenesis in the femoral head, which fails to nourish the bone tissue and leads to osteonecrosis. Restoring angiogenesis is not only essential for vessel formation, but also crucial for osteogenesis. Poly (L-lactic acid) (PLLA) is commonly used in the bone tissue engineering field. Panax notoginseng saponins (PNS) and osteopractic total flavone (OTF) promote angiogenesis and osteogenesis, respectively. We designed a sequentially releasing PLLA scaffold including PLLA loaded with OTF (inner layer) and PLLA loaded with PNS (outer layer). We assessed the osteogenic effect of angiogenesis in this scaffold by comparing it with the one-layered scaffold (PLLA embedded with OTF and PNS) in vivo. Results from the micro-CT showed that the data of bone mineral density (BMD), bone volume (BV), and percent bone volume (BV/TV) in the PO-PP group were significantly higher than those in the POP group $(p < 0.01)$ . Histological analyses show that the PO-PP scaffold exhibits better angiogenic and osteogenic effects compared with the one-layered scaffold. These might result from the different structures between them, where the sequential release of a bi-layer scaffold achieves the osteogenic effect of vascularization by initially releasing PNS in the outer layer. We further explored the possible mechanism by an immunohistochemistry analysis and an immunofluorescence assay. The results showed that the protein expressions of vascular endothelial growth factor (VEGF) and platelet endothelial cell adhesion molecule-1(CD31) in the PO-PP scaffold were significantly higher than those in the POP scaffold $(p < 0.01)$ ; the protein expressions of osteocalcin (OCN), osteopontin (OPN), and alkaline phosphatase (ALP) in the PO-PP scaffold were significantly higher than those in the POP scaffold $(p < 0.05)$ . Upregulating the expressions of angiogenic and osteogenic proteins might be the possible mechanism. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,789,939,817]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords: Poly (L-lactic acid); osteopractic total flavone; panax notoginseng saponins; osteogenesis; angiogenesis \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,59,115,100]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"# \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[119,60,221,98]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Journal of Functional Biomaterials \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[865,61,942,97]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"MDPI \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[58,967,445,978]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31. https://doi.org/10.3390/jfb14010031 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[774,967,939,978]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"www.mdpi.com/journal/jfb \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[278,108,391,122]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,125,939,277]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Glucocorticoid-associated osteonecrosis of the femoral head (GONFH) is known as bone death, resulting from the use of chronic glucocorticoids [1]. Glucocorticoids could directly damage endothelial cells, impair new blood vessels formation, and decrease the vascular supply to bone tissue [2]. The vascular supply to the femoral head involves lateral and medial circumflex femoral arteries, which penetrates the femoral head and supply most of the femoral epiphysis. Vascular damage easily leads to the inability to deliver nutrients to the bone tissue of the femoral head, leading to femoral head necrosis [1]. Restoring angiogenesis is not only essential for vessel formation, but also crucial for osteogenesis [3,4]. Angiogenesis and osteogenesis, closely related to each other, are both crucial for bone repair. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,279,939,460]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bone tissue engineering is a promising strategy for bone repair [5,6]. Biomaterials represent a key element in bone tissue engineering. Poly (L-lactic acid) (PLLA), a highly versatile biodegradable aliphatic polyester, is commonly used as degradable biomaterial [7]. It has been widely used in biomedical and surgical fields, such as surgical equipment and fusion substance, tissue engineering scaffolds, and drug carriers including orthopedic treatment, owing to its safety and low toxicity [8-10]. The most commonly used PLLA in tissue engineering has been approved by FDA [11]. Recent medical studies confirmed that the implant of an electrospun, silicon-doped vaterite/PLLA membrane used to treat the calvaria defect in the rabbit model could enhance the proliferation of pre-osteoblasts and promote new bone formation [12]. Implanted PLLA also could improve the attachment, proliferation, and differentiation of pre-osteoblasts, and lead to the effective osteo-conductivity contributing to new bone formation in vitro and in vivo [13,14]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,461,939,583]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In addition to biomaterials, drug delivery systems comprise another key element in bone tissue engineering. The sequential treatment of collagen scaffolds with bone morphogenetic protein-2 (BMP-2) following stromal cell derived factor-1 (SDF-1) showed superior new bone regeneration compared with their simultaneous treatment [15]. The initial release of BMP-2, followed by the sequential release of alendronate (ALN) in collagen-hydroxyapatite composite scaffold, significantly increases osteogenic activity owing to the synergistic effect of BMP-2 and ALN. In this study, we aim to obtain a sequentially releasing PLLA scaffold, which enhances the osteogenic effect by restoring angiogenesis. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,584,939,795]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Panax notoginseng saponins (PNS) is the main ingredient of Panax Notoginseng (a medicinal herb). Treatment by PNS promotes angiogenesis, as PNS activates the VEGF-KDR and PI3K-Akt-eNOS pathways, which contribute to angiogenesis [16]. Furthermore, PNS induces the production of VEGF-A, VEGFR-1, and VEGFR-2 in bone marrow mesenchymal stem cells (BMSCs) [17]. OsteopRACTic total flavone (OTF) is the main ingredient of Drynariae Rhizoma (a medicinal herb). Several studies confirmed that OTF could improve the formation of new bone. OTF in vivo experiments could significantly promote the formation of epiphyses and bone trabeculae, increase bone mineralization, accelerate the differentiation and mineralization of osteoblasts, and inhibit the bone resorption of osteoclasts [18-21]. In this study, we constructed a bi-layered PLLA scaffold including PLLA embedded with OTF (PO, inner layer) and PLLA embedded with PNS (PP, outer layer). We hypothesize that PNS in the outer layer would be released initially, followed by the OTF release of the inner layer. We further presumed that this sequential release model could enhance the osteogenic effect by promoting angiogenesis. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,796,939,903]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We designed this sequentially released PO-PP scaffold to adopt the advantage of this sequential release model: PNS being initially released, followed by the release of OTF. We also assessed the osteogenic effect of angiogenesis (a key advantage of sequential release) in the PO-PP scaffold by comparing it with a one-layered PLLA scaffold (PLLA embedded with OTF and PNS) in vivo. We hypothesized that this sequentially released scaffold (PO-PP) could enhance the osteogenic effect by restoring angiogenesis at the basis of initially released PNS-promoting angiogenesis. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,62,233,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[897,61,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 of 17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[277,108,465,122]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. Material and Methods \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[277,125,600,142]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.1. Preparation of One- and Bi-Layer Scaffolds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[277,144,939,312]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Poly (L-lactic acid) (PLLA, molecular weight: $2.00 \\\\times 10^{5}$ ) was purchased from Evonik Industries AG (Essen, German). 1,4-Dioxane (molecular formula: $\\\\mathrm{C_4H_8O_2}$ , molecular weight: 88.10, standard quality), Tetrahydrofuran (THF, molecular formula: $\\\\mathrm{C_4H_8O}$ , molecular weight: 72.11, standard quality), and N, N-Dimethylformamide (DMF, molecular formula: $\\\\mathrm{C_3H_7NO}$ , molecular weight: 73.10, standard quality) were purchased from Macklin Biochemical Co., Ltd. (Shanghai, China). PNS was purchased from Yunnan Baiyao Group Wenshan Qihua Co., Ltd. (Kunming, China). OTF was purchased from Beijing Qihuang Pharmaceutical Co., Ltd. (Beijing, China). The purities of purchased PNS and OTF were detected by high performance liquid chromatography (HPLC) analysis, and the results showed the purities of purchased OTF and PNS reached the standards (Supplementary Figures S1 and S2). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[277,313,939,461]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The one-layer scaffold, named POP (PLLA+OTF+PNS), was created through thermally induced phase separation (TIPS). At first, $10\\\\%$ PLLA was dissolved in 1,4-dioxane, and water was then added into this solution, stirred at $60^{\\\\circ}\\\\mathrm{C}$ for $1\\\\mathrm{h}$ (1,4-dioxane used in former step:water = 87:13). Then, OTF $(8\\\\mathrm{mg})$ and PNS $(5\\\\mathrm{mg})$ were added into the solution and stirred at $60^{\\\\circ}\\\\mathrm{C}$ for $20\\\\mathrm{min}$ . When OTF:PNS = 8:5, the release concentration of both OTF and PNS reached $100~\\\\mu \\\\mathrm{g / mL}$ so that the proliferation and differentiation of osteoblasts and the proliferation of vascular endothelial cells were promoted in the concentration of $100~\\\\mu \\\\mathrm{g / mL}$ [22]. The solution was poured into a plastic mold (4 mm diameter) and placed at $30^{\\\\circ}\\\\mathrm{C}$ for $1\\\\mathrm{h}$ , and $-20^{\\\\circ}\\\\mathrm{C}$ for $1\\\\mathrm{h}$ , after which the solution was freeze-dried for $24\\\\mathrm{h}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[277,465,939,600]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The bi-layer scaffold, named PO-PP, included two layers (inner layer: PLLA+ OTF; outer layer: PLLA+ PNS). The inner layer (PO) was prepared by the same method, and the solution was poured into a plastic mold (3 mm diameter). OTF was added into this layer. The outer layer (PP) was created by an electrospun membrane. The $10\\\\%$ PLLA was dissolved in THF and DMF at $60^{\\\\circ}\\\\mathrm{C}$ for 1 h. PNS $(5\\\\mathrm{mg})$ was added into the solution. The resulting solution was electrospun into fibers (voltage: +13, -2 kV; extrusion speed: 0.0025 mm/s). The inner layer was rolled to receive the electrospun membrane by a receiving device (running at 10 rpm) at receiving distance 20 cm. All scaffolds were sterilized by $^{60}\\\\mathrm{Co}$ irradiation and stored at room temperature before use. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[277,616,621,631]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2. Characterization of PO-PP and POP Scaffolds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[277,635,939,846]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The microstructures of PO-PP and POP scaffolds were observed by field emission-scanning electron microscopy (FE-SEM, FEI Quanta 250 FEG, Hillsboro, OR, USA). The round surfaces of scaffolds were placed on the platform and the pressing speed was $1\\\\mathrm{mm / min}$ , pressing to $80\\\\%$ length of scaffolds. The maximum compressive strength of the PO layer ( $3\\\\mathrm{mm}$ diameter, $16\\\\mathrm{mm}$ height) and POP scaffold ( $4\\\\mathrm{mm}$ diameter, $16\\\\mathrm{mm}$ height) was assessed through a universal testing machine (EZ-LX, SHIMADZU, Kyoto, Japan). The pycnometer method was used to assess the porosity of the porous structure (PO layer and POP scaffold). The mass $m_{1}(\\\\mathrm{g})$ of the scaffolds was weighed. The pycnometer filled with deionized water also likewise weighed $m_{2}(\\\\mathrm{g})$ ; the sample was placed into the pycnometer and soaked adequately. Then, the pycnometer was weighed $m_{3}(\\\\mathrm{g})$ after filling up with water. The mass of the remaining water and the pycnometer weighed $m_{4}(\\\\mathrm{g})$ , which were assessed after the soaked sample taken out. The porosity of the scaffolds $(\\\\varepsilon)$ was calculated based on foregoing data, as shown in the equation $\\\\varepsilon = (m_{3} - m_{4} - m_{1}) / (m_{2} - m_{4}) \\\\times 100\\\\%$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[277,847,939,908]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The loaded OTF and PNS were evaluated by a Fourier transform infrared spectroscope (FTIR, FTIR-7600, Lambda Scientific, Rüsselsheim, Germany). The sheared scaffold and potassium bromide were mixed at a ratio of 1: 100, pressed, and scanned by FTIR at a scanning range of $4000 - 400\\\\mathrm{cm}^{-1}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,61,233,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[897,61,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 of 17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[278,106,939,229]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The release medium was PBS $(37^{\\\\circ}\\\\mathrm{C})$ and shaken at $100\\\\mathrm{rpm}$ . The cumulative release rate of OTF was measured at $283\\\\mathrm{nm}$ in quartz plate by ultraviolet spectrophotometry equipped with a microplate reader (Thermo Multiskan FC, Waltham, MA, USA). The cumulative release rate of PNS was measured through the vanillin-perchloric acid chromogenic method. After the solvent evaporated, prepared $5\\\\%$ vanillin-glacial acetic acid $(0.2\\\\mathrm{mL})$ and perchloric acid $(0.8\\\\mathrm{mL})$ were added. The mixture was heated at $60^{\\\\circ}\\\\mathrm{C}$ for $20\\\\mathrm{min}$ in a thermostatic water bath, cooled for $5\\\\mathrm{min}$ in a cold water bath, and then $5\\\\mathrm{mL}$ glacial acetic acid was added. Finally, PNS was measured at $545\\\\mathrm{nm}$ by a microplate reader. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,243,519,260]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.3. Animal Grouping and Surgery \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,262,939,397]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Thirty rabbits were used in this experiment. All rabbits were administered $10~\\\\mu \\\\mathrm{g / kg}$ of lipopolysaccharide (LPS, Sigma-Aldrich, St. Louis, MO, USA) fluid intravenously. Each rabbit was then given an intramuscular injection of $20\\\\mathrm{mg / kg}$ methylprednisolone (MPS, Macklin, Shanghai, China) every $24\\\\mathrm{h}$ , which was repeated three times. And then, two weeks later, a femoral head necrosis model was successfully induced, which was reported in our previously published article [23]. The thirty rabbits were randomly divided into three groups 12 weeks after modeling: the PO-PP group, imbedded with PO-PP scaffold; the POP group, imbedded with POP scaffold; and the control group, imbedded with no scaffold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,399,939,581]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The surgery was conducted two weeks after the third MPS injection. Femoral head perforation was only performed on the right side. A $25\\\\mathrm{mm}$ skin incision was made to expose the right hip joint. A bone tunnel (4 mm diameter, 16 mm depth) from the hole to the femoral head through the mid-axis of femoral neck was created by guiding a C-arm fluoroscope. The PO-PP and POP scaffolds (PO-PP group, $4\\\\mathrm{mm} \\\\times 16\\\\mathrm{mm}$ PO-PP scaffold; POP group, $4\\\\mathrm{mm} \\\\times 16\\\\mathrm{mm}$ POP scaffold; control group, no scaffold) were inserted into the bone tunnels. These rabbits were intramuscularly injected with 20 wu of penicillin twice a day for two weeks. After 12 weeks post-surgery, the experimental rabbits were sacrificed by intravenously injecting $150\\\\mathrm{mg/kg}$ bodyweight sodium pentobarbital. The whole right femur was resected, and the soft tissues were removed. Finally, the whole right femurs were fixed in $10\\\\%$ neutral formalin for micro-CT scanning, as well as histological, immunohistochemistry, and immunofluorescence analyses. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,596,440,612]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.4. Micro-CT Analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,614,939,705]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The harvested femoral head samples were assessed by a micro-CT scanner (Skyscan1076, Bruker, Karlsruhe, Germany) at $40\\\\mathrm{kVP}$ , $250~\\\\mu \\\\mathrm{A}$ for $800~\\\\mathrm{ms}$ . The region of interest (ROI, $4\\\\mathrm{mm}$ diameter in the center of the bone tunnel) was selected and reconstructed by a CT-analyzer. The bone mineral density (BMD, g/cc), bone surface density (BS/TV, 1/mm), percent bone volume (BV/TV, %), trabecular number (Tb.N, 1/mm), and bone volume (BV, mm³) were calculated based on the data from the micro-CT scans. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,720,453,736]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.5. Histological Analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,739,939,830]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The femoral heads were decalcified with $14\\\\%$ EDTA solution (pH7.2) for 28 days. The decalcified samples were embedded in paraffin and cut into $4\\\\mu \\\\mathrm{m}$ sections. The Hematoxylin and Eosin (H&E) (Servicebio, Wuhan, China), Masson trichrome (MT) (Servicebio), Goldner's trichrome (Servicebio), and Tartrate-resistant acid phosphatase (TRAP) (Servicebio) stainings were executed on the sections. The histological sections were observed with a light microscope (Olympus, Tokyo, Japan). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,845,574,861]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.6. Immunohistochemistry (IHC) Analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,864,939,923]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The quantity and distribution of the vascular endothelial growth factor (VEGF), osteopontin (OPN), and osteocalcin (OCN) in the bone samples were detected through IHC analysis. The sections were incubated for heat-induced epitope retrieval in citrate buffer at $60^{\\\\circ}\\\\mathrm{C}$ , then blocked with $1\\\\%$ hydrogen peroxide/methanol (Sigma-Aldrich, St. Louis, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,61,233,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[897,61,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 of 17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[278,106,939,274]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"MO, USA) for $30\\\\mathrm{min}$ at $25^{\\\\circ}\\\\mathrm{C}$ . After blocking, the sections were incubated with primary antibodies VEGF (1: 100; Novus NBP2-45235, Bio-Techne Corporation, Minneapolis, MN, USA), OPN (1: 100; Novus NB110-89062, Bio-Techne Corporation, Minneapolis, MN, USA) and OCN (1: 100; Novus NBP2-89037, Bio-Techne Corporation, Minneapolis, MN, USA) overnight at $4^{\\\\circ}\\\\mathrm{C}$ . Next, the samples were incubated with goat anti-mouse secondary antibodies (1: 200; Servicebio G23301, Wuhan servicebio technology CO., LTD, Wuhan, China) for $1\\\\mathrm{h}$ at room temperature. The positive staining sites were colored by $3,3^{\\\\prime}$ -diaminobenzidine (Dako Liquid DAB, Carpinteria, CA, USA) color solution. Finally, the sections were counterstained with hematoxylin. The sections were then visualized with a light microscope, and the positive staining sites were assessed by Image J software (Image J, V1.8.0, Bethesda, MD, USA). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,290,541,304]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.7. Immunofluorescence Assay (IMA) \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,308,939,490]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IMA was performed to detect the quantity and distribution of the platelet endothelial cell adhesion molecule-1 (PECAM-1/CD31) and alkaline phosphatase (ALP). After heat-induced epitope retrieval and blockage, the slides were incubated with primary antibodies for CD31 (1:20; Novus NB600-562, Bio-Techne Corporation, Minneapolis, MN, USA) and ALP (1: 100; Novus NB600-540, Bio-Techne Corporation, Minneapolis, MN, USA) overnight at $4^{\\\\circ}\\\\mathrm{C}$ . Subsequently, the samples were incubated with goat anti-rabbit secondary antibodies (CD31:1:400, Servicebio GB25303, Wuhan servicebio technology CO., LTD, Wuhan, China; ALP:1:300, Servicebio GB21303, Wuhan servicebio technology CO., LTD, Wuhan, China) for 1 h at room temperature. Ultimately, the samples were counterstained with DAPI (Servicebio) for 10 min at room temperature. Images were captured with a Leitz/Leica TCSSP2 microscope (Leica Lasertechnik GmbH, Heidelberg, Germany) and assessed by ImageJ software (Image J, V1.8.0, Bethesda, MD, USA). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,505,440,520]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.8. Statistical Analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,524,939,584]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"All data were presented as mean $\\\\pm$ standard deviation (SD). Differences among the groups were assessed using one-way analysis of variance (ANOVA) with LSD for multiple comparisons in this study. A $p$ value $< 0.05$ was considered as statistically significant. All data analyses were performed using the SPSS 23.0 software (IBM, Armonk, NY, USA). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,600,351,612]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3. Results \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,618,500,633]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.1. Characterization of Scaffolds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,636,939,925]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The PO-PP scaffold is a bi-layer scaffold including PLLA embedded with OTF (PO, inner layer) and PLLA embedded with PNS (PP, outer layer), which achieves sequential delivery of PNS and OTF; PNS of the outer layer is initially released, followed by the release of OTF from the inner layer. The initially released PNS could promote new vessel formation, which would further enhance new bone formation by providing sufficient nutrition to the bone tissue. The inner OTF was released later, which could promote new bone formation. In this manner, better and faster new bone formation is obtained. Enhancing the osteogenic effect of vascularization (PNS in the outer layer) is the key technique employed by our scaffold. The PO layer has a porous structure, and the PP layer has an electrospinning net. The POP scaffold has a one-layer scaffold, including PLLA embedded with OTF and PNS (POP), which did not obtain sequential release and released OTF and PNS simultaneously. It only had a porous structure (Figure 1A,B). For the PO-PP scaffold, the pore diameter was about $10 - 38\\\\mu \\\\mathrm{m}$ (PO layer), and the electrospinning diameter was about $0.7 - 1.7\\\\mu \\\\mathrm{m}$ (PP layer). For the POP scaffold, the pore diameter ranged within 14-31 $\\\\mu \\\\mathrm{m}$ . Table 1 indicates that porous materials (PO layer, POP scaffold) meet the minimum strength requirement of cancellous bone repair (1 MPa) and the minimum porosity requirement of vessel formation (50%) [24,25]. OTF was successfully loaded into the PO layer of the PO-PP scaffold with hydroxyl groups $(3470~\\\\mathrm{cm}^{-1})$ and benzene ring (1654 $\\\\mathrm{cm}^{-1}$ ), which was the characteristic absorption peak of OTF in the FTIR image (Figure 2A). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,61,233,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[897,61,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 of 17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[275,106,939,244]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PNS with hydroxyl structures $(3470~\\\\mathrm{cm}^{-1})$ and a carbon-carbon double bond $(1639~\\\\mathrm{cm}^{-1})$ was successfully loaded into PP layer of PO-PP scaffold in the FTIR image (Figure 2A). In this manner, OTF and PNS were both successfully loaded into the POP scaffold. For cumulative drug release, no difference was found between the PO-PP and POP scaffolds in OTF (Figure 2B). The PNS cumulative release was faster in the PO-PP scaffold than in the POP scaffold, which not only achieves better vessel formation, but also enhances the osteogenic effect of vascularization of the PO-PP scaffold to some degree (Figure 2B). The weight-averaged molecular weight (Mw) of PLLA was about $2\\\\times 10^{5}$ and slowly degraded in vivo, which was reported in our previous published article [26]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[270,258,643,378]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"A \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0043/caaa87863a6120893a980d94e4d1fde7c7d0994647c0322e2e5bf896d268e47c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[694,277,813,378]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0533/15a91e7c44b7cd59cf50e6fd563ee3de7235ca0fce07aee6d33d5277982cb2d4.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[176,391,403,520]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"B \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0532/0b4eab84a6bfb6923b988774b84782bf006801bb5d7f054cdc641a78e12a3e61.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[406,391,633,520]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0919/5decaad71ec4b108a5eb1a850f8c273ae64d5065086a7b05008f88ea7bf0a80a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[636,391,863,520]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 1. Scaffold characteristics. (A) Structural diagrams of PO-PP and POP scaffolds. PO-PP scaffold is a bi-layer scaffold, including inner layer (PO layer: PLLA embedded with OTF) and outer layer (PP layer: PLLA embedded with PNS). POP scaffold is a one-layer scaffold (PLLA embedded with OTF and PNS). (B) SEM images of PO-PP and POP scaffolds. PO layer exhibits porous structure at magnifications of $4500\\\\\\\\times$ ; PP layer shows electrospinning net at magnifications of $2500\\\\\\\\times$ ; POP scaffold has a porous structure at magnifications of $4500\\\\\\\\times$ . \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0138/ae5fc31579a32780a9c6e913023dbfc4d9420640ce19ac0e15a33b3f24b36a49.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[57,61,233,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[897,61,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 of 17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[300,110,507,241]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"A \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0287/2e102b79367aa42884c1e7f1539a54db01a392d23a03c6c2de8b4d1f89148e27.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[512,111,722,243]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0490/de01a740d271bbf4bdc0d6a21a3b5f8cb41ef1a0916195f36a3936659ada9f5e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[722,111,924,242]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0392/23be232e017c3df9b621bec556f966443ae76f1629a4a0791115ff3de75da1c9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[337,289,566,418]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"B \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0941/b3bfe358cccb12b11289ed3be1ac6fe2406a6814036680cac277a8954c2ca59e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[660,288,887,421]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 2. Scaffold characteristics. (A) FTIR diagram of scaffolds including PO layer, PP layer, and POP scaffold. (B) Cumulative release profiles of OTF and PNS from POP and PO-PP scaffolds at different times; data are presented as mean $\\\\\\\\pm$ SD, $n = 10$ . \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0602/0198750b637bc564c474c5029cb346fb5c4b3e0ecd4e227b15d6c672fb7f6eb1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[275,508,939,595]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0020/0a27103b75f1ffc5211e03bb1fed19896ff490be76dc4c6f5a8bd49d4e68b39d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Scaffold</td><td>Porosity (%)</td><td>Maximum Compressive Strength (MPa)</td></tr><tr><td>PO layer in PO-PP scaffold</td><td>89.56 ± 2.15</td><td>1.46 ± 0.13</td></tr><tr><td>POP scaffold</td><td>91.06 ± 2.13</td><td>1.34 ± 0.12</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 1. Porosity and maximum compressive strength of PO(PO-PP) and POP. \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[275,609,803,625]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.2. Micro-CT Analysis for Bone Formation Treated by Scaffold Implantation \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[275,628,939,825]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The micro-CT analysis was conducted twelve weeks after surgery. As shown in Figure 3, the scaffolds were successfully implanted into the femoral head of animals. The reconstructed 3D image indicates that the new bone formation in the PO-PP and POP groups was significantly better than that in the control group; and further PO-PP scaffold exhibits more new bone formation than the POP scaffold. The BMD, BV, and BV/TV of the PO-PP and POP groups were significantly higher than those of the control group $(p < 0.05)$ ; furthermore, the data of BMD, BV, and BV/TV in the PO-PP group show significant differences compared with those in the POP group $(p < 0.01)$ . The data of Tb.N and BS/TV in the PO-PP and POP groups show significant differences compared with those in the control group $(p < 0.05)$ . These results indicate that PO-PP and POP scaffolds all have a good osteogenesis effect. The osteogenic capacity of the PO-PP scaffold is superior to that of the POP scaffold, which indicates the advantage of sequential delivery (the key technique of PO-PP scaffold). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,61,233,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[897,61,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7 of 17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[137,109,875,470]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"A \\\",\\\"Figure 3. New bone formation in bone tunnel. (A) Scanning image (black): the focus of the red and green lines depicts the scaffold, (1) Coronal plane, (2) Sagittal plane, (3) Cross-section. Representative 3D micro-CT images (blue) within a ROI of central $3\\\\\\\\mathrm{mm}$ in diameter of the bone tunnel, (4) Coronal plane, (5) Cross-section. (B) Quantitative analysis of new bone in bone tunnel: BMD, Tb.N, BV, BV/TV, BS/TV. $\\\\\\\\mathfrak{n} = 10$ ; Error bar represents SD. ANOVA test with LSD analysis is used for statistical analysis; $^* p < 0.05$ vs. control group or POP group; $^{**}p < 0.01$ vs. control group or POP group. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0185/b974dd787ddd412f625b1327f260944854cb6adf2b11ebf8c35bb7143b4e7bd6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[273,583,559,598]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.3. Histological Analysis of Bone Repair \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,600,941,783]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the process of repairing GONFH, vessels delivered nutrients to the femoral head and improved the new bone formation in the necrotic area. In this experiment, newly formed vessel and bone around the tunnel were assessed by HE staining. As shown in Figure 4A, there were more new vessels and trabecular bones filled at the injury site of the femoral head of the PO-PP and POP groups as compared with that of the control group. The repair effect of the PO-PP group was superior to that of the POP group. MT staining was commonly used to identify collagen (blue) and osteoid tissue (red). A mount of collagen fibers and osteoid tissue were filled around the defect regions in the treatment groups (Figure 4B). The therapeutic effect of the PO-PP group was superior to that of the POP group. The results of HE and Masson staining indicated better angiogenic and osteogenic effects of the PO-PP scaffold. It is the contribution of the PO-PP scaffold's sequential release that enhances the osteogenic effect of vascularization. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,60,233,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[897,60,941,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"8 of 17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[163,112,391,296]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"A \\\",\\\"Control \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0489/8a31bf0a65c34dd674ba0947321271de50c4274719705e8cf1e5be1a5894ab41.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[398,112,628,296]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"PO-PP \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0735/e4c15c7480b1e900af704d7bf08922dfb4ac8b09d8ef7dab236630c77bdf6c5f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[638,112,865,296]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"POP \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0717/7d7302224936bea7903dd54c4bdc098efcca4d1d8b1d4302cd971c233e0334c2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[163,318,391,500]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"B \\\",\\\"Control \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0664/21f6fb2ad594d137fc70b110463ffa7115837b593e4799bd80edde2d82a5bba3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[400,319,626,500]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"PO-PP \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0502/5d361ef1b6823ee8ef26fc8003f6a2483f63af55ac989ba554e60cced7b0508c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[636,319,863,406]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0774/a2ea538fb413f9128c76f9f5e5995000fc89b607aba3845f6ccb7ad14bb6cda0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[636,406,863,500]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"POP \\\",\\\"Figure 4. Histological analysis of bone repair after scaffold implantation. (A) H&E staining of decalcified sections (bone tunnel, black circle) at magnifications of $20 \\\\\\\\times$ and $200 \\\\\\\\times$ . The $200 \\\\\\\\times$ images represent higher magnifications of areas within black boxes. White arrow indicates trabecular bone, blue arrow indicates vessel, bars = 500 and $50 \\\\\\\\mu \\\\\\\\mathrm{m}$ for $20 \\\\\\\\times$ and $200 \\\\\\\\times$ , respectively. (B) Masson trichrome staining of decalcified sections (bone tunnel: black circle) at magnifications of $20 \\\\\\\\times$ and $200 \\\\\\\\times$ . $200 \\\\\\\\times$ images are higher magnifications of the areas within black boxes. White arrow indicates collagen (blue) and osteoid tissue (red), bars = 500 and $50 \\\\\\\\mu \\\\\\\\mathrm{m}$ for $20 \\\\\\\\times$ and $200 \\\\\\\\times$ , respectively. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0641/36401d9d832a3d9aecbc07bf0c1492996805804ea8e81c1353f3808c1c3b4da6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[272,630,939,934]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bone, a dynamic tissue, remodels continuously throughout life, which constitutes an important process for bone repair [27]. Bone remodeling is tightly regulated by the balance between osteoblasts and osteoclasts [27]. Osteoid, a collagen-rich matrix, is produced by osteoblasts and mineralized extracellularly by nanosized calcium phosphate (CaP) [28]. The formed osteoid in the mineralized bone represents the osteogenic capacity. Goldner staining was conducted to distinguish between osteoid tissue (red) and mineralized bone tissue (green). TRAP staining, used to reveal the osteoclast activity, was also performed to assess the osteoclastic bone resorption [29]. The results of Goldner staining (Figure 5A) show a large amount of osteoid tissue around the tunnel, appearing near the implantation site of the treatment groups. The amount of osteoid tissue in the PO-PP group was more than those in the POP group, which indicates better promoting effect of osteogenic activity in the PO-PP scaffold. As shown in Figure 5B, a large quantity of TRAP-positive red cells was detected near the tunnel of the control group, while the defect regions in the PO-PP and POP groups were surrounded with a small quantity of TRAP-positive red cells after implantation. The quantity of TRAP-positive red cells was lower in the PO-PP group than in the POP group, which indicates a better inhibiting effect of osteoclastic activity in the PO-PP scaffold. In summary, the PO-PP scaffold has better effects in promoting osteogenic activity and inhibiting osteoclastic activity, owing to the sequential release of the PO-PP scaffold. Initially released PNS not only promotes vessel formation, but also enhances new bone formation by providing sufficient nutrition to bone tissue. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,60,233,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[897,60,941,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"9 of 17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[171,117,394,292]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"A \\\",\\\"Control \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0866/a3bbf1e41e129d514f5ac1066bed65ba53a61c73028e0d6b89fd598dac1dc0f4.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[405,117,628,293]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"PO-PP \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0134/530d6a58986fb520fedba5df7bc8360b402acee2da8cfda58a8ba89f59d785fa.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[638,116,863,292]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"POP \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0354/7d3bf6200510c9532b0162c4913c9bcb285afe0b17f7374add46c5a96789b661.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[169,304,394,477]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"B \\\",\\\"Control \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0711/cfd169b210e5c81b992615feec930d716e798ae6c0cc36bcd6c29312c241f746.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[403,305,626,476]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"PO-PP \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0678/6c4c47c795534e725a2fb2854451fd5972676ee625e12b21f421ab6df0d7c542.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[638,305,860,477]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"POP \\\",\\\"Figure 5. Histological analysis of bone repair after scaffold implantation. (A) Goldner staining of decalcified sections (bone tunnel: black circle) at magnifications of $20\\\\\\\\times$ and $200\\\\\\\\times$ ; $200\\\\\\\\times$ images are higher magnifications of areas within black boxes. White arrow indicates osteoid tissue, bars = 500 and $50~{\\\\\\\\mu\\\\\\\\mathrm{m}}$ for $20\\\\\\\\times$ and $200\\\\\\\\times$ , respectively. (B) TRAP staining of decalcified sections at magnifications of $20\\\\\\\\times$ and $200\\\\\\\\times$ , $200\\\\\\\\times$ images are higher magnifications of areas within black boxes. Red arrow indicates osteoclast, bars = 500 and $50~{\\\\\\\\mu\\\\\\\\mathrm{m}}$ for $20\\\\\\\\times$ and $200\\\\\\\\times$ , respectively. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0886/07ec784a3902441f1e58dacac59c78ba7111170f3525dcdcb7353042705d9389.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[275,602,776,616]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.4. Distribution and Quantity of VEGF and CD31 in Implantation Sites \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[275,619,939,832]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Vascularization during bone repair supports the recruitment of circulating osteoblast-lineage cells for osteogenesis [30]. This is the key to the sequential release of the PO-PP scaffold. Therefore, to investigate the effector mechanism of vascularization, VEGF and CD31 stainings were performed (Figures 6A and 7A). Positive cells were stained brown for VEGF (IHC), and green for CD31(IMA). The distribution and quantity of VEGF proteins was significantly increased in the treatment groups compared with those in the control group $(p < 0.01)$ , and furthermore, the PO-PP scaffold showed much higher VEGF protein expression when compared with the POP scaffold $(p < 0.01)$ (Figure 6B). The same observation was made on the CD31 protein in the treatment groups, compared with that of the control group around the tunnel after implantation $(p < 0.01)$ . The PO-PP scaffold also showed significantly higher CD31 protein expression compared with the POP scaffold $(p < 0.01)$ (Figure 7B). The higher protein expressions of VEGF and CD31 in the PO-PP scaffold might be attributed to the sequential release of the PO-PP scaffold, where PNS is released first and at a faster release rate. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,61,233,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[890,61,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10 of 17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[109,106,633,410]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"B \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0522/3f73e41954e71340874d3ad112e0e70c61bae014efe87ab0d9ab9def8259517c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[663,191,878,355]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0447/deb5af8008bccd667601af3925c3bc6fac35fe0a29900ffedee8c22d4c73b1db.jpg\",\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[115,529,640,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 6. Immunohistochemistry staining of VEGF after scaffold implantation. (A) Distribution and quantity of VEGF in decalcified sections (bone tunnel, black circle) at magnifications of $20\\\\\\\\times$ and $200\\\\\\\\times$ . $200\\\\\\\\times$ images are higher magnifications of areas within black boxes. Red arrow indicates VEGF positive cells, bars = 500 and $50~{\\\\\\\\mu\\\\\\\\mathrm{m}}$ for $20\\\\\\\\times$ and $200\\\\\\\\times$ , respectively. (B) Quantitative analysis of VEGF expression. $\\\\\\\\mathfrak{n} = 10$ in each group, error bar represents SD. ANOVA test with LSD analysis was used for statistical analysis, $^{*}p < 0.05$ vs. control group or POP group; $^{**}p < 0.01$ vs. control group or POP group. \\\",\\\"B \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0644/b573f5eb729c2325e71f36da6fab1dcfe55468874fff782bcfd27ff82488807b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[663,612,870,771]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 7. Immunofluorescence staining of CD31 after scaffold implantation. (A) Distribution and quantity of CD31 in decalcified sections (bone tunnel: blue circle) at magnifications of $20\\\\\\\\times$ and $200\\\\\\\\times$ . $200\\\\\\\\times$ images are higher magnifications of areas within blue boxes. White arrow indicates CD31 positive cells, bars = 500 and $50~{\\\\\\\\mu\\\\\\\\mathrm{m}}$ for $20\\\\\\\\times$ and $200\\\\\\\\times$ , respectively. (B) Quantitative analysis of CD31 expression. $\\\\\\\\mathfrak{n} = 10$ in each group, error bar represents SD. ANOVA test with LSD analysis was used for statistical analysis; $*p < 0.05$ vs. control group or POP group; \\\\\\\\*\\\\\\\\* $p < 0.01$ vs. control group or POP group. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0368/d42c762ed1d468f4c4ffd3cf0715d137e027ab89b0ad5b70a83cd8d4690edf65.jpg\",\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[55,60,233,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[890,60,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"11 of 17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[277,108,811,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.5. Distribution and Quantity of OCN, OPN, and ALP in Implantation Sites \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[277,127,939,263]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To explore the osteogenesis mechanism, the distribution and quantity of OCN, OPN, and ALP around implantation sites were assessed (Figures 8-10). The positive cells were stained brown for OCN and OPN (IHC), and red for ALP (IMA). The distribution and quantity of OCN, OPN, and ALP in the treatment groups were increased significantly compared to those in the control group $(p < 0.01)$ . Furthermore, the data of OCN, OPN, and ALP expression in the PO-PP scaffold showed significant differences when compared with those in the POP scaffold $(p < 0.05)$ . This represented the advantage of the sequential release of the PO-PP scaffold, which not only provided the osteogenic effect of vascularization (PP layer), but also the osteogenic effect of OTF (PO layer). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[275,276,709,517]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0426/23eb6f47cd838a7d11a6da8ea5e7827e7d9e05fd55597a846fe0aaf2afbf2288.jpg\",\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[729,274,910,480]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0814/4dd6e7a62275819fc7dde755d5e59e01b6ed44c6e9e0647c2daa96f8d26a92b3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[161,631,633,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 8. Immunohistochemistry staining of OCN after scaffold implantation. (A) Distribution and quantity of OCN in decalcified sections (bone tunnel: black circle) at magnifications of $20\\\\\\\\times$ and $200\\\\\\\\times$ . $200\\\\\\\\times$ images are higher magnifications of areas within black boxes. Red arrow indicates OCN positive cells, bars = 500 and $50~{\\\\\\\\mu\\\\\\\\mathrm{m}}$ for $20\\\\\\\\times$ and $200\\\\\\\\times$ , respectively. (B) Quantitative analysis of OCN expression. $n = 10$ in each group, error bar represents SD. ANOVA test with LSD analysis was used for statistical analysis, $^{*}p < 0.05$ vs. control group or POP group; $^{**}p < 0.01$ vs. control group or POP group. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0253/526a5a2e3b84123b44d385bac37928d80828fa0ea702f50566e3d2a3254bfde1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[640,630,830,847]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 9. Immunohistochemistry staining of OPN after scaffold implantation. (A) Distribution and quantity of OPN in decalcified sections (bone tunnel, black circle) at magnifications of $20\\\\\\\\times$ and $200\\\\\\\\times$ . \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0907/5742e661e2cbe9406f0548c2dcc19d336a9ae6ea15e3d9bfebf138448e5b51d1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[57,61,233,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[890,61,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"12 of 17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[272,106,941,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$200\\\\times$ images are higher magnifications of areas within black boxes. Red arrow indicates OPN positive cells, bars $= 500$ and $50~\\\\mu \\\\mathrm{m}$ for $20\\\\times$ and $200\\\\times$ , respectively. (B) Quantitative analysis of OPN expression. $n = 10$ in each group, error bar represents SD. ANOVA test with LSD analysis was used for statistical analysis, $^{*}p < 0.05$ vs. control group or POP group, $^{**}p < 0.01$ vs. control group or POP group. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,190,623,461]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0196/5113ce52c44d094872778a9f32f6357e23d3f92e97c753115d66d597ef4340c1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[638,188,860,426]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 10. Immunofluorescence staining of ALP after scaffold implantation. (A) Distribution and quantity of ALP expression in decalcified sections (bone tunnel: blue circle) at magnifications of $20\\\\\\\\times$ and $200\\\\\\\\times$ . $200\\\\\\\\times$ images are higher magnifications of areas within blue boxes. White arrow indicates ALP positive cells, bars = 500 and $50~{\\\\\\\\mu\\\\\\\\mathrm{m}}$ for $20\\\\\\\\times$ and $200\\\\\\\\times$ , respectively. (B) Quantitative analysis of ALP expression. $\\\\\\\\mathfrak{n} = 10$ in each group, error bar represents SD. ANOVA test with LSD analysis was used for statistical analysis, $^{*}p < 0.05$ vs. control group or POP group; $^{**}p < 0.01$ vs. control group or POP group. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0631/0afa736ab7e3e7436397dfc22da0cc952dc42009fbff26c08a49af1c501aa7fd.jpg\",\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[273,579,381,593]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4. Discussion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,598,939,703]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"GONFH, characterized by progressive deterioration of the hip joint [31], is a serious complication occurring after long-term or excess administration of clinical glucocorticoids [32]. With the development of GONFH, the femoral head will collapse, ultimately requiring total hip arthroplasty (THA) [33,34]. About 30-60,000 new cases of GONFH occur in the United States every year [35], and approximately 8.12 million people suffer from GONFH in China [36]. GONFH can create substantial financial, physical, and emotional burdens for patients and their families. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,705,941,934]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Glucocorticoids have an inhibitory function on angiogenesis in the femoral head, such that synthetic glucocorticoid treatment causes vascularity disruption at the site [37]. Osteogenesis requires an adequate blood supply [38], and vascular damage easily results in an inability to deliver nutrients to the femoral head [1]. Angiogenesis and osteogenesis are closely related to femoral head repair [4]. To achieve the osteogenic effect of vascularization, we aim to design a sequentially drug-releasing scaffold. PLLA, a biodegradable and biocompatible scaffold material, has been widely used as carrier for delivering drugs or bioactive factors in orthopedic treatment [39]. PNS, the main ingredient of Panax Notoginseng, is able to promote angiogenesis. It thus prevents the loss of vascular endothelial cells [40]. OTF, the major component of Drynariae Rhizoma, promotes osteogenesis. Several studies showed that OTF promotes bone formation in large tibial defects (LTDs) [18], and thus increases the bone density and mechanical strength of tail suspension rats [41]. In this study, we designed a sequentially released bi-layer PLLA scaffold, where the inner layer contained OTF, and the outer layer contained PNS. PNS in the outer layer was initially released to promote vessel formation and, more importantly, to enhance the \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,59,233,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[890,59,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"13 of 17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[278,108,939,198]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"osteogenic effect of vascularization (the key technique employed in our study). The OTF in the inner layer was released later to promote new bone formation. We also made a one-layer PLLA scaffold containing OTF and PNS, which were successfully loaded into scaffolds. The PO-PP and POP scaffolds released PNS and OTF steadily and durably. The PNS release was faster in the PO-PP scaffold than in the POP scaffold, which also supported the osteogenic effect of vascularization. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,200,939,471]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"After scaffold implantation, we assessed new bone formation through a micro-CT. The results of reconstructed 3D image showed that the new bone formation in the treatment groups was significantly better than that in the control group; and the PO-PP scaffold had more new bone formation than the POP scaffold. The data of BMD, BV, and BV/TV in the treatment groups were significantly higher than those in the control group with $p < 0.05$ ; and the data of BMD, BV, and BV/TV in the PO-PP group were significantly higher than those in the POP group with $p < 0.01$ . These results indicated that both scaffolds had good osteogenesis effect, while the osteogenic effect of the PO-PP scaffold was superior to that of the POP scaffold. We observed the formation of new vessels and new bone via HE staining. The results showed that both scaffolds exhibited more vessel and bone formation around the defect when compared with the control group, and more vessel and bone formation was observed in the PO-PP scaffold than in the POP scaffold. Because bone formation is tightly regulated by the balance between osteoblasts and osteoclasts [27], we observed the osteogenic activity by Masson staining, Goldner staining, and the osteoclastic activity by TRAP staining. The results of these stainings indicate that both scaffolds promote osteogenic activity and inhibit osteoclastic activity. Interestingly, superior effects (promoting osteogenic activity and inhibiting osteoclastic activity) were found in the PO-PP scaffold compared with the POP scaffold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,475,939,624]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the process of promoting bone repair through angiogenesis and osteogenesis, the PO-PP scaffold not only promotes more vessel formation via the faster-released PNS, but the PNS in the outer layer enhanced the osteogenic effect of vascularization based on the sequential release system. Therefore, sequential release, which represent the key technology in our study, not only achieved more angiogenesis, but also indirectly promoted more new bone formation. Moreover, the OTF in the inner layer also promoted new bone formation. The one-layer scaffold (POP) obtained a slow PNS release. Furthermore, the simultaneous release of PNS and OTF lost the advantage of the sequential release, thus depriving PNS of its role in indirect osteogenesis. Therefore, a sequential release is the key technology in the PO-PP scaffold to achieve better bone repair. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,625,939,928]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To explore the possible mechanisms of angiogenesis and osteogenesis, the distribution and quantity of key proteins were evaluated through IHC and IMA. VEGF (a homodimeric disulfide bound glycoprotein) promotes endothelial growth [42]. CD31 (the most abundant constitutive co-signaling receptor glycoprotein on endothelial cells) is highly expressed at endothelial cell-cell junctions [43], and forms the first-line of cellular contact with the blood [44]. The results in our study showed that the protein expressions of VEGF and CD31 in the treatment groups were significantly higher than those in the control group with $p < 0.01$ , and the protein expressions of VEGF and CD31 in the PO-PP scaffold were significantly higher than those in the POP scaffold with $p < 0.01$ . The enhancing effect of the PO-PP scaffold might be attributed to the sequential release of the PO-PP scaffold where PNS is released first and at a faster release rate. OCN synthesized by osteoblasts is known for its strong binding affinity to bone minerals [45]. OPN, which binds to extracellular calcium via acidic (aspartate and glutamate) and phosphorylated (serine and threonine) residues, plays a key role in nucleating bone minerals [46]. ALP (a homodimer anchored to the membrane of osteoblasts) is essential for biomineralization [47]. The results indicated that the protein expressions of OCN, OPN, and ALP in the treatment groups were significantly higher than those in the control group with $p < 0.01$ , and the protein expressions of OCN, OPN, and ALP in the PO-PP scaffold were significantly higher than those in the POP scaffold with $p < 0.05$ . The promotion function of the PO-PP scaffold was better than that of the POP scaffold. Based on the sequential release \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,62,233,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[890,61,939,70]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"14 of 17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[272,106,939,156]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"advantage of the bi-layer scaffold, the PO-PP scaffold had a better angiogenic and osteogenic effect. Research into the mechanisms further found that the PO-PP scaffold achieved the above effects by promoting the expression of angiogenic and osteogenic proteins. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,168,391,182]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5. Conclusions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,186,944,369]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We fabricated a sequentially released bi-layer scaffold called a PO-PP scaffold. In this PO-PP scaffold, PNS was initially released, followed by the release of OTF to obtain the osteogenic effect of angiogenesis (the key advantage of our scaffold). Compared with the POP scaffold (with concomitant release of OTF and PNS), the PO-PP scaffold had better effects in promoting new bone formation and blood vessel formation, promoting osteoid tissue formation, and inhibiting osteoclastic activity. Upregulating the expression of angiogenic proteins (VEGF and CD31) and osteogenic proteins (OCN, OPN and ALP) might be the possible mechanism. Therefore, the sequentially released PO-PP scaffold proposes a new therapy for bone repair. Based on the osteogenic effect of angiogenesis of the sequentially released PO-PP scaffold, we want to apply this theory to osteoporotic bone defect in rats (different disease and different species) in future to explore the possible effects and mechanisms. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,382,941,494]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Supplementary Materials: The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/jfb14010031/s1, Figure S1: Purity identification of osteoptractic total flavone (OTF). A, HPLC ion chromatogram of ATF (Reference substance): Neoeriocitrin and Naringin. B, HPLC ion chromatogram of ATF (Sample): Neoeriocitrin and Naringin. C, the molecular formulas of Neoeriocitrin and Naringin; Figure S2: Purity identification of panax notoginseng saponins (PNS). A, HPLC ion chromatogram of PNS (Reference substance): Ginsenoside Rg 1 and Ginsenoside Re. B, HPLC ion chromatogram of PNS (Sample): Ginsenoside Rg 1 and Ginsenoside Re. C, the molecular formulas of Ginsenoside Rg 1 and Ginsenoside Re \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,505,941,562]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Author Contributions: Writing—original draft, G.F.; Conceptualization, L.C.; X.N.; F.Y.; J.L.; D.L. and X.Z.; Supervision, L.C.; Writing—review & editing, L.C.; Resources, Y.Y.; X.N.; J.H.; S.S. and W.W.; Visualization, P.Z. All authors have read and agreed to the published version of the manuscript. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,574,941,605]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Funding: This research was financially supported by the National Natural Science Foundation of China (82074463). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,610,941,654]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Institutional Review Board Statement: The study was conducted according to the guidelines of the Declaration of Helsinki, and approved by the Ethics Committee of Beijing University of Aeronautics and Astronautics (protocol code: BM20220026, date of approval: 2022.03.04). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,659,594,674]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Informed Consent Statement: Not applicable. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,679,939,709]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data Availability Statement: The data presented in this study are available on request from the corresponding author. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,714,707,728]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conflicts of Interest: The authors declare no conflict of interest. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,743,147,756]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,761,941,926]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. Chang, C.; Greenspan, A.; Gershwin, M.E. The pathogenesis, diagnosis and clinical manifestations of steroid-induced osteonecrosis. J. Autoimmun. 2020, 110, 102460. \\\",\\\"2. Zalavras, C.; Shah, S.; Birnbaum, M.J.; Frenkel, B. Role of apoptosis in glucocorticoid-induced osteoporosis and osteonecrosis. Crit. Rev. Eukaryot. Gene Expr. 2003, 13, 221-235. \\\",\\\"3. Li, J.; Fan, L.; Yu, Z.; Dang, X.; Wang, K. The effect of deferoxamine on angiogenesis and bone repair in steroid-induced osteonecrosis of rabbit femoral heads. Exp. Biol. Med. 2015, 240, 273-280. \\\",\\\"4. Liao, Y.; Zhang, P.; Yuan, B.; Li, L.; Bao, S. Pravastatin Protects Against Avascular Necrosis of Femoral Head via Autophagy. Front. Physiol. 2018, 9, 307. \\\",\\\"5. Tang, D.; Tare, R.S.; Yang, L.Y.; Williams, D.F.; Ou, K.L.; Oreffo, R.O. Biofabrication of bone tissue: Approaches, challenges and translation for bone regeneration. Biomaterials 2016, 83, 363-382. \\\",\\\"6. Farokhi, M.; Mottaghitalab, F.; Shokrgozar, M.A.; Ou, K.L.; Mao, C.; Hosseinkhani, H. Importance of dual delivery systems for bone tissue engineering. J. Control Release 2016, 225, 152-169. \\\"]\",\n \"page_idx\": \"14\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[55,60,233,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[890,60,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"15 of 17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[53,106,941,916]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"7. Yao, C.H.; Lai, Y.H.; Chen, Y.W.; Cheng, C.H. Bone Morphogenetic Protein-2-Activated 3D-Printed Polylactic Acid Scaffolds to Promote Bone Regrowth and Repair. Macromol. Biosci. 2020, 20, e2000161. \\\",\\\"8. Fairag, R.; Rosenzweig, D.H.; Ramirez-Garcialuna, J.L.; Weber, M.H.; Haglund, L. Three-Dimensional Printed Polylactic Acid Scaffolds Promote Bone-like Matrix Deposition in Vitro. ACS Appl. Mater. Interfaces 2019, 11, 15306-15315. \\\",\\\"9. Ritz, U.; Gerke, R.; Götz, H.; Stein, S.; Rommens, P.M. A New Bone Substitute Developed from 3D-Prints of Polylactide (PLA) Loaded with Collagen I: An In Vitro Study. Int. J. Mol. Sci. 2017, 18, 2569. \\\",\\\"10. Hu, J.; Liu, X.; Ma, P.X. Induction of osteoblast differentiation phenotype on poly(L-lactic acid) nanofibrous matrix. Biomaterials 2008, 29, 3815-3821. \\\",\\\"11. Iqbal, N.K.; Khan, A.S.; Asif, A.; Yar, M.; Haycock, J.W.; Rehman, I.U. Recent concepts in biodegradable polymers for tissue engineering paradigms: A critical review. Int. Mater. Rev. 2018, 64, 91-126. \\\",\\\"12. Obata, A.; Hotta, T.; Wakita, T.; Ota, Y.; Kasuga, T. Electrospun microfiber meshes of silicon-doped vaterite/poly(lactic acid) hybrid for guided bone regeneration. Acta Biomater. 2010, 6, 1248-1257. \\\",\\\"13. Jang, T.S.; Lee, E.J.; Jo, J.H.; Jeon, J.M.; Kim, M.Y.; Kim, H.E.; Koh, Y.H. Fibrous membrane of nano-hybrid poly-L-lactic acid/silica xerogel for guided bone regeneration. J. Biomed. Mater. Res. Part B Appl. Biomater. 2012, 100, 321-330. \\\",\\\"14. Hwang, C.; Park, S.; Kang, I.G.; Kim, H.E.; Han, C.M. Tantalum-coated polylactic acid fibrous membranes for guided bone regeneration. Mater. Sci. Eng. C Mater. Biol. Appl. 2020, 115, 111112. \\\",\\\"15. Hwang, H.D.; Lee, J.T.; Koh, J.T.; Jung, H.M.; Lee, H.J.; Kwon, T.G. Sequential Treatment with SDF-1 and BMP-2 Potentiates Bone Formation in Calvarial Defects. Tissue Eng. Part A 2015, 21, 2125-2135. \\\",\\\"16. Hong, S.J.; Wan, J.B.; Zhang, Y.; Hu, G.; Lin, H.C.; Seto, S.W.; Kwan, Y.W.; Lin, Z.X.; Wang, Y.T.; Lee, S.M. Angiogenic effect of saponin extract from Panax notoginseng on HUVECs in vitro and zebrafish in vivo. Phytother. Res. 2009, 23, 677-686. \\\",\\\"17. Zheng, H.; Liu, C.; Ou, Y.; Zhang, Y.; Fu, X. Total saponins of Panax notoginseng enhance VEGF and relative receptors signals and promote angiogenesis derived from rat bone marrow mesenchymal stem cells. J. Ethnopharmacol. 2013, 147, 595-602. \\\",\\\"18. Sun, W.; Li, M.; Zhang, Y.; Huang, Y.; Zhan, Q.; Ren, Y.; Dong, H.; Chen, J.; Li, Z.; Fan, C.; et al. Total flavonoids of rhizoma drynariae ameliorates bone formation and mineralization in BMP-Smad signaling pathway induced large tibial defect rats. Biomed. Pharmacother. 2021, 138, 111480. \\\",\\\"19. Wong, R.W.; Rabie, B.; Bendeus, M.; Hägg, U. The effects of Rhizoma Curculiginis and Rhizoma Drynariae extracts on bones. Chin. Med. 2007, 2, 13. \\\",\\\"20. Chen, L.L.; Lei, L.H.; Ding, P.H.; Tang, Q.; Wu, Y.M. Osteogenic effect of Drynariae rhizoma extracts and Naringin on MC3T3-E1 cells and an induced rat alveolar bone resorption model. Arch. Oral Biol. 2011, 56, 1655-1662. \\\",\\\"21. Jeong, J.C.; Kang, S.K.; Youn, C.H.; Jeong, C.W.; Kim, H.M.; Lee, Y.C.; Chang, Y.C.; Kim, C.H. Inhibition of Drynariae Rhizoma extracts on bone resorption mediated by processing of cathepsin K in cultured mouse osteoclasts. Int. Immunopharmacol. 2003, 3, 1685-1697. \\\",\\\"22. Li, J.; Jiang, J.; Zhao, X.; Sun, Q.; Zheng, C.; Bai, C.; Liu, C.; Jia, Y. Effects of osteopractic total flavone on proliferation of vascular endothelial cells of osteoblasts—Vascular endothelial cells co-culture system in nano-hydroxyapatite collagen composite. Chin. Med. Zhenggu 2019, 31, 1-8. \\\",\\\"23. Yang, F.G.; Li, J.; Niu, X. Improved modeling strategy and result evaluation of steroid-induced femoral head necrosis in rabbits. J. Beijing Univ. Aeronaut. Astronaut. 2022, 1-8. \\\",\\\"24. Lai, Y.; Li, Y.; Cao, H.; Long, J.; Wang, X.; Li, L.; Li, C.; Jia, Q.; Teng, B.; Tang, T.; et al. Osteogenic magnesium incorporated into PLGA/TCP porous scaffold by 3D printing for repairing challenging bone defect. Biomaterials 2019, 197, 207-219. \\\",\\\"25. Fiume, E.; Tulyaganov, D.; Ubertalli, G.; Verné, E.; Baino, F. Dolomite-Foamed Bioactive Silicate Scaffolds for Bone Tissue Repair. Materials 2020, 13, 628. \\\",\\\"26. Yang, F.; Niu, X.; Gu, X.; Xu, C.; Wang, W.; Fan, Y. Biodegradable Magnesium-Incorporated Poly(l-lactic acid) Microspheres for Manipulation of Drug Release and Alleviation of Inflammatory Response. ACS Appl Mater. Interfaces 2019, 11, 23546-23557. \\\",\\\"27. Kim, J.M.; Lin, C.; Stavre, Z.; Greenblatt, M.B.; Shim, J.H. Osteoblast-Osteoclast Communication and Bone Homeostasis. Cells 2020, 9, 2073. \\\",\\\"28. Kalia, P.; Vizcay-Barrena, G.; Fan, J.P.; Warley, A.; Di Silvio, L.; Huang, J. Nanohydroxyapatite shape and its potential role in bone formation: An analytical study. J. R. Soc. Interface 2014, 11, 20140004. \\\",\\\"29. Nemoto, Y.; Higuchi, K.; Baba, O.; Takano, Y. Multinucleate osteoclasts in medaka as evidence of active bone remodeling. Bone 2007, 40, 399-408. \\\",\\\"30. Eghbali-Fatourechi, G.Z.; Lamsam, J.; Fraser, D.; Nagel, D.; Riggs, B.L.; Khosla, S. Circulating osteoblast-lineage cells in humans. N. Engl. J. Med. 2005, 352, 1959-1966. \\\",\\\"31. Jilka, R.L.; Weinstein, R.S.; Bellido, T.; Roberson, P.; Parfitt, A.M.; Manolagas, S.C. Increased bone formation by prevention of osteoblast apoptosis with parathyroid hormone. J. Clin. Investig. 1999, 104, 439-446. \\\",\\\"32. Chen, L.; Wang, B.Z.; Xie, J.; Zhang, R.Y.; Jin, C.; Chen, W.K.; Fang, K.H.; Hong, C.X.; Xu, T.H.; Huang, C.B.; et al. Therapeutic effect of SIRT3 on glucocorticoid-induced osteonecrosis of the femoral head via intracellular oxidative suppression. Free Radic. Biol. Med. 2021, 176, 228-240. \\\",\\\"33. Weinstein, R.S. Glucocorticoid-induced osteonecrosis. Endocrine 2012, 41, 183-190. \\\",\\\"34. Jing, X.; Du, T.; Yang, X.; Zhang, W.; Wang, G.; Liu, X.; Li, T.; Jiang, Z. Desferoxamine protects against glucocorticoid-induced osteonecrosis of the femoral head via activating HIF-1 $\\\\\\\\alpha$ expression. J. Cell. Physiol. 2020, 235, 9864–9875. \\\"]\",\n \"page_idx\": \"15\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[55,60,233,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[890,60,941,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"16 of 17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[53,106,941,491]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"35. Fukushima, W.; Fujioka, M.; Kubo, T.; Tamakoshi, A.; Nagai, M.; Hirota, Y. Nationwide epidemiologic survey of idiopathic osteonecrosis of the femoral head. Clin. Orthop. Relat. Res. 2010, 468, 2715-2724. \\\",\\\"36. Zhao, D.W.; Yu, M.; Hu, K.; Wang, W.; Yang, L.; Wang, B.J.; Gao, X.H.; Guo, Y.M.; Xu, Y.Q.; Wei, Y.S.; et al. Prevalence of Nontraumatic Osteonecrosis of the Femoral Head and its Associated Risk Factors in the Chinese Population: Results from a Nationally Representative Survey. Chin. Med. J. 2015, 128, 2843-2850. \\\",\\\"37. Weinstein, R.S.; Hogan, E.A.; Borrelli, M.J.; Liachenko, S.; O'Brien, C.A.; Manolagas, S.C. The Pathophysiological Sequence of Glucocorticoid-Induced Osteonecrosis of the Femoral Head in Male Mice. Endocrinology 2017, 158, 3817-3831. \\\",\\\"38. Liu, X.; Chai, Y.; Liu, G.; Su, W.; Guo, Q.; Lv, X.; Gao, P.; Yu, B.; Ferbeyre, G.; Cao, X.; et al. Osteoclasts protect bone blood vessels against senescence through the angiogenin/plexin-B2 axis. Nat. Commun. 2021, 12, 1832. \\\",\\\"39. Kanczler, J.M.; Ginty, P.J.; Barry, J.J.; Clarke, N.M.; Howdle, S.M.; Shakesheff, K.M.; Oreffo, R.O. The effect of mesenchymal populations and vascular endothelial growth factor delivered from biodegradable polymer scaffolds on bone formation. Biomaterials 2008, 29, 1892-1900. \\\",\\\"40. Hu, H.; Chen, Y.; Zou, Z.; Li, L.; Wei, F.; Liu, C.; Ling, Z.; Zou, X. Panax Notoginseng Saponins Prevent Bone Loss by Promoting Angiogenesis in an Osteoporotic Mouse Model. Biomed. Res. Int. 2020, 2020, 8412468. \\\",\\\"41. Song, S.; Gao, Z.; Lei, X.; Niu, Y.; Zhang, Y.; Li, C.; Lu, Y.; Wang, Z.; Shang, P. Total Flavonoids of Drynariae Rhizoma Prevent Bone Loss Induced by Hindlimb Unloading in Rats. Molecules 2017, 22, 1033. \\\",\\\"42. Theis, V.; Theiss, C. VEGF-A Stimulus for Neuronal Development and Regeneration in the CNS and PNS. Curr. Protein Pept. Sci. 2018, 19, 589-597. \\\",\\\"43. Lertkiatmongkol, P.; Liao, D.; Mei, H.; Hu, Y.; Newman, P.J. Endothelial functions of platelet/endothelial cell adhesion molecule-1 (CD31). Curr. Opin. Hematol. 2016, 23, 253-259. \\\",\\\"44. Caligiuri, G. Mechanotransduction, immunoregulation, and metabolic functions of CD31 in cardiovascular pathophysiology. Cardiovasc. Res. 2019, 115, 1425-1434. \\\",\\\"45. Dowd, T.L.; Rosen, J.F.; Li, L.; Gundberg, C.M. The three-dimensional structure of bovine calcium ion-bound osteocalcin using 1H NMR spectroscopy. Biochemistry 2003, 42, 7769-7779. \\\",\\\"46. Gericke, A.; Qin, C.; Spevak, L.; Fujimoto, Y.; Butler, W.T.; Sorensen, E.S.; Boskey, A.L. Importance of phosphorylation for osteopontin regulation of biomineralization. Calcif. Tissue Int. 2005, 77, 45-54. \\\",\\\"47. Nizet, A.; Cavalier, E.; Stenvinkel, P.; Haarhaus, M.; Magnusson, P. Bone alkaline phosphatase: An important biomarker in chronic kidney disease-mineral and bone disorder. Clin. Chim. Acta 2020, 501, 198-206. \\\"]\",\n \"page_idx\": \"16\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[53,505,941,548]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Disclaimer/Publisher's Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,61,233,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2023, 14, 31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[890,60,941,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"17 of 17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.3390/jfb14010031"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Medicine and Health Sciences"},"sha256":{"kind":"string","value":"12b30878a2bf98dc943aad012d49d79f67da82f9756be9a9fdaf32df9fd2ee09"},"title":{"kind":"string","value":"Sequential Release of Panax Notoginseng Saponins and Osteopractic Total Flavone from Poly (L-Lactic Acid) Scaffold for Treating Glucocorticoid-Associated Osteonecrosis of Femoral Head"}}},{"rowIdx":42,"cells":{"abstract":{"kind":"string","value":""},"author":{"kind":"string","value":"Nao Tahara|Kentaro Hayashi|Yukie Kanai|Yoshie Kato|Satoshi Abe|Shingo Mitaki|Atsushi Nagai"},"content_list":{"kind":"list like","value":[{"bbox":"[178,149,818,168]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"頸部血管造影動態攝影で診断し得たbowhunter症候群の1例","text_format":null,"text_level":"1","type":"text"},{"bbox":"[272,192,722,210]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"田原 奈生 林 健太郎 金井由貴枝 加藤 芳惠","text_format":null,"text_level":null,"type":"text"},{"bbox":"[331,212,665,227]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"安部 哲史 2 三灌 真悟1 長井 答1","text_format":null,"text_level":null,"type":"text"},{"bbox":"[141,256,853,395]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"要旨：症例は71歳女性．後頸部に一過性に疼痛が出現た．その翌日，自家用車運転中に頭部を右に回旋た際，複視と恶心を自覺た．左に回旋ても症状の出現はなかた．救急外来受診時には症状が持続たが，徐々に改善た．身体所見に特記所見はなかた．頭部MRI拋散強調像で両側小脑に梗塞巢を認た．頭頸部CTA,頭頸部MRAは右椎骨動脈の低形成と蛇行を認たが，明かな異常は指摘くださいます．Bowhunter症候群を疑い，頭部を回旋し頸動脈エロー行った，流速低下は認うなかた．左椎骨動脈造影動態撮影で第1頸椎-第2頸椎移行部で血流が途絕し，bowhunter症候群と診断た．臨床所見上，bowhunter症候群が疑わった場合には，血管造影動態撮影が有用oled.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[142,414,678,429]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Key words: digital subtraction angiography, kymography, bow hunter's syndrome 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3 CT angiography \"]","image_footnote":"[\"A：頭部CTangiography正面像は右後下小腦動脈はみらむ，左後下小腦動脈が両側小脑下方を灌流んだ。\",\"B：頸部CTangiography左前斜位像は右椎骨動脈は低形成%=，V3部で屈曲蛇行を認た.\",\"C：頸部CTangiography右前斜位像は左椎骨動脈に閉塞性病变はみらをたか。\",\"D：頭部CTangiography 以左椎骨動脈壁不整（矢印）為認，解離が疑わた。\"]","img_path":"images/0745/0bc3277b6ebf648f2c9dad2262ffb15a26a441eabf1a23c9be6e919666b92181.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[75,575,480,831]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"約30度すつ回旋て評価了一份，血流障害は検出くださいます。左椎骨動脈解離の可能性のは，1カ月後にMRAで壁不整が進行,Thいを確認した後，頸部血管造影を施行。左椎骨動脈造影は頸部正中位で壁不整を疑う所見等の異常はなかたが(Fig.4A)，頭部を右側に約60度回旋た際に，第1頸椎-第2頸椎移行部で血流は途絕し，遠位部は上行頸動脈の側副血行に則遲延て描出さた(Fig.4B,C)．右椎骨動脈造影は右側に60度回旋た際に閉塞性病変はみらしおかった，遠位部の血流は遲延た(Fig.4D)．小腦梗塞の原因为，血行力学の機序のみはなく，頭部右回旋に則左椎骨動脈に微細な損傷と血栓形成，そのによる塞栓症と診断た．病態にて說明し，頭部回旋に留意て自宅に退院,ThUp","text_format":null,"text_level":null,"type":"text"},{"bbox":"[75,835,480,868]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"本症例報告に際し，患者よう診療情報を症例報告へ用いとごの同意を得た.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[240,881,315,896]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"考察 ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[92,909,480,923]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"BHSは頭位を回旋さするとに用椎骨動脈が血流","text_format":null,"text_level":null,"type":"text"},{"bbox":"[514,575,922,925]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"障害来形容，椎骨脑底動脈循環不全に険る病態oled。頭部回旋は主に環椎と軸椎の閲節が担うた，頭部回旋は亀椎骨動脈が血流障害来形容部位も，C1-2レルにみらるごて多い3。その原因は，頭部回旋運動時には回旋側の環椎軸椎関節部は比較的固定せているのに対て，反対側の環椎は軸椎に対て前下方に変位するた。椎骨動脈が同部で進展・压迫さる点にある4(Fig.5).循環不全の機序は一般的に，椎骨動脈の閉塞性変化による血流低下の场合と，動脈損傷部の血栓形成による塞栓症のは。本例は頸椎の骨奇形や骨極形成は不孕のの，鞘帶の硬化等诸多加齟性変化に碍生理のな压迫が增強した状態が存在し，その状況を右へ頭部回旋たにと作用微細な動脈解離来形容し，付着た血栓が脑塞栓を来ったと考えた。画像所見は両側小脑に点状の梗塞を認たが，いども左後下小脑動脈の灌流域ename，左椎骨動脈を責任血管AVE。ま，右椎骨動脈は低形成て，前方循環からの側副血行も不十分くださいます。右回旋に用左椎骨動脈の血流が低下し，複視と倒了脑幹症状来形容た上物と考えら","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,52,134,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"44:390 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[403,49,591,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"脑卒中 44卷4号（2022：7）","text_format":null,"text_level":null,"type":"header"},{"bbox":"[154,110,357,351]","code_body":null,"code_caption":null,"image_caption":"[\"A \"]","image_footnote":"[]","img_path":"images/0304/d1c6a0fb1041f4de5c7389ef88f301531a30ac421d79e07b31565116b3d71de0.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[359,111,527,351]","code_body":null,"code_caption":null,"image_caption":"[\"B \"]","image_footnote":"[]","img_path":"images/0142/5ec30d903adef9cb9cbdc0b397041f8ce8fc437780f893fcb02e95828cd976b2.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[529,111,705,351]","code_body":null,"code_caption":null,"image_caption":"[\"C \"]","image_footnote":"[]","img_path":"images/0410/8d4f1fd9566527512e0aac1243e8b7620ab6a9935dc58ce3e877f4ea9d282e40.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[707,111,843,351]","code_body":null,"code_caption":null,"image_caption":"[\"D \",\"Fig.4 左椎骨動脈造影\"]","image_footnote":"[\"A：頭部正中位は左椎骨動脈に閉塞性病変はみらをた。\",\"B,C：頭部右回旋動態攝影正面像(B)，側面像(C)与左椎骨動脈はC1-2レルで血流が途絶し（矢印)，遠位部は上行頸動脈からの側副血行に作用描出た(矢頭).\",\"D：頭部右回旋動態攝影正面像與右椎骨動脈遠位部の描出は遲延た（矢印）\"]","img_path":"images/0609/7aefbda5ed9713a2af2e21905efbe89fe307ccb1a6e38ccf554047f712370c2f.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[191,476,473,771]","code_body":null,"code_caption":null,"image_caption":"[\"A \"]","image_footnote":"[]","img_path":"images/0321/2b5172c126271086f23807b8efda56ee0f1baf1b8c152c65a1f0a777a3d69a4f.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[492,476,806,771]","code_body":null,"code_caption":null,"image_caption":"[\"B \",\"Fig.5 項椎·椎骨動脈模型の左側かの觀察 \"]","image_footnote":"[\"A：頭部正中位\",\"B：頭部右回旋は環椎は軸椎に対て前下方に変位するた，左椎骨動脈が同部で進展·压迫さる.\"]","img_path":"images/0084/30410c3c95ff6cf1552410885cc52e0643f7b7d033ba725b2cb9ca134d8db6bc.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[73,853,480,925]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"た。右回旋時の血管造影は左上行頸動脈から左椎骨動脈閉塞部の遠位部に側副血行を認，遠位部は遲延て描出た。BHSの症状は，意識消失をはじ，耳鳴りやまい，視野異常んだ後方循環系の血流不全","text_format":null,"text_level":null,"type":"text"},{"bbox":"[514,853,921,923]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"に起因する症状が知らてい。BHSに前庭神経系の障害を起ごことも報告さており，内耳性てまい之称別で注意を要す．た，BHSの報告の多は，一側椎骨動脈の低形成か，頭部回旋に優位","text_format":null,"text_level":null,"type":"text"},{"bbox":"[337,49,658,63]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"頸部血管造影及診断はbowhunter症候群の1例","text_format":null,"text_level":null,"type":"header"},{"bbox":"[862,52,917,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"44:391 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[75,93,480,256]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"側の椎骨動脈の血流障害を来すのoledが，低形成側の椎骨動脈の血流障害にや眼振な的症状を来すとも有。BHSは頸部回旋にや優位側椎骨動脈の閉塞に乗的血行力学的機序和動脈損傷に乗的血栓塞栓症の機序のは，本例は両機序が合併ていのに加え，椎骨動脈閉塞部の遠位部へ側副血行の発達や対側椎骨動脈の屈曲蛇行,No，樣々要因が閲与ており，複視で癒症た点から非典型的なBHSか，多元のに檢討する必要くださ。","text_format":null,"text_level":null,"type":"text"},{"bbox":"[75,260,485,684]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"檢查方法とて，近年，超音波檢查の進步にようBHSを椎骨動脈超音波檢查や絶頭蓋ドフ拉檢查で診斷する報告8)也有．BHSを評価するには頸部を回旋て椎骨動脈の血流の変化を検出する必要,Noが，特別な姿勢下ごの検査とり，超音波檢查は検者的技量に右左右さること。本例は超音波检查時の頭部の回旋が不十分くださいますとあい，椎骨動脈の血流低下は検出くださいます．絶頭蓋ドフ拉檢查は主に前方循環の評価に使用さるが，後方循環系の血流を評価し，BHSを觀察する報告もさて．当院は絶頭蓋ドフ拉検査を实施くださす，本例は評価くださいますた．BHSの病態評価のたに，他的デバイスとてMRangiography或頸部CTangiography也施行，後交通動脈の発達程度や安静時の右椎骨動脈の屈曲蛇行等各种評価了一，原因を同定得なかた，頸部血管造影を施行了．頸部血管造影はは，正中位で椎骨動脈を造影し，形態を評価了上ご，頭部を回旋させて造影するとて血流障害の有無和部位を同定するとが可能のは．本例はは，椎骨動脈の閉塞に加え，左上行頸動脈から左椎骨動脈への側副血行も評価するとがきた閉塞の評価以外に側副血行の評価也越来越点で頸部血管造影動態撮影は有用と考えた．本例はは，抗血小板薬の内服と病態を説明するとて頸部回旋による頸動脈損傷","text_format":null,"text_level":null,"type":"text"},{"bbox":"[514,93,919,146]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"の注意を促し，再発なく經過てい．今後は，新規梗塞の出現や椎骨動脈の壁不整哪家変化をMRIで追跡評価する予定uable.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[680,158,754,173]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"結語","text_format":null,"text_level":"1","type":"text"},{"bbox":"[515,186,919,237]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"小腦梗塞と脑幹症状を呈たbowhunter症候群の1例を報告た．診断には椎骨動脈造影動態摄影が有用FOUNDATED.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[682,250,752,266]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"利益相反","text_format":null,"text_level":"1","type":"text"},{"bbox":"[515,279,919,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"著者は日本脑卒中学会のCOI自己申告を完了くださいます，本論文の発表に関係て，開示お願い COIはなり。","text_format":null,"text_level":null,"type":"text"},{"bbox":"[684,325,752,338]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"参考文献","text_format":null,"text_level":"1","type":"text"},{"bbox":"[522,353,921,681]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1) Sorensen BF: Bow hunter's stroke. Neurosurgery 2: 259-261, 1978 \",\"2) 小杉晴香，前田陽子，小林妙子ら：頸動脈エコ一檢查が診断の契機とんだ Bow hunter 症候群の一症例．島根医检45:34-38,2017\",\"3）林健太郎，松尾孝之，栗原正紀ら：若年性Bowhunter's strokeの1例．脑と神経52:431-434,2000\",\"4) Hanakita J, Miyake H, Nagayasu S, et al.: Angiographic examination and surgical treatment of bow hunter's stroke. Neurosurgery 23: 228-232, 1988 \",\"5) Jost GF, Dailey AT: Bow hunter's syndrome revisited: 2 new cases and literature review of 124 cases. Neurosurg Focus 38: E7, 2015 doi: 10.3171/2015.1.FOCUS14791 \",\"6）平林源希，武富弘敬，安藤裕史：前庭障害来し的Bowhunter症候群の1例．耳鼻展望60:281-285,2017\",\"7) Matsuyama T, Morimoto T, Sakaki T: Bow hunter's stroke caused by a nondominant vertebral artery occlusion: case report. Neurosurgery 41: 1393-1395, 1997 \",\"8）田中将平，稻富雄一郎，米原敏郎：自然輕快たbow hunter症候群の1例．臨神經52:34-37,2012\",\"9) Iguchi Y, Kimura K, Shibazaki K, et al.: Transcranial doppler and carotid duplex ultrasonography findings in Bow hunter's syndrome. J Neuroimaging 16: 278-280, 2006 \"]","page_idx":"4","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[77,52,134,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"44:392 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[403,49,591,63]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"脑卒中 44卷4号（2022：7）","text_format":null,"text_level":null,"type":"header"},{"bbox":"[463,93,532,105]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[168,123,828,139]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A case of bow hunter's syndrome diagnosed by digital subtraction angiography kymograph ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[258,154,739,184]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Nao Tahara, $^{1}$ Kentaro Hayashi, $^{2}$ Yukie Kanai, $^{1}$ Yoshie Kato, $^{1}$ Satoshi Abe, $^{2}$ Shingo Mitaki, $^{1}$ and Atsushi Nagai $^{1}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[332,196,663,223]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$^{1}$ Department of Neurology, Shimane University Hospital $^{2}$ Shimane University Hospital Advanced Stroke Center ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[136,237,860,363]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A 71-year-old woman presented to our care facility with complaints of diplopia and nausea when she rotated her head to the right side. She was neurologically intact and denied experiencing the symptoms when she rotated her head to the left side. Head MRI diffusion-weighted image showed a high-intensity spot in the cerebellum. MR angiography showed hypoplasia of the right vertebral artery (VA). However, the causes of the cerebellar infarction were not detected. We considered the possibility of bow hunter's syndrome and performed ultrasonography during head rotation. However, this failed to detect the flow impairment of VA. Finally, digital subtraction angiography was performed and it was found that the left VA was occluded at the C1-2 level when she rotated the head to the right side. Digital subtraction angiography kymography was useful for the diagnosis of bow hunter's syndrome. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[161,375,668,388]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Key words: digital subtraction angiography, kymography, bow hunter's syndrome ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[650,399,858,413]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(Jpn J Stroke 44:388-393,2022) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[337,49,658,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"頸部血管造影成診断了bowhunter症候群の1例","text_format":null,"text_level":null,"type":"header"},{"bbox":"[862,52,919,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"44:393 ","text_format":null,"text_level":null,"type":"page_number"}],"string":"[\n {\n \"bbox\": \"[178,149,818,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"頸部血管造影動態攝影で診断し得たbowhunter症候群の1例\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,192,722,210]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"田原 奈生 林 健太郎 金井由貴枝 加藤 芳惠\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[331,212,665,227]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"安部 哲史 2 三灌 真悟1 長井 答1\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[141,256,853,395]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"要旨：症例は71歳女性．後頸部に一過性に疼痛が出現た．その翌日，自家用車運転中に頭部を右に回旋た際，複視と恶心を自覺た．左に回旋ても症状の出現はなかた．救急外来受診時には症状が持続たが，徐々に改善た．身体所見に特記所見はなかた．頭部MRI拋散強調像で両側小脑に梗塞巢を認た．頭頸部CTA,頭頸部MRAは右椎骨動脈の低形成と蛇行を認たが，明かな異常は指摘くださいます．Bowhunter症候群を疑い，頭部を回旋し頸動脈エロー行った，流速低下は認うなかた．左椎骨動脈造影動態撮影で第1頸椎-第2頸椎移行部で血流が途絕し，bowhunter症候群と診断た．臨床所見上，bowhunter症候群が疑わった場合には，血管造影動態撮影が有用oled.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[142,414,678,429]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Key words: digital subtraction angiography, kymography, bow hunter's syndrome \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[240,473,315,489]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"緒言\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,502,482,665]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bowhunter症候群（bowhunter'ssyndrome:BHS）は頭位回旋による椎骨脑底動脈領域の循環不全が原因とせ，最初の報告例が一チル一競技中に複視と上下肢脱力が出現たから名付けらた1．診断に頸部血管超音波検査が有用と報告せてい的が2），頭部回旋時の血流障害的程度や障害部位の同定は困難;o。今回我せは，頭部回旋後に小脑梗塞来形容し，頸部血管造影動態撮影にてBHSを指摘得た症例を經驗しぃた則，文献的考察を加え報告する.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[242,677,315,693]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"症例提示\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,705,230,720]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"患者：71歲，女性\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,724,226,738]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"主訴：複視·惡心\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,743,226,757]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"既往歷：骨粗鬆症\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,760,480,794]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"現病歴：20XX年Y月某日，掃除を了後から後頸部に疼痛が出現了一、徐々に改善たて樣子を見てい\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,464,922,645]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7. 翌日，自家用車を車庫接入するたに頭部のみを60度から90度右に回旋し，後方を確認しぃうとし例に，複視と気分不良を自覺た。左に回旋いたします同様の症状はなかた。同日当院を受診た。当院到着時，ふらっは持續,leftいが複視は改善,leftい。救急外来の待合室で椅子に座り，頭部を右に60度から90度程度回旋ると，すに複視が再燃た。その時，頭位を正中位に尻しぃた左に回旋,leftい。複視は改善,leftい。脑卒中的可能性を疑われ，当科に診療依賴がた。\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,649,919,775]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"来院時現症：身長 $147~\\\\mathrm{cm}$ ，体重 $49.9\\\\mathrm{kg}$ ，体温36.4度，血压 $158 / 128\\\\mathrm{mmHg}$ ，脈拍 $67\\\\mathrm{bpm}$ ·整，酸素飽和度 $98\\\\%$ （室内氣），JCS0．一般身体所見に特記事項はなかた．神經學的所見は，頭部のみを右に60度程度回旋た際，複視と恶心の增悪はなかた．腦神經，運動系，感覺系，協調運動に異常はなか，步行障害もみられなかた.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,778,919,812]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"十二誘導心電円：心拍の不整や明かが異常はなかた. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,816,917,850]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"頸椎レntトgren：骨極之称の変形性の変化はみら�能なかた(Fig.1).\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[529,853,820,868]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"頭部CT：出血性變化はみらんかた. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,872,921,925]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"頭部MRI：擴散強調画像で両側小腦半球に点状の高信号域を認わた(Fig.2A).FLAIR像のは同部位に信号変化はなく，新鮮梗塞の所見くださった．頭部MRA て\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,52,134,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"44:388 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[78,92,206,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\spadesuit$ 症例报告 $\\\\spadesuit$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[652,74,919,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J-STAGE 早期公開2022年2月7日\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[742,93,919,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"脑卒中44:388-393,2022 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[92,832,324,845]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1島根大学医学部附属病院脑神絃内科\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[92,846,374,857]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2島根大学医学部附属病院高度脑卒中七夕一\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[92,859,391,871]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"責任著者：T693-8501 岛根県出雲市壇冶町89-1\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[92,872,376,884]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"島根大学医学部附属病院脑神經内科 田原奈生\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[92,887,307,898]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E-mail: nnishida@med.shimane-u.ac.jp \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[92,898,379,910]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(2021年10月7日受付，2021年11月4日受理)\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[92,913,240,923]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"doi: 10.3995/jstroke.10974 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[132,95,425,411]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig.1 頭椎單純レntトgren摄影 個面像のは骨極形成之称の変形は認なかた。\\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0450/fd28121477b7668e72785bc5aa4ec6791a5fc59ff70732504d745ea6105232fb.jpg\",\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[514,93,919,145]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"は脑主幹動脈に異常はみらんかた(Fig.2B).右後交通動脈はみらむ，細い左後交通動脈を認た(Fig. 2C).\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,149,919,237]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"入院後經過：病歴からBHSを疑った。当院到着から約3時間30分た特点でアスリン100mgを内服た．内服後から症状の改善を自覺た．入院後はるに右回旋しおいょう指導しぃつ，アスリン100mgの内服治療を開始た.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,242,919,294]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"經胸壁心臟超音波檢查のはは心拍出能は保たれ，心擴大はなかた．Holル一心電円ても心房細動い的不整脈はみらし，心原性脑塞栓症は否定のFOUND。\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,297,919,442]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"頭部CTangiographyは脑主幹動脈に異常を認ねなかた(Fig.3A).右後下小腦動脈はみらむ，左後下小腦動脈は太く両側性に灌流ているものと考えらた．頸部CTangiographyは右椎骨動脈は低形成,No，V3レルで屈曲·蛇行ていた(Fig.3B).左椎骨動脈に閉塞性病変はみらむかた(Fig.3C).左斜位像は左椎骨動脈V3部に壁不整を認た(Fig.3D).BHSを疑い，椎骨動脈超音波検査を施行た．左右に\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[228,460,457,648]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0838/a60397b9527b0599875c7c27773e456b7ff01a169e697de637a9c076730588db.jpg\",\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[463,460,769,648]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0900/0f8643c9fe5acd0f0a79af3dfa815743c54dca686d59b9758b6716bd5994ea29.jpg\",\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[438,655,769,858]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig.2 頭部MRI\\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0701/a0f5f13099a438c3f84136349e53d5246253a828d2d8a46a4477baca7906a720.jpg\",\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[171,882,766,923]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"A：批散強調像にて両側小腦半球に点状の高信号域を認た. \\\",\\\"B：MR angiography 正面像は脑主幹動脈に異常を認うた。\\\",\\\"C：MR angiography 轴位像は右後交通動脈はみらむ，細い左後交通動脈を認た（矢印）.\\\"]\",\n \"page_idx\": \"1\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[337,49,658,63]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"頸部血管造影及診断はbowhunter症候群の1例\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[862,52,919,63]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"44:389 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[151,95,401,263]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0156/b06d21c09eb8c8e163af6f6c6066ed2136c87e9626dafc615bad32224ca43cab.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[438,96,623,432]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0501/7a870260614dfa2eb767d065038f8a88f6ef159f26c2e3e320628372bf232337.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[638,96,847,432]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0851/5f4192361bca8f799ba662f0cb42e9d8a50059ca032bc93755b97d5b15cf0ce7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[152,268,406,448]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 3 CT angiography \\\"]\",\n \"image_footnote\": \"[\\\"A：頭部CTangiography正面像は右後下小腦動脈はみらむ，左後下小腦動脈が両側小脑下方を灌流んだ。\\\",\\\"B：頸部CTangiography左前斜位像は右椎骨動脈は低形成%=，V3部で屈曲蛇行を認た.\\\",\\\"C：頸部CTangiography右前斜位像は左椎骨動脈に閉塞性病变はみらをたか。\\\",\\\"D：頭部CTangiography 以左椎骨動脈壁不整（矢印）為認，解離が疑わた。\\\"]\",\n \"img_path\": \"images/0745/0bc3277b6ebf648f2c9dad2262ffb15a26a441eabf1a23c9be6e919666b92181.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[75,575,480,831]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"約30度すつ回旋て評価了一份，血流障害は検出くださいます。左椎骨動脈解離の可能性のは，1カ月後にMRAで壁不整が進行,Thいを確認した後，頸部血管造影を施行。左椎骨動脈造影は頸部正中位で壁不整を疑う所見等の異常はなかたが(Fig.4A)，頭部を右側に約60度回旋た際に，第1頸椎-第2頸椎移行部で血流は途絕し，遠位部は上行頸動脈の側副血行に則遲延て描出さた(Fig.4B,C)．右椎骨動脈造影は右側に60度回旋た際に閉塞性病変はみらしおかった，遠位部の血流は遲延た(Fig.4D)．小腦梗塞の原因为，血行力学の機序のみはなく，頭部右回旋に則左椎骨動脈に微細な損傷と血栓形成，そのによる塞栓症と診断た．病態にて說明し，頭部回旋に留意て自宅に退院,ThUp\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,835,480,868]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"本症例報告に際し，患者よう診療情報を症例報告へ用いとごの同意を得た.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[240,881,315,896]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"考察 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,909,480,923]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"BHSは頭位を回旋さするとに用椎骨動脈が血流\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,575,922,925]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"障害来形容，椎骨脑底動脈循環不全に険る病態oled。頭部回旋は主に環椎と軸椎の閲節が担うた，頭部回旋は亀椎骨動脈が血流障害来形容部位も，C1-2レルにみらるごて多い3。その原因は，頭部回旋運動時には回旋側の環椎軸椎関節部は比較的固定せているのに対て，反対側の環椎は軸椎に対て前下方に変位するた。椎骨動脈が同部で進展・压迫さる点にある4(Fig.5).循環不全の機序は一般的に，椎骨動脈の閉塞性変化による血流低下の场合と，動脈損傷部の血栓形成による塞栓症のは。本例は頸椎の骨奇形や骨極形成は不孕のの，鞘帶の硬化等诸多加齟性変化に碍生理のな压迫が增強した状態が存在し，その状況を右へ頭部回旋たにと作用微細な動脈解離来形容し，付着た血栓が脑塞栓を来ったと考えた。画像所見は両側小脑に点状の梗塞を認たが，いども左後下小脑動脈の灌流域ename，左椎骨動脈を責任血管AVE。ま，右椎骨動脈は低形成て，前方循環からの側副血行も不十分くださいます。右回旋に用左椎骨動脈の血流が低下し，複視と倒了脑幹症状来形容た上物と考えら\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,52,134,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"44:390 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[403,49,591,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"脑卒中 44卷4号（2022：7）\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[154,110,357,351]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"A \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0304/d1c6a0fb1041f4de5c7389ef88f301531a30ac421d79e07b31565116b3d71de0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[359,111,527,351]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"B \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0142/5ec30d903adef9cb9cbdc0b397041f8ce8fc437780f893fcb02e95828cd976b2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[529,111,705,351]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"C \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0410/8d4f1fd9566527512e0aac1243e8b7620ab6a9935dc58ce3e877f4ea9d282e40.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[707,111,843,351]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"D \\\",\\\"Fig.4 左椎骨動脈造影\\\"]\",\n \"image_footnote\": \"[\\\"A：頭部正中位は左椎骨動脈に閉塞性病変はみらをた。\\\",\\\"B,C：頭部右回旋動態攝影正面像(B)，側面像(C)与左椎骨動脈はC1-2レルで血流が途絶し（矢印)，遠位部は上行頸動脈からの側副血行に作用描出た(矢頭).\\\",\\\"D：頭部右回旋動態攝影正面像與右椎骨動脈遠位部の描出は遲延た（矢印）\\\"]\",\n \"img_path\": \"images/0609/7aefbda5ed9713a2af2e21905efbe89fe307ccb1a6e38ccf554047f712370c2f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[191,476,473,771]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"A \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0321/2b5172c126271086f23807b8efda56ee0f1baf1b8c152c65a1f0a777a3d69a4f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[492,476,806,771]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"B \\\",\\\"Fig.5 項椎·椎骨動脈模型の左側かの觀察 \\\"]\",\n \"image_footnote\": \"[\\\"A：頭部正中位\\\",\\\"B：頭部右回旋は環椎は軸椎に対て前下方に変位するた，左椎骨動脈が同部で進展·压迫さる.\\\"]\",\n \"img_path\": \"images/0084/30410c3c95ff6cf1552410885cc52e0643f7b7d033ba725b2cb9ca134d8db6bc.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[73,853,480,925]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"た。右回旋時の血管造影は左上行頸動脈から左椎骨動脈閉塞部の遠位部に側副血行を認，遠位部は遲延て描出た。BHSの症状は，意識消失をはじ，耳鳴りやまい，視野異常んだ後方循環系の血流不全\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,853,921,923]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"に起因する症状が知らてい。BHSに前庭神経系の障害を起ごことも報告さており，内耳性てまい之称別で注意を要す．た，BHSの報告の多は，一側椎骨動脈の低形成か，頭部回旋に優位\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[337,49,658,63]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"頸部血管造影及診断はbowhunter症候群の1例\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[862,52,917,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"44:391 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[75,93,480,256]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"側の椎骨動脈の血流障害を来すのoledが，低形成側の椎骨動脈の血流障害にや眼振な的症状を来すとも有。BHSは頸部回旋にや優位側椎骨動脈の閉塞に乗的血行力学的機序和動脈損傷に乗的血栓塞栓症の機序のは，本例は両機序が合併ていのに加え，椎骨動脈閉塞部の遠位部へ側副血行の発達や対側椎骨動脈の屈曲蛇行,No，樣々要因が閲与ており，複視で癒症た点から非典型的なBHSか，多元のに檢討する必要くださ。\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,260,485,684]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"檢查方法とて，近年，超音波檢查の進步にようBHSを椎骨動脈超音波檢查や絶頭蓋ドフ拉檢查で診斷する報告8)也有．BHSを評価するには頸部を回旋て椎骨動脈の血流の変化を検出する必要,Noが，特別な姿勢下ごの検査とり，超音波檢查は検者的技量に右左右さること。本例は超音波检查時の頭部の回旋が不十分くださいますとあい，椎骨動脈の血流低下は検出くださいます．絶頭蓋ドフ拉檢查は主に前方循環の評価に使用さるが，後方循環系の血流を評価し，BHSを觀察する報告もさて．当院は絶頭蓋ドフ拉検査を实施くださす，本例は評価くださいますた．BHSの病態評価のたに，他的デバイスとてMRangiography或頸部CTangiography也施行，後交通動脈の発達程度や安静時の右椎骨動脈の屈曲蛇行等各种評価了一，原因を同定得なかた，頸部血管造影を施行了．頸部血管造影はは，正中位で椎骨動脈を造影し，形態を評価了上ご，頭部を回旋させて造影するとて血流障害の有無和部位を同定するとが可能のは．本例はは，椎骨動脈の閉塞に加え，左上行頸動脈から左椎骨動脈への側副血行も評価するとがきた閉塞の評価以外に側副血行の評価也越来越点で頸部血管造影動態撮影は有用と考えた．本例はは，抗血小板薬の内服と病態を説明するとて頸部回旋による頸動脈損傷\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,93,919,146]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"の注意を促し，再発なく經過てい．今後は，新規梗塞の出現や椎骨動脈の壁不整哪家変化をMRIで追跡評価する予定uable.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[680,158,754,173]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"結語\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[515,186,919,237]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"小腦梗塞と脑幹症状を呈たbowhunter症候群の1例を報告た．診断には椎骨動脈造影動態摄影が有用FOUNDATED.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[682,250,752,266]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"利益相反\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[515,279,919,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"著者は日本脑卒中学会のCOI自己申告を完了くださいます，本論文の発表に関係て，開示お願い COIはなり。\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[684,325,752,338]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"参考文献\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[522,353,921,681]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1) Sorensen BF: Bow hunter's stroke. Neurosurgery 2: 259-261, 1978 \\\",\\\"2) 小杉晴香，前田陽子，小林妙子ら：頸動脈エコ一檢查が診断の契機とんだ Bow hunter 症候群の一症例．島根医检45:34-38,2017\\\",\\\"3）林健太郎，松尾孝之，栗原正紀ら：若年性Bowhunter's strokeの1例．脑と神経52:431-434,2000\\\",\\\"4) Hanakita J, Miyake H, Nagayasu S, et al.: Angiographic examination and surgical treatment of bow hunter's stroke. Neurosurgery 23: 228-232, 1988 \\\",\\\"5) Jost GF, Dailey AT: Bow hunter's syndrome revisited: 2 new cases and literature review of 124 cases. Neurosurg Focus 38: E7, 2015 doi: 10.3171/2015.1.FOCUS14791 \\\",\\\"6）平林源希，武富弘敬，安藤裕史：前庭障害来し的Bowhunter症候群の1例．耳鼻展望60:281-285,2017\\\",\\\"7) Matsuyama T, Morimoto T, Sakaki T: Bow hunter's stroke caused by a nondominant vertebral artery occlusion: case report. Neurosurgery 41: 1393-1395, 1997 \\\",\\\"8）田中将平，稻富雄一郎，米原敏郎：自然輕快たbow hunter症候群の1例．臨神經52:34-37,2012\\\",\\\"9) Iguchi Y, Kimura K, Shibazaki K, et al.: Transcranial doppler and carotid duplex ultrasonography findings in Bow hunter's syndrome. J Neuroimaging 16: 278-280, 2006 \\\"]\",\n \"page_idx\": \"4\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[77,52,134,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"44:392 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[403,49,591,63]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"脑卒中 44卷4号（2022：7）\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[463,93,532,105]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[168,123,828,139]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A case of bow hunter's syndrome diagnosed by digital subtraction angiography kymograph \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[258,154,739,184]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Nao Tahara, $^{1}$ Kentaro Hayashi, $^{2}$ Yukie Kanai, $^{1}$ Yoshie Kato, $^{1}$ Satoshi Abe, $^{2}$ Shingo Mitaki, $^{1}$ and Atsushi Nagai $^{1}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[332,196,663,223]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$^{1}$ Department of Neurology, Shimane University Hospital $^{2}$ Shimane University Hospital Advanced Stroke Center \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[136,237,860,363]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A 71-year-old woman presented to our care facility with complaints of diplopia and nausea when she rotated her head to the right side. She was neurologically intact and denied experiencing the symptoms when she rotated her head to the left side. Head MRI diffusion-weighted image showed a high-intensity spot in the cerebellum. MR angiography showed hypoplasia of the right vertebral artery (VA). However, the causes of the cerebellar infarction were not detected. We considered the possibility of bow hunter's syndrome and performed ultrasonography during head rotation. However, this failed to detect the flow impairment of VA. Finally, digital subtraction angiography was performed and it was found that the left VA was occluded at the C1-2 level when she rotated the head to the right side. Digital subtraction angiography kymography was useful for the diagnosis of bow hunter's syndrome. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[161,375,668,388]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Key words: digital subtraction angiography, kymography, bow hunter's syndrome \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[650,399,858,413]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(Jpn J Stroke 44:388-393,2022) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[337,49,658,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"頸部血管造影成診断了bowhunter症候群の1例\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[862,52,919,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"44:393 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.3995/jstroke.10974"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Medicine and Health Sciences"},"sha256":{"kind":"string","value":"3d77c22603b410507894aaa29380924089a36afe9e20245198d1c5cbfe35ccbc"},"title":{"kind":"string","value":"A case of bow hunter’s syndrome diagnosed by digital subtraction angiography kymograph"}}},{"rowIdx":43,"cells":{"abstract":{"kind":"string","value":"A medium chain-length polyhydroxyalkanoate (PHA) was produced by Pseudomonas mendocina CH50 using a cheap carbon substrate, sugarcane molasses. A PHA yield of 14.2% dry cell weight was achieved. Chemical analysis confirmed that the polymer produced was a medium chain-length PHA, a copolymer of 3-hydroxyoctanoate and 3-hydroxydecanoate, P(3HO-co-3HD). Lime oil, an essential oil with known antimicrobial activity, was used as an additive to P(3HO-co-3HD) to confer antibacterial properties to this biodegradable polymer. The incorporation of lime oil induced a slight decrease in crystallinity of P(3HO-co-3HD) films. The antibacterial properties of lime oil were investigated using ISO 20776 against Staphylococcus aureus 6538P and Escherichia coli 8739, showing a higher activity against the Gram-positive bacteria. The higher activity of the oil against S. aureus 6538P defined the higher efficiency of loaded polymer films against this strain. The effect of storage on the antimicrobial properties of the loaded films was investigated. After one-year storage, the content of lime oil in the films decreased, causing a reduction of the antimicrobial activity of the materials produced. However, the films still possessed antibacterial activity against S. aureus 6538P."},"author":{"kind":"string","value":"Basnett, Pooja|Marcello, Elena|Lukasiewicz, Barbara|Nigmatullin, Rinat|Paxinou, Alexandra|Ahmad, Muhammad Haseeb|Gurumayum, Bhavana|Roy, Ipsita"},"content_list":{"kind":"list like","value":[{"bbox":"[126,117,174,130]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Article ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,131,847,204]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Antimicrobial Materials with Lime Oil and a Poly(3-hydroxyalkanoate) Produced via Valorisation of Sugar Cane Molasses ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[127,222,845,254]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Pooja Basnett 1, Elena Marcello 1, Barbara Lukasiewicz 1, Rinat Nigmatullin $^ { 1 , 2 } \\oplus _ { i }$ Alexandra Paxinou 1, Muhammad Haseeb Ahmad 1, Bhavana Gurumayum 1 and Ipsita Roy $^ { 3 , * }$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[134,260,873,375]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 Faculty of Science and Technology, University of Westminster, London W1W 6UW, UK; P.Basnett@westminster.ac.uk (P.B.); w1614733@my.westminster.ac.uk (E.M.); barbara.lukasiewicz@gmail.com (B.L.); rn17541@bristol.ac.uk (R.N.); w1621777@my.westminster.ac.uk (A.P.); Mhaseebahmadd@gmail.com (M.H.A.); gurumayumbhavana@gmail.com (B.G.) \n2 Bristol Composites Institute (ACCIS), University of Bristol, Bristol BS8 1TR, UK \n3 Department of Material Science and Engineering, Faculty of Engineering, University of Sheffield, Sheffield S1 3JD, UK Correspondence: I.Roy@sheffield.ac.uk; Tel.: +44-114-2225962 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[142,392,628,406]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Received: 9 March 2020; Accepted: 6 April 2020; Published: 10 April 2020 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[136,419,873,626]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract: A medium chain-length polyhydroxyalkanoate (PHA) was produced by Pseudomonas mendocina CH50 using a cheap carbon substrate, sugarcane molasses. A PHA yield of $1 4 . 2 \\%$ dry cell weight was achieved. Chemical analysis confirmed that the polymer produced was a medium chain-length PHA, a copolymer of 3-hydroxyoctanoate and 3-hydroxydecanoate, P(3HO-co-3HD). Lime oil, an essential oil with known antimicrobial activity, was used as an additive to $\\mathrm { P ( 3 H O - } c o { - 3 } \\mathrm { H D ) }$ to confer antibacterial properties to this biodegradable polymer. The incorporation of lime oil induced a slight decrease in crystallinity of P(3HO-co-3HD) films. The antibacterial properties of lime oil were investigated using ISO 20776 against Staphylococcus aureus 6538P and Escherichia coli 8739, showing a higher activity against the Gram-positive bacteria. The higher activity of the oil against S. aureus 6538P defined the higher efficiency of loaded polymer films against this strain. The effect of storage on the antimicrobial properties of the loaded films was investigated. After one-year storage, the content of lime oil in the films decreased, causing a reduction of the antimicrobial activity of the materials produced. However, the films still possessed antibacterial activity against S. aureus 6538P. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[134,643,821,657]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords: polyhydroxyalkanotes; sugarcane molasses; antibacterial materials; essential oils ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,703,243,718]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,726,873,917]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Polyhydroxyalkanoates (PHAs) are a family of natural polymers produced by bacterial fermentation using renewable carbon sources under nutrient-limiting conditions. PHAs can be broadly classified into short chain length (SCL: $C _ { 3 } – C _ { 5 } \\mathrm { , }$ ) and medium-chain length (MCL: $C _ { 6 } – C _ { 1 4 } )$ ) based on the number of carbon atoms within their monomer units [1]. SCL PHAs have a high melting temperature and are brittle, whereas MCL PHAs are elastomeric and have a low-melting and glass-transition temperature. PHAs can match the functional properties of common petroleum-based plastics. For example, the mechanical properties of poly(3-hydroxybutyrate) (P(3HB)) are similar to those of polystyrene, while its copolymers with 3-hydroxyvalerate (P(3HB-co-3HV)) are characterised by an improved ductility resulting in a combination of mechanical properties close to polypropylene. One the other hand, MCL-PHAs can emulate the properties of soft thermoplastic elastomers such as thermoplastic polyurethanes, olefinic thermoplastic elastomers. The functional properties of PHA-based materials can be further modified and tuned via blending with other polymers [2]. Thanks to the presence of hydrolysable ester linkage connecting the monomer units of PHAs, PHAs are biodegradable in nature unlike most of conventional plastics. Their degradation rate is affected by the chemical composition of the polymer, and by environmental factors [3]. Based on life-cycle assessment (LCA) analysis, PHAs have been recognised as one of the safest bioplastics for the environment [4]. PHAs are also known to be biocompatible and have been widely investigated for various biomedical applications [5]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,57,295,98]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Journal of Functional Biomaterials ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[796,64,870,97]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"MDPI ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[127,941,447,952]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2020, 11, 24; doi:10.3390/jfb11020024 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[709,942,870,952]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"www.mdpi.com/journal/jfb ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[783,382,870,404]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"\u0001-\u0003\u0001\u0004\u0005\u0006\u0007\b \u0001-\u0003\u0004\u0005\u0006\u0007 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[126,103,873,199]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,200,873,473]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Although PHAs represent a family of sustainable polymers with attractive properties, they are yet to replace petroleum based conventional plastics used in most industrial applications. One of the major factors limiting the commercialisation of PHAs is its high production cost. Several cheap carbon sources, waste materials, genetically modified bacterial strains, fermentation conditions and recovery methods have been employed either on their own or in combination, to allow economical production of PHAs [3]. Since the cost of carbon accounts for $3 1 \\%$ of the overall PHA production cost, the use of inexpensive substrates is one of the effective strategies to produce PHAs in an economical manner [6,7]. Simple sugars, triglycerides (vegetable oils) are the most widely used carbon sources for the microbial production of PHAs [8]. In addition to their high cost, the utilization of such raw materials can be in competition with the use of these resources as food. Therefore, the valorisation of cheap agri-food by-products can address both issues; the cost of carbon source and the competition with food. Although potentially biomass rich in polysaccharides can be used for generating simple sugars, this requires additional chemical and enzymatic treatment. Molasses and whey, on the other hand, contain up to 60 and $4 . 5 \\ \\mathrm { w t . \\% }$ of simple sugars, respectively. Currently, a significant part of molasses is used in bioethanol production. The production of PHAs with the use of molasses as the carbon feedstock is a feasible alternative for the valorisation of molasses. Many studies demonstrated that molasses is a good substrate for microbial production of poly(3-hydroxybutyrate) [9]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,474,875,633]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PHAs have diverse physical properties which makes them suitable for various applications such as packaging, medical devices, bio-implants and tissue engineering [10]. For all these applications, microbial contamination is a major issue. Bacterial adhesion and colonisation on the polymer surface form a reservoir for pathogens which leads to various nosocomial infections, skin related issues, degradation of food and cosmetic products, and food contamination [11]. For several decades, antibiotics have been used to treat and prevent such infections. However, with the rise in the number of multidrug-resistant pathogens, antibiotic resistance has now become a global health risk. Annual costs of treating antibiotic resistant infections is estimated to be between $\\$ 21,000$ and $\\$ 34,000$ million in the United States alone and around €1500 million in Europe [12]. In such a scenario, there is a high demand for natural antibacterial agents to prevent or mitigate bacterial contamination. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,634,875,809]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Traditionally, essential oils, plant extracts and other natural products were used to treat infectious diseases. According to the World Health Organization (WHO), the majority of the world population is dependent on treatment using natural products as a primary care. Various essential oils have shown to possess antibacterial, antifungal and antioxidant properties [13–15]. These oils are now being screened to find alternative remedies to fight bacterial infections in a wide range of applications. Generally, a higher antimicrobial activity has been observed for oxygenated constituents of essential oils [16]. However, as antimicrobial additives to polymer materials, non-polar terpene hydrocarbons can bring other benefits such as improved compatibility with hydrophobic polymer matrices, enhanced water vapour barrier properties and plasticising effect [17]. Lime oil is an example of essential oils rich of terpene hydrocarbons such as limonene (ca. $4 0 \\%$ ), $\\beta$ -pinene (ca. $2 5 \\%$ ), and $\\gamma$ -terpinene (ca. $1 0 \\%$ ) [18]. The antimicrobial activity of lime oil is well documented [18–20]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,810,873,907]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The aim of this study was the development of natural antimicrobial polymer films via the incorporation of an essential oil into biodegradable PHA produced from agri-food by-products. We report a feasible way of valorisation of sugarcane molasses for the production of PHA by Pseudomonas mendocina CH50, a bacterial strain which has been previously confirmed as an efficient producer of MCL PHAs [5,20,21]. It was demonstrated that in the presence of sugarcane molasses $P .$ mendocina CH50 was able to accumulate an MCL PHA, which was identified as a copolymer of 3-hydroxyoctanoate and 3-hydroxydecanoate, or P(3HO-co-3HD). Lime oil was incorporated into P(3HO-co-3HD) films to produce a natural antimicrobial material. The oil content in the films was monitored during long storage, and loss of oil was observed during the storage. Both fresh and aged films exhibited antimicrobial activity against Gram-positive S. aureus. The films were less efficient against Gram-negative E. coli reflecting lower activity of lime oil against this bacterial strain. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,66,300,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2020, 11, 24 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[830,66,872,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 of 14 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[126,103,873,183]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,196,317,210]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. Material and Methods ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[129,223,438,237]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.1. Bacterial Strains and Culture Conditions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,247,873,342]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Pseudomonas mendocina CH50 was obtained from the University of Westminster culture collection. The strain was cultured at $3 0 ^ { \\circ } \\mathrm { C }$ for $1 6 \\mathrm { { h } }$ in a shaking incubator $( 2 0 0 \\mathrm { r p m } )$ . After incubation, the strain was stored as a glycerol stock at $- 8 0 ^ { \\circ } \\mathrm { C }$ . For the antibacterial characterization, Staphylococcus aureus 6538P and Escherichia coli 8739 were bought from the American Type Culture Collection (ATCC). They were cultured in sterile nutrient broth at $3 7 ^ { \\circ } \\mathrm { C }$ and $2 0 0 \\mathrm { r p m }$ for $1 6 \\mathrm { { h } }$ in a shaking incubator and stored as a glycerol stock at $- 8 0 ^ { \\circ } \\mathrm { C }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,354,228,368]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2. Chemicals ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,378,872,424]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"All the chemicals for the production and characterization of PHAs were purchased from Sigma Aldrich Ltd., England and VWR international, England. Lime oil used as an antibacterial agent was purchased from Holland and Barrett, London, UK. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[124,437,872,464]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.3. Production of Polyhydroxyalkanoates by Pseudomonas mendocina CH50 Using Sugarcane Molasses as the Sole Carbon Source ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,473,873,650]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Production of PHAs by Pseudomonas mendocina CH50 using $2 0 ~ \\mathrm { g / L }$ of sugarcane molasses was carried out in $1 5 \\mathrm { L }$ bioreactors, with $1 0 \\mathrm { ~ L ~ }$ working volume (Applikon Biotechnology, Tewkesbury, UK). Batch fermentation was carried out in two stages. The seed culture was prepared using a single colony of $P .$ mendocina CH50 to inoculate sterile nutrient broth. This was incubated for $1 6 \\mathrm { { h } }$ at $3 0 ~ ^ { \\circ } \\mathrm { C }$ and at $2 0 0 \\mathrm { r p m }$ . Inoculum of $1 0 \\ \\mathrm { v o l . \\% }$ of the final volume was used to inoculate the second stage seed culture (mineral salt medium-MSM) which was incubated at $3 0 ~ ^ { \\circ } \\mathrm { C } ,$ , $2 0 0 \\mathrm { r p m }$ for $2 4 \\mathrm { { h } }$ . Second stage seed culture was used to inoculate the final PHA production media [21,22]. The culture was grown for $4 8 \\mathrm { h } ,$ at $3 0 ^ { \\circ } C ,$ $2 0 0 \\mathrm { r p m }$ and air flow rate of 1 vvm. Temporal profile of the production of PHAs by $P .$ mendocina CH50 using $2 0 \\mathrm { g / L }$ sugarcane molasses was obtained by monitoring parameters such as optical density, biomass, nitrogen and glucose concentrations, $\\mathrm { p H }$ and dissolved oxygen tension (DOT) at regular intervals during the fermentation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,662,284,677]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.4. Analytical Studies ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,686,873,764]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bacterial growth was measured by monitoring the optical density (OD) at $4 5 0 \\mathrm { n m }$ . Biomass yield was determined by weighing the dried pellet of the cells obtained after centrifugation of $1 \\mathrm { m L }$ at 12,000 rpm for $1 0 \\mathrm { m i n }$ (Heraeus Pico 17 Centrifuge, Thermofisher Scientific, MA, USA). pH of the supernatant was measured using Seven Compact $\\mathrm { p H }$ meter (Mettler Toledo Ltd., Leicester, UK). Nitrogen in the form of ammonium ions was estimated by the phenol-hypochlorite method [21,22]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,766,873,860]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Polymer was extracted from the dried biomass using the soxhlet extraction method. To remove organic soluble impurities, dried biomass was refluxed in methanol for $2 4 \\mathrm { { h } }$ followed by refluxing in chloroform for $4 8 \\mathrm { h }$ to extract the polymer. Polymer solution in chloroform was concentrated using a rotary evaporator. Polymer was precipitated using ice-cold methanol under continuous stirring and dried at room temperature. Polymer yield was calculated as a percentage of dry cell weight (DCW), using the formula: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[371,859,626,890]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0177/9c5a12acf1c7bab3a2b37f9728b82a2480090b2680ceee151b6bc57ddc7efcf8.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\% \\mathrm { { D C W } = \\frac { \\mathrm { { P o l y m e r ~ m a s s } } } { \\mathrm { { M a s s ~ o f ~ b i o m a s s } } } \\times 1 0 0 }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[126,66,300,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2020, 11, 24 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[831,66,872,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 of 14 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[127,104,351,118]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.5. Preparation of Film Samples ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,128,873,271]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Solvent cast films were prepared by pouring 5 wt. $\\%$ polymer solution in chloroform in $6 0 \\mathrm { m m }$ diameter glass Petri dishes. The polymer was dried in the fume hood at the room temperature until the solvent evaporated completely. For oil-incorporated films, the oils were added to 5 wt. $\\%$ polymer solution in chloroform. The loaded films were stored at room temperature in sealed Petri dishes for one year at room temperature and identified as aged samples. Quantification of the oil content in the films after complete solvent evaporation was carried out by thermogravimetric analysis (TGA) using the STA 449 F3 Jupiter® instrument (Netzsch, Germany); $1 0 { - } 1 5 \\mathrm { m g }$ film samples were heated at 10 degrees per minute from 30 to $1 5 0 ^ { \\circ } \\mathrm { C }$ under nitrogen flow and kept at $1 5 0 ^ { \\circ } \\mathrm { C }$ for $2 \\mathrm { { h } }$ . The cumulative mass loss in dynamic and isothermal steps was defined as oil amount left in the films after solvent evaporation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[127,284,332,298]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.6. Polymer Characterization ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,306,873,386]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The monomeric composition of polymer was determined using gas chromatography–mass spectrometry (GC–MS) analysis and $^ { 1 3 } { \\cal C }$ , $^ 1 \\mathrm { H }$ nuclear magnetic resonance (NMR) spectroscopy. For GC–MS analysis (Chrompack CP-3800 gas chromatograph and Saturn 200 MS/MS block), $2 0 \\mathrm { m g }$ of the polymer sample was subjected to methanolysis as described in Basnett et al. [5]. For NMR analysis, the polymer was dissolved in deuterated chloroform $( 2 0 \\mathrm { m g / m L } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,387,873,449]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Thermal analysis of polymers was conducted using a differential scanning calorimeter (DSC) 214 Polyma (Netzsch, Germany) equipped with Intracooler IC70 cooling system. $5 \\mathrm { m g }$ of polymer was heated from $- 5 0$ to $1 0 0 ^ { \\circ } \\mathrm { C }$ at a heating rate of $2 0 ~ ^ { \\circ } \\mathrm { C }$ per min. DSC thermograms were analysed using Proteus 7.0 software. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[127,462,364,478]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.7. Antibacterial Characterization ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[129,489,848,505]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.7.1. MIC (Minimum Inhibition Concentration) and MBC (Minimum Bactericidal Concentration) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,513,873,689]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The minimum inhibition concentration (MIC) was estimated according to the ISO 20776 against Staphylococcus aureus 6538P and Escherichia coli 8739. Stock solution of the lime oil was prepared in Mueller Hinton Broth and $1 \\%$ dimethyl sulfoxide (added as a co-solvent which enables mixing the oils with aqueous medium). A range of concentrations from 100 to $5 ~ { \\mu \\mathrm { L } } / { \\mathrm { m L } }$ essential oils was tested in a 96-well plate. Each well was inoculated to a final volume of $1 0 0 \\mu \\mathrm { L }$ using a microbial suspension adjusted to achieve a final concentration of $5 \\times 1 0 ^ { 5 } \\mathrm { C F U / m L }$ and incubated at $3 7 ^ { \\circ } \\mathrm { C }$ for $2 4 \\mathrm { ~ h ~ }$ at 100 rpm. Studies were performed in triplicate. The MIC was determined by measuring the absorbance at $6 0 0 \\mathrm { n m }$ . Following the MIC test, the entire volume of the MIC well and the concentrations above were removed and 10-fold dilution was performed in PBS and plated onto agar plates. The plates were incubated at $3 7 ^ { \\circ } \\mathrm { C }$ for $2 4 \\mathrm { { h } }$ . The minimum bactericidal concentration (MBC) was determined as the lowest concentration of the compound that induced $9 9 . 9 \\%$ killing of the bacteria. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,701,243,715]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.7.2. Halo Test ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,725,873,901]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The films loaded with essential oils were tested for antibacterial activity using the halo test against Staphylococcus aureus 6538P and Escherichia coli 8739. The test was conducted according to the European Committee on Antimicrobial Susceptibility Testing (EUCAST) Disc Diffusion Test Methodology. A single colony was used to prepare a suspension of 0.5 McFarland turbidity standard (approx. $1 0 ^ { 8 }$ colony-forming units $\\mathrm { ( C F U ) / m L }$ ). A sterile cotton swab was immersed into the suspension and was spread evenly on the agar plates. The agar plates as well as the antimicrobial discs were kept at room temperature prior to inoculation. Polymer films of $0 . 7 \\mathrm { c m }$ diameter containing active agents, sterilized under ultraviolet (UV) light for $3 0 \\mathrm { m i n }$ (15 min each side) were placed on the inoculated agar surface. The plates were incubated at $3 7 ^ { \\circ } \\mathrm { C }$ for $1 8 \\mathrm { { h } }$ . The inhibition zones formed were measured and noted. For the antibacterial test, both freshly prepared samples and aged samples were investigated. Each material was tested in triplicates. Films without active agents were used as negative controls, antibiotic discs containing oxacillin (1 µg/disc) and streptomycin J. Funct. Biomater. 2020, 11, x FOR PEER REVIEW J. Funct. Biomater. 2020, 11, x FOR PEER REVIEWFunct. Biomater. 2020, 11, x FOR PEER REVIEW $( 3 0 0 ~ \\mu \\mathrm { g / d i s c } )$ were used as positive controls2 of 142 of 142 of 14 against Staphylococcus aureus 6538P and Escherichia coli 8739, respectively. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,66,300,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2020, 11, 24 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[831,66,872,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 of 14 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[122,104,870,135]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,147,324,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3. Results and Discussion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[127,174,860,203]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.1. Temporal Profile of Polyhydroxyalkanoate (PHA) Production by Pseudomonas mendocina CH50 UsingSugarcane Molasses as the Sole Carbon Source3.1. Temporal Profile of Polyhydroxyalkanoate (PHA) Production by Pseudomonas mendocina CH50 using3.1. Temporal Profile of Polyhydroxyalkanoate (PHA) Production by Pseudomonas mendocina CH50 using 3.1. Temporal Profile of Polyhydroxyalkanoate (PHA) Production by Pseudomonas mendocina CH50 using .1. Temporal Profile of Polyhydroxyalkanoate (PHA) Production by Pseudomonas mendocina CH50 using Sugarcane Molasses as the Sole Carbon SourceSugarcane Molasses as the Sole Carbon SourceSugarcane Molasses as the Sole Carbon Sourceugarcane Molasses as the Sole Carbon Sourceugarcane Molasses as the Sole Carbon Sourceugarcane Molasses as the Sole Carbon Source ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,211,873,259]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Figure 1 represents the fermentation profile of the PHA production by Figure 1 represents the fermentation profile of the PHA production byFigure 1 represents the fermentation profile of the PHA production by Figure 1 represents the fermentation profile of the PHA production by Pigure 1 represents the fermentation profile of the PHA production by Pigure 1 represents the fermentation profile of the PHA production by P. $P .$ mendocina CH50 using . mendocina CH50 using mendocina CH50 using mendocina CH50 using mendocina CH50 using endocina CH50 using sugarcane molasses as the carbon source in asugarcane molasses as the carbon sourcesugarcane molasses as the carbon source sugarcane molasses as the carbon source ugarcane molasses as the carbon source iugarcane molasses as the carbon source i $1 5 \\mathrm { L }$ bioreactor. Fermentation was performed as described a 15 L bioreactor. Fermentation was performed as 15 L bioreactor. Fermentation was performed as 15 L bioreactor. Fermentation was performed as 15 L bioreactor. Fermentation was performed as 5 L bioreactor. Fermentation was performed as in Section 2.3., and OD,described in Section 2described in Section 2.described in Section 2.3escribed in Section 2.3.escribed in Section 2.3. $\\mathrm { p H } , \\%$ DOT, biomass, nitrogen concentration and polymer yield were monitored.d OD, pH, % DOT, biomass, nitrogen concentration and polymer yield OD, pH, % DOT, biomass, nitrogen concentration and polymer yield OD, pH, % DOT, biomass, nitrogen concentration and polymer yield OD, pH, % DOT, biomass, nitrogen concentration and polymer yield OD, pH, % DOT, biomass, nitrogen concentration and polymer yield ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[181,279,798,550]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 1. Temporal profile of P(3HO-co-3HD) production byFigure 1. Temporal profile of P(3HO-co-3HD) production by Figure 1. Temporal profile of P(3HO-co-3HD) production byFigure 1. Temporal profile of P(3HO-co-3HD) production by Figure 1. Temporal profile of P(3HO-co-3HD) production by Pigure 1. Temporal profile of P(3HO-co-3HD) production by Pigure 1. Temporal profile of P(3HO-co-3HD) production by P. $P .$ . mendocina CH50 using sugarcane mendocina CH50 using sugarcane. mendocina CH50 using sugarcane mendocina CH50 using sugarcane mendocina CH50 using sugarcane mendocina CH50 using sugarcane endocina CH50 using sugarcane molasses as the sole carbon source. The profiling illustrates the polyhydroxyalkanoate (PHA) yield (molasses as the sole carbon source. The profiling illustrates the polyhydroxyalkanoate (PHA) yieldmolasses as the sole carbon source. The profiling illustrates the polyhydroxyalkanoate (PHA) yield (molasses as the sole carbon source. The profiling illustrates the polyhydroxyalkanoate (PHA) yield (molasses as the sole carbon source. The profiling illustrates the polyhydroxyalkanoate (PHA) yield (olasses as the sole carbon source. The profiling illustrates the polyhydroxyalkanoate (PHA) yield (olasses as the sole carbon source. The profiling illustrates the polyhydroxyalkanoate (PHA) yield ( ), optical density( ), optical density), optical density), optical density ), optical density ( ), optical density ( ), optical density ( $( \\\\cdots ) , \\\\mathrm { { p H } ( \\\\not { p } \\\\not { q } \\\\mathrm { { \\\\Gamma } } ^ { } \\\\not { q } \\\\mathrm { { \\\\Gamma } } ^ { } ) , \\\\mathrm { { D O T } } ^ { \\\\circ } / \\\\mathrm { { \\\\Gamma } } ^ { } ( \\\\not { q } \\\\mathrm { { \\\\Gamma } } ^ { } \\\\not { q } \\\\mathrm { { \\\\Gamma } } ^ { } ) , }$ ), bioma), bioma), bioma, biomass, biomass biomass biomass $( \\\\twoheadleftarrow )$ ) and the nitrogen concentration) and the nitrogen concentration) and the nitrogen concentration ) and the nitrogen concentration and the nitrogen concentration and the nitrogen concentration and the nitrogen concentration in in the form of the ammonium ions throughin the form of the ammonium ions throughin the form of the ammonium ions throughon the form of the ammonium ions throughoun the form of the ammonium ions throughou the form of the ammonium ions throughoutthe form of the ammonium ions throughout $4 8 \\\\mathrm { h }$ 8 h of fermentati8 h of fermentati h of fermentatioh of fermentatioh of fermentation of fermentation. of fermentation. \"]","image_footnote":"[]","img_path":"images/0795/5f5af7ee949750e1dd287a8e247b21bd6285b3ef3bcd300aa77b964fabbdcf7d.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[124,634,875,908]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The optical density increased gradually until the firstThe optical density increased gradually until the firstThe optical density increased gradually until the first he optical density increased gradually until the first he optical density increased gradually until the first 3he optical density increased gradually until the first 3The optical density increased gradually until the first $^ { 3 \\mathrm { ~ h ~ } }$ (lag phase) of the fermentation and(lag phase) of the fermentation and(lag phase) of the fermentation and (lag phase) of the fermentation and lag phase) of the fermentation and ag phase) of the fermentation and (lag phase) of the fermentation and continued to increase steadily until the end of process, reaching a value of 4.7. Biomass concentrationcontinued to increase steadily until the end of process, reaching a value of 4.7. Biomass concentrationcontinued to increase steadily until the end of process, reaching a value of 4.7. Biomass concentration ontinued to increase steadily until the end of process, reaching a value of 4.7. Biomass concentration ontinued to increase steadily until the end of process, reaching a value of 4.7. Biomass concentration ontinued to increase steadily until the end of process, reaching a value of 4.7. Biomass concentration continued to increase steadily until the end of process, reaching a value of 4.7. Biomass concentration increased untiincreased untiincreased untilncreased until ncreased until 4creased until 4increased until $4 8 \\mathrm { h }$ reaching a value ofreaching a value oreaching a value ofeaching a value of 1eaching a value of 1.aching a value of 1. reaching a value of $1 . 9 \\mathrm { g / L }$ . Prior to the fermentation,. Prior to the fermentation, Prior to the fermentation,Prior to the fermentation, Prior to the fermentation, prior to the fermentation, p. Prior to the fermentation, $\\mathrm { p H }$ of the production media of the production media of the production media f the production media f the production media the production media of the production media was set to 7.0. Thewas set to 7.0. Thewas set to 7.0. The was set to 7.0. The pwas set to 7.0. The pas set to 7.0. The pwas set to 7.0. The $\\mathrm { p H }$ of the culture did not change significantly during the fermentation process of the culture did not change significantly during the fermentation processof the culture did not change significantly during the fermentation process f the culture did not change significantly during the fermentation process f the culture did not change significantly during the fermentation process f the culture did not change significantly during the fermentation process of the culture did not change significantly during the fermentation process whereas thwhereas thwhereas thewhereas the whereas the hereas the whereas the $\\mathrm { D O T \\% }$ % reduced fro% reduced fro reduced fro reduced fromreduced fromreduced from reduced from $1 0 0 \\%$ % %% t t to to $0 \\%$ 0% within few hours of the fermentation and remained0% within few hours of the fermentation and remained% within few hours of the fermentation and remained % within few hours of the fermentation and remained within few hours of the fermentation and remained within few hours of the fermentation and remained within few hours of the fermentation and remained very very low until the end of the process, maintaining an oxygen-limiting environment. Nutrient very low until the end of the process, maintaining an oxygen-limiting environment. Nutrientvery low until the end of the process, maintaining an oxygen-limiting environment. Nutrient very low until the end of the process, maintaining an oxygen-limiting environment. Nutrient ery low until the end of the process, maintaining an oxygen-limiting environment. Nutrient ery low until the end of the process, maintaining an oxygen-limiting environment. Nutrient low until the end of the process, maintaining an oxygen-limiting environment. Nutrient limitation limitation plays an important role in the production of PHAs. Furthermore, it is known that nutrientlimitation plays an important role in the production of PHAs. Furthermore, it is known that nutrientimitation plays an important role in the production of PHAs. Furthermore, it is known that nutrientimitation plays an important role in the production of PHAs. Furthermore, it is known that nutrientmitation plays an important role in the production of PHAs. Furthermore, it is known that nutrientplays an important role in the production of PHAs. Furthermore, it is known that nutrient limitation limitation coupled with an excess of carbon provides an ideal environment to produce PHAs [23]. In limitation coupled with an excess of carbon provides an ideal environment to produce PHAs [23]. Inlimitation coupled with an excess of carbon provides an ideal environment to produce PHAs [23]. In imitation coupled with an excess of carbon provides an ideal environment to produce PHAs [23]. In imitation coupled with an excess of carbon provides an ideal environment to produce PHAs [23]. In mitation coupled with an excess of carbon provides an ideal environment to produce PHAs [23]. In coupled with an excess of carbon provides an ideal environment to produce PHAs [23]. In this study, this study, nitrogen was the limiting factor. As illustrated in Figure 1, the nitrogen concentrationthis study, nitrogen was the limiting factor. As illustrated in Figure 1, the nitrogen concentrationthis study, nitrogen was the limiting factor. As illustrated in Figure 1, the nitrogen concentration this study, nitrogen was the limiting factor. As illustrated in Figure 1, the nitrogen concentration his study, nitrogen was the limiting factor. As illustrated in Figure 1, the nitrogen concentration his study, nitrogen was the limiting factor. As illustrated in Figure 1, the nitrogen concentration nitrogen was the limiting factor. As illustrated in Figure 1, the nitrogen concentration decreased from $0 . 5 8 ~ \\mathrm { g / L }$ ed ed d f fr frfro to $0 . 1 4 ~ \\mathrm { g / L }$ 8 g/L to 0.14 g8 g/L to 0.14 g g/L to 0.14 gg/L to 0.14 g/g/L to 0.14 g/L/L to 0.14 g/L at the end of $4 8 – \\mathrm { h }$ the end of 48-h fermentation. T the end of 48-h fermentation. Tthe end of 48-h fermentation. The end of 48-h fermentation. Thhe end of 48-h fermentation. The end of 48-h fermentation. The fermentation. The PHA yield $\\%$ PHA yi PHA yiPHA yieHA yielHA yielHA yield dcw) at $^ { 4 8 \\mathrm { ~ h ~ } }$ % dcw) at 48 h% dcw) at 48 h dcw) at 48 hdcw) at 48 hdcw) at 48 hdcw) at 48 h was found to was fowas fowas fowas fouwas fouas founbe was $1 4 . 2 2 \\%$ be was 14.22%. As mentioned above, optical density and biomass concentration be was 14.22%. As mentioned above, optical density and biomass concentrationbe was 14.22%. As mentioned above, optical density and biomass concentration e was 14.22%. As mentioned above, optical density and biomass concentration e was 14.22%. As mentioned above, optical density and biomass concentration e was 14.22%. As mentioned above, optical density and biomass concentration . As mentioned above, optical density and biomass concentration increased steadily increased steadily during the fermentation indicating cell growth in the presence of nitrogen in theincreased steadily during the fermentation indicating cell growth in the presence of nitrogen in theincreased steadily during the fermentation indicating cell growth in the presence of nitrogen in the ncreased steadily during the fermentation indicating cell growth in the presence of nitrogen in the ncreased steadily during the fermentation indicating cell growth in the presence of nitrogen in the creased steadily during the fermentation indicating cell growth in the presence of nitrogen in the during the fermentation indicating cell growth in the presence of nitrogen in the media. The absence media. The absence of nitrogen limitation could have influenced the overall accumulation of PHAs.media. The absence of nitrogen limitation could have influenced the overall accumulation of PHAs.media. The absence of nitrogen limitation could have influenced the overall accumulation of PHAs. media. The absence of nitrogen limitation could have influenced the overall accumulation of PHAs. edia. The absence of nitrogen limitation could have influenced the overall accumulation of PHAs. edia. The absence of nitrogen limitation could have influenced the overall accumulation of PHAs. of nitrogen limitation could have influenced the overall accumulation of PHAs. Sugarcane molasses Sugarcane molasses have been previously used as a carbon substrate to produce SCL-PHA by BacillusSugarcane molasses have been previously used as a carbon substrate to produce SCL-PHA by BacillusSugarcane molasses have been previously used as a carbon substrate to produce SCL-PHA by Bacillus Sugarcane molasses have been previously used as a carbon substrate to produce SCL-PHA by Bacillus ugarcane molasses have been previously used as a carbon substrate to produce SCL-PHA by Bacillus ugarcane molasses have been previously used as a carbon substrate to produce SCL-PHA by Bacillus have been previously used as a carbon substrate to produce SCL-PHA by Bacillus cereus SPV with a cereus SPV wcereus SPV wcereus SPV wereus SPV wiereus SPV witreus SPV witPHA yield of $5 1 . 3 7 \\%$ HA yield of 51.37% dcw [24]. In another study, Naheed and Jamil obtained a HA yield of 51.37% dcw [24]. In another study, Naheed and Jamil obtained aHA yield of 51.37% dcw [24]. In another study, Naheed and Jamil obtained a A yield of 51.37% dcw [24]. In another study, Naheed and Jamil obtained a A yield of 51.37% dcw [24]. In another study, Naheed and Jamil obtained a A yield of 51.37% dcw [24]. In another study, Naheed and Jamil obtained a dcw [24]. In another study, Naheed and Jamil obtained a maximum PHA yield of $8 0 . 5 \\%$ mum PHA yield of 80.5% dcw by Enterobacter sp. SEL2, a bacterium isolated from soil ofmum PHA yield of 80.5% dcw by Enterobacter sp. SEL2, a bacterium isolated from soil ofum PHA yield of 80.5% dcw by Enterobacter sp. SEL2, a bacterium isolated from soil of um PHA yield of 80.5% dcw by Enterobacter sp. SEL2, a bacterium isolated from soil of um PHA yield of 80.5% dcw by Enterobacter sp. SEL2, a bacterium isolated from soil of m PHA yield of 80.5% dcw by Enterobacter sp. SEL2, a bacterium isolated from soil of dcw by Enterobacter sp. SEL2, a bacterium isolated from soil of contaminated sites, with the use $3 \\%$ taminated sites, with the use 3% of sugarcantaminated sites, with the use 3% of sugarcataminated sites, with the use 3% of sugarcanaminated sites, with the use 3% of sugarcanaminated sites, with the use 3% of sugarcaneminated sites, with the use 3% of sugarcane of sugarcane molasses after fermentation for $2 4 \\mathrm { h }$ olololalasasass at $\\mathsf { p H } 5 . 0$ ter fermentation for 24 h at pH 5.0 [25].ter fermentation for 24 h at pH 5.0 [25].ter fermentation for 24 h at pH 5.0 [25]. er fermentation for 24 h at pH 5.0 [25]. r fermentation for 24 h at pH 5.0 [25]. r fermentation for 24 h at pH 5.0 [25]. [25]. Chaijamrus and Udpuay reported a $4 3 \\%$ yield of P(3HB) by Bacillus megaterium ATCC 6748 after $4 5 \\mathrm { h }$ of fermentation when $4 \\%$ sugarcane molasses was used as the carbon source [26]. With regards to MCL-PHA production using sugarcane molasses, a limited investigation has been conducted. This is due to the fact that the majority of theJ. Funct. Biomater. 2020, 11, x FOR PEER REVIEW 2 of 14 Pseudomonas species (i.e., the main producer of MCL-PHAs) cannot utilize sucrose, which is one of the main components of sugarcane molasses along with glucose and fructose [27]. Therefore, the substrate is usually modified in order to obtain equimolar concentrations of glucose and fructose by either ionic exchange or acid hydrolysis [27]. To date, only two species of the Pseudomonas family have been shown to metabolize sucrose directly, P. fluorescens A2a5 and P. corrugata 388. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,65,302,77]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2020, 11, 24 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[830,66,872,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 of 14 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[126,103,873,231]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,233,873,328]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"P. fluorescens A2a5 produced P(3HB) with a yield of $7 0 \\%$ dcw when cultured using sugarcane liquor [28]. In contrast, Solaiman et al. showed that P. corrugata 388 was able to consume sucrose (present in soy molasses used as carbon source) producing an MCL-PHA with a yield of $1 7 \\%$ dcw [27]. P. mendocina CH50, the strain used in this study is not able to utilize the sucrose present in the media. The PHA produced by P. mendocina CH50 using sugarcane molasses as the carbon source wasmedia. The PHA produced by P. mendocina CH50 using sugarcane molasses as the carbon source was further characterised. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[127,341,331,355]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.2. Polymer Characterisation ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[129,368,403,382]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.2.1. Polymer Composition Analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,391,873,551]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As can be seen from Figure 2, GC–MS chromatograms of the methanolysed polymer demonstrated two peaks identified using the National Institute of Standards and Technology (NIST) library. The retention peak at 7.7 and 9.2 min were identified as the methyl ester of 3-hydroxyoctanoic acid and methyl ester of 3-hydroxydecanoic acid respectively. Peak at $6 . 6 4 \\mathrm { m i n }$ corresponded to the methyl benzoate, used as an internal standard. Thus, the polymer produced by methyl benzoate, used as an internal standard. Thus, the polymer produc $P .$ mendocina CH50 using by P. mendocina CH50 sugarcane molasses as the sole carbon source was identified as the copolymer of 3-hydroxyoctanoate and 3-hydroxydecanoate, Poly(3-hydroxyoctanoate-co-3-hydroxydecanoate) or P(3HO-co-3HD). The mole percentage of 3-HO and 3-HD was found to beP(3HO-co-3HD). The mole percentage of 3-HO an $2 5 . 6 \\%$ andD w $7 4 . 4 \\%$ respectively. P(3HO-co-3HD)nd to be 25.6% and 74.4% copolymers with 3-HD-dominant composition are usually produced by Pseudomonas species whenrespectively. P(3HO-co-3HD) copolymers with 3-HD-dominant composition are usually produced grown on structurally unrelated simple sugars [29].by Pseudomonas species when grown on structurally ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[136,568,860,772]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 2. Gas chromatography–mass spectrometry (GC–MS) data of the PHA copolymer produced Figure 2. Gas chromatography–mass spectrometry (GC–MS) data of the PHA copolymer produced with sugarcane molasses as the carbon source: (A) Gas chromatogram, (B) Mass spectrum of a peak with sugarcane molasses as the carbon source: (A) Gas chromatogram, (B) Mass spectrum of a peak witwith $\\\\mathrm { R } _ { \\\\mathrm { t } } = 7 . 7$ min identified using the National Institute of Standards and Technology (NIST) library min identified using the National Institute of Standards and Technology (NIST) library as as methyl ester of 3-hydroxyoctanoic acid, and (C) mass spectrum of a pemethyl ester of 3-hydroxyoctanoic acid, and (C) mass spectrum of a peak with $\\\\mathrm { R _ { t } } = 9 . 2$ Rt = 9.2 min min identified identified using NIST library as methyl ester of 3-hydroxydecanoic acid. Methyl benzoate was used using NIST library as methyl ester of 3-hydroxydecanoic acid. Methyl benzoate was used as an internal as an internal ststandard (STD). \"]","image_footnote":"[]","img_path":"images/0591/a9fd3ca4adbb7947f7dd7ad1ea510fc98d732b866bdb89b12480aa00dde73187.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[124,887,873,917]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In order to confirm the structure of synthesised PHA, NMR analysis has been performed. In order to confirm the structure of synthesised PHA, NMR analysis has been performed. $^ 1 \\mathrm { H }$ nd and $^ { 1 3 } \\mathrm { C }$ NMR spectra are presented in Figure 3. All peaks expected for P(3HO-co-3HD) where observed NMR spectra are presented in Figure 3. All peaks expected for P(3HO-co-3HD) where observed in $^ 1 \\mathrm { H }$ and $^ { 1 3 } \\mathrm { C }$ NMR spectroscopy. The chemical shift at 169, 70.9, 39.1 ppm corresponds to $C _ { 1 }$ $\\scriptstyle ( C = \\mathbf { O } )$ , $C _ { 3 }$ (–CH), $C _ { 2 }$ $\\mathrm { ( - C H } _ { 2 } )$ respectively, forming the backbone of the copolymer. The resonance in the range between $2 3 \\mathrm { - } 3 5 \\mathrm { p p m }$ related to olefinic methylene groups: $\\mathrm { C _ { 4 } , ^ { * } C _ { 4 } }$ (33.86, 33.92 ppm), ${ C } _ { 5 } , { ^ { * } } { C } _ { 5 }$ $( 2 4 . 8 4 , 2 5 . 2 0 \\mathrm { p p m } )$ , ${ C _ { 6 } } , { ^ { * } } C _ { 6 }$ (31.64, 29.48 ppm), ${ C _ { 7 } } , { { ^ * } } { C _ { 7 } }$ (22.63, 29.32 ppm), $^ { * } C _ { 8 }$ $( 3 1 . 9 1 \\mathrm { p p m } )$ , $^ { * } { \\cal C } _ { 9 }$ (22.76 ppm) (carbon atoms of 3-hydroxydecanoate market with asterisk). The terminal methyl groups $\\left( \\mathrm { - C H } _ { 3 } \\right)$ of $\\mathrm { C } _ { 8 } , { } ^ { * } \\mathrm { C } _ { 1 0 }$ were observed at 14.12 and $1 4 . 2 2 \\mathrm { p p m }$ , respectively. All observed peaks in the $^ { 1 3 } \\mathrm { C }$ spectrum (Figure 3A) are in agreement with spectral characteristic reported for 3-hydroxyoctanoate and 3-hydroxydecanoate [30].J. Funct. Biomater. 2020, 11, x FOR PE ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,66,300,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2020, 11, 24 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[831,66,872,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 of 14 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[124,103,877,231]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[210,248,781,752]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 3.Figure 3. $^ { 1 3 } \\\\mathrm { C }$ nuclear magnetic resonance (NMR) (A), and nuclear magnetic resonance (NMR) (A), and $\\\\mathrm { ^ { 1 } H N M R }$ (B) spectra of PHA copolymer (B) spectra of PHA copolymer produced with sugarcane molasses as carbon source. The structure ofproduced with sugarcane molasses as carbon source. The structure of $\\\\mathrm { P ( 3 H O - } c o { - 3 } \\\\mathrm { H D ) }$ is shown as an is shown as an insert within the spectra.insert within the spectra. \"]","image_footnote":"[]","img_path":"images/0548/8db28bd000f64b2c945147fc946facd813017e170534e3cf4a6299f3ed6eac18.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[124,821,873,919]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":". ProThe $^ 1 \\mathrm { H }$ ties of Copolymer and Films Containing Lime Oil NMR spectrum (Figure 3B) shows 5 groups of overlapping peaks corresponding to 5 different proton environments characteristic for medium chain 3-hydroxyalakanoates. Three different environments of the olefinic proton manifested at chemical shifts 0.8, 1.2, and $1 . 5 \\mathrm { p p m }$ . The chemical properties of MCL-PHAs [5]. With glass-transition temperature ca. −45 °C, the amorphous phase ofshift at 0.8 corresponds to the methyl groups of both monomers. The protons of the methylene group at carbon atoms 5, 6, 7 of 3-hydroxyoctanoate and carbon atoms 5, 6, 7, 8, 9 of 3-hydroxydecanoate were observed at $1 . 2 \\ { \\mathrm { p p m } } ,$ while proton resonance of the methylene group at carbon 4 for both monomers shifted further downfield to $1 . 5 \\mathrm { p p m }$ . Two types of backbone protons for both monomers generated signals at chemical shifts of 2.52 (methylene group of carbon 2) and 5.2 (methine group of carbon 3) ppm. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,66,302,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2020, 11, 24 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[831,66,872,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7 of 14 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[126,104,872,151]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,153,875,311]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The monomeric composition of PHAs is strictly influenced by the carbon source utilized. In presence of structurally related carbon sources (e.g., fatty acids), the final composition of the polymer reflects that of the substrate. When unrelated substrates (e.g., carbohydrates) are utilized, the monomeric unit of the PHAs are independent of the structure of the carbon source [1–3]. As mentioned before, sugarcane molasses is composed of up to $6 0 \\%$ of sugars which are classified as unrelated substrates. Pseudomonas sp. can convert sugars into PHAs through the de novo fatty acid pathway. One of the enzymes involved in this pathway has a high affinity to the substrate containing 10 carbon atoms [31]. For this reason, Pseudomonas sp. have been shown to produce PHAs with a high content of the 3-HD monomer when cultured using glucose or glycerol [5,31]. This study is in agreement with previous studies, as the MCL-PHA produced showed a high content of 3-HD. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,324,589,340]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.2.2. Properties of Copolymer and Films Containing Lime Oil. Funct. Biomater. 2020, 11, x FOR PEER REVIEW ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,348,875,492]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The DSC ofual amount $\\mathrm { P ( 3 H O - } c o { - 3 } \\mathrm { H D ) }$ produced with sugarcane molasses as the carbon source revealedP(3HO-co-3HD) films were determined from TGA experiments both low glass transition and melting temperature (Figure 4A, Table 1) which are characteristic(Figure 4B) and found to be 6.9 wt.% for the new film and 2.8 wt.% for the aged one. The enthalpy of properties of MCL-PHAs [5]. With glass-transition temperature ca. melting of the P(3HO-co-3HD) crystalline phase decreased from 18.2 $- 4 5 ^ { \\circ } \\mathrm { C } ,$ the amorphous phase of4.7 and 15.8 J/g for fresh this semi-crystalline polymer is in a rubbery state at room temperature and polymer is expected toand aged films, respectively. Thus, the addition of lime oil induced a decrease in crystallinity of the be a flexible and stretchable material. Also, such low glass transition would prevent transition of thecopolymer. This implies that lime oil components had a plasticising effect on the copolymer. Slight polymer into a brittle material at sub-zero temperatures which can be required for a packaging material.increase in enthalpy of polymer melting (and thereby crystallinity degree) for aged films compared The melting temperature is aroundwith the fresh one might be explain $5 5 ^ { \\circ } \\mathrm { C }$ . This could be beneficial for melt processing at relatively lowcontinuation of crystallisation during 1-year storage and by temperatures. Melting is not observed in a DSC thermogram of second heating demonstrating slowpromotion of crystallisation with the loss of plasticising lime oil. Also, with storage, enthalpy of kinetics of the crystallisation process for this copolymer.melting shifted to higher temperatures indicating rigidifi ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[156,510,843,711]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 4. Thermal analysis of P(3Figure 4. Thermal analysis of $\\\\mathrm { P ( 3 H O - } c o { - 3 } \\\\mathrm { H D ) }$ at film and films with incorporated lime oil. (A) neat film and films with incorporated lime oil. Differential scanning calorimetry (DSC) thermograms: neat film (black lines), fresh film with lime oil (A) Differential scanning calorimetry (DSC) thermograms: neat film (black lines), fresh film with (red lines), and aged films with lime oil (blue lines). First heating—solid line, second heating—dashed lime oil (red lines), and aged films with lime oil (blue lines). First heating—solid line, second line. Thermograms were shifted vertically for better visibility. (B) Thermogravimetric analysis (TGA)heating—dashed line. Thermograms were shifted vertically for better visibility. (B) Thermogravimetric curves for determination of oil content after isothermal evaporation at 150 °C for 2 h.analysis (TGA) curves for determination of oil content after isothermal evaporation at $1 5 0 ^ { \\\\circ } \\\\mathrm { C }$ for $^ { 2 \\\\mathrm { h } }$ . \"]","image_footnote":"[]","img_path":"images/0586/8295aa3a5fcd5d888d80d89da2cc11c56294afca0311eb44656e95bc6fd0668f.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[126,66,300,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2020, 11, 24 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[831,66,872,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"8 of 14 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[252,123,746,223]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0956/6187a8589ec68862a5fea82832d6ecba08fde26715697aa4a6308c0eaa77e4f3.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":"<table><tr><td rowspan=\"2\">Sample</td><td colspan=\"3\">Thermal Properties</td></tr><tr><td>Tg,C</td><td>Tm℃a</td><td>△H, J/g</td></tr><tr><td>Neat film</td><td>-45.6</td><td>55.6</td><td>18.2</td></tr><tr><td>Fresh film with lime oil</td><td>-44.3</td><td>51.4</td><td>14.7</td></tr><tr><td>Aged film with lime oil</td><td>-40.8</td><td>50.5</td><td>15.8</td></tr></table>","table_caption":"[\"Table 1. Thermal properties of P(3HO-co-3HD) neat films and films with incorporated lime oil. \"]","table_footnote":"[\"a Determined as the end of melting peak. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[126,254,873,494]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Incorporating lime oil into P(3HO-co-3HD) films decreased the melting temperature by approximately $4 ~ ^ { \\circ } \\mathrm { C }$ . For an evaluation of the effect of lime oil on the crystallinity of the copolymer, an observed enthalpy of melting of P(3HO-co-3HD) in films containing lime oil needs to be normalised to the actual mass fraction of the copolymer. The real content of oil in the films does not correspond to oil/polymer composition in the casting solution due to oil loss as a result of evaporation along with solvent during the film formation as well as during the storage of the prepared film. Residual amounts of lime oil in $\\mathrm { P ( 3 H O - } c o { - 3 } \\mathrm { H D ) }$ films were determined from TGA experiments (Figure 4B) and found to be $6 . 9 \\mathrm { w t . \\% }$ for the new film and $2 . 8 \\mathrm { w t . \\% }$ for the aged one. The enthalpy of melting of the P(3HO-co-3HD) crystalline phase decreased from $1 8 . 2 \\mathrm { J / g }$ to 14.7 and $1 5 . 8 \\mathrm { J } / \\mathrm { g }$ for fresh and aged films, respectively. Thus, the addition of lime oil induced a decrease in crystallinity of the copolymer. This implies that lime oil components had a plasticising effect on the copolymer. Slight increase in enthalpy of polymer melting (and thereby crystallinity degree) for aged films compared with the fresh one might be explained by continuation of crystallisation during 1-year storage and by promotion of crystallisation with the loss of plasticising lime oil. Also, with storage, enthalpy of melting shifted to higher temperatures indicating rigidification of the amorphous phase. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[127,506,364,521]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.3. Antibacterial Characterization ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[129,535,848,549]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.3.1. MIC (Minimum Inhibition Concentration) and MBC (Minimum Bactericidal Concentration) ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,557,875,733]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lime oil was tested against S. aureus 6538P and E. coli 8739, chosen as an example of Gram-positive and Gram-negative bacteria, respectively. The values obtained are shown in Table 2. The MIC and MBC values against S. aureus 6538P were 60 and $1 0 0 ~ \\mu \\mathrm { L / m L }$ respectively. While against E. coli 8739, MIC value of $8 0 ~ \\mu \\mathrm { L / m L }$ was obtained, no MBC could be determined at the concentration investigated. The MBC is defined as the lowest concentration of compound inducing $9 9 . 9 \\%$ killing of the bacteria, corresponding to 3 log reduction in CFU. Therefore, the results obtained for lime oil showed that the compound had a bacteriostatic rather than bactericidal effect against E. coli 8739 at the concentrations investigated. In literature, MIC and MBC values for lime oil and generally for all essential oils comprise a very large range. For lime oil MIC values were reported from $0 . 1 ~ \\mathrm { m g / m L }$ [18] to few hundreds $\\mathrm { m g / m L }$ [32]. Such wide variations can be caused by plant sources, time of harvesting, methods of oil extraction, etc. Lime oil used in this study was of relatively low antimicrobial activity. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[196,783,801,846]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/1023/a136c6094f3d54a6c2b91b4f600787191e7c5aa57deb139f701c52484d27e065.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":"<table><tr><td colspan=\"2\">S.aureus 6538P</td><td colspan=\"2\">E coli 8739</td></tr><tr><td>MIC (μL/mL)</td><td>MBC (μL/mL)</td><td>MIC (μL/mL)</td><td>MBC (μL/mL)</td></tr><tr><td>60</td><td>100</td><td>80</td><td>1</td></tr></table>","table_caption":"[\"Table 2. Minimum inhibition concentration (MIC) and minimum bactericidal concentration (MBC) values of lime against S. aureus 6538P and E. coli 8739. \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[127,864,872,912]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The antibacterial mechanism of such essential oils is usually related to the activity of their main components, terpenes and terpenoids. Lime oil (Citrus aurantifolia) is composed mainly of limonene [18,20]. The mechanism of action of such organic compounds is linked to their ability to modify the integrity of the bacterial cell wall, increasing its permeability and causing membrane disruption [15,33]. Moreover, a higher effect against Gram-positive bacteria compared to negative onesTable 2. Minimum inhibition concentration (MIC) and minimum bactericidal concentration (MBC) is usually reported, due to protection of Gram-negative bacteria by the outer membrane composed ofvalues of lime against S. aureus 6538P and E. coli 8739. lipopolysaccharides [34]. Such a response was confirmed in this study as overall lime oil showed a higher effect against S. aureus 6538P than E. coli 8739. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,65,302,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2020, 11, 24 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[830,66,872,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"9 of 14 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[126,104,873,183]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,196,242,210]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.3.2. Halo Test ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,219,875,362]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The antimicrobial materials were tested against both S. aureus 6538P and E. coli 8739. Freshly3.3.2. Halo Test prepared and aged films were analysed to investigate whether the material was able to preserve theirThe antimicrobial materials were tested against both S. aureus 6538P and E. coli 8739. Freshly antimicrobial efficacy. As it was discussed earlier, the content of lime oil decreased fromprepared and aged films were analysed to investigate whether the material was able to preserv $7 4 \\mathrm { m g } / \\mathrm { g }$ of copolymer for fresh film toantimicrobial efficacy. A $3 0 \\mathrm { m g / g }$ for the aged sample. The freshly prepared samples showed activitydiscussed earlier, the content of lime oil decreased from 74 mg/g of against S. aureus 6538P, inducing the formation of inhibition zones as shown in Figure 5 and Table 3.copolymer for fresh film to 30 mg/g for the aged sample. The freshly prepared samples showed Moreover, the samples obtained after a one-year storage period still possessed activity against theactivity against S. aureus 6538P, inducing the formation of inhibition zones as shown in Figure 5 and microorganism, even though aTable 3. Moreover, the sam $1 5 \\%$ reduction of the zone of inhibition could be detected. By contrast,obtained after a one-year storage period still possessed activity both the freshly prepared and aged films did not show activity against E. coli 8739, as reported in Table 3. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[163,379,840,682]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 5. Halo test showing the antibacterial activity of aged lime oil containing P(3HO-co-3HD) films Figure 5. Halo test showing the antibacterial activity of aged lime oil containing P(3HO-co-3HD) films (a–c) and fresh lime oil containing films (d–f) against S. aureus 6538P (control: antibiotic disc containing $1 \\\\mu \\\\mathrm { g }$ of oxacillin, left hand side). \"]","image_footnote":"[]","img_path":"images/0718/5eeb92c2c5b7dc1e71bcfe87c0fdb503b86b0d31dc366b0f17a96346a4e8d11d.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[171,783,826,896]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0699/52fd6de078eea814f2c05aaee3f8662e4357add39dc1c07011d62358621c9a3f.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":"<table><tr><td rowspan=\"2\"> Sample</td><td colspan=\"2\">Zone of Inhibition (ZOI) (cm)</td></tr><tr><td>S. aureus 6538P</td><td>E. coli 8739</td></tr><tr><td>Aged P(3HO-co-3HD)/lime films</td><td>1.06 ± 0.02</td><td>1</td></tr><tr><td>Fresh P(3HO-co-3HD)/lime films</td><td>1.23 ± 0.05</td><td>0.13 ± 0.04</td></tr><tr><td rowspan=\"2\">Control</td><td>3.13 ± 0.10</td><td>1.99 ± 0.09</td></tr><tr><td>(oxacillin)</td><td>(streptomycin)</td></tr></table>","table_caption":"[\"Table 3. Halo diameters showing the effect of P(3HO-co-3HD) films containing lime oil against S. aureus 6538P and E. coli 8739. \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[126,66,300,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2020, 11, 24 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[825,66,872,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10 of 14 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[126,104,873,376]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Limited research has been conducted on the incorporation of essential oils into PHAs and the studies conducted have focused only on SCL-PHAs. Films for antibacterial food packaging were produced using P(3HB-co-3HV) and two different essential oils such as oregano and clove, showing antibacterial activity against common food pathogens Listeria innocua 910 and E. coli 101 [35]. In another study, P(3HB-co-3HV) films were loaded with coconut fibres impregnated with oregano essential oil to produce composite antibacterial packaging materials, showing activity against S. aureus 6538P [36]. Alternatively, electrospinning has been investigated to produce fibrous mats based on P(3HB) or P(3HB-co-3HV) fibres containing essential oil. The natural compounds were either adsorbed onto the fibres after the spinning process (cinnamon, clove, oregano and oak bark) [37] or incorporated into the solution prior to electrospinning (rosemary and green tea extracts) [38]. The material produced showed activity against Micrococcus luteus 1569, Serratia marcescens 8587, S. aureus 6538P and E. coli 25922. Finally, Mendelez-Rodriguez et al. produced P(3HB-co-3HV) antibacterial fibres by loading eugenol essential oil onto mesoporous silica nanoparticles, followed by their incorporation into the fibres by electrospinning. The encapsulation was employed to maintain and protect the antibacterial effect of the agent, as essential oils are volatile compounds. The films obtained showed antibacterial activity against the S. aureus 6538P and E. coli 25922 up to 15 days after the day of fabrication and storage in a tightly closed system [39]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,388,242,403]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4. Conclusions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,412,873,732]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This study demonstrated the valorisation of sugarcane molasses, an agri-food by-product, through the production of a high-value MCL-PHA. The use of sugarcane molasses as the single carbon source and Pseudomonas mendocina CH50, a known MCL PHA producer, resulted in the synthesis of P(3HO-co-3HD) copolymer. P(3HO-co-3HD) is a polyester with low glass-transition and melting temperature, high elasticity and flexibility. In order to expand its potential applications, in this study, we explored the possibility of making P(3HO-co-3HD) antimicrobial through the incorporation of lime oil, an essential oil known for its antimicrobial activity. The antibacterial properties of the lime oil were investigated using the ISO 20776 against two bacterial strains, Gram-positive S. aureus 6538P and Gram-negative E. coli 8739. In line with general observations on the activity of essential oils, the lime oil used in this study was more active against S. aureus 6538P than E. coli 8739. Incorporation of the lime oil into the P(3HO-co-3HD) film had a minor effect on the material properties, slightly decreasing its crystallinity. As expected, the content of lime oil decreased with the storage due to migration out and evaporation from the polymer films. After one-year storage, the content of lime oil in the films decreased from ca 7 wt.% to $2 . 9 \\mathrm { w t . \\% }$ . Although the antimicrobial activity decreased after prolonged storage, both fresh and aged films of P(3HO-co-3HD) with incorporated lime oil showed activity against S. aureus 6538P. On the contrary, no antimicrobial activity was observed against E. coli 8739 for both fresh and aged films. The feasibility of imparting microbial activity to PHAs can broaden the application of this biodegradable polymer in several sectors, including coatings, adhesives, biodegradable rubbers, implantable materials. In the last sector, the developed P(3HO-co-3HD)/lime oil films could be employed as an antibacterial material for the regeneration of soft tissues, like skin. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,744,873,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Author Contributions: Conceptualization, P.B. and I.R.; methodology, P.B., E.M., R.N. and A.P.; formal analysis, P.B., E.M., R.N., A.P., B.G., M.H.A.; investigation, P.B., E.M., R.N., A.P., B.G., M.H.A.; resources, I.R.; data curation, P.B., E.M., R.N., A.P., B.G., M.H.A.; writing—P.B., E.M., R.N., A.P., B.G., M.H.A., B.L.; writing—review and editing, I.R.; supervision, I.R. and P.B.; project administration, I.R.; funding acquisition, I.R. All authors have read and agreed to the published version of the manuscript. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,812,872,850]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Funding: This research was funded by POLYBIOSKIN (H2020), Grant Agreement Number 745839, EM was funded by HyMedPoly, Marie Sklodowska-Curie Grant Agreement No 643050. The APC was funded by the University of Sheffield using the POLYBIOSKIN (H2020), Grant Agreement Number 745839. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[124,854,872,881]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acknowledgments: The authors will like to acknowledge technical support by Neville Antonio and Zhi Song at the University of Westminster. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,887,552,900]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conflicts of Interest: The authors declare no conflict of interest. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,66,300,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2020, 11, 24 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[825,66,872,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"11 of 14 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[126,104,213,118]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[124,128,875,851]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Khanna, S.; Srivastava, A.K. Recent advances in microbial polyhydroxyalkanoates. Process Biochem. 2005, 40, 607–619. [CrossRef] \n2. Li, Z.; Yang, J.; Loh, X. Polyhydroxyalkanoates: Opening doors for a sustainable future. NPG Asia Mater. 2016, 8, 265. [CrossRef] \n3. Raza, Z.A.; Abid, S.; Banat, I.M. Polyhydroxyalkanoates: Characteristics, production, recent developments and applications. Int. Biodeterior. Biodegrad. 2018, 126, 45–56. [CrossRef] \n4. Álvarez-Chávez, C.R.; Edwards, S.; Moure-Eraso, R.; Geiser, K. Sustainability of bio-based plastics: General comparative analysis and recommendations for improvement. J. Clean. Prod. 2012, 23, 47–56. [CrossRef] \n5. Basnett, P.; Lukasiewicz, B.; Marcello, E.; Gura, H.K.; Knowles, J.C.; Roy, I. Production of a novel medium chain length poly(3-hydroxyalkanoate) using unprocessed biodiesel waste and its evaluation as a tissue engineering scaffold. Microb. Biotechnol. 2017, 10, 1384–1399. [CrossRef] [PubMed] \n6. Singh, A.K.; Mallick, N. Exploitation of inexpensive substrates for production of a novel SCL-LCL-PHA co-polymer by Pseudomonas aeruginosa MTCC 7925. J. Ind. Microbiol. Biotechnol. 2009, 36, 347–354. [CrossRef] \n7. Chee, J.; Yoga, S.; Lau, N.; Ling, S.; Abed, R.M.M. Bacterially produced Polyhydroxyalkanoate (PHA): Converting renewable resources into bioplastics. In Current Research, Technology and Education Topics in Applied Microbiology and Microbial Biotechnology; Méndez-Vilas, A., Ed.; Formatex Research Center: Badajoz, Spain, 2010; pp. 1395–1404. \n8. Jiang, G.; Hill, D.J.; Kowalczuk, M.; Johnston, B.; Adamus, G.; Irorere, V.; Radecka, I. Carbon sources for Polyhydroxyalkanoates and an integrated biorefinery. Int. J. Mol. Sci. 2016, 17, 1157. [CrossRef] \n9. Nikodinovic-Runic, J.; Guzik, M.; Kenny, S.T.; Babu, R.; Werker, A.; Connor, K.E. Carbon-rich wastes as feedstocks for biodegradable polymer (Polyhydroxyalkanoate) production using bacteria. In Advances in Applied Microbiology, 1st ed.; Sariaslani, S., Gadd, G.M., Eds.; Academic Press: Cambridge, MA, USA, 2013; Volume 84, pp. 139–200. \n10. Basnett, P.; Ravi, S.; Roy, I. Natural bacterial biodegradable medical polymers: Polyhydroxyalkanoates. In Science and Principles of Biodegradable and Bioresorbable Medical Polymers; Zhang, X., Ed.; Woodhead Publishing: Cambridge, UK, 2017; pp. 257–277. \n11. Paris, J.B.; Seyer, D.; Jouenne, T.; Thébault, P. Elaboration of antibacterial plastic surfaces by a combination of antiadhesive and biocidal coatings of natural products. Colloids Surf. B 2017, 156, 186–193. [CrossRef] \n12. Roca, I.; Akova, M.; Baquero, F.; Carlet, J.; Cavaleri, M.; Coenen, S.; Cohen, J.; Findlay, D.; Gyssens, I.; Heure, O.E.; et al. The global threat of antimicrobial resistance: Science for intervention. NMNI 2015, 6, 22–29. [CrossRef] \n13. Prabuseenivasan, S.; Jayakumar, M.; Ignacimuthu, S. In vitro antibacterial activity of some plant essential oils. BMC Complement. Altern. Med. 2006, 6, 39. [CrossRef] \n14. Kalemba, D.; Kunicka, A. Antibacterial and Antifungal Properties of Essential Oils. Curr. Med. Chem. 2003, 10, 813–829. [CrossRef] [PubMed] \n15. Hyldgaard, M.; Mygind, T.; Meyer, R.L. Essential oils in food preservation: Mode of action, synergies, and interactions with food matrix components. Front. Microbiol. 2012, 3, 12. [CrossRef] [PubMed] \n16. Di Pasqua, R.; Hoskins, N.; Betts, G.; Mauriello, G. Changes in membrane fatty acids composition of microbial cells induced by addiction of thymol, carvacrol, limonene, cinnamaldehyde, and eugenol in the growing media. J. Agric. Food Chem. 2006, 54, 2745–2749. [CrossRef] [PubMed] \n17. Arrieta, M.P.; López, J.; Hernández, A.; Rayón, E. Ternary PLA–PHB–Limonene blends intended for biodegradable food packaging applications. Eur. Polym. J. 2014, 50, 255–270. [CrossRef] \n18. Hsouna, A.B.; Halima, N.B.; Smaoui, S.; Hamdi, N. Citrus lemon essential oil: Chemical composition, antioxidant and antimicrobial activities with its preservative effect against Listeria monocytogenes inoculated in minced beef meat. Lipids Health Dis. 2017, 16, 146. [CrossRef] \n19. H ˛ac-Wydro, K.; Flasi ´nski, M.; Roma ´nczuk, K. Essential oils as food eco-preservatives: Model system studies on the effect of temperature on limonene antibacterial activity. Food Chem. 2017, 235, 127–135. [CrossRef] \n20. Costa, R.; Bisignano, C.; Filocamo, A.; Grasso, E.; Occhiuto, F.; Spadaro, F. Antimicrobial activity and chemical composition of Citrus aurantifolia (Christm.) Swingle essential oil from Italian organic crops. J. Essent. Oil Res. 2014, 26, 400–408. [CrossRef] \n21. Basnett, P.; Marcello, E.; Lukasiewicz, B.; Panchal, B.; Nigmatullin, R.; Roy, I. Biosynthesis and characterization of novel, biocompatible medium chain length polyhydroxyalkanoates by Pseudomonas mendocina CH50 using coconut oil as the carbon source. J. Mater. Sci. Mater. Med. 2018, 29, 179. [CrossRef] \n22. Rai, R.; Yunos, D.M.; Boccaccini, A.R.; Knowles, J.C.; Barker, I.A.; Howdle, S.M.; Tredwell, G.D.; Keshavarz, T.; Roy, I. Poly-3-hydroxyoctanoate P(3HO), a Medium Chain Length Polyhydroxyalkanoate Homopolymer from Pseudomonas mendocina. Biomacromolecules 2001, 12, 2126–2136. [CrossRef] \n23. Gumel, A.M.; Annuar, M.S.M.; Heidelberg, T. Growth kinetics, effect of carbon substrate in biosynthesis of MCL-PHA by Pseudomonas putida Bet001. Braz. J. Microbiol. 2014, 45, 427–438. [CrossRef] \n24. Akaraonye, E.; Moreno, C.; Knowles, J.C.; Keshavarz, T.; Roy, I. Poly(3-hydroxybutyrate) production by Bacillus cereus SPV using sugarcane molasses as the main carbon source. Biotechnol. J. 2012, 7, 293–303. [CrossRef] [PubMed] \n25. Naheed, N.; Jamil, N. Optimization of biodegradable plastic production on sugar cane molasses in Enterobacter sp. SEL2. Braz. J. Microbiol. 2014, 45, 417–426. [CrossRef] [PubMed] \n26. Chaijamrus, S.; Udpuay, N. Production and characterization of polyhydroxybutyrate from molasses and corn steep liquor produced by Bacillus megaterium ATCC 6748. Agric. Eng. Int. CIGR J. 2008, 10, 1–12. \n27. Solaiman, D.K.Y.; Ashby, R.D.; Hotchkiss, A.T.; Foglia, T.A. Biosynthesis of medium-chain-length Poly(hydroxyalkanoates) from soy molasses. Biotechnol. Lett. 2006, 28, 157–162. [CrossRef] \n28. Jiang, Y.; Song, X.; Gong, L.; Li, P.; Dai, C.; Shao, W. High poly (β-hydroxybutyrate) production by Pseudomonas fluorescens A2a5 from inexpensive substrates. Enzym. Microb. Technol. 2008, 42, 167–172. [CrossRef] \n29. Chen, Y.-J.; Huang, Y.-C.; Lee, C.-Y. Production and characterization of medium-chain-length polyhydroxyalkanoates by Pseudomonas mosselii TO7. J. Biosci. Bioeng. 2014, 118, 145–152. [CrossRef] \n30. De Rijk, T.C.; Van de Meer, P.; Eggink, G.; Weusthuis, R.A. Methods for Analysis of Poly(3-hydroxyalkanoate) Composition. In Biopolymers: Polyesters. II Properties and Chemical Synthesis; Doi, Y., Steinbuchel, A., Eds.; Wiley-Blackwell: Hoboken, NJ, USA, 2005; Volume 3b, pp. 1–12. \n31. Poblete-Castro, I.; Rodriguez, A.L.; Chi Lam, C.M.; Kessler, W. Improved production of medium-chain-length polyhydroxyalkanoates in glucose-based fed-batch cultivations of metabolically engineered Pseudomonas putida strains. J. Microbiol. Biotechnol. 2014, 24, 59–69. [CrossRef] \n32. Aibinu, I.; Adenipekun, T.; Adelowotan, T.; Ogunsanya, T.; Odugbemi, T. Evaluation of the antimicrobial properties of different parts of Citrus Aurantifolia (Lime Fruit) as used locally. Afr. J. Tradit. Complement. Altern. Med. 2007, 4, 185–190. \n33. Khameneh, B.; Iranshahy, M.; Soheili, V.; Fazly Bazzaz, B.S. Review on plant antimicrobials: A mechanistic viewpoint. Antimicrob. Resist. Infect. Control 2019, 8, 118. [CrossRef] \n34. Nazzaro, F.; Fratianni, F.; De Martino, L.; Coppola, R.; De Feo, V. Effect of essential oils on pathogenic bacteria. Pharmaceuticals 2013, 6, 1451–1474. [CrossRef] \n35. Requena, R.; Jiménez, A.; Vargas, M.; Chiralt, A. Poly [(3-hydroxybutyrate)-co-(3-hydroxyvalerate)] active bilayer films obtained by compression moulding and applying essential oils at the interface. Polym. Int. 2016, 65, 883–891. [CrossRef] \n36. Torres-Giner, S.; Hilliou, L.; Melendez-Rodriguez, B.; Figueroa-Lopez, K.J.; Madalena, D.; Cabedo, L.; Covas, J.A.; Vicente, A.A.; Lagaron, J.M. Melt processability, characterization, and antibacterial activity of compression-molded green composite sheets made of poly(3-hydroxybutyrate-co-3-hydroxyvalerate) reinforced with coconut fibers impregnated with oregano essential oil. Food Packag. Shelf Life 2018, 17, 39–49. [CrossRef] \n37. Kundrat, V.; Matouskova, P.; Marova, I. Facile Preparation of Porous Microfiber from Poly-3-(R)-Hydroxybutyrate and Its Application. Materials 2019, 13, 86. [CrossRef] [PubMed] ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,65,302,77]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2020, 11, 24 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[825,65,872,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"12 of 14 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[124,105,873,826]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,65,302,77]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2020, 11, 24 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[825,65,872,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"13 of 14 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[126,104,873,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"38. Figueroa-Lopez, K.J.; Vicente, A.A.; Reis, M.A.; Torres-Giner, S.; Lagaron, J.M. Antimicrobial and antioxidant performance of various essential oils and natural extracts and their incorporation into biowaste derived poly (3-hydroxybutyrate-co-3-hydroxyvalerate) layers made from electrospun ultrathin fibers. Nanomaterials 2019, 9, 144. [CrossRef] ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[124,166,873,211]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"39. Melendez-Rodriguez, B.; Figueroa-Lopez, K.J.; Bernardos, A.; Martínez-Máñez, R.; Cabedo, L.; Torres-Giner, S.; M Lagaron, J. Electrospun antimicrobial films of poly(3-hydroxybutyrate-co-3-hydroxyvalerate) containing eugenol essential oil encapsulated in mesoporous silica nanoparticles. Nanomaterials 2019, 9, 227. [CrossRef] $©$ 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[277,223,873,268]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,65,302,77]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Funct. Biomater. 2020, 11, 24 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[825,66,872,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"14 of 14 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[126,224,268,262]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"}],"string":"[\n {\n \"bbox\": \"[126,117,174,130]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Article \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,131,847,204]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Antimicrobial Materials with Lime Oil and a Poly(3-hydroxyalkanoate) Produced via Valorisation of Sugar Cane Molasses \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[127,222,845,254]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Pooja Basnett 1, Elena Marcello 1, Barbara Lukasiewicz 1, Rinat Nigmatullin $^ { 1 , 2 } \\\\oplus _ { i }$ Alexandra Paxinou 1, Muhammad Haseeb Ahmad 1, Bhavana Gurumayum 1 and Ipsita Roy $^ { 3 , * }$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,260,873,375]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 Faculty of Science and Technology, University of Westminster, London W1W 6UW, UK; P.Basnett@westminster.ac.uk (P.B.); w1614733@my.westminster.ac.uk (E.M.); barbara.lukasiewicz@gmail.com (B.L.); rn17541@bristol.ac.uk (R.N.); w1621777@my.westminster.ac.uk (A.P.); Mhaseebahmadd@gmail.com (M.H.A.); gurumayumbhavana@gmail.com (B.G.) \\n2 Bristol Composites Institute (ACCIS), University of Bristol, Bristol BS8 1TR, UK \\n3 Department of Material Science and Engineering, Faculty of Engineering, University of Sheffield, Sheffield S1 3JD, UK Correspondence: I.Roy@sheffield.ac.uk; Tel.: +44-114-2225962 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[142,392,628,406]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Received: 9 March 2020; Accepted: 6 April 2020; Published: 10 April 2020 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[136,419,873,626]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract: A medium chain-length polyhydroxyalkanoate (PHA) was produced by Pseudomonas mendocina CH50 using a cheap carbon substrate, sugarcane molasses. A PHA yield of $1 4 . 2 \\\\%$ dry cell weight was achieved. Chemical analysis confirmed that the polymer produced was a medium chain-length PHA, a copolymer of 3-hydroxyoctanoate and 3-hydroxydecanoate, P(3HO-co-3HD). Lime oil, an essential oil with known antimicrobial activity, was used as an additive to $\\\\mathrm { P ( 3 H O - } c o { - 3 } \\\\mathrm { H D ) }$ to confer antibacterial properties to this biodegradable polymer. The incorporation of lime oil induced a slight decrease in crystallinity of P(3HO-co-3HD) films. The antibacterial properties of lime oil were investigated using ISO 20776 against Staphylococcus aureus 6538P and Escherichia coli 8739, showing a higher activity against the Gram-positive bacteria. The higher activity of the oil against S. aureus 6538P defined the higher efficiency of loaded polymer films against this strain. The effect of storage on the antimicrobial properties of the loaded films was investigated. After one-year storage, the content of lime oil in the films decreased, causing a reduction of the antimicrobial activity of the materials produced. However, the films still possessed antibacterial activity against S. aureus 6538P. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,643,821,657]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords: polyhydroxyalkanotes; sugarcane molasses; antibacterial materials; essential oils \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,703,243,718]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,726,873,917]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Polyhydroxyalkanoates (PHAs) are a family of natural polymers produced by bacterial fermentation using renewable carbon sources under nutrient-limiting conditions. PHAs can be broadly classified into short chain length (SCL: $C _ { 3 } – C _ { 5 } \\\\mathrm { , }$ ) and medium-chain length (MCL: $C _ { 6 } – C _ { 1 4 } )$ ) based on the number of carbon atoms within their monomer units [1]. SCL PHAs have a high melting temperature and are brittle, whereas MCL PHAs are elastomeric and have a low-melting and glass-transition temperature. PHAs can match the functional properties of common petroleum-based plastics. For example, the mechanical properties of poly(3-hydroxybutyrate) (P(3HB)) are similar to those of polystyrene, while its copolymers with 3-hydroxyvalerate (P(3HB-co-3HV)) are characterised by an improved ductility resulting in a combination of mechanical properties close to polypropylene. One the other hand, MCL-PHAs can emulate the properties of soft thermoplastic elastomers such as thermoplastic polyurethanes, olefinic thermoplastic elastomers. The functional properties of PHA-based materials can be further modified and tuned via blending with other polymers [2]. Thanks to the presence of hydrolysable ester linkage connecting the monomer units of PHAs, PHAs are biodegradable in nature unlike most of conventional plastics. Their degradation rate is affected by the chemical composition of the polymer, and by environmental factors [3]. Based on life-cycle assessment (LCA) analysis, PHAs have been recognised as one of the safest bioplastics for the environment [4]. PHAs are also known to be biocompatible and have been widely investigated for various biomedical applications [5]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,57,295,98]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Journal of Functional Biomaterials \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[796,64,870,97]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"MDPI \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[127,941,447,952]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2020, 11, 24; doi:10.3390/jfb11020024 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[709,942,870,952]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"www.mdpi.com/journal/jfb \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[783,382,870,404]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\\u0001-\\u0003\\u0001\\u0004\\u0005\\u0006\\u0007\\b \\u0001-\\u0003\\u0004\\u0005\\u0006\\u0007 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[126,103,873,199]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,200,873,473]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Although PHAs represent a family of sustainable polymers with attractive properties, they are yet to replace petroleum based conventional plastics used in most industrial applications. One of the major factors limiting the commercialisation of PHAs is its high production cost. Several cheap carbon sources, waste materials, genetically modified bacterial strains, fermentation conditions and recovery methods have been employed either on their own or in combination, to allow economical production of PHAs [3]. Since the cost of carbon accounts for $3 1 \\\\%$ of the overall PHA production cost, the use of inexpensive substrates is one of the effective strategies to produce PHAs in an economical manner [6,7]. Simple sugars, triglycerides (vegetable oils) are the most widely used carbon sources for the microbial production of PHAs [8]. In addition to their high cost, the utilization of such raw materials can be in competition with the use of these resources as food. Therefore, the valorisation of cheap agri-food by-products can address both issues; the cost of carbon source and the competition with food. Although potentially biomass rich in polysaccharides can be used for generating simple sugars, this requires additional chemical and enzymatic treatment. Molasses and whey, on the other hand, contain up to 60 and $4 . 5 \\\\ \\\\mathrm { w t . \\\\% }$ of simple sugars, respectively. Currently, a significant part of molasses is used in bioethanol production. The production of PHAs with the use of molasses as the carbon feedstock is a feasible alternative for the valorisation of molasses. Many studies demonstrated that molasses is a good substrate for microbial production of poly(3-hydroxybutyrate) [9]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,474,875,633]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PHAs have diverse physical properties which makes them suitable for various applications such as packaging, medical devices, bio-implants and tissue engineering [10]. For all these applications, microbial contamination is a major issue. Bacterial adhesion and colonisation on the polymer surface form a reservoir for pathogens which leads to various nosocomial infections, skin related issues, degradation of food and cosmetic products, and food contamination [11]. For several decades, antibiotics have been used to treat and prevent such infections. However, with the rise in the number of multidrug-resistant pathogens, antibiotic resistance has now become a global health risk. Annual costs of treating antibiotic resistant infections is estimated to be between $\\\\$ 21,000$ and $\\\\$ 34,000$ million in the United States alone and around €1500 million in Europe [12]. In such a scenario, there is a high demand for natural antibacterial agents to prevent or mitigate bacterial contamination. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,634,875,809]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Traditionally, essential oils, plant extracts and other natural products were used to treat infectious diseases. According to the World Health Organization (WHO), the majority of the world population is dependent on treatment using natural products as a primary care. Various essential oils have shown to possess antibacterial, antifungal and antioxidant properties [13–15]. These oils are now being screened to find alternative remedies to fight bacterial infections in a wide range of applications. Generally, a higher antimicrobial activity has been observed for oxygenated constituents of essential oils [16]. However, as antimicrobial additives to polymer materials, non-polar terpene hydrocarbons can bring other benefits such as improved compatibility with hydrophobic polymer matrices, enhanced water vapour barrier properties and plasticising effect [17]. Lime oil is an example of essential oils rich of terpene hydrocarbons such as limonene (ca. $4 0 \\\\%$ ), $\\\\beta$ -pinene (ca. $2 5 \\\\%$ ), and $\\\\gamma$ -terpinene (ca. $1 0 \\\\%$ ) [18]. The antimicrobial activity of lime oil is well documented [18–20]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,810,873,907]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The aim of this study was the development of natural antimicrobial polymer films via the incorporation of an essential oil into biodegradable PHA produced from agri-food by-products. We report a feasible way of valorisation of sugarcane molasses for the production of PHA by Pseudomonas mendocina CH50, a bacterial strain which has been previously confirmed as an efficient producer of MCL PHAs [5,20,21]. It was demonstrated that in the presence of sugarcane molasses $P .$ mendocina CH50 was able to accumulate an MCL PHA, which was identified as a copolymer of 3-hydroxyoctanoate and 3-hydroxydecanoate, or P(3HO-co-3HD). Lime oil was incorporated into P(3HO-co-3HD) films to produce a natural antimicrobial material. The oil content in the films was monitored during long storage, and loss of oil was observed during the storage. Both fresh and aged films exhibited antimicrobial activity against Gram-positive S. aureus. The films were less efficient against Gram-negative E. coli reflecting lower activity of lime oil against this bacterial strain. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,66,300,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2020, 11, 24 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[830,66,872,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 of 14 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[126,103,873,183]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,196,317,210]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. Material and Methods \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[129,223,438,237]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.1. Bacterial Strains and Culture Conditions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,247,873,342]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Pseudomonas mendocina CH50 was obtained from the University of Westminster culture collection. The strain was cultured at $3 0 ^ { \\\\circ } \\\\mathrm { C }$ for $1 6 \\\\mathrm { { h } }$ in a shaking incubator $( 2 0 0 \\\\mathrm { r p m } )$ . After incubation, the strain was stored as a glycerol stock at $- 8 0 ^ { \\\\circ } \\\\mathrm { C }$ . For the antibacterial characterization, Staphylococcus aureus 6538P and Escherichia coli 8739 were bought from the American Type Culture Collection (ATCC). They were cultured in sterile nutrient broth at $3 7 ^ { \\\\circ } \\\\mathrm { C }$ and $2 0 0 \\\\mathrm { r p m }$ for $1 6 \\\\mathrm { { h } }$ in a shaking incubator and stored as a glycerol stock at $- 8 0 ^ { \\\\circ } \\\\mathrm { C }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,354,228,368]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2. Chemicals \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,378,872,424]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"All the chemicals for the production and characterization of PHAs were purchased from Sigma Aldrich Ltd., England and VWR international, England. Lime oil used as an antibacterial agent was purchased from Holland and Barrett, London, UK. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[124,437,872,464]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.3. Production of Polyhydroxyalkanoates by Pseudomonas mendocina CH50 Using Sugarcane Molasses as the Sole Carbon Source \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,473,873,650]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Production of PHAs by Pseudomonas mendocina CH50 using $2 0 ~ \\\\mathrm { g / L }$ of sugarcane molasses was carried out in $1 5 \\\\mathrm { L }$ bioreactors, with $1 0 \\\\mathrm { ~ L ~ }$ working volume (Applikon Biotechnology, Tewkesbury, UK). Batch fermentation was carried out in two stages. The seed culture was prepared using a single colony of $P .$ mendocina CH50 to inoculate sterile nutrient broth. This was incubated for $1 6 \\\\mathrm { { h } }$ at $3 0 ~ ^ { \\\\circ } \\\\mathrm { C }$ and at $2 0 0 \\\\mathrm { r p m }$ . Inoculum of $1 0 \\\\ \\\\mathrm { v o l . \\\\% }$ of the final volume was used to inoculate the second stage seed culture (mineral salt medium-MSM) which was incubated at $3 0 ~ ^ { \\\\circ } \\\\mathrm { C } ,$ , $2 0 0 \\\\mathrm { r p m }$ for $2 4 \\\\mathrm { { h } }$ . Second stage seed culture was used to inoculate the final PHA production media [21,22]. The culture was grown for $4 8 \\\\mathrm { h } ,$ at $3 0 ^ { \\\\circ } C ,$ $2 0 0 \\\\mathrm { r p m }$ and air flow rate of 1 vvm. Temporal profile of the production of PHAs by $P .$ mendocina CH50 using $2 0 \\\\mathrm { g / L }$ sugarcane molasses was obtained by monitoring parameters such as optical density, biomass, nitrogen and glucose concentrations, $\\\\mathrm { p H }$ and dissolved oxygen tension (DOT) at regular intervals during the fermentation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,662,284,677]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.4. Analytical Studies \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,686,873,764]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bacterial growth was measured by monitoring the optical density (OD) at $4 5 0 \\\\mathrm { n m }$ . Biomass yield was determined by weighing the dried pellet of the cells obtained after centrifugation of $1 \\\\mathrm { m L }$ at 12,000 rpm for $1 0 \\\\mathrm { m i n }$ (Heraeus Pico 17 Centrifuge, Thermofisher Scientific, MA, USA). pH of the supernatant was measured using Seven Compact $\\\\mathrm { p H }$ meter (Mettler Toledo Ltd., Leicester, UK). Nitrogen in the form of ammonium ions was estimated by the phenol-hypochlorite method [21,22]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,766,873,860]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Polymer was extracted from the dried biomass using the soxhlet extraction method. To remove organic soluble impurities, dried biomass was refluxed in methanol for $2 4 \\\\mathrm { { h } }$ followed by refluxing in chloroform for $4 8 \\\\mathrm { h }$ to extract the polymer. Polymer solution in chloroform was concentrated using a rotary evaporator. Polymer was precipitated using ice-cold methanol under continuous stirring and dried at room temperature. Polymer yield was calculated as a percentage of dry cell weight (DCW), using the formula: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[371,859,626,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0177/9c5a12acf1c7bab3a2b37f9728b82a2480090b2680ceee151b6bc57ddc7efcf8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\% \\\\mathrm { { D C W } = \\\\frac { \\\\mathrm { { P o l y m e r ~ m a s s } } } { \\\\mathrm { { M a s s ~ o f ~ b i o m a s s } } } \\\\times 1 0 0 }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[126,66,300,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2020, 11, 24 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[831,66,872,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 of 14 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[127,104,351,118]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.5. Preparation of Film Samples \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,128,873,271]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Solvent cast films were prepared by pouring 5 wt. $\\\\%$ polymer solution in chloroform in $6 0 \\\\mathrm { m m }$ diameter glass Petri dishes. The polymer was dried in the fume hood at the room temperature until the solvent evaporated completely. For oil-incorporated films, the oils were added to 5 wt. $\\\\%$ polymer solution in chloroform. The loaded films were stored at room temperature in sealed Petri dishes for one year at room temperature and identified as aged samples. Quantification of the oil content in the films after complete solvent evaporation was carried out by thermogravimetric analysis (TGA) using the STA 449 F3 Jupiter® instrument (Netzsch, Germany); $1 0 { - } 1 5 \\\\mathrm { m g }$ film samples were heated at 10 degrees per minute from 30 to $1 5 0 ^ { \\\\circ } \\\\mathrm { C }$ under nitrogen flow and kept at $1 5 0 ^ { \\\\circ } \\\\mathrm { C }$ for $2 \\\\mathrm { { h } }$ . The cumulative mass loss in dynamic and isothermal steps was defined as oil amount left in the films after solvent evaporation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[127,284,332,298]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.6. Polymer Characterization \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,306,873,386]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The monomeric composition of polymer was determined using gas chromatography–mass spectrometry (GC–MS) analysis and $^ { 1 3 } { \\\\cal C }$ , $^ 1 \\\\mathrm { H }$ nuclear magnetic resonance (NMR) spectroscopy. For GC–MS analysis (Chrompack CP-3800 gas chromatograph and Saturn 200 MS/MS block), $2 0 \\\\mathrm { m g }$ of the polymer sample was subjected to methanolysis as described in Basnett et al. [5]. For NMR analysis, the polymer was dissolved in deuterated chloroform $( 2 0 \\\\mathrm { m g / m L } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,387,873,449]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Thermal analysis of polymers was conducted using a differential scanning calorimeter (DSC) 214 Polyma (Netzsch, Germany) equipped with Intracooler IC70 cooling system. $5 \\\\mathrm { m g }$ of polymer was heated from $- 5 0$ to $1 0 0 ^ { \\\\circ } \\\\mathrm { C }$ at a heating rate of $2 0 ~ ^ { \\\\circ } \\\\mathrm { C }$ per min. DSC thermograms were analysed using Proteus 7.0 software. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[127,462,364,478]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.7. Antibacterial Characterization \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[129,489,848,505]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.7.1. MIC (Minimum Inhibition Concentration) and MBC (Minimum Bactericidal Concentration) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,513,873,689]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The minimum inhibition concentration (MIC) was estimated according to the ISO 20776 against Staphylococcus aureus 6538P and Escherichia coli 8739. Stock solution of the lime oil was prepared in Mueller Hinton Broth and $1 \\\\%$ dimethyl sulfoxide (added as a co-solvent which enables mixing the oils with aqueous medium). A range of concentrations from 100 to $5 ~ { \\\\mu \\\\mathrm { L } } / { \\\\mathrm { m L } }$ essential oils was tested in a 96-well plate. Each well was inoculated to a final volume of $1 0 0 \\\\mu \\\\mathrm { L }$ using a microbial suspension adjusted to achieve a final concentration of $5 \\\\times 1 0 ^ { 5 } \\\\mathrm { C F U / m L }$ and incubated at $3 7 ^ { \\\\circ } \\\\mathrm { C }$ for $2 4 \\\\mathrm { ~ h ~ }$ at 100 rpm. Studies were performed in triplicate. The MIC was determined by measuring the absorbance at $6 0 0 \\\\mathrm { n m }$ . Following the MIC test, the entire volume of the MIC well and the concentrations above were removed and 10-fold dilution was performed in PBS and plated onto agar plates. The plates were incubated at $3 7 ^ { \\\\circ } \\\\mathrm { C }$ for $2 4 \\\\mathrm { { h } }$ . The minimum bactericidal concentration (MBC) was determined as the lowest concentration of the compound that induced $9 9 . 9 \\\\%$ killing of the bacteria. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,701,243,715]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.7.2. Halo Test \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,725,873,901]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The films loaded with essential oils were tested for antibacterial activity using the halo test against Staphylococcus aureus 6538P and Escherichia coli 8739. The test was conducted according to the European Committee on Antimicrobial Susceptibility Testing (EUCAST) Disc Diffusion Test Methodology. A single colony was used to prepare a suspension of 0.5 McFarland turbidity standard (approx. $1 0 ^ { 8 }$ colony-forming units $\\\\mathrm { ( C F U ) / m L }$ ). A sterile cotton swab was immersed into the suspension and was spread evenly on the agar plates. The agar plates as well as the antimicrobial discs were kept at room temperature prior to inoculation. Polymer films of $0 . 7 \\\\mathrm { c m }$ diameter containing active agents, sterilized under ultraviolet (UV) light for $3 0 \\\\mathrm { m i n }$ (15 min each side) were placed on the inoculated agar surface. The plates were incubated at $3 7 ^ { \\\\circ } \\\\mathrm { C }$ for $1 8 \\\\mathrm { { h } }$ . The inhibition zones formed were measured and noted. For the antibacterial test, both freshly prepared samples and aged samples were investigated. Each material was tested in triplicates. Films without active agents were used as negative controls, antibiotic discs containing oxacillin (1 µg/disc) and streptomycin J. Funct. Biomater. 2020, 11, x FOR PEER REVIEW J. Funct. Biomater. 2020, 11, x FOR PEER REVIEWFunct. Biomater. 2020, 11, x FOR PEER REVIEW $( 3 0 0 ~ \\\\mu \\\\mathrm { g / d i s c } )$ were used as positive controls2 of 142 of 142 of 14 against Staphylococcus aureus 6538P and Escherichia coli 8739, respectively. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,66,300,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2020, 11, 24 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[831,66,872,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 of 14 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[122,104,870,135]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,147,324,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3. Results and Discussion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[127,174,860,203]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.1. Temporal Profile of Polyhydroxyalkanoate (PHA) Production by Pseudomonas mendocina CH50 UsingSugarcane Molasses as the Sole Carbon Source3.1. Temporal Profile of Polyhydroxyalkanoate (PHA) Production by Pseudomonas mendocina CH50 using3.1. Temporal Profile of Polyhydroxyalkanoate (PHA) Production by Pseudomonas mendocina CH50 using 3.1. Temporal Profile of Polyhydroxyalkanoate (PHA) Production by Pseudomonas mendocina CH50 using .1. Temporal Profile of Polyhydroxyalkanoate (PHA) Production by Pseudomonas mendocina CH50 using Sugarcane Molasses as the Sole Carbon SourceSugarcane Molasses as the Sole Carbon SourceSugarcane Molasses as the Sole Carbon Sourceugarcane Molasses as the Sole Carbon Sourceugarcane Molasses as the Sole Carbon Sourceugarcane Molasses as the Sole Carbon Source \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,211,873,259]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Figure 1 represents the fermentation profile of the PHA production by Figure 1 represents the fermentation profile of the PHA production byFigure 1 represents the fermentation profile of the PHA production by Figure 1 represents the fermentation profile of the PHA production by Pigure 1 represents the fermentation profile of the PHA production by Pigure 1 represents the fermentation profile of the PHA production by P. $P .$ mendocina CH50 using . mendocina CH50 using mendocina CH50 using mendocina CH50 using mendocina CH50 using endocina CH50 using sugarcane molasses as the carbon source in asugarcane molasses as the carbon sourcesugarcane molasses as the carbon source sugarcane molasses as the carbon source ugarcane molasses as the carbon source iugarcane molasses as the carbon source i $1 5 \\\\mathrm { L }$ bioreactor. Fermentation was performed as described a 15 L bioreactor. Fermentation was performed as 15 L bioreactor. Fermentation was performed as 15 L bioreactor. Fermentation was performed as 15 L bioreactor. Fermentation was performed as 5 L bioreactor. Fermentation was performed as in Section 2.3., and OD,described in Section 2described in Section 2.described in Section 2.3escribed in Section 2.3.escribed in Section 2.3. $\\\\mathrm { p H } , \\\\%$ DOT, biomass, nitrogen concentration and polymer yield were monitored.d OD, pH, % DOT, biomass, nitrogen concentration and polymer yield OD, pH, % DOT, biomass, nitrogen concentration and polymer yield OD, pH, % DOT, biomass, nitrogen concentration and polymer yield OD, pH, % DOT, biomass, nitrogen concentration and polymer yield OD, pH, % DOT, biomass, nitrogen concentration and polymer yield \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[181,279,798,550]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 1. Temporal profile of P(3HO-co-3HD) production byFigure 1. Temporal profile of P(3HO-co-3HD) production by Figure 1. Temporal profile of P(3HO-co-3HD) production byFigure 1. Temporal profile of P(3HO-co-3HD) production by Figure 1. Temporal profile of P(3HO-co-3HD) production by Pigure 1. Temporal profile of P(3HO-co-3HD) production by Pigure 1. Temporal profile of P(3HO-co-3HD) production by P. $P .$ . mendocina CH50 using sugarcane mendocina CH50 using sugarcane. mendocina CH50 using sugarcane mendocina CH50 using sugarcane mendocina CH50 using sugarcane mendocina CH50 using sugarcane endocina CH50 using sugarcane molasses as the sole carbon source. The profiling illustrates the polyhydroxyalkanoate (PHA) yield (molasses as the sole carbon source. The profiling illustrates the polyhydroxyalkanoate (PHA) yieldmolasses as the sole carbon source. The profiling illustrates the polyhydroxyalkanoate (PHA) yield (molasses as the sole carbon source. The profiling illustrates the polyhydroxyalkanoate (PHA) yield (molasses as the sole carbon source. The profiling illustrates the polyhydroxyalkanoate (PHA) yield (olasses as the sole carbon source. The profiling illustrates the polyhydroxyalkanoate (PHA) yield (olasses as the sole carbon source. The profiling illustrates the polyhydroxyalkanoate (PHA) yield ( ), optical density( ), optical density), optical density), optical density ), optical density ( ), optical density ( ), optical density ( $( \\\\\\\\cdots ) , \\\\\\\\mathrm { { p H } ( \\\\\\\\not { p } \\\\\\\\not { q } \\\\\\\\mathrm { { \\\\\\\\Gamma } } ^ { } \\\\\\\\not { q } \\\\\\\\mathrm { { \\\\\\\\Gamma } } ^ { } ) , \\\\\\\\mathrm { { D O T } } ^ { \\\\\\\\circ } / \\\\\\\\mathrm { { \\\\\\\\Gamma } } ^ { } ( \\\\\\\\not { q } \\\\\\\\mathrm { { \\\\\\\\Gamma } } ^ { } \\\\\\\\not { q } \\\\\\\\mathrm { { \\\\\\\\Gamma } } ^ { } ) , }$ ), bioma), bioma), bioma, biomass, biomass biomass biomass $( \\\\\\\\twoheadleftarrow )$ ) and the nitrogen concentration) and the nitrogen concentration) and the nitrogen concentration ) and the nitrogen concentration and the nitrogen concentration and the nitrogen concentration and the nitrogen concentration in in the form of the ammonium ions throughin the form of the ammonium ions throughin the form of the ammonium ions throughon the form of the ammonium ions throughoun the form of the ammonium ions throughou the form of the ammonium ions throughoutthe form of the ammonium ions throughout $4 8 \\\\\\\\mathrm { h }$ 8 h of fermentati8 h of fermentati h of fermentatioh of fermentatioh of fermentation of fermentation. of fermentation. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0795/5f5af7ee949750e1dd287a8e247b21bd6285b3ef3bcd300aa77b964fabbdcf7d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[124,634,875,908]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The optical density increased gradually until the firstThe optical density increased gradually until the firstThe optical density increased gradually until the first he optical density increased gradually until the first he optical density increased gradually until the first 3he optical density increased gradually until the first 3The optical density increased gradually until the first $^ { 3 \\\\mathrm { ~ h ~ } }$ (lag phase) of the fermentation and(lag phase) of the fermentation and(lag phase) of the fermentation and (lag phase) of the fermentation and lag phase) of the fermentation and ag phase) of the fermentation and (lag phase) of the fermentation and continued to increase steadily until the end of process, reaching a value of 4.7. Biomass concentrationcontinued to increase steadily until the end of process, reaching a value of 4.7. Biomass concentrationcontinued to increase steadily until the end of process, reaching a value of 4.7. Biomass concentration ontinued to increase steadily until the end of process, reaching a value of 4.7. Biomass concentration ontinued to increase steadily until the end of process, reaching a value of 4.7. Biomass concentration ontinued to increase steadily until the end of process, reaching a value of 4.7. Biomass concentration continued to increase steadily until the end of process, reaching a value of 4.7. Biomass concentration increased untiincreased untiincreased untilncreased until ncreased until 4creased until 4increased until $4 8 \\\\mathrm { h }$ reaching a value ofreaching a value oreaching a value ofeaching a value of 1eaching a value of 1.aching a value of 1. reaching a value of $1 . 9 \\\\mathrm { g / L }$ . Prior to the fermentation,. Prior to the fermentation, Prior to the fermentation,Prior to the fermentation, Prior to the fermentation, prior to the fermentation, p. Prior to the fermentation, $\\\\mathrm { p H }$ of the production media of the production media of the production media f the production media f the production media the production media of the production media was set to 7.0. Thewas set to 7.0. Thewas set to 7.0. The was set to 7.0. The pwas set to 7.0. The pas set to 7.0. The pwas set to 7.0. The $\\\\mathrm { p H }$ of the culture did not change significantly during the fermentation process of the culture did not change significantly during the fermentation processof the culture did not change significantly during the fermentation process f the culture did not change significantly during the fermentation process f the culture did not change significantly during the fermentation process f the culture did not change significantly during the fermentation process of the culture did not change significantly during the fermentation process whereas thwhereas thwhereas thewhereas the whereas the hereas the whereas the $\\\\mathrm { D O T \\\\% }$ % reduced fro% reduced fro reduced fro reduced fromreduced fromreduced from reduced from $1 0 0 \\\\%$ % %% t t to to $0 \\\\%$ 0% within few hours of the fermentation and remained0% within few hours of the fermentation and remained% within few hours of the fermentation and remained % within few hours of the fermentation and remained within few hours of the fermentation and remained within few hours of the fermentation and remained within few hours of the fermentation and remained very very low until the end of the process, maintaining an oxygen-limiting environment. Nutrient very low until the end of the process, maintaining an oxygen-limiting environment. Nutrientvery low until the end of the process, maintaining an oxygen-limiting environment. Nutrient very low until the end of the process, maintaining an oxygen-limiting environment. Nutrient ery low until the end of the process, maintaining an oxygen-limiting environment. Nutrient ery low until the end of the process, maintaining an oxygen-limiting environment. Nutrient low until the end of the process, maintaining an oxygen-limiting environment. Nutrient limitation limitation plays an important role in the production of PHAs. Furthermore, it is known that nutrientlimitation plays an important role in the production of PHAs. Furthermore, it is known that nutrientimitation plays an important role in the production of PHAs. Furthermore, it is known that nutrientimitation plays an important role in the production of PHAs. Furthermore, it is known that nutrientmitation plays an important role in the production of PHAs. Furthermore, it is known that nutrientplays an important role in the production of PHAs. Furthermore, it is known that nutrient limitation limitation coupled with an excess of carbon provides an ideal environment to produce PHAs [23]. In limitation coupled with an excess of carbon provides an ideal environment to produce PHAs [23]. Inlimitation coupled with an excess of carbon provides an ideal environment to produce PHAs [23]. In imitation coupled with an excess of carbon provides an ideal environment to produce PHAs [23]. In imitation coupled with an excess of carbon provides an ideal environment to produce PHAs [23]. In mitation coupled with an excess of carbon provides an ideal environment to produce PHAs [23]. In coupled with an excess of carbon provides an ideal environment to produce PHAs [23]. In this study, this study, nitrogen was the limiting factor. As illustrated in Figure 1, the nitrogen concentrationthis study, nitrogen was the limiting factor. As illustrated in Figure 1, the nitrogen concentrationthis study, nitrogen was the limiting factor. As illustrated in Figure 1, the nitrogen concentration this study, nitrogen was the limiting factor. As illustrated in Figure 1, the nitrogen concentration his study, nitrogen was the limiting factor. As illustrated in Figure 1, the nitrogen concentration his study, nitrogen was the limiting factor. As illustrated in Figure 1, the nitrogen concentration nitrogen was the limiting factor. As illustrated in Figure 1, the nitrogen concentration decreased from $0 . 5 8 ~ \\\\mathrm { g / L }$ ed ed d f fr frfro to $0 . 1 4 ~ \\\\mathrm { g / L }$ 8 g/L to 0.14 g8 g/L to 0.14 g g/L to 0.14 gg/L to 0.14 g/g/L to 0.14 g/L/L to 0.14 g/L at the end of $4 8 – \\\\mathrm { h }$ the end of 48-h fermentation. T the end of 48-h fermentation. Tthe end of 48-h fermentation. The end of 48-h fermentation. Thhe end of 48-h fermentation. The end of 48-h fermentation. The fermentation. The PHA yield $\\\\%$ PHA yi PHA yiPHA yieHA yielHA yielHA yield dcw) at $^ { 4 8 \\\\mathrm { ~ h ~ } }$ % dcw) at 48 h% dcw) at 48 h dcw) at 48 hdcw) at 48 hdcw) at 48 hdcw) at 48 h was found to was fowas fowas fowas fouwas fouas founbe was $1 4 . 2 2 \\\\%$ be was 14.22%. As mentioned above, optical density and biomass concentration be was 14.22%. As mentioned above, optical density and biomass concentrationbe was 14.22%. As mentioned above, optical density and biomass concentration e was 14.22%. As mentioned above, optical density and biomass concentration e was 14.22%. As mentioned above, optical density and biomass concentration e was 14.22%. As mentioned above, optical density and biomass concentration . As mentioned above, optical density and biomass concentration increased steadily increased steadily during the fermentation indicating cell growth in the presence of nitrogen in theincreased steadily during the fermentation indicating cell growth in the presence of nitrogen in theincreased steadily during the fermentation indicating cell growth in the presence of nitrogen in the ncreased steadily during the fermentation indicating cell growth in the presence of nitrogen in the ncreased steadily during the fermentation indicating cell growth in the presence of nitrogen in the creased steadily during the fermentation indicating cell growth in the presence of nitrogen in the during the fermentation indicating cell growth in the presence of nitrogen in the media. The absence media. The absence of nitrogen limitation could have influenced the overall accumulation of PHAs.media. The absence of nitrogen limitation could have influenced the overall accumulation of PHAs.media. The absence of nitrogen limitation could have influenced the overall accumulation of PHAs. media. The absence of nitrogen limitation could have influenced the overall accumulation of PHAs. edia. The absence of nitrogen limitation could have influenced the overall accumulation of PHAs. edia. The absence of nitrogen limitation could have influenced the overall accumulation of PHAs. of nitrogen limitation could have influenced the overall accumulation of PHAs. Sugarcane molasses Sugarcane molasses have been previously used as a carbon substrate to produce SCL-PHA by BacillusSugarcane molasses have been previously used as a carbon substrate to produce SCL-PHA by BacillusSugarcane molasses have been previously used as a carbon substrate to produce SCL-PHA by Bacillus Sugarcane molasses have been previously used as a carbon substrate to produce SCL-PHA by Bacillus ugarcane molasses have been previously used as a carbon substrate to produce SCL-PHA by Bacillus ugarcane molasses have been previously used as a carbon substrate to produce SCL-PHA by Bacillus have been previously used as a carbon substrate to produce SCL-PHA by Bacillus cereus SPV with a cereus SPV wcereus SPV wcereus SPV wereus SPV wiereus SPV witreus SPV witPHA yield of $5 1 . 3 7 \\\\%$ HA yield of 51.37% dcw [24]. In another study, Naheed and Jamil obtained a HA yield of 51.37% dcw [24]. In another study, Naheed and Jamil obtained aHA yield of 51.37% dcw [24]. In another study, Naheed and Jamil obtained a A yield of 51.37% dcw [24]. In another study, Naheed and Jamil obtained a A yield of 51.37% dcw [24]. In another study, Naheed and Jamil obtained a A yield of 51.37% dcw [24]. In another study, Naheed and Jamil obtained a dcw [24]. In another study, Naheed and Jamil obtained a maximum PHA yield of $8 0 . 5 \\\\%$ mum PHA yield of 80.5% dcw by Enterobacter sp. SEL2, a bacterium isolated from soil ofmum PHA yield of 80.5% dcw by Enterobacter sp. SEL2, a bacterium isolated from soil ofum PHA yield of 80.5% dcw by Enterobacter sp. SEL2, a bacterium isolated from soil of um PHA yield of 80.5% dcw by Enterobacter sp. SEL2, a bacterium isolated from soil of um PHA yield of 80.5% dcw by Enterobacter sp. SEL2, a bacterium isolated from soil of m PHA yield of 80.5% dcw by Enterobacter sp. SEL2, a bacterium isolated from soil of dcw by Enterobacter sp. SEL2, a bacterium isolated from soil of contaminated sites, with the use $3 \\\\%$ taminated sites, with the use 3% of sugarcantaminated sites, with the use 3% of sugarcataminated sites, with the use 3% of sugarcanaminated sites, with the use 3% of sugarcanaminated sites, with the use 3% of sugarcaneminated sites, with the use 3% of sugarcane of sugarcane molasses after fermentation for $2 4 \\\\mathrm { h }$ olololalasasass at $\\\\mathsf { p H } 5 . 0$ ter fermentation for 24 h at pH 5.0 [25].ter fermentation for 24 h at pH 5.0 [25].ter fermentation for 24 h at pH 5.0 [25]. er fermentation for 24 h at pH 5.0 [25]. r fermentation for 24 h at pH 5.0 [25]. r fermentation for 24 h at pH 5.0 [25]. [25]. Chaijamrus and Udpuay reported a $4 3 \\\\%$ yield of P(3HB) by Bacillus megaterium ATCC 6748 after $4 5 \\\\mathrm { h }$ of fermentation when $4 \\\\%$ sugarcane molasses was used as the carbon source [26]. With regards to MCL-PHA production using sugarcane molasses, a limited investigation has been conducted. This is due to the fact that the majority of theJ. Funct. Biomater. 2020, 11, x FOR PEER REVIEW 2 of 14 Pseudomonas species (i.e., the main producer of MCL-PHAs) cannot utilize sucrose, which is one of the main components of sugarcane molasses along with glucose and fructose [27]. Therefore, the substrate is usually modified in order to obtain equimolar concentrations of glucose and fructose by either ionic exchange or acid hydrolysis [27]. To date, only two species of the Pseudomonas family have been shown to metabolize sucrose directly, P. fluorescens A2a5 and P. corrugata 388. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,65,302,77]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2020, 11, 24 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[830,66,872,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 of 14 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[126,103,873,231]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,233,873,328]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"P. fluorescens A2a5 produced P(3HB) with a yield of $7 0 \\\\%$ dcw when cultured using sugarcane liquor [28]. In contrast, Solaiman et al. showed that P. corrugata 388 was able to consume sucrose (present in soy molasses used as carbon source) producing an MCL-PHA with a yield of $1 7 \\\\%$ dcw [27]. P. mendocina CH50, the strain used in this study is not able to utilize the sucrose present in the media. The PHA produced by P. mendocina CH50 using sugarcane molasses as the carbon source wasmedia. The PHA produced by P. mendocina CH50 using sugarcane molasses as the carbon source was further characterised. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[127,341,331,355]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.2. Polymer Characterisation \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[129,368,403,382]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.2.1. Polymer Composition Analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,391,873,551]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As can be seen from Figure 2, GC–MS chromatograms of the methanolysed polymer demonstrated two peaks identified using the National Institute of Standards and Technology (NIST) library. The retention peak at 7.7 and 9.2 min were identified as the methyl ester of 3-hydroxyoctanoic acid and methyl ester of 3-hydroxydecanoic acid respectively. Peak at $6 . 6 4 \\\\mathrm { m i n }$ corresponded to the methyl benzoate, used as an internal standard. Thus, the polymer produced by methyl benzoate, used as an internal standard. Thus, the polymer produc $P .$ mendocina CH50 using by P. mendocina CH50 sugarcane molasses as the sole carbon source was identified as the copolymer of 3-hydroxyoctanoate and 3-hydroxydecanoate, Poly(3-hydroxyoctanoate-co-3-hydroxydecanoate) or P(3HO-co-3HD). The mole percentage of 3-HO and 3-HD was found to beP(3HO-co-3HD). The mole percentage of 3-HO an $2 5 . 6 \\\\%$ andD w $7 4 . 4 \\\\%$ respectively. P(3HO-co-3HD)nd to be 25.6% and 74.4% copolymers with 3-HD-dominant composition are usually produced by Pseudomonas species whenrespectively. P(3HO-co-3HD) copolymers with 3-HD-dominant composition are usually produced grown on structurally unrelated simple sugars [29].by Pseudomonas species when grown on structurally \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[136,568,860,772]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 2. Gas chromatography–mass spectrometry (GC–MS) data of the PHA copolymer produced Figure 2. Gas chromatography–mass spectrometry (GC–MS) data of the PHA copolymer produced with sugarcane molasses as the carbon source: (A) Gas chromatogram, (B) Mass spectrum of a peak with sugarcane molasses as the carbon source: (A) Gas chromatogram, (B) Mass spectrum of a peak witwith $\\\\\\\\mathrm { R } _ { \\\\\\\\mathrm { t } } = 7 . 7$ min identified using the National Institute of Standards and Technology (NIST) library min identified using the National Institute of Standards and Technology (NIST) library as as methyl ester of 3-hydroxyoctanoic acid, and (C) mass spectrum of a pemethyl ester of 3-hydroxyoctanoic acid, and (C) mass spectrum of a peak with $\\\\\\\\mathrm { R _ { t } } = 9 . 2$ Rt = 9.2 min min identified identified using NIST library as methyl ester of 3-hydroxydecanoic acid. Methyl benzoate was used using NIST library as methyl ester of 3-hydroxydecanoic acid. Methyl benzoate was used as an internal as an internal ststandard (STD). \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0591/a9fd3ca4adbb7947f7dd7ad1ea510fc98d732b866bdb89b12480aa00dde73187.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[124,887,873,917]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In order to confirm the structure of synthesised PHA, NMR analysis has been performed. In order to confirm the structure of synthesised PHA, NMR analysis has been performed. $^ 1 \\\\mathrm { H }$ nd and $^ { 1 3 } \\\\mathrm { C }$ NMR spectra are presented in Figure 3. All peaks expected for P(3HO-co-3HD) where observed NMR spectra are presented in Figure 3. All peaks expected for P(3HO-co-3HD) where observed in $^ 1 \\\\mathrm { H }$ and $^ { 1 3 } \\\\mathrm { C }$ NMR spectroscopy. The chemical shift at 169, 70.9, 39.1 ppm corresponds to $C _ { 1 }$ $\\\\scriptstyle ( C = \\\\mathbf { O } )$ , $C _ { 3 }$ (–CH), $C _ { 2 }$ $\\\\mathrm { ( - C H } _ { 2 } )$ respectively, forming the backbone of the copolymer. The resonance in the range between $2 3 \\\\mathrm { - } 3 5 \\\\mathrm { p p m }$ related to olefinic methylene groups: $\\\\mathrm { C _ { 4 } , ^ { * } C _ { 4 } }$ (33.86, 33.92 ppm), ${ C } _ { 5 } , { ^ { * } } { C } _ { 5 }$ $( 2 4 . 8 4 , 2 5 . 2 0 \\\\mathrm { p p m } )$ , ${ C _ { 6 } } , { ^ { * } } C _ { 6 }$ (31.64, 29.48 ppm), ${ C _ { 7 } } , { { ^ * } } { C _ { 7 } }$ (22.63, 29.32 ppm), $^ { * } C _ { 8 }$ $( 3 1 . 9 1 \\\\mathrm { p p m } )$ , $^ { * } { \\\\cal C } _ { 9 }$ (22.76 ppm) (carbon atoms of 3-hydroxydecanoate market with asterisk). The terminal methyl groups $\\\\left( \\\\mathrm { - C H } _ { 3 } \\\\right)$ of $\\\\mathrm { C } _ { 8 } , { } ^ { * } \\\\mathrm { C } _ { 1 0 }$ were observed at 14.12 and $1 4 . 2 2 \\\\mathrm { p p m }$ , respectively. All observed peaks in the $^ { 1 3 } \\\\mathrm { C }$ spectrum (Figure 3A) are in agreement with spectral characteristic reported for 3-hydroxyoctanoate and 3-hydroxydecanoate [30].J. Funct. Biomater. 2020, 11, x FOR PE \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,66,300,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2020, 11, 24 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[831,66,872,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 of 14 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[124,103,877,231]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[210,248,781,752]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 3.Figure 3. $^ { 1 3 } \\\\\\\\mathrm { C }$ nuclear magnetic resonance (NMR) (A), and nuclear magnetic resonance (NMR) (A), and $\\\\\\\\mathrm { ^ { 1 } H N M R }$ (B) spectra of PHA copolymer (B) spectra of PHA copolymer produced with sugarcane molasses as carbon source. The structure ofproduced with sugarcane molasses as carbon source. The structure of $\\\\\\\\mathrm { P ( 3 H O - } c o { - 3 } \\\\\\\\mathrm { H D ) }$ is shown as an is shown as an insert within the spectra.insert within the spectra. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0548/8db28bd000f64b2c945147fc946facd813017e170534e3cf4a6299f3ed6eac18.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[124,821,873,919]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \". ProThe $^ 1 \\\\mathrm { H }$ ties of Copolymer and Films Containing Lime Oil NMR spectrum (Figure 3B) shows 5 groups of overlapping peaks corresponding to 5 different proton environments characteristic for medium chain 3-hydroxyalakanoates. Three different environments of the olefinic proton manifested at chemical shifts 0.8, 1.2, and $1 . 5 \\\\mathrm { p p m }$ . The chemical properties of MCL-PHAs [5]. With glass-transition temperature ca. −45 °C, the amorphous phase ofshift at 0.8 corresponds to the methyl groups of both monomers. The protons of the methylene group at carbon atoms 5, 6, 7 of 3-hydroxyoctanoate and carbon atoms 5, 6, 7, 8, 9 of 3-hydroxydecanoate were observed at $1 . 2 \\\\ { \\\\mathrm { p p m } } ,$ while proton resonance of the methylene group at carbon 4 for both monomers shifted further downfield to $1 . 5 \\\\mathrm { p p m }$ . Two types of backbone protons for both monomers generated signals at chemical shifts of 2.52 (methylene group of carbon 2) and 5.2 (methine group of carbon 3) ppm. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,66,302,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2020, 11, 24 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[831,66,872,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7 of 14 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[126,104,872,151]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,153,875,311]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The monomeric composition of PHAs is strictly influenced by the carbon source utilized. In presence of structurally related carbon sources (e.g., fatty acids), the final composition of the polymer reflects that of the substrate. When unrelated substrates (e.g., carbohydrates) are utilized, the monomeric unit of the PHAs are independent of the structure of the carbon source [1–3]. As mentioned before, sugarcane molasses is composed of up to $6 0 \\\\%$ of sugars which are classified as unrelated substrates. Pseudomonas sp. can convert sugars into PHAs through the de novo fatty acid pathway. One of the enzymes involved in this pathway has a high affinity to the substrate containing 10 carbon atoms [31]. For this reason, Pseudomonas sp. have been shown to produce PHAs with a high content of the 3-HD monomer when cultured using glucose or glycerol [5,31]. This study is in agreement with previous studies, as the MCL-PHA produced showed a high content of 3-HD. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,324,589,340]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.2.2. Properties of Copolymer and Films Containing Lime Oil. Funct. Biomater. 2020, 11, x FOR PEER REVIEW \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,348,875,492]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The DSC ofual amount $\\\\mathrm { P ( 3 H O - } c o { - 3 } \\\\mathrm { H D ) }$ produced with sugarcane molasses as the carbon source revealedP(3HO-co-3HD) films were determined from TGA experiments both low glass transition and melting temperature (Figure 4A, Table 1) which are characteristic(Figure 4B) and found to be 6.9 wt.% for the new film and 2.8 wt.% for the aged one. The enthalpy of properties of MCL-PHAs [5]. With glass-transition temperature ca. melting of the P(3HO-co-3HD) crystalline phase decreased from 18.2 $- 4 5 ^ { \\\\circ } \\\\mathrm { C } ,$ the amorphous phase of4.7 and 15.8 J/g for fresh this semi-crystalline polymer is in a rubbery state at room temperature and polymer is expected toand aged films, respectively. Thus, the addition of lime oil induced a decrease in crystallinity of the be a flexible and stretchable material. Also, such low glass transition would prevent transition of thecopolymer. This implies that lime oil components had a plasticising effect on the copolymer. Slight polymer into a brittle material at sub-zero temperatures which can be required for a packaging material.increase in enthalpy of polymer melting (and thereby crystallinity degree) for aged films compared The melting temperature is aroundwith the fresh one might be explain $5 5 ^ { \\\\circ } \\\\mathrm { C }$ . This could be beneficial for melt processing at relatively lowcontinuation of crystallisation during 1-year storage and by temperatures. Melting is not observed in a DSC thermogram of second heating demonstrating slowpromotion of crystallisation with the loss of plasticising lime oil. Also, with storage, enthalpy of kinetics of the crystallisation process for this copolymer.melting shifted to higher temperatures indicating rigidifi \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[156,510,843,711]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 4. Thermal analysis of P(3Figure 4. Thermal analysis of $\\\\\\\\mathrm { P ( 3 H O - } c o { - 3 } \\\\\\\\mathrm { H D ) }$ at film and films with incorporated lime oil. (A) neat film and films with incorporated lime oil. Differential scanning calorimetry (DSC) thermograms: neat film (black lines), fresh film with lime oil (A) Differential scanning calorimetry (DSC) thermograms: neat film (black lines), fresh film with (red lines), and aged films with lime oil (blue lines). First heating—solid line, second heating—dashed lime oil (red lines), and aged films with lime oil (blue lines). First heating—solid line, second line. Thermograms were shifted vertically for better visibility. (B) Thermogravimetric analysis (TGA)heating—dashed line. Thermograms were shifted vertically for better visibility. (B) Thermogravimetric curves for determination of oil content after isothermal evaporation at 150 °C for 2 h.analysis (TGA) curves for determination of oil content after isothermal evaporation at $1 5 0 ^ { \\\\\\\\circ } \\\\\\\\mathrm { C }$ for $^ { 2 \\\\\\\\mathrm { h } }$ . \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0586/8295aa3a5fcd5d888d80d89da2cc11c56294afca0311eb44656e95bc6fd0668f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[126,66,300,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2020, 11, 24 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[831,66,872,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"8 of 14 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[252,123,746,223]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0956/6187a8589ec68862a5fea82832d6ecba08fde26715697aa4a6308c0eaa77e4f3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=\\\"2\\\">Sample</td><td colspan=\\\"3\\\">Thermal Properties</td></tr><tr><td>Tg,C</td><td>Tm℃a</td><td>△H, J/g</td></tr><tr><td>Neat film</td><td>-45.6</td><td>55.6</td><td>18.2</td></tr><tr><td>Fresh film with lime oil</td><td>-44.3</td><td>51.4</td><td>14.7</td></tr><tr><td>Aged film with lime oil</td><td>-40.8</td><td>50.5</td><td>15.8</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 1. Thermal properties of P(3HO-co-3HD) neat films and films with incorporated lime oil. \\\"]\",\n \"table_footnote\": \"[\\\"a Determined as the end of melting peak. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[126,254,873,494]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Incorporating lime oil into P(3HO-co-3HD) films decreased the melting temperature by approximately $4 ~ ^ { \\\\circ } \\\\mathrm { C }$ . For an evaluation of the effect of lime oil on the crystallinity of the copolymer, an observed enthalpy of melting of P(3HO-co-3HD) in films containing lime oil needs to be normalised to the actual mass fraction of the copolymer. The real content of oil in the films does not correspond to oil/polymer composition in the casting solution due to oil loss as a result of evaporation along with solvent during the film formation as well as during the storage of the prepared film. Residual amounts of lime oil in $\\\\mathrm { P ( 3 H O - } c o { - 3 } \\\\mathrm { H D ) }$ films were determined from TGA experiments (Figure 4B) and found to be $6 . 9 \\\\mathrm { w t . \\\\% }$ for the new film and $2 . 8 \\\\mathrm { w t . \\\\% }$ for the aged one. The enthalpy of melting of the P(3HO-co-3HD) crystalline phase decreased from $1 8 . 2 \\\\mathrm { J / g }$ to 14.7 and $1 5 . 8 \\\\mathrm { J } / \\\\mathrm { g }$ for fresh and aged films, respectively. Thus, the addition of lime oil induced a decrease in crystallinity of the copolymer. This implies that lime oil components had a plasticising effect on the copolymer. Slight increase in enthalpy of polymer melting (and thereby crystallinity degree) for aged films compared with the fresh one might be explained by continuation of crystallisation during 1-year storage and by promotion of crystallisation with the loss of plasticising lime oil. Also, with storage, enthalpy of melting shifted to higher temperatures indicating rigidification of the amorphous phase. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[127,506,364,521]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.3. Antibacterial Characterization \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[129,535,848,549]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.3.1. MIC (Minimum Inhibition Concentration) and MBC (Minimum Bactericidal Concentration) \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,557,875,733]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lime oil was tested against S. aureus 6538P and E. coli 8739, chosen as an example of Gram-positive and Gram-negative bacteria, respectively. The values obtained are shown in Table 2. The MIC and MBC values against S. aureus 6538P were 60 and $1 0 0 ~ \\\\mu \\\\mathrm { L / m L }$ respectively. While against E. coli 8739, MIC value of $8 0 ~ \\\\mu \\\\mathrm { L / m L }$ was obtained, no MBC could be determined at the concentration investigated. The MBC is defined as the lowest concentration of compound inducing $9 9 . 9 \\\\%$ killing of the bacteria, corresponding to 3 log reduction in CFU. Therefore, the results obtained for lime oil showed that the compound had a bacteriostatic rather than bactericidal effect against E. coli 8739 at the concentrations investigated. In literature, MIC and MBC values for lime oil and generally for all essential oils comprise a very large range. For lime oil MIC values were reported from $0 . 1 ~ \\\\mathrm { m g / m L }$ [18] to few hundreds $\\\\mathrm { m g / m L }$ [32]. Such wide variations can be caused by plant sources, time of harvesting, methods of oil extraction, etc. Lime oil used in this study was of relatively low antimicrobial activity. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[196,783,801,846]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/1023/a136c6094f3d54a6c2b91b4f600787191e7c5aa57deb139f701c52484d27e065.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td colspan=\\\"2\\\">S.aureus 6538P</td><td colspan=\\\"2\\\">E coli 8739</td></tr><tr><td>MIC (μL/mL)</td><td>MBC (μL/mL)</td><td>MIC (μL/mL)</td><td>MBC (μL/mL)</td></tr><tr><td>60</td><td>100</td><td>80</td><td>1</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 2. Minimum inhibition concentration (MIC) and minimum bactericidal concentration (MBC) values of lime against S. aureus 6538P and E. coli 8739. \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[127,864,872,912]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The antibacterial mechanism of such essential oils is usually related to the activity of their main components, terpenes and terpenoids. Lime oil (Citrus aurantifolia) is composed mainly of limonene [18,20]. The mechanism of action of such organic compounds is linked to their ability to modify the integrity of the bacterial cell wall, increasing its permeability and causing membrane disruption [15,33]. Moreover, a higher effect against Gram-positive bacteria compared to negative onesTable 2. Minimum inhibition concentration (MIC) and minimum bactericidal concentration (MBC) is usually reported, due to protection of Gram-negative bacteria by the outer membrane composed ofvalues of lime against S. aureus 6538P and E. coli 8739. lipopolysaccharides [34]. Such a response was confirmed in this study as overall lime oil showed a higher effect against S. aureus 6538P than E. coli 8739. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,65,302,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2020, 11, 24 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[830,66,872,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"9 of 14 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[126,104,873,183]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,196,242,210]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.3.2. Halo Test \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,219,875,362]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The antimicrobial materials were tested against both S. aureus 6538P and E. coli 8739. Freshly3.3.2. Halo Test prepared and aged films were analysed to investigate whether the material was able to preserve theirThe antimicrobial materials were tested against both S. aureus 6538P and E. coli 8739. Freshly antimicrobial efficacy. As it was discussed earlier, the content of lime oil decreased fromprepared and aged films were analysed to investigate whether the material was able to preserv $7 4 \\\\mathrm { m g } / \\\\mathrm { g }$ of copolymer for fresh film toantimicrobial efficacy. A $3 0 \\\\mathrm { m g / g }$ for the aged sample. The freshly prepared samples showed activitydiscussed earlier, the content of lime oil decreased from 74 mg/g of against S. aureus 6538P, inducing the formation of inhibition zones as shown in Figure 5 and Table 3.copolymer for fresh film to 30 mg/g for the aged sample. The freshly prepared samples showed Moreover, the samples obtained after a one-year storage period still possessed activity against theactivity against S. aureus 6538P, inducing the formation of inhibition zones as shown in Figure 5 and microorganism, even though aTable 3. Moreover, the sam $1 5 \\\\%$ reduction of the zone of inhibition could be detected. By contrast,obtained after a one-year storage period still possessed activity both the freshly prepared and aged films did not show activity against E. coli 8739, as reported in Table 3. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[163,379,840,682]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 5. Halo test showing the antibacterial activity of aged lime oil containing P(3HO-co-3HD) films Figure 5. Halo test showing the antibacterial activity of aged lime oil containing P(3HO-co-3HD) films (a–c) and fresh lime oil containing films (d–f) against S. aureus 6538P (control: antibiotic disc containing $1 \\\\\\\\mu \\\\\\\\mathrm { g }$ of oxacillin, left hand side). \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0718/5eeb92c2c5b7dc1e71bcfe87c0fdb503b86b0d31dc366b0f17a96346a4e8d11d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[171,783,826,896]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0699/52fd6de078eea814f2c05aaee3f8662e4357add39dc1c07011d62358621c9a3f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=\\\"2\\\"> Sample</td><td colspan=\\\"2\\\">Zone of Inhibition (ZOI) (cm)</td></tr><tr><td>S. aureus 6538P</td><td>E. coli 8739</td></tr><tr><td>Aged P(3HO-co-3HD)/lime films</td><td>1.06 ± 0.02</td><td>1</td></tr><tr><td>Fresh P(3HO-co-3HD)/lime films</td><td>1.23 ± 0.05</td><td>0.13 ± 0.04</td></tr><tr><td rowspan=\\\"2\\\">Control</td><td>3.13 ± 0.10</td><td>1.99 ± 0.09</td></tr><tr><td>(oxacillin)</td><td>(streptomycin)</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 3. Halo diameters showing the effect of P(3HO-co-3HD) films containing lime oil against S. aureus 6538P and E. coli 8739. \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[126,66,300,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2020, 11, 24 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[825,66,872,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10 of 14 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[126,104,873,376]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Limited research has been conducted on the incorporation of essential oils into PHAs and the studies conducted have focused only on SCL-PHAs. Films for antibacterial food packaging were produced using P(3HB-co-3HV) and two different essential oils such as oregano and clove, showing antibacterial activity against common food pathogens Listeria innocua 910 and E. coli 101 [35]. In another study, P(3HB-co-3HV) films were loaded with coconut fibres impregnated with oregano essential oil to produce composite antibacterial packaging materials, showing activity against S. aureus 6538P [36]. Alternatively, electrospinning has been investigated to produce fibrous mats based on P(3HB) or P(3HB-co-3HV) fibres containing essential oil. The natural compounds were either adsorbed onto the fibres after the spinning process (cinnamon, clove, oregano and oak bark) [37] or incorporated into the solution prior to electrospinning (rosemary and green tea extracts) [38]. The material produced showed activity against Micrococcus luteus 1569, Serratia marcescens 8587, S. aureus 6538P and E. coli 25922. Finally, Mendelez-Rodriguez et al. produced P(3HB-co-3HV) antibacterial fibres by loading eugenol essential oil onto mesoporous silica nanoparticles, followed by their incorporation into the fibres by electrospinning. The encapsulation was employed to maintain and protect the antibacterial effect of the agent, as essential oils are volatile compounds. The films obtained showed antibacterial activity against the S. aureus 6538P and E. coli 25922 up to 15 days after the day of fabrication and storage in a tightly closed system [39]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,388,242,403]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4. Conclusions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,412,873,732]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This study demonstrated the valorisation of sugarcane molasses, an agri-food by-product, through the production of a high-value MCL-PHA. The use of sugarcane molasses as the single carbon source and Pseudomonas mendocina CH50, a known MCL PHA producer, resulted in the synthesis of P(3HO-co-3HD) copolymer. P(3HO-co-3HD) is a polyester with low glass-transition and melting temperature, high elasticity and flexibility. In order to expand its potential applications, in this study, we explored the possibility of making P(3HO-co-3HD) antimicrobial through the incorporation of lime oil, an essential oil known for its antimicrobial activity. The antibacterial properties of the lime oil were investigated using the ISO 20776 against two bacterial strains, Gram-positive S. aureus 6538P and Gram-negative E. coli 8739. In line with general observations on the activity of essential oils, the lime oil used in this study was more active against S. aureus 6538P than E. coli 8739. Incorporation of the lime oil into the P(3HO-co-3HD) film had a minor effect on the material properties, slightly decreasing its crystallinity. As expected, the content of lime oil decreased with the storage due to migration out and evaporation from the polymer films. After one-year storage, the content of lime oil in the films decreased from ca 7 wt.% to $2 . 9 \\\\mathrm { w t . \\\\% }$ . Although the antimicrobial activity decreased after prolonged storage, both fresh and aged films of P(3HO-co-3HD) with incorporated lime oil showed activity against S. aureus 6538P. On the contrary, no antimicrobial activity was observed against E. coli 8739 for both fresh and aged films. The feasibility of imparting microbial activity to PHAs can broaden the application of this biodegradable polymer in several sectors, including coatings, adhesives, biodegradable rubbers, implantable materials. In the last sector, the developed P(3HO-co-3HD)/lime oil films could be employed as an antibacterial material for the regeneration of soft tissues, like skin. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,744,873,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Author Contributions: Conceptualization, P.B. and I.R.; methodology, P.B., E.M., R.N. and A.P.; formal analysis, P.B., E.M., R.N., A.P., B.G., M.H.A.; investigation, P.B., E.M., R.N., A.P., B.G., M.H.A.; resources, I.R.; data curation, P.B., E.M., R.N., A.P., B.G., M.H.A.; writing—P.B., E.M., R.N., A.P., B.G., M.H.A., B.L.; writing—review and editing, I.R.; supervision, I.R. and P.B.; project administration, I.R.; funding acquisition, I.R. All authors have read and agreed to the published version of the manuscript. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,812,872,850]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Funding: This research was funded by POLYBIOSKIN (H2020), Grant Agreement Number 745839, EM was funded by HyMedPoly, Marie Sklodowska-Curie Grant Agreement No 643050. The APC was funded by the University of Sheffield using the POLYBIOSKIN (H2020), Grant Agreement Number 745839. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[124,854,872,881]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acknowledgments: The authors will like to acknowledge technical support by Neville Antonio and Zhi Song at the University of Westminster. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,887,552,900]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conflicts of Interest: The authors declare no conflict of interest. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,66,300,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2020, 11, 24 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[825,66,872,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"11 of 14 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[126,104,213,118]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[124,128,875,851]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Khanna, S.; Srivastava, A.K. Recent advances in microbial polyhydroxyalkanoates. Process Biochem. 2005, 40, 607–619. [CrossRef] \\n2. Li, Z.; Yang, J.; Loh, X. Polyhydroxyalkanoates: Opening doors for a sustainable future. NPG Asia Mater. 2016, 8, 265. [CrossRef] \\n3. Raza, Z.A.; Abid, S.; Banat, I.M. Polyhydroxyalkanoates: Characteristics, production, recent developments and applications. Int. Biodeterior. Biodegrad. 2018, 126, 45–56. [CrossRef] \\n4. Álvarez-Chávez, C.R.; Edwards, S.; Moure-Eraso, R.; Geiser, K. Sustainability of bio-based plastics: General comparative analysis and recommendations for improvement. J. Clean. Prod. 2012, 23, 47–56. [CrossRef] \\n5. Basnett, P.; Lukasiewicz, B.; Marcello, E.; Gura, H.K.; Knowles, J.C.; Roy, I. Production of a novel medium chain length poly(3-hydroxyalkanoate) using unprocessed biodiesel waste and its evaluation as a tissue engineering scaffold. Microb. Biotechnol. 2017, 10, 1384–1399. [CrossRef] [PubMed] \\n6. Singh, A.K.; Mallick, N. Exploitation of inexpensive substrates for production of a novel SCL-LCL-PHA co-polymer by Pseudomonas aeruginosa MTCC 7925. J. Ind. Microbiol. Biotechnol. 2009, 36, 347–354. [CrossRef] \\n7. Chee, J.; Yoga, S.; Lau, N.; Ling, S.; Abed, R.M.M. Bacterially produced Polyhydroxyalkanoate (PHA): Converting renewable resources into bioplastics. In Current Research, Technology and Education Topics in Applied Microbiology and Microbial Biotechnology; Méndez-Vilas, A., Ed.; Formatex Research Center: Badajoz, Spain, 2010; pp. 1395–1404. \\n8. Jiang, G.; Hill, D.J.; Kowalczuk, M.; Johnston, B.; Adamus, G.; Irorere, V.; Radecka, I. Carbon sources for Polyhydroxyalkanoates and an integrated biorefinery. Int. J. Mol. Sci. 2016, 17, 1157. [CrossRef] \\n9. Nikodinovic-Runic, J.; Guzik, M.; Kenny, S.T.; Babu, R.; Werker, A.; Connor, K.E. Carbon-rich wastes as feedstocks for biodegradable polymer (Polyhydroxyalkanoate) production using bacteria. In Advances in Applied Microbiology, 1st ed.; Sariaslani, S., Gadd, G.M., Eds.; Academic Press: Cambridge, MA, USA, 2013; Volume 84, pp. 139–200. \\n10. Basnett, P.; Ravi, S.; Roy, I. Natural bacterial biodegradable medical polymers: Polyhydroxyalkanoates. In Science and Principles of Biodegradable and Bioresorbable Medical Polymers; Zhang, X., Ed.; Woodhead Publishing: Cambridge, UK, 2017; pp. 257–277. \\n11. Paris, J.B.; Seyer, D.; Jouenne, T.; Thébault, P. Elaboration of antibacterial plastic surfaces by a combination of antiadhesive and biocidal coatings of natural products. Colloids Surf. B 2017, 156, 186–193. [CrossRef] \\n12. Roca, I.; Akova, M.; Baquero, F.; Carlet, J.; Cavaleri, M.; Coenen, S.; Cohen, J.; Findlay, D.; Gyssens, I.; Heure, O.E.; et al. The global threat of antimicrobial resistance: Science for intervention. NMNI 2015, 6, 22–29. [CrossRef] \\n13. Prabuseenivasan, S.; Jayakumar, M.; Ignacimuthu, S. In vitro antibacterial activity of some plant essential oils. BMC Complement. Altern. Med. 2006, 6, 39. [CrossRef] \\n14. Kalemba, D.; Kunicka, A. Antibacterial and Antifungal Properties of Essential Oils. Curr. Med. Chem. 2003, 10, 813–829. [CrossRef] [PubMed] \\n15. Hyldgaard, M.; Mygind, T.; Meyer, R.L. Essential oils in food preservation: Mode of action, synergies, and interactions with food matrix components. Front. Microbiol. 2012, 3, 12. [CrossRef] [PubMed] \\n16. Di Pasqua, R.; Hoskins, N.; Betts, G.; Mauriello, G. Changes in membrane fatty acids composition of microbial cells induced by addiction of thymol, carvacrol, limonene, cinnamaldehyde, and eugenol in the growing media. J. Agric. Food Chem. 2006, 54, 2745–2749. [CrossRef] [PubMed] \\n17. Arrieta, M.P.; López, J.; Hernández, A.; Rayón, E. Ternary PLA–PHB–Limonene blends intended for biodegradable food packaging applications. Eur. Polym. J. 2014, 50, 255–270. [CrossRef] \\n18. Hsouna, A.B.; Halima, N.B.; Smaoui, S.; Hamdi, N. Citrus lemon essential oil: Chemical composition, antioxidant and antimicrobial activities with its preservative effect against Listeria monocytogenes inoculated in minced beef meat. Lipids Health Dis. 2017, 16, 146. [CrossRef] \\n19. H ˛ac-Wydro, K.; Flasi ´nski, M.; Roma ´nczuk, K. Essential oils as food eco-preservatives: Model system studies on the effect of temperature on limonene antibacterial activity. Food Chem. 2017, 235, 127–135. [CrossRef] \\n20. Costa, R.; Bisignano, C.; Filocamo, A.; Grasso, E.; Occhiuto, F.; Spadaro, F. Antimicrobial activity and chemical composition of Citrus aurantifolia (Christm.) Swingle essential oil from Italian organic crops. J. Essent. Oil Res. 2014, 26, 400–408. [CrossRef] \\n21. Basnett, P.; Marcello, E.; Lukasiewicz, B.; Panchal, B.; Nigmatullin, R.; Roy, I. Biosynthesis and characterization of novel, biocompatible medium chain length polyhydroxyalkanoates by Pseudomonas mendocina CH50 using coconut oil as the carbon source. J. Mater. Sci. Mater. Med. 2018, 29, 179. [CrossRef] \\n22. Rai, R.; Yunos, D.M.; Boccaccini, A.R.; Knowles, J.C.; Barker, I.A.; Howdle, S.M.; Tredwell, G.D.; Keshavarz, T.; Roy, I. Poly-3-hydroxyoctanoate P(3HO), a Medium Chain Length Polyhydroxyalkanoate Homopolymer from Pseudomonas mendocina. Biomacromolecules 2001, 12, 2126–2136. [CrossRef] \\n23. Gumel, A.M.; Annuar, M.S.M.; Heidelberg, T. Growth kinetics, effect of carbon substrate in biosynthesis of MCL-PHA by Pseudomonas putida Bet001. Braz. J. Microbiol. 2014, 45, 427–438. [CrossRef] \\n24. Akaraonye, E.; Moreno, C.; Knowles, J.C.; Keshavarz, T.; Roy, I. Poly(3-hydroxybutyrate) production by Bacillus cereus SPV using sugarcane molasses as the main carbon source. Biotechnol. J. 2012, 7, 293–303. [CrossRef] [PubMed] \\n25. Naheed, N.; Jamil, N. Optimization of biodegradable plastic production on sugar cane molasses in Enterobacter sp. SEL2. Braz. J. Microbiol. 2014, 45, 417–426. [CrossRef] [PubMed] \\n26. Chaijamrus, S.; Udpuay, N. Production and characterization of polyhydroxybutyrate from molasses and corn steep liquor produced by Bacillus megaterium ATCC 6748. Agric. Eng. Int. CIGR J. 2008, 10, 1–12. \\n27. Solaiman, D.K.Y.; Ashby, R.D.; Hotchkiss, A.T.; Foglia, T.A. Biosynthesis of medium-chain-length Poly(hydroxyalkanoates) from soy molasses. Biotechnol. Lett. 2006, 28, 157–162. [CrossRef] \\n28. Jiang, Y.; Song, X.; Gong, L.; Li, P.; Dai, C.; Shao, W. High poly (β-hydroxybutyrate) production by Pseudomonas fluorescens A2a5 from inexpensive substrates. Enzym. Microb. Technol. 2008, 42, 167–172. [CrossRef] \\n29. Chen, Y.-J.; Huang, Y.-C.; Lee, C.-Y. Production and characterization of medium-chain-length polyhydroxyalkanoates by Pseudomonas mosselii TO7. J. Biosci. Bioeng. 2014, 118, 145–152. [CrossRef] \\n30. De Rijk, T.C.; Van de Meer, P.; Eggink, G.; Weusthuis, R.A. Methods for Analysis of Poly(3-hydroxyalkanoate) Composition. In Biopolymers: Polyesters. II Properties and Chemical Synthesis; Doi, Y., Steinbuchel, A., Eds.; Wiley-Blackwell: Hoboken, NJ, USA, 2005; Volume 3b, pp. 1–12. \\n31. Poblete-Castro, I.; Rodriguez, A.L.; Chi Lam, C.M.; Kessler, W. Improved production of medium-chain-length polyhydroxyalkanoates in glucose-based fed-batch cultivations of metabolically engineered Pseudomonas putida strains. J. Microbiol. Biotechnol. 2014, 24, 59–69. [CrossRef] \\n32. Aibinu, I.; Adenipekun, T.; Adelowotan, T.; Ogunsanya, T.; Odugbemi, T. Evaluation of the antimicrobial properties of different parts of Citrus Aurantifolia (Lime Fruit) as used locally. Afr. J. Tradit. Complement. Altern. Med. 2007, 4, 185–190. \\n33. Khameneh, B.; Iranshahy, M.; Soheili, V.; Fazly Bazzaz, B.S. Review on plant antimicrobials: A mechanistic viewpoint. Antimicrob. Resist. Infect. Control 2019, 8, 118. [CrossRef] \\n34. Nazzaro, F.; Fratianni, F.; De Martino, L.; Coppola, R.; De Feo, V. Effect of essential oils on pathogenic bacteria. Pharmaceuticals 2013, 6, 1451–1474. [CrossRef] \\n35. Requena, R.; Jiménez, A.; Vargas, M.; Chiralt, A. Poly [(3-hydroxybutyrate)-co-(3-hydroxyvalerate)] active bilayer films obtained by compression moulding and applying essential oils at the interface. Polym. Int. 2016, 65, 883–891. [CrossRef] \\n36. Torres-Giner, S.; Hilliou, L.; Melendez-Rodriguez, B.; Figueroa-Lopez, K.J.; Madalena, D.; Cabedo, L.; Covas, J.A.; Vicente, A.A.; Lagaron, J.M. Melt processability, characterization, and antibacterial activity of compression-molded green composite sheets made of poly(3-hydroxybutyrate-co-3-hydroxyvalerate) reinforced with coconut fibers impregnated with oregano essential oil. Food Packag. Shelf Life 2018, 17, 39–49. [CrossRef] \\n37. Kundrat, V.; Matouskova, P.; Marova, I. Facile Preparation of Porous Microfiber from Poly-3-(R)-Hydroxybutyrate and Its Application. Materials 2019, 13, 86. [CrossRef] [PubMed] \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,65,302,77]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2020, 11, 24 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[825,65,872,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"12 of 14 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[124,105,873,826]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,65,302,77]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2020, 11, 24 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[825,65,872,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"13 of 14 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[126,104,873,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"38. Figueroa-Lopez, K.J.; Vicente, A.A.; Reis, M.A.; Torres-Giner, S.; Lagaron, J.M. Antimicrobial and antioxidant performance of various essential oils and natural extracts and their incorporation into biowaste derived poly (3-hydroxybutyrate-co-3-hydroxyvalerate) layers made from electrospun ultrathin fibers. Nanomaterials 2019, 9, 144. [CrossRef] \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[124,166,873,211]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"39. Melendez-Rodriguez, B.; Figueroa-Lopez, K.J.; Bernardos, A.; Martínez-Máñez, R.; Cabedo, L.; Torres-Giner, S.; M Lagaron, J. Electrospun antimicrobial films of poly(3-hydroxybutyrate-co-3-hydroxyvalerate) containing eugenol essential oil encapsulated in mesoporous silica nanoparticles. Nanomaterials 2019, 9, 227. [CrossRef] $©$ 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[277,223,873,268]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,65,302,77]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Funct. Biomater. 2020, 11, 24 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[825,66,872,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"14 of 14 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[126,224,268,262]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.3390/jfb11020024"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Materials Science and Engineering/Medicine and Health Sciences"},"sha256":{"kind":"string","value":"72cebe97131d67ba3c9d06302aef7dfc851a63067a6acaf333d42be350f4814b"},"title":{"kind":"string","value":"Antimicrobial Materials with Lime Oil and a Poly(3-hydroxyalkanoate) Produced via Valorisation of Sugar Cane Molasses"}}},{"rowIdx":44,"cells":{"abstract":{"kind":"string","value":"The sluggish kinetics of sulfur conversions have long been hindering the implementation of fast and efficient sulfur electrochemistry in lithium-sulfur (Li-S) batteries. In this regard, herein the unique chromium boride (CrB) is developed via a well-confined mild-temperature thermal reaction to serve as an advanced sulfur electrocatalyst. Its interstitial-alloy nature features excellent conductivity, while the nano-lamination architecture affords abundant active sites for host-guest interactions. More importantly, the CrB nanocatalyst demonstrates a dual sulphophilicity with simultaneous Cr─S and B─S bondage for establishing strong interactions with the intermediate polysulfides. As a result, significant stabilization and promotion of sulfur redox behavior can be achieved, enabling an excellent Li-S cell cyclability with a minimum capacity fading rate of 0.0176% per cycle over 2000 cycles and a favorable rate capability up to 7 C. Additionally, a high areal capacity of 5.2 mAh cm-2 , and decent cycling and rate performances are still attainable under high sulfur loading and low electrolyte dosage. This work offers a facile approach and instructive insights into metal boride sulfur electrocatalyst, holding a good promise for pursuing high-efficiency sulfur electrochemistry and high-performance Li-S batteries."},"author":{"kind":"string","value":"Hongyang Li|Guxian Chen|Kailong Zhang|Liangbiao Wang|Gaoran Li"},"content_list":{"kind":"list like","value":[{"bbox":"[85,93,805,180]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Dually Sulphophilic Chromium Boride Nanocatalyst Boosting Sulfur Conversion Kinetics Toward High-Performance Lithium-Sulfur Batteries ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[78,199,811,221]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hongyang Li, Guxian Chen, Kailong Zhang, Liangbiao Wang,\\* and Gaoran Li\\* ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[82,263,610,561]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The sluggish kinetics of sulfur conversions have long been hindering the implementation of fast and efficient sulfur electrochemistry in lithium-sulfur (Li-S) batteries. In this regard, herein the unique chromium boride (CrB) is developed via a well-confined mild-temperature thermal reaction to serve as an advanced sulfur electrocatalyst. Its interstitial-alloy nature features excellent conductivity, while the nano-lamination architecture affords abundant active sites for host-guest interactions. More importantly, the CrB nanocatalyst demonstrates a dual sulphophilicity with simultaneous Cr-S and B-S bondage for establishing strong interactions with the intermediate polysulfides. As a result, significant stabilization and promotion of sulfur redox behavior can be achieved, enabling an excellent Li-S cell cyclability with a minimum capacity fading rate of $0 . 0 1 7 6 \\%$ per cycle over 2000 cycles and a favorable rate capability up to 7 C. Additionally, a high areal capacity of $5 . 2 ~ \\mathsf { m A h } ~ \\mathsf { c m } ^ { - 2 }$ , and decent cycling and rate performances are still attainable under high sulfur loading and low electrolyte dosage. This work offers a facile approach and instructive insights into metal boride sulfur electrocatalyst, holding a good promise for pursuing high-eficiency sulfur electrochemistry and high-performance Li-S batteries. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[645,251,763,267]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[645,277,919,626]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The global target of carbon neutralization necessitates the active development and deployment of green energies such as solar, wind,and hydro.[1] Electrochemical energy storage hereto affords an attractive solution to the efficient and convenient utilization of these sustainable but intermittent energies,and has attracted tremendous research enthusiasm during the past few decades.[2] The currently mainstream lithium-ion battery (LIB) technology plays a critical role in electric vehicles and grid energy storage due to its high energy，durability,and technical maturity. However, energy density and affordability bottlenecks increasingly challenge their future implementation in these fields.3] In this context,lithium-sulfur (Li-S) batteries emerge as a promising nextgeneration energy storage systemattributed to their theoretically high energy density,almost five times that of the conventional LIB (2600 vs $5 0 0 \\ \\mathrm { W h \\ k g ^ { - 1 } }$ ),as well as the lowcost that nearly half-cut that of LIBs benefiting from the abundant reserve and wide geometrical distribution of sulfur.[4] ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,627,919,794]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Despite these intriguing merits,Li-S batteries struggle with several critical and characteristic challenges that obstruct their entrance into the practical markets.[5] Specifically, the insulating nature of sulfur and sulfides,the multi-electron transfer reaction pathway,and the liquid-solid phase-conversion mechanism collectively determine the intrinsic sluggish kinetics of sulfur reactions,rendering the difficult realization of fast and efficient sulfur electrochemistry.Meanwhile,the high solubility and mobility of the intermediate lithium polysulfides (LiPS) trigger the notorious shuttle effect accompanied by persistent sulfur loss,coulombic inefficiency,and anode passivation,resulting in fast capacity fading and battery failure.[6] ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,795,919,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Given these intractable problems,catalytic agents are desired to stabilize and promote sulfur conversions.In the past few years,researchers worldwide have devoted significant efforts to developing advanced sulfur electrocatalysts for alleviating the kinetic limits and inhibiting shutle behaviors.[7] Among them, chemically polar inorganics such as metal compounds have attracted particular attention due to their strong interactions with LiPS,which are expected to chemically immobilize sulfur species and facilitate their electrochemical conversions.8] Liang et al. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[82,604,485,817]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"H. Li, G. Chen, G. Li MIIT Key Laboratory of Advanced Display Materials and Devices School of Materials Science and Engineering Nanjing University of Science and Technology Nanjing,Jiangsu 210094,P.R. China E-mail: gaoranli@njust.edu.cn K. Zhang KeyLaboratory for Palygorskite Science and Applied Technology ofJiangsu National& Local Joint Engineering Research Center forMineral Salt Deep Utilization SchoolofChemical Engineering Huaiyin InstituteofTechnology Huaian,Jiangsu 223003,P.R.China L.Wang SchoolofChemistry and Chemical Engineering Jiangsu University ofTechnology Changzhou,Jiangsu 2130o1,P.R.China E-mail: Ibwang@jsut.edu.cn ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[84,951,216,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Adv. Sci. 2023,10,2303830 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[84,34,297,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RESEARCH ARTICLE ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[443,950,919,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2303830 (1 of 10) $\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[746,61,919,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"www.advancedscience.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[84,855,487,901]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution License,which permits use,distribution and reproduction in any medium,provided the original work is properly cited. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[84,828,489,852]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The ORClD identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/advs.202303830 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[786,0,998,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates ADVANCED SCIENCE OpenAccess ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[84,909,275,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DOI: 10.1002/advs.202303830 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[78,94,485,276]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"presented unique $\\mathrm { V O } _ { \\mathrm { x } }$ hollow spheres,as representative metal oxide,for chemically adsorbing LiPS following a“Goldilocks\" principle to resist the shutling behavior[9] Cui's group systematically studied the catalytic oxidation of $\\mathrm { L i } _ { 2 } \\mathrm { S }$ on different polar sulfide surfaces, demonstrating that $\\mathrm { V S } _ { 2 }$ ， $\\mathrm { T i } \\mathsf { S } _ { 2 }$ ,and ${ \\mathrm { C o } } { \\mathrm { S } } _ { 2 }$ deliver relatively superior catalytic activity due to their strong interaction with LiPS and the great facilitation of Li-ion transport.10l Apart from that,metal-organic frameworks (MOFs) are also potential host matrixes for improving sulfur electrochemistry. Huang's group recently reported an ultralight electroconductive Ni-TABQ, which efficiently promotes the multistep sulfur conversions benefiting from the dual catalytic centers within the framework,i.e., Ni-N4 and quinone.[11] ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,276,485,753]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Despite the encouraging progress, common metal compounds encounter some substantial shortcomings,represented by their poor electrical conductivity and limited access to the active sites, struggling to boost battery performance further. Given this, metal borides (MBs) recently become a rising star in this field due to their intrinsically high conductivity and unique chemical structures.[12] Metal-rich MB with alloy lattice structure inherits good metallicity, endowing high electrical conductivity up to $1 0 ^ { 7 } \\mathrm { ~ S ~ m ^ { - 1 } }$ .Additionally,MBs deliver a controllable transition from superconductor to semiconductor and further into insulator along with the increase of metal-to-boron ratio in stoichiometry,enabling tailorable chemical and electronic structures that afford attractive opportunities to meet specific demands in extensive areas.[13] Given the challenges in the Li-S system, MBs have shown considerable promise in regulating sulfur electrochemistry. Qian's group pioneered a layer-structure and highly conductive titanium diboride $\\left( \\mathrm { T i B } _ { 2 } \\right)$ as the polar sulfur host ma-terial in Li-S batteries.[14] A surface passivation mechanism was proposed for enhancing the polysulfide anchoring effect of $\\mathrm { T i B } _ { 2 }$ ， thus contributing to a strong sulfur immobilization with stable cell cycling.Li et al.compared $\\mathrm { T i B } _ { 2 }$ with other Ti-based compounds, i.e., $\\mathrm { T i O } _ { 2 }$ and TiC,and revealed the highest LiPS adsorption strength by boride that mostly favors the catalytic sulfur conversions.[15] Recently, MoB was also explored as an electrocatalyst for sulfur electrochemistry.16] Combining high conductivity,rich catalytic sites,and good hydrophilicity,the MoB catalyst enables a capacity of $1 2 5 3 \\mathrm { \\ m A h \\ g ^ { - 1 } }$ ,a long lifespan of over 1000 cycles,and reliable sulfur redox under high sulfur loading and limited electrolyte.These advances have demonstrated the great potential of MBs for catalyzing sulfur conversions and improving Li-S battery performance.However, this family of electrocatalysts is still in its infancy,with their structural potentialities to be further exploited,nanostructure to be rationally designed,and underlying functioning mechanisms to be deeply probed into. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,754,485,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this contribution,chromium boride (CrB) nanocatalyst was developed for the first time to power superior Li-S battery chemistry. The $\\mathrm { C r B }$ catalyst with unique laminated nanostructure was prepared via a well-confined,mild-temperature,and solid-state thermal method.Benefiting from its 2D interstitial-alloy latice structure,CrB inherits good metallicity with a high conductivity of $3 5 7 \\mathrm { S } \\mathrm { m } ^ { - 1 }$ .More importantly, the unique dual-sulphophilic character of $\\mathrm { C r B }$ imposes simultaneous p-d and p-p hybridizations with LiPS,which is distinct from the Lewis acid-base mode in conventional catalyst scenarios,and enables strong sulfur immobilization against the shuttle effect and lowered energy barriers for multistep sulfur conversions.As a result, fast,efficient, and durable sulfur redox electrochemistry was realized,instantiated by the excellent cyclability over 20oo cycles,superb rate capability up to 7 C,and decent energy output even under the highloading (sulfur loading $: = 5 . 0 \\mathrm { m g } \\mathrm { c m } ^ { - 2 }$ ),and lean-electrolyte (E/S ratio $= 5 . 5 \\mathrm { m L g ^ { - 1 } }$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,94,914,164]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,181,714,198]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. Results and Discussion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[507,207,914,332]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The $\\mathrm { C r B }$ nanocatalyst was prepared via a well-confined,mildtemperature，solid-state method. $\\mathrm { C r } { \\mathrm { O } } _ { 3 }$ was utilized as the chromium source,while $\\mathrm { N a B H } _ { 4 }$ was simultaneously the boron source and reductive agent.The as-builthigh-pressure and highly reductive environment enables the mild-temperature boronization to yield the well-dispersed nanometric $\\mathrm { C r B }$ .It should be noted that the one-pot synthesis with the simple setup and mild conditions endows this method with good scalability for potential practical implementation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,326,914,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The morphology and microstructure of the as-prepared $\\mathrm { C r B }$ were first investigated by scanning electron microscopy (SEM). It is observed that the $\\mathrm { C r B }$ delivers a nanometric lamination structure with a lateral size of ${ \\approx } 2 0 0 { \\mathrm { - } } 5 0 0 \\ \\mathrm { n m }$ (Figure 1A,B). The welldefined edges and flat surfaces can be noticed, signifying a layered lattice structure. Transmission electron microscope (TEM) characterization was carried out for a more profound structural study as shown in Figure 1C.As expected,stacked layers can be noted with some exfoliated nanosheets scattering around, likely due to their peeled-off upon the ultrasonic process, which strongly indicates the 2D lattice structure. The corresponding selected area electron diffraction (SAED) pattern is depicted in Figure 1D.The as-developed CrB exhibits a somewhat singlecrystalline character, showing bright spots in an ordered arangement. The marked spots can be assigned to the (111),(110), and (130) crystal faces of the orthorhombic CrB,respectively. Meanwhile,the high-resolution TEM (HRTEM) and inverse fast Fourier transform (IFFT) reveal an explicit interplanar space of $0 . 2 7 7 \\mathrm { n m }$ assigned to the (110) facet of CrB (Figure 1E; Figure S1, Supporting Information),while the high-angle annular dark field (HAADF) scanning transmission electron microscopy (STEM) image and energy dispersive X-ray spectroscopy (EDS) mapping in Figure 1F confirm the uniform distribution of Cr and B in the layered product. In addition, the crystallographic structure of the as-developed CrB was further confirmed by X-Ray Diffrac-tion (XRD, Figure 1G). The resulting pattern shows sharp peaks at 32.3,38.3,45.0,and 46.2 degrees corresponding to the (110), (021),(111),and (130) facets,respectively,of the orthorhombic CrB in Cmcm space group (PDF# 32-0277),which is well con-sistent with the SAED and HRTEM results. Therefore,it is rea-sonable to conclude that well-crystallized $\\mathrm { C r B }$ was successfully obtained in a typical 2D lattice structure,where the slabs of face-sharing $\\mathrm { C r } _ { 6 } \\mathrm { B }$ trigonal prisms extend along the (010) plane and stack in the b-direction,and the boron atoms are covalently bonded with each other to form zigzag chains parallel to the (001) direction.What should be further noted is that the interstitially alloyed structure with well-inherited metal bonds in the matrix endows $\\mathrm { C r B }$ with good metallicity and high electrical conductiv-ity that significantly outperforms its oxide counterpart (357 vs $4 . 5 \\times 1 0 ^ { - 3 } \\ \\mathrm { S \\ m ^ { - 1 } }$ ,Figure S2, Supporting Information). Beyond that, the coexistence of ${ \\mathrm { C r } } { - } { \\mathrm { C r } }$ (metallic)，B-B (covalent)，and $\\mathrm { C r - B }$ (ionic) bonds could trigger unique chemical features for potential host-guest interactions with LiPS and thus regulates the sulfur electrochemical behaviors. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,26,282,74]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCENEWS www.advancedsciencenews.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[78,951,211,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Adv.Sci. 2023,10,2303830 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[440,950,914,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2303830 (2 of 10) $\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[783,20,912,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCE OpenAccess ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[742,61,914,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"www.advancedscience.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[206,94,806,514]","code_body":null,"code_caption":null,"image_caption":"[\"Figure1.A,B)Saes,C)age,D)paernE)Hge,F)HDD-STgeamentalapg,G) projected crystalline structure (inset),high-resolution XPS H) $C r 2 \\\\mathsf { p }$ and I) B 1s spectra of CrB. \"]","image_footnote":"[]","img_path":"images/0034/2d182cb2010eac968873925ce0b30500d876a7c1e40f33981fa0517bd71a9f38.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[84,571,489,598]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,599,489,892]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Given this,the surface chemistry of the obtained $\\mathrm { C r B }$ was studied by X-ray photoelectron spectroscopy (XPS).All the spectra were calibrated according to the $\\mathrm { C - C }$ peak position of $2 8 4 . 8 \\ \\mathrm { e V }$ (Figure S3,Supporting Information).Figure 1H depicts the Cr $2 \\mathrm { p }$ spectrum of $\\mathrm { C r B }$ ，which witnesses three subpeaks at 574.1, 575.6,and $5 7 7 . 1 \\ \\mathrm { e V }$ ascribed to the $\\mathrm { C r } ^ { 0 }$ ， $\\mathrm { C r } ^ { 3 + }$ ,and Cr hydroxide species,respectively[17] The strong $\\mathrm { C r } ^ { 0 }$ signal confirms the metallic character of $\\mathrm { C r B }$ ,while the rest species suggest the ionic feature as well as the potential partial oxidization/hydroxidization at the surface.Meanwhile,the B 1s spectrum exhibits characteristic peaks at 187.5 and $1 8 8 . 5 \\ \\mathrm { e V }$ assigned to the $\\mathrm { B - C r }$ and $\\mathrm { B - B }$ ,respectively (Figure 1I)[18] manifesting the coexistence of local ionic and covalent features in the $\\mathrm { C r B }$ lattice.Oxidized B species such as ${ \\bf B } _ { 2 } \\mathrm { O } _ { 3 }$ also exist,which is commonly observed in the literature.[19] These results confirm the successful acquisition of $\\mathrm { C r B }$ based on mild-temperature thermal synthesis.The alloy structure well inherits the metallic character that would greatly benefit the electron conduction,while the as-established local chemical polarity is expected to impose favorable chemical affinity to LiPS for achieving reliable sulfur immobilization and catalyzation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,893,489,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In view of these potential benefits,the Li-S battery performance based on the CrB nanocatalyst was evaluated in coin cells (see details in Experimental Section).The $\\mathrm { C r B }$ dosage in the electrodes was first optimized to $1 0 \\mathrm { w t . } \\%$ as shown in Figure S4 (Supporting Information).Skimpy addition seems hardly fulfill the catalytic advantages of $\\mathrm { C r B }$ ,while excessive $\\mathrm { C r B }$ likely densifies the electrode with potential structural nonuniformity and thus deteriorates the electrochemical performance.Figure 2A shows the CV curves of the cell based on the $\\mathrm { C r B }$ catalyst (denoted as w CrB),while the one based on super $\\mathrm { P }$ (SP) was employed as a contrast sample (denoted as w/o CrB).Distinctly, the $\\mathrm { C r B }$ cell exhibits a much higher current response with a much sharper peak shape and smaller redox overpotential gap compared with those of the SP cell (61 vs $9 9 ~ \\mathrm { m V }$ for Gap I,248 vs $2 9 3 ~ \\mathrm { m V }$ for Gap II),suggesting the considerably enhanced conversion kinetics by the CrB catalyst. This advance can also be confirmed by the Tafel plots as shown in Figure S5 (Supporting Information), the redox peaks referring to the $\\operatorname { L i } _ { 2 } \\mathrm { S }$ conversion,which forms the Gap II,are specifically studied as it is generally considered the most kinetic-limiting process in sulfur electrochemistry. During both the cathodic and anodic scanning,the CrB cell shows a much smaller Tafel slope $( - 3 3 . 4 $ and $6 5 . 1 \\mathrm { m V } \\mathrm { d e c ^ { - 1 } }$ ) compared with those without the CrB catalyst $( - 6 7 . 3$ and $7 1 . 3 ~ \\mathrm { m V ~ d e c ^ { - 1 } }$ ）， indicating the lowered energy barrier and facilitated $\\operatorname { L i } _ { 2 } \\mathrm { S }$ redox by the CrB catalyst. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,571,919,892]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,893,919,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Correspondingly, the galvanostatic charge-discharge profiles (Figure 2B)of the CrB cell demonstrate a much higher capacity (1260.9 vs $1 0 0 2 . 7 \\mathrm { \\ m A h \\ g ^ { - 1 } } .$ ）and smaller charge-discharge potential hysteresis (denoted as $\\Delta \\mathrm { E }$ ）compared with its SP counterpart (121 vs $1 6 5 \\ \\mathrm { m V } ,$ Figure S6, Supporting Information). Apart from that, the capacity ratio of the low-potential plateau to the high-potential plateau in the discharge curve $( \\mathrm { Q _ { L } } / \\mathrm { Q _ { H } } )$ is another critical indicator of the sulfur redox efficiency.The much higher $\\mathrm { Q _ { L } } / \\mathrm { Q _ { H } }$ (1.975) of the $\\mathrm { C r B }$ cell over its w/o CrB counterpart (1.594) illustrates that the sulfur species dissolved during the high-plateau discharging can be reutilized and precipitated back on the cathode surface upon the low-plateau stage.These results consistently and collectively demonstrate the much faster and more efficient sulfur redox behaviors on the $\\mathrm { C r B }$ surface. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[82,26,287,74]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCENEWS www.advancedsciencenews.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[84,951,216,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Adv. Sci. 2023,10,2303830 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[443,950,917,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2303830 (3 of 10) $\\circledcirc$ 2023 The Authors. Advanced Science published by Wiley-VCH GmbH ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[788,20,919,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCE Open Access ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[746,61,919,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"www.advancedscience.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[189,94,794,457]","code_body":null,"code_caption":null,"image_caption":"[\"Figurte ofLiSestal; H) long-term cycling performance of CrB cell at $_ { 2 \\\\subset }$ \"]","image_footnote":"[]","img_path":"images/0975/de1c08473c1cc1b6c6b24cf3543259b71db0e90115340768493e9110e3a88fa4.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[78,529,485,695]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,698,485,864]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The kinetic improvement can also be perceived from the EIS analysis.As shown in Figure 2C,the Nyquist plots of both cells consist of a suppressed semicircle at the high-medium frequency range,which is assigned to the charge-transfer resistance $\\mathrm { ( R _ { c t } ) }$ ,as well as an oblique line at the low-frequency range referring to the Warburg resistance.The inset depicts the corresponding equivalent circuit. It can be noticed that the $\\mathrm { C r B }$ cell delivers a much lower $\\mathrm { R _ { c t } }$ compared with the w/o CrB cell, while the much higher slope of the oblique line suggests the faster $\\mathrm { L i ^ { + } }$ diffusion on the $\\mathrm { C r B }$ surface.These results confirm the more facile charge supplies in the CrB-based configuration and support its faster kinetic behaviors. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,865,485,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The galvanostatic cycling behaviors of different cells at $0 . 2 ~ \\mathrm { { C } }$ were compared in Figure 2D.The CrB cell retains a capacity of $9 5 4 . 7 \\mathrm { ~ m A h ~ g ^ { - 1 } }$ after 50 cycles, corresponding to a capacity retention of $7 5 . 7 \\%$ .By contrast, the SP cell experiences much faster capacity decay, showing a capacity of $6 4 4 . 4 \\ \\mathrm { m A h \\ g ^ { - 1 } }$ at the $5 0 ^ { \\mathrm { t h } }$ cycle with alower retention of $6 4 . 2 \\%$ .Apart from that,the rate capability of different cells was also compared by cycling at varied current rates,as shown in Figure 2E.The $\\mathrm { C r B }$ electrode retains a favorably high capacity of $7 7 5 \\mathrm { m A h g ^ { - 1 } }$ even under a high rate up to $7 \\mathrm { C }$ ,which recovers to $1 1 1 2 . 4 \\mathrm { ~ m A h ~ g ^ { - 1 } }$ as the applied current returns to $0 . 2 \\ : \\mathrm { C }$ .In comparison, the control cell undergoes fast capacity decaying along with the increase of current rate and retains only a capacity of $2 2 8 . 4 \\ : \\mathrm { m A h \\ : g ^ { - 1 } }$ at7C.Sucha distinctive comparison demonstrates the excellent reaction kinetic realized by the $\\mathrm { C r B }$ electrocatalyst, which significantly promotes the sulfur conversions toward a superior rate capability. This improvement can be more vividly evidenced by the evolution of chargedischarge curves upon the multi-rate cycling. Figure 2F shows the voltage profiles of the $\\mathrm { C r B }$ cell at different current rates.It is noticed that the potential gap between the charge and discharge plateaus is gradually expanded along the rate increase from 0.2 to 7 C,yet the two-platform discharge profile can be maintained. Whereas for the control cell without CrB,the severe polarization deforms the charge-discharge profiles significantly, failing to maintain the two-plateau profile at 7 C (Figure S7, Supporting Information).The potential gap evolutions along the current rate increase are summarized in Figure 2G.It reveals the persistently lower overpotential of the CrB cellat the whole rate range compared with its w/o CrB counterpart,strongly confirming its much superior kinetic properties. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,530,914,892]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,893,915,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Additionally, the cycling behaviors of cells based on $\\mathrm { C r B }$ and $\\mathrm { C r } _ { 2 } \\mathrm { O } _ { 3 }$ were compared (Figure S8, Supporting Information). The $\\mathrm { C r } _ { 2 } \\mathrm { O } _ { 3 }$ cell shows higher capacity，stability，and rate capability than the w/o $\\mathrm { C r B }$ cell but is inferior to the $\\mathrm { C r B }$ cell. The electrochemical improvement by $\\mathrm { C r } _ { 2 } \\mathrm { O } _ { 3 }$ could be attributed to the decent sulfur confinement as illustrated by the static adsorption test (Figure S9, Supporting Information),whereas the lack of conductivity and catalytic activity limit the sulfur redox effciency[20] Such a comparison further highlights the balanced adsorption and catalyzation by the $\\mathrm { C r B }$ design,which enables fast and reversible sulfur electrochemistry. Given these advantages,a longterm cycling evaluation was performed for the $\\mathrm { C r B }$ cell. As depicted in Figure 2H,a stable cycling behavior can be achieved with a minimum capacity fading rate of $0 . 0 1 7 6 \\%$ per cycle over a prolonged 2000 cycles at $2 \\mathrm { C }$ Meanwhile,a high coulombic effciency close to unity can also be sustained, indicating excellent sulfur redox reversibility. These electrochemical performances are highly competitive among those reported in recent literature (Table S1, Supporting Information),further illustrating the superiorities of the $\\mathrm { C r B }$ catalyst design. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,26,282,74]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCENEWS www.advancedsciencenews.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[78,951,211,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Adv.Sci. 2023,10,2303830 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[440,950,914,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2303830 (4 of 10) $\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[783,20,914,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCE Open Access ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[742,61,914,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"www.advancedscience.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[193,94,805,517]","code_body":null,"code_caption":null,"image_caption":"[\"Figure3A)UVisectadticalagsfluldsobdreles;Botefsly) surface; C) geometrical configuration,D) bond lengths, XPS $S 2 \\\\mathsf { p }$ spectra of ${ \\\\dot { \\\\mathsf { L i } } } _ { 2 } { \\\\mathsf { S } } _ { 4 }$ E) before and F)after adsorption on CrB; G) energy profiles of sulfur conversion with or without CrB catalyst; H) DOS pattern of CrB; I) DOS pattern of sulfur in ${ \\\\dot { \\\\mathsf { L i } } } _ { 2 } { \\\\mathsf { S } } _ { 4 }$ before and after adsorption. \"]","image_footnote":"[]","img_path":"images/0532/9e608d47a6b599da6684ee311bf747f74fe83d1bdb8fe93695b35eebf1354f8e.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[84,584,489,836]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,837,489,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To understand these electrochemical improvements,the chemical interaction between the $\\mathrm { C r B }$ catalyst and LiPS was studied first by static adsorption test.As presented in the inset of Figure 3A,the blank $\\mathrm { L i } _ { 2 } \\mathrm { S } _ { 4 } /$ /tetrahydrofuran (THF) solution shows a typical brownish color. When immersed with the same amount of $\\mathrm { S P }$ and $\\mathrm { C r B }$ and let stand for $^ { 1 2 \\mathrm { ~ h ~ } }$ ，the solution with SP turns slightly lighter, while the one with $\\mathrm { C r B }$ undergoes a strong decoloration,resulting in an almost transparent and colorless supernatant. Such a contrastive comparison strongly confirms the much superior LiPS adsorbability of $\\mathrm { C r B }$ ,which can be further verified by the lowest UV-vis absorbance of its corresponding supernatant (Figure 3A).Further probe into the interactive mechanism was carried out by DFT calculation.The $\\mathrm { C r B }$ (111) facet was selected as a typical platform for the DFT study as it is the strongest exposed facet in the XRD result.A variety of sulfur species were simulated anchoring on the $\\mathrm { C r B }$ (111) surface,contributing to adsorption energies of $- 2 . 7 1$ ,-5.37,-5.19, $- 5 . 0 2$ ， $- 5 . 1 6$ ，and $- 5 . 1 9 \\ \\mathrm { e V } ,$ respectively (Figure 3B). The corresponding geometrically stable configurations are presented in Figure S10 (Supporting Information). These values are among the relatively high level in literature based on conventional carbon or metal compound substrates,21] confirming the strong chemical interactions between $\\mathrm { C r B }$ and LiPS.Figure 3C shows specifically the geometry of $\\mathrm { L i } _ { 2 } \\mathrm { S } _ { 4 }$ adsorption on $\\mathrm { C r B }$ .Interestingly,both chromium and boron are available and preferential to bond with terminal sulfur, forming a $\\mathrm { C r - } { \\mathsf { S } } { \\mathrm { - } } { \\mathsf { B } }$ bridging structure.This distinguishes from what is commonly observed in conventional metal compounds scenarios,such as oxides and sulfide,where the Lewis acid-base principle governs the interactions with one end (e.g.,metal sites) sulphophilic and the other (e.g.,oxygen or sulfur sites) lithiophilic.[22] The bond lengths of ${ \\mathrm { C r } } { - } S$ are 2.333 and $2 . 4 0 9 \\mathrm { ~ \\AA ~ }$ ,while the B-S lengths are 1.819 and $1 . 9 5 0 \\mathring { \\mathrm { A } }$ , suggesting a strong bonding strength. Such a dual sulfur-bonding mode contributes to favorably high adsorption energy of $- 5 . 0 2 \\mathrm { e V } ,$ while the lessened bondage with lithium may facilitate the ion transfer, as demonstrated above.Beyond that,a considerable distortion of polysulfide geometry can be noticed upon the interaction.As shown in Figure 3D and Figure S11 (Supporting Information),the bond lengths of S1-S2 and S2-S3 decrease significantly,accompanied by the drastic increase of the S3-S4 bond,referring to the compression of the $\\mathrm { L i } _ { 2 } \\mathrm { S } _ { 4 }$ molecule at one side and the stretch at the other. Such a distortion is also supported by the consistent variations of the Li-S bonds (Figure S12,Supporting Information).As a result, the lengthened S-S bond tends to be more easily cleaved,contributing to the facilitated conversion kinetics. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,585,919,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[82,26,287,74]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCENEWS www.advancedsciencenews.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[84,951,216,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Adv. Sci. 2023,10,2303830 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[443,950,919,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2303830 (5 of 10) $\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[788,20,919,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCE Open Access ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[746,61,919,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"www.advancedscience.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[78,94,485,318]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,319,485,765]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Furthermore,the interactive behaviors between $\\mathrm { C r B }$ and polysulfide were analyzed by XPS.Figure 3E,F shows the S 2p XPS spectra of $\\mathrm { L i } _ { 2 } \\mathsf { S } _ { 4 }$ before and after the adsorption.Typically,the bare $\\mathrm { L i } _ { 2 } \\mathrm { S } _ { 4 }$ exhibits two pairs of peaks assigned to the terminal $ { ( \\mathrm { S _ { T } } ^ { - 1 } }$ ， $1 6 1 . 6 \\ \\mathrm { e V } )$ and bridging sulfur $( \\mathrm { S _ { B } } ^ { 0 } , 1 6 3 . 5 \\ : \\mathrm { e V } )$ , respectively.[23] A small peak can be detected at a relatively higher binding energy (BE),which could be ascribed to the slight oxidization upon the sampling.By contrast, the adsorption of $\\mathrm { L i } _ { 2 } \\mathrm { S } _ { 4 }$ on the $\\mathrm { C r B }$ surface witnesses the peak shift to lower BE,particularly for the ${ \\mathrm { S } _ { \\mathrm { B } } } ^ { 0 }$ species, suggesting the electron-accepting behavior of sulfur that increases the electron density and lowers the valence state.This well accords with the lengthened S-S bond as well as the formation of ${ \\mathrm { C r } } { - } S$ bonds in the DFT result. Accordingly, the $\\mathrm { C r ~ } 2 \\mathrm { p }$ and B 1s XPS spectra were also compared. As shown in Figure S13A,B (Supporting Information), the Cr 2p XPS signal becomes drastically weaker after the adsorption, likely due to the strong polysulfide adsorption and coverage thereon.More importantly, the relative content of $\\mathrm { C r }$ (III) species significantly increases after the adsorption,confirming the establishment of the ${ \\mathrm { C r } } { - } S$ bonding that oxidizes $\\mathrm { C r }$ and reduces S.The decrease of Cr hydroxide is likely due to its consumption by LiPS to form the sulfate-related species,which is consistent with the S 2p spectrum.Meanwhile,a considerable peak shift to lower BE can be noticed in B 1s spectra (Figure S13C,D,Supporting Information),reflecting the electron transfer from sulfur to the electrondeficient boron upon the $\\mathtt { B - S }$ bonding. This behavior seems mainly occur at the terminal sulfur with higher electron density and partially offset the electron gained from Cr, thus resulting in the limited peak shift for ${ \\mathrm { S _ { T } } } ^ { - 1 }$ . Overall,the XPS results are highly consistent with the DFT calculation,collectively confirming the strong and unique chemical interactions between $\\mathrm { C r B }$ and polysulfides. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,767,485,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Given these interesting interactions,the kinetic behaviors of sulfur conversions on the $\\mathrm { C r B }$ surface were studied as depicted in Figure 3G.The energy profile for bare sulfur conversion was also calculated as a control. It can be recognized that the ratedetermining step for sulfur conversions is the redox of $\\operatorname { L i } _ { 2 } \\mathrm { S }$ in both cases,which is consistent with that commonly and experimentally observed.Notably, the $\\mathrm { C r B }$ catalyst effectively lowers the energy barrier for $\\operatorname { L i } _ { 2 } \\mathrm { S }$ redox from 0.936 to $0 . 8 9 9 \\ \\mathrm { e V } .$ Meanwhile,a significantly lower energy barrier of $0 . 7 2 9 \\mathrm { e V }$ (vs 0.849 for the control) can be also achieved for another kinetic-limiting step from $\\mathrm { L i } _ { 2 } \\mathrm { S } _ { 4 }$ to $\\mathrm { L i } _ { 2 } \\mathrm { S } _ { 2 }$ . These results strongly illustrate the substantial promotion of sulfur conversion kinetics by the highly catalytic $\\mathrm { C r B }$ .The density of states (DOS)of CrB was further calculated to unveil the underlying mechanism (Figure 3H). Strong and broad DOS can be observed around the Femilevel, which is mainly contributed by the 3d electrons of Cr, indicating its excellent metallic character for rapid electron conduction.By contrast,the S pDOs of $\\mathrm { L i } _ { 2 } \\mathrm { S } _ { 4 }$ shows a discrete and cleaved pattern with several isolated bands along the energy axis and no distribution at the Fermi level (Figure 3I; Figure S14,Supporting Information),which well accords with its insulating nature.Interestingly,after the adsorption,the S pDOS converts into a continuous pattern with consideration distribution near the Femi level (Figure 3I).Such a drastic change in the S pDOS pattern strongly indicates that the adsorption on $\\mathrm { C r B }$ provides sulfur with significantly enhanced electron transfer for fast reaction kinetics.Apart from that,strong overlapping between S and Cr as well as between S and B pDOS patterns can be witnessed (Figure S15,Supporting Information),further testifying the dual-site interaction via strong S-Cr (p-d) and S-B (p-p)hybridizations that expedite the charge transfer for fast sulfur electrochemistry. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,94,914,373]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,377,914,751]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Inspired by the above computational results,the practical catalytic effect of $\\mathrm { C r B }$ was further investigated by symmetric cell characterizations.Figure 4A shows the CV profiles of polysulfide conversions at a scanning rate of $5 0 ~ \\mathrm { m V ~ s ^ { - 1 } }$ based on symmetric electrodes with or without the $\\mathrm { C r B }$ catalyst.As expected, the electrode shows undetectable redox peaks without LiPS as the active species.Notably, the $\\mathrm { C r B }$ catalyst enables a much higher current response with narrowed redox polarization compared with that without $\\mathrm { C r B }$ (0.18 vs 0.45 V),manifesting the promoted polysulfide conversion on the $\\mathrm { C r B }$ surface.In addition, CV curves at varied scanning rates were also recorded.The profle is well maintained upon the increase of scanning rate for the $\\mathrm { C r B }$ cell, while serious profile deformation can be observed for that without $\\mathrm { C r B }$ (Figure S16, Supporting Information).This can be more quantitively comprehended via the overpotential comparison as depicted in Figure 4B,where $\\mathrm { C r B }$ enables much lower overpotential as well as its slower growth along the increase of scanning rate. Such a distinct comparison further confirms the significant kinetic promotion by the highly catalytic $\\mathrm { C r B }$ ,which can also be supported by the much smaller internal resistance of the CrB-based symmetric cell (Figure 4C). Beyond that, the $\\mathrm { L i ^ { + } }$ diffusion property is also a critical factor governing kinetic behaviors.Based on the Randles-Sevcik equation, the symmetric cells' peak currents versus the scanning rate's square root were fitted, as shown in Figure 4D.It is noticed that the $\\mathrm { C r B }$ cell delivers a more significant slope $( - 3 . 9 9 1$ VS $- 2 . 0 6 6 ]$ ,indicating its higher $\\mathrm { L i ^ { + } }$ diffusion coefficient that facilitates sulfur conversions. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,754,914,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In addition to the polysulfide redox,the liquid-solid $\\operatorname { L i } _ { 2 } \\mathrm { S }$ precipitation, which is considered the most kinetic-limiting process, was specifically studied via the potentiostatic measurement.As shown in Figure 4E,F,the potentiostatic LiPS reduction initiates a current downslope followed by an upward peak,and gradually approaches the current limit after $^ { 8 \\mathrm { ~ h ~ } }$ .The downslope and level-off current (dark region) correspond to the reduction of residue $\\mathrm { L i } _ { 2 } \\mathrm S _ { 8 }$ and $\\mathrm { L i } _ { 2 } \\mathrm { S } _ { 6 } ,$ respectively, while the left-over upward peak part (light region) can be assigned to the $\\operatorname { L i } _ { 2 } \\mathrm { S }$ formation and deposition.[24] By comparison, the $\\mathrm { C r B }$ catalyst enables a much earlier peak position (2800 vs 7240 s) and sharper peak shape compared with its counterpart, suggesting a lower energy barrier for $\\operatorname { L i } _ { 2 } \\mathrm { S }$ nucleation and accumulation growth.As such, the CrB electrode obtains a higher precipitation capacity than that without $\\mathrm { C r B }$ (303.8 vs $2 5 7 . 1 \\mathrm { m A h } \\ \\mathrm { g } ^ { - 1 }$ ). Such an improvement can be also verified by the more uniform and smoother $\\operatorname { L i } _ { 2 } \\mathrm { S }$ precipitation on the CrB-loaded carbon paper compared with that without CrB (Figure S17, Supporting Information).Based on these specific studies,Galvanostatic Intermittent Titration Technique (GITT)and operando impedance tests further probed the overall sulfur kinetic behaviors.Figure S18 (Supporting Information) shows the GITT profles of cells with and without the CrB catalyst. The corresponding IR drop evolutions upon the chargedischarge process were compared in Figure 4G.The IR drop refers to the gap between working potential and equilibrium potential at certain reaction status,which reflects the dynamic polarization upon the cell charge and discharge process.It is worth noting that the CrB cell shows persistently lower IR drop along the whole discharge-charge process, strongly confirming the alleviated polarization and the enhanced reaction kinetics by the $\\mathrm { C r B }$ catalyst. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,26,282,74]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCENEWS www.advancedsciencenews.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[78,951,211,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Adv. Sci. 2023,10,2303830 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[440,950,914,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2303830 (6 of 10) $\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[783,20,912,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCE OpenAccess ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[742,61,914,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"www.advancedscience.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[194,95,805,462]","code_body":null,"code_caption":null,"image_caption":"[\"Figure4)CfpoeldserilititCrB; peak current versus square root of scanning rate; $\\\\mathsf { L i } _ { 2 } \\\\mathsf { S }$ precipitation profiles on electrode E) without and F) with CrB; G) IR drops in GITT profiles,H operando resistances and their first-derivative of Li-S cells with or without CrB. \"]","image_footnote":"[]","img_path":"images/0293/5c9e2714ca446885230bf39e9ee4cd9581e36ecb009656930a896777bc73e7ef.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[84,530,489,794]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,795,489,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Furthermore,the operando resistance profles were recorded for different cells as shown in Figure 4H.Upon the voltammetric scanning from 2.6 to $1 . 8 \\mathrm { V }$ at $0 . 2 \\mathrm { m V s ^ { - 1 } }$ ,both the cells begin with a fairly steady resistance and encounter a sudden drop at ${ \\approx } 2 . 3 \\mathrm { V } ,$ which results from the initiation of polysulfide dissolution that expose the conductive surfaces.After that,a steep resistance rise can be noted at ${ \\approx } 2 . 2 \\ \\mathrm { V } ,$ owing to the polysulfide accumulation that increases the viscosity and deteriorates the ion transportation.Further cathodic scanning witnesses the drastic decline of resistance at ${ \\approx } 2 . 1 \\ \\mathrm { V } ,$ where $\\operatorname { L i } _ { 2 } \\mathrm { S }$ starts to precipitate,accompanied by the recovery of electrolyte conductivity. Interestingly,at the end section of discharge, the $\\mathrm { C r B }$ cell shows a continuous decrease of resistance,while the control cell undergoes a re-rise of resistance,likely due to the electrode surface passivation by $\\operatorname { L i } _ { 2 } \\mathbf { S } .$ This contrast further suggests that the $\\mathrm { C r B }$ could regulate the $\\operatorname { L i } _ { 2 } \\mathrm { S }$ precipitation behavior that facilitates the reaction kinetics.Apart from that, the operando profile of the CrB cell exhibits persistently lower resistance upon the discharging process compared with the w/o CrB counterpart.The first derivation of the profile more intuitively distinguishes the hysteretic and steadier resistance fluctuation for the CrB-based configuration.Based on these results,it is reasonable to conclude that the unique CrB catalyst delivers a great capability of lowering the electrochemical resistance and promoting sulfur conversion,which contributes to excellent cycling and rate performance. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,530,919,751]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,754,919,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Based on these encouraging superiorities,Li-S cells based on the $\\mathrm { C r B }$ catalyst were further examined under high sulfur loading and limited electrolyte for potential practical application. Figure 5A shows the rate performance of the $\\mathrm { C r B }$ cell at a sulfur loading of $5 . 0 \\mathrm { m g } \\mathrm { c m } ^ { - 2 }$ and an electrolyte-to-sulfur (E/S) ratio of $5 . 5 \\mathrm { m L g ^ { - 1 } }$ .An areal capacity of $5 . 2 7 \\mathrm { m A h } \\mathrm { c m } ^ { - 2 }$ can be obtained at $0 . 1 \\mathrm { C }$ ,which retains at $4 . 5 3 \\mathrm { \\ m A h \\ c m ^ { - 2 } }$ even under a high rate of $1 \\mathrm { { C } }$ .What should also be noted is that the high-loading CrB cell can basically maintain the two-plateau discharging profile even under $1 . 0 ~ \\mathrm { { C } }$ rate despite certain deformation (Figure 5B). These results strongly demonstrate the excellent reaction effciency and kinetics realized by the $\\mathrm { C r B }$ catalyst even under the high-loading configuration.Moreover, the cycling performance was also evaluated.As shown in Figure 5C,a stable cycling behavior can be achieved by the $\\mathrm { C r B }$ cell with a highly reversible capacity of $3 . 9 8 \\mathrm { \\ m A h \\ c m ^ { - 2 } }$ after 50 cycles at $0 . 2 \\textrm { C }$ Meanwhile, high coulombic efficiency close to unity can be obtained.These favorable results should be attributed to the strong sulfur adsorption and effective catalyzation of $\\mathrm { C r B }$ ,which inhibits the shuttle effect and promote the conversion reaction toward fast and durable sulfur electrochemistry. On this basis,the Li-S pouch cell with $\\mathrm { C r B }$ electrode was further fabricated (Figure S19, Supporting Information),which is capable of successfully charging a smartphone,as well as powering electric clock and electric fan as depicted in Figure 5D and Figure S20 (Supporting Information), demonstrating an intriguing potential for practical application. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[82,26,285,74]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCENEWS www.advancedsciencenews.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[84,951,216,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Adv. Sci. 2023,10,2303830 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[443,950,919,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2303830 (7 of 10) $\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[788,20,919,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCE Open Access ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[746,61,919,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"www.advancedscience.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[191,95,801,488]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 5.A) Rate performance, B) voltage profiles at diferent curent rates,and C) cycling performance at $_ { 0 . 2 \\\\textsf { C } }$ of CrB cells under sulfur loading ol $5 ~ \\\\mathrm { { m g } } \\\\thinspace \\\\mathrm { { c m } } ^ { - 2 }$ and E/S ratio of $5 . 5 ~ \\\\mathsf { m L } ~ \\\\mathsf { g } ^ { - 1 }$ ; D) Optical image of CrB-based Li-S pouch cell charging a cellphone. \"]","image_footnote":"[]","img_path":"images/0293/87e0d37c711af554cc2d47d445644a9816115d28ee6d0358ef070006841b5f82.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[78,543,485,739]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,757,188,772]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3. Conclusion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[78,782,485,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In summary, this work developed a unique CrB nanocatalyst for boosting sulfur electrochemistry.A well-confined,mild-temperature,solid-state thermal method was designed to prepare $\\mathrm { C r B }$ with lamination nanostructure.The highly conductive nature ensures a fast charge supply for electrochemical reactions, while more importantly, the dual-sulphophilic feature of $\\mathrm { C r B }$ establishes strong interactions with LiPS via simultaneous p-d and p-p hybridizations.Such a unique interaction imposes strong sulfur adsorption against the shuttle effect,and lowers energy barriers for multistep sulfur conversions,thus contributing to fast and efficient sulfur electrochemistry.As a result,Li-S cells based on $\\mathrm { C r B }$ catalyst realize excellent cyclability over 2ooo cycles, rate capability up to 7 C,and decent performance under high sulfur loading and limited electrolyte,demonstratinga good promise for the development of high-performance and practically viable Li-S batteries. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,543,914,625]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,667,697,682]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4. Experimental Section ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[507,690,914,800]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Preparation ofCrB:CrB was prepared via a confined solid thermal reaction.All the chemicals were used directly without further purification.Typically, $0 . 3 \\mathrm { g }$ chromium oxide $\\left( \\mathsf { C r O } _ { 3 } \\right)$ and $2 . 0 { \\ g }$ sodium borohydride $( N a B H _ { 4 } )$ were mixed and sealed in a $2 0 ~ \\mathsf { m L }$ stainless high-pressure reactor under an Aratmosphere,and then subjected to the thermal treatment at $5 0 0 ^ { \\circ } \\mathsf { C }$ for $\\mathsf { \\Delta } \\rceil 0 \\mathrm { ~ h ~ }$ .After cooling,the product was washed with diluted hydrochloric acid to remove the by-product and $\\mathsf { N a B H } _ { 4 }$ residue,and then washed repeatedly with deionized water and ethanol.The final CrB product was collected after vacuum drying for over $1 2 \\mathrm { ~ h ~ }$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,800,914,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Preparation ofSulfur Electrodes:Sulfur electrodes were prepared by the conventional casting method.Typically, the slurry was prepared by dispersing elemental sulfur, SP,CrB,and PVDF with a weight ratio of 5:3:i:1 in NMP solvent.The homogeneous slurry was then cast onto the carboncoated aluminum foil and vacuum dried at $6 0 ~ ^ { \\circ } \\mathsf { C }$ for over $^ { 1 2 \\mathrm { ~ h ~ } }$ .Thedried electrode was then cut into discs with a diameter of $1 2 \\mathsf { m m }$ for use.Sulfur loadings of 1.5 and $5 . 0 \\mathsf { m g } \\mathsf { c m } ^ { - 2 }$ were applied for regular and high-loading electrode evaluations with the electrode thickness of 33 and $9 6 ~ { \\mu \\mathrm { m } }$ respectively.Sulfur electrode containing elemental sulfur,SP,and PVDF in a weight ratio of 5:4:1 was also prepared for comparison. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,26,282,74]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCENEWS www.advancedsciencenews.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[78,951,211,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Adv.Sci. 2023,10,2303830 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[440,950,914,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2303830 (8 of 10) $\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[783,20,914,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCE Open Access ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[742,61,914,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"www.advancedscience.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[84,95,489,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Electrochemical Evaluation: The electrochemical performance was evaluated in coin-cell configuration. The cells were assembled using the prepared sulfur electrode as the cathode,Li foil as the anode, and Celgard 240O membrane as the separator in an Ar-filled glove box $( \\mathrm { O } _ { 2 } { < } 0 . 1$ ppm, $H _ { 2 } O < 0 . 1$ ppm).The electrolyte contains 1 M lithium bis(trifluoromethane)sulfonimide (LiTFSl) in the mixed solvent of 1,2- dimethoxyethane (DME) and 1,3-dioxolane (DOL) (1:l in volume ratio) with 2 wt. $\\%$ lithium nitrate $( \\mathsf { L N O } _ { 3 } )$ additive.The electrolyte-to-sulfur (E/S) ratio was controlled at 12 and $5 . 5 ~ \\mathsf { m L } ~ \\mathsf { g } ^ { - 1 }$ ,respectively，for regular and high-loading configurations.The pouch cell was assembled using the CrBbased sulfur cathode with a sulfur loading of $4 . 4 \\ : \\mathsf { m g c m } ^ { - 2 }$ and a thickness of $8 5 ~ { \\mu \\mathrm { m } }$ ,lithium foil anode with a thickness of $5 0 ~ { \\mu \\mathrm { m } }$ ,and a controlled E/S ratio of $4 . 0 \\mathsf { m L } \\mathsf { g } ^ { - 1 }$ .Galvanostatic charge/discharge was performed by LAND testers (CT3002A).The current was calculated based on the mass of sulfur ( $\\mathsf { I } ~ { \\mathsf { C } } = 1 6 7 5 ~ \\mathsf { m A } ~ \\mathsf { g } ^ { - 1 } ,$ ).The cyclic voltammetry of Li-S cells was recorded by CHl604E electrochemical workstation with a scanning rate of $0 . 1 \\mathrm { \\ m V \\ } \\mathsf { s } ^ { - 1 }$ ,while the electrochemical impedance spectra were collected within the frequency range of $0 . 7 { - } 7 0 0 \\mathrm { ~ k ~ } \\mathsf { H z }$ under an amplitude of $5 \\mathsf { m V } .$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,313,489,373]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Polysulfide Adsorption Test:The ${ \\dot { \\mathsf { L i } } } _ { 2 } { \\mathsf { S } } _ { 4 }$ solution $( 5 0 \\ : \\mathrm { m M } )$ was prepared by dissolving elemental sulfur and $\\mathsf { L i } _ { 2 } \\mathsf { S }$ in a molar ratio of3:1in THF under vigorous stirring.After that,the same amount of absorbent,i.e.,SP or CrB, was added to the as-prepared homogenous ${ \\dot { \\mathsf { L i } } } _ { 2 } { \\mathsf { S } } _ { 4 }$ solution and let stand for $1 2 \\mathrm { ~ h ~ }$ to observe the color variation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,374,489,554]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Kinetic Characterizations:The symmetric cells were fabricated by using identical electrodes as cathode and anode with an electrolyte containing 0.1 M ${ \\dot { \\mathsf { L } } } { \\dot { \\mathsf { I } } } _ { 2 } { \\mathsf { S } } _ { 6 }$ .The electrodes were prepared by loading the same amount $( \\approx 1 ~ \\mathsf { m g } ~ \\mathsf { c m } ^ { - 2 } )$ of SP or CrB on carbon paper. The corresponding CV measurement was conducted at varied scanning rates from 10 to $5 0 \\mathsf { m } \\mathsf { V } \\mathsf { s } ^ { - 1 }$ ,while the ElS spectra were recorded as described above.Moreover, the $\\mathsf { L i } _ { 2 } \\mathsf { S }$ deposition profiles were collected in asymmetric cells with CrB (or SP) cathode and Li foil anode.The $\\mathsf { L i } _ { 2 } \\mathsf { S } _ { 8 }$ /tetraglyme $( 0 . 2 5 ~ \\mathsf { M } )$ solution was specifically employed as the electrolyte according to the previous report.[24a,25] The cells were galvanostatically discharged to $2 . 0 6 \\mathrm { ~ V ~ }$ and then subjected to potentiostatically discharge at $2 . 0 5 \\lor$ for $\\mathsf { L i } _ { 2 } \\mathsf { S }$ nucleation and precipitation until the current was lower than ${ 7 0 ^ { - 5 } } \\mathsf { A }$ In addition, operando resistance was recorded upon the voltammetric scanning from 2.6 to $\\phantom { - } 7 . 8 \\ : \\mathsf { V }$ at a rate of $0 . 1 \\ : \\mathrm { m V } \\ : \\mathsf { s } ^ { - 1 }$ .The applied amplitude was $5 \\mathsf { m V } ,$ and the constant frequency was set at $1 5 \\mathsf { H z }$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,556,489,713]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Material Characterizations:The phase and crystalline characters were studied by X-ray diffraction (XRD, Bruker-AXS D8 Advances).Scanning electron microscopy (SEM,FEl Quanta 250F) and transmission electron microscopy (TEM,FEl Talos F200X G2) were applied to collect the morphological and microstructure information of the product.The chemical status was measured by X-ray photoelectron spectroscopy (XPS,Thermo Scientific K-Alpha).The UV-vis spectra were recorded on a UV-3600 SHl-MADZU spectrophotometer.The conductivities were obtained via the twoelectrode measurement.The CrB powder was sealed and densified in a plastic tube with tabs at each end connecting to the workstation.The conductivity (σ) was calculated based on the equation $\\sigma = \\mathsf { L } / ( \\mathsf { A } { \\cdot } \\mathsf { R } )$ ,where L, A,and R refer to the length,sectional area,and measured resistance,respectively,of the sample cylinder. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,713,489,858]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theoretical Calculations:The Vienna Ab-lnitio Simulation Package (VASP) was employed for density functional theory (DFT) calculations.[26] The Perdew-Burke-Ernzerhof (PBE) formulation was used within the generalized gradient approximation (GGA),[27] while the projected augmented wave (PAW) potentials were applied to describe the ionic cores,[28] The self-consistency of electronic energy was determined by the energy change smaller than $1 0 ^ { - 5 } \\thinspace \\thinspace \\mathrm { e V } .$ The geometryoptimization was converged to the energy change smaller than $0 . \\mathsf { \\bar { O } } 2 \\mathsf { e } \\mathsf { V } \\mathring { \\mathsf { A } } ^ { - 1 }$ .The vacuum spacing perpendicular to the plane was set to be $1 8 { \\mathring { \\mathsf { A } } } .$ The weak interaction was described by $\\mathsf { D F T } + \\mathsf { D } 3$ method using empirical correction in Grimme's scheme,[29] Additionally, the Gibbs free energy was calculated based on the equation below: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,870,223,886]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0670/83129f5f913ff145e71c3584e1842a1bb3ac1f345862f79a103b1967278f7063.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathsf { C } = E _ { e l e c } + E _ { Z P E } - \\mathsf { \\Delta } T S\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[84,896,489,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\mathsf { E } _ { \\mathrm { e } | \\mathrm { e } \\mathrm { c } }$ refers to the electronic energy at $0 \\mathrm { ~ K ~ }$ calculated by DFT, $\\mathsf { E } _ { \\mathsf { Z P E } }$ is assigned to the zero-point energy term. S and $\\intercal$ are the entropy and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,95,919,131]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"the absolute temperature (i.e.,298.15 K in this case),respectively.The adsorption energies were determined as the Gibbs energy differences of the projected system before and after relaxation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[514,157,700,173]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Supporting Information ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,181,919,205]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Supporting Information is available from the Wiley Online Library or from the author. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[514,231,668,248]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acknowledgements ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,255,919,364]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The authors thank the financial support from the National Natural Science Foundation of China (22109072,52372286),Natural Science Foundation of Jiangsu Province (BK20210349),Fundamental Research Funds for the Central Universities (30922010304),Foundation of Key Laboratory for Palygorskite Science and Applied Technology of Jiangsu Province (HPK202102),Open Project Foundation of National & Local Joint Engineering Research Center for Mineral Salt Deep Utilization (SF202105),the 2021 Qinlan Project of Jiangsu Province,the Jiangsu University of Technology Zhongwu Youth Innovative Talent Support Program. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,390,662,406]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conflict of Interest ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[514,414,747,425]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The authors declare no conflict of interest. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[514,451,729,468]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Data Availability Statement ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,475,919,512]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The data that support the findings of this study are available on request from the corresponding author.The data are not publicly available due to privacy or ethical restrictions. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[514,538,591,553]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[510,562,917,586]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"chromium boride,dual sulphophilicity,electrocatalysis,lithium-sulfur batteries ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[707,602,919,638]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Received: June 12,2023 \nRevised: August 5,2023 \nPublished online: September 25,2023 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,951,216,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Adv. Sci. 2023,10,2303830 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[788,20,917,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCE OpenAccess ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[443,950,919,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2303830 (9 of 10) $\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[84,26,282,74]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCENEWS www.advancedsciencenews.con ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[517,679,919,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"[1] C. Steven,A. Majumdar,Nature 20l2,488, 294. [2] a) M.Zhao,B.Q.Li, H.J.Peng,H.Yuan,J.Y.Wei,J.Q.Huang, Angew. Chem.,Int.Ed.2020,59,12636;b) M.Armand,J.M.Tarascon,Nature 2008,451, 652. [3] a)S.Chu,Y. Cui,N. Liu,Nat.Mater. 2017,16,16; b)J. M.Tarascon, M.Armand,Nature 200l,414,359. [4] a) Y.Chen,T. Y.Wang,H.J.Tian, D.W.Su,Q.Zhang, G.X.Wang, Adv. Mater.2021,33,2003666; b) S.Feng,Z. H.Fu,X.Chen,Q.Zhang, Infomat 2022,4, e12304. [5] a) G.R.Li,S.Wang,Y.N.Zhang,M.Li,Z.W.Chen,J. Lu,Adv. Mater. 2018,30,1705590; b) C.Y.Li, Y.Y. Li,Y.X. Fan,F.Wang,T.Y. Lei,W. Chen,A.J.Hu,L.J. Xue,J. W. Huang,C.T. Yang,Y. Hu,Y.T.Yan,Small 2022,18,2106657. [6] a) M.Jana, R.Xu,X. B.Cheng,J.S.Yeon,J. M.Park,J. Q. Huang, Q. Zhang,H.S.Park,Energy Environ.Sci.2020,13,1049;b) Z.Zhuang, Q. Kang,D.Wang,Y.LiNano Res.2020,13,1856;c)J.Pu,W.B.Gong, Z.X. Shen,L.T. Wang,Y. G.Yao,G. Hong,Adv.Sci. 2022,9,2104375; d) N.Sapkota,S.Chiluwal,P.Parajuli,A. Rowland,R.Podila,Adv. Sci. 2023,10,2206901. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[747,61,917,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"www.advancedscience.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[78,95,485,478]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"[7]a)F.Y. Shi,J.Y.Yu,C.H.Chen, S.P. Lau,W. Lv,Z.L. Xu,J.Mater. Chem.A 2022,10,19412; b) H.Tang,W.L.Li,L.M.Pan, C.P.Cullen, Y. Liu,A. Pakdel,D.H. Long,J. Yang,N. McEvoy, G.S. Duesberg, V. Nicolosi, C.F. Zhang,Adv.Sci.2018,5,1800502. \n[8] a) M.L. Wang, Z.T. Sun,H.N.Ci,Z.X. Shi,L.Shen,C. H.Wei,Y.D. Yi,X.Z.Yang,J.Y. Sun,Angew. Chem.,Int.Ed.2021,60,24558; b) D. Q.Cai,J.L.Yang,T. Liu,S.X. Zhao,G. Z. Cao,Nano Energy 2021,86, 106452; c) K.Y. Zou,X.X. Chen,W.T. Jing,X. Dai,P.F.Wang,Y. Liu, R. Qiao,M.Shi,Y.Z.Chen,J.J.Sun,Y.N. Liu,Energy Storage Mater. 2022,48,133;d) K. Xi,W.Wang,S.Liu,G.R.Li, X.P.Gao,Energy Storage Mater 2022,45,1229. \n[9] X.Liang,C.Y. Kwok,L.M.Fernanda,Q. Pang,C. Marine,H. Huang, C.J.Hart,D.Houtarde, K. Kaup,H.Sommer,Adv.Energy Mater.2016, 6,1501636. \n[10] G. M. Zhou,H.Z. Tian,Y. Jin,X.Y. Tao,B.F. Liu,R.F. Zhang,W.S. Zhi, D. Zhuo,Y.Y.Liu,J.Sun,j.Zhao,C.X.Zu,D.S.Wu,Q.F. Zhang, Y. Cui, PANS 2017,114,840. \n[11] Y. B. Xiao, Y. C. Xiang,S.J. Guo,J. Wang, Y. Ouyang, D. X. Li, Q. H. Zeng,W. Gong,L.Y. Gan,Q. Zhang, Energy Storage Mater.2022,51, 882. \n[12] a) G.R.Li, H.Y.Li, H. B. Zeng,J. Inorg. Mater. 2021,32,152; b) B. Wang,L.Wang,B. Zhang,Z. Kong,S.Y. Zeng,M.W. Zhao,Y.T. Qian, L. Q.Xu,Energy Storage Mater.202,45,130;c) B.Wang,L.Wang, Y. Y. Kong,F. B. Wang, Z.X.Jing, X. F. Yang,Y.T. Qian,M. Chen,L. Q.Xu,Adv. Energy Mater. 2023,13,2300590;d) B.Wang,L.Wang, B. Zhang,S.Y. Zeng,F.Tian,J. M. Dou,Y.T. Qian,L. Q. Xu,ACS Nano 2022, 16,4947. \n[13] a) S.Carenco,D. Portehault,C. Boissiere,N.Mezailles,C.Sanchez, Chem.Rev.2013,113,7981; b)J.P.Scheifers,Y.M. Zhang,B.P.T. Fokwa,Acc.Chem.Res.2017,50,2317. \n[14] C. C. Li,X.B. Liu,L. Zhu,R.Z. Huang,M.W. Zhao,L. Q.Xu,Y.T. Qian,Chem.Mater 2018,30,6969. \n[15] Z.J.Li, H.R.Jiang,N.C.Lai,T.S. Zhao,Y.C.Lu,Chem. Mater. 2019, 31,10186. \n[16] J. R. He,A. Bhargav,A. Manthiram,Adv. Mater. 2020,32,2004741. \n[17] V.Natu,S.S.Kota,M.W. Barsoum,J. Eur. Ceram. Soc.2020,40,305. \n[18] S.L. Ma, K. Bao,Q.Tao,P.W. Zhu,T. Ma,B. Liu,Y.Z. Liu,T.Cui, Sci. Rep.2017,7,43759. \n[19] X.Y. Liu,Y. Gong,ACS Appl. Nano Mater. 2022,5,10183. \n[20] S.Yu,W.Cai,L.Chen,L.Song,Y. Song,J.Energy Chem.2021,55, 533. \n[21] X. R. Zhang,Z. H.Shen,Y. Wen,Q.Y. He,J.Yao, H.Y. Cheng,T. Gao, X.M.Wang,H.G. Zhang,H. Jiao,ACS Appl. Mater. Interfaces 2023, 15,21040. \n[22] a) X. Z. Zhang,C.Q. Shang,E.M.Akinoglu, X.Wang, G.F. Zhou, Adv. Sci. 2020,7,2002037; b) J.Y. Zhang,G.B.Xu,Q. Zhang,X. Li,Y. Yang,L.W. Yang,J.Y. Huang,G.M. Zhou,Adv. Sci.2022,9,2201579. \n[23] a) G. R. Li, W. Lei, D. Luo,Y.P.Deng,Z. P. Deng,D.L. Wang,A. P. Yu, Z.W. Chen, Energy Environ. Sci.2018,11,2372; b) W.L.Qiu,G.R.Li, D. Luo,Y.G. Zhang, Y. Zhao,G.F. Zhou,L. L. Shui, X. Wang, Z. W. Chen,Adv. Sci.2021,8,2003400. \n[24] a) F.Y. Fan,W.C. Carter,Y. M. Chiang,Adv. Mater. 2015,27,5203;b) K. Kim,J. Kim,J.H.Moon,Adv.Sci.2023,10,2206057. \n[25] Z.J. Li,Y. C. Zhou,Y. Wang,Y. C. Lu,Adv. Energy Mater. 2019,9, 1802207. \n[26] G. Kresse,J.Furthmuller, Comput. Mater. Sci.1996,6,15. \n[27] J.P.Perdew,K. Burke,M.Ernzerhof,Phys.Rev. Lett.1996,77,3865. \n[28] P.E. Blochl,Phys.Rev. B1994,50,17953. \n[29] S.Grimme,J.Antony,S.Ehrlich,H. Krieg,J. Chem.Phys.2010,132, 154104. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,97,914,476]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,26,282,75]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCENEWS www.advancedsciencenews.com ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[78,951,211,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Adv.Sci.2023,10,2303830 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[435,950,914,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2303830 (10 of 10) $\\circledcirc$ 2023 The Authors. Advanced Science published by Wiley-VCH GmbH ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[783,20,914,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADVANCED SCIENCE Open Access ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[742,61,914,72]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"www.advancedscience.com ","text_format":null,"text_level":null,"type":"discarded"}],"string":"[\n {\n \"bbox\": \"[85,93,805,180]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Dually Sulphophilic Chromium Boride Nanocatalyst Boosting Sulfur Conversion Kinetics Toward High-Performance Lithium-Sulfur Batteries \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,199,811,221]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hongyang Li, Guxian Chen, Kailong Zhang, Liangbiao Wang,\\\\* and Gaoran Li\\\\* \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,263,610,561]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The sluggish kinetics of sulfur conversions have long been hindering the implementation of fast and efficient sulfur electrochemistry in lithium-sulfur (Li-S) batteries. In this regard, herein the unique chromium boride (CrB) is developed via a well-confined mild-temperature thermal reaction to serve as an advanced sulfur electrocatalyst. Its interstitial-alloy nature features excellent conductivity, while the nano-lamination architecture affords abundant active sites for host-guest interactions. More importantly, the CrB nanocatalyst demonstrates a dual sulphophilicity with simultaneous Cr-S and B-S bondage for establishing strong interactions with the intermediate polysulfides. As a result, significant stabilization and promotion of sulfur redox behavior can be achieved, enabling an excellent Li-S cell cyclability with a minimum capacity fading rate of $0 . 0 1 7 6 \\\\%$ per cycle over 2000 cycles and a favorable rate capability up to 7 C. Additionally, a high areal capacity of $5 . 2 ~ \\\\mathsf { m A h } ~ \\\\mathsf { c m } ^ { - 2 }$ , and decent cycling and rate performances are still attainable under high sulfur loading and low electrolyte dosage. This work offers a facile approach and instructive insights into metal boride sulfur electrocatalyst, holding a good promise for pursuing high-eficiency sulfur electrochemistry and high-performance Li-S batteries. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[645,251,763,267]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[645,277,919,626]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The global target of carbon neutralization necessitates the active development and deployment of green energies such as solar, wind,and hydro.[1] Electrochemical energy storage hereto affords an attractive solution to the efficient and convenient utilization of these sustainable but intermittent energies,and has attracted tremendous research enthusiasm during the past few decades.[2] The currently mainstream lithium-ion battery (LIB) technology plays a critical role in electric vehicles and grid energy storage due to its high energy，durability,and technical maturity. However, energy density and affordability bottlenecks increasingly challenge their future implementation in these fields.3] In this context,lithium-sulfur (Li-S) batteries emerge as a promising nextgeneration energy storage systemattributed to their theoretically high energy density,almost five times that of the conventional LIB (2600 vs $5 0 0 \\\\ \\\\mathrm { W h \\\\ k g ^ { - 1 } }$ ),as well as the lowcost that nearly half-cut that of LIBs benefiting from the abundant reserve and wide geometrical distribution of sulfur.[4] \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,627,919,794]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Despite these intriguing merits,Li-S batteries struggle with several critical and characteristic challenges that obstruct their entrance into the practical markets.[5] Specifically, the insulating nature of sulfur and sulfides,the multi-electron transfer reaction pathway,and the liquid-solid phase-conversion mechanism collectively determine the intrinsic sluggish kinetics of sulfur reactions,rendering the difficult realization of fast and efficient sulfur electrochemistry.Meanwhile,the high solubility and mobility of the intermediate lithium polysulfides (LiPS) trigger the notorious shuttle effect accompanied by persistent sulfur loss,coulombic inefficiency,and anode passivation,resulting in fast capacity fading and battery failure.[6] \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,795,919,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Given these intractable problems,catalytic agents are desired to stabilize and promote sulfur conversions.In the past few years,researchers worldwide have devoted significant efforts to developing advanced sulfur electrocatalysts for alleviating the kinetic limits and inhibiting shutle behaviors.[7] Among them, chemically polar inorganics such as metal compounds have attracted particular attention due to their strong interactions with LiPS,which are expected to chemically immobilize sulfur species and facilitate their electrochemical conversions.8] Liang et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,604,485,817]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"H. Li, G. Chen, G. Li MIIT Key Laboratory of Advanced Display Materials and Devices School of Materials Science and Engineering Nanjing University of Science and Technology Nanjing,Jiangsu 210094,P.R. China E-mail: gaoranli@njust.edu.cn K. Zhang KeyLaboratory for Palygorskite Science and Applied Technology ofJiangsu National& Local Joint Engineering Research Center forMineral Salt Deep Utilization SchoolofChemical Engineering Huaiyin InstituteofTechnology Huaian,Jiangsu 223003,P.R.China L.Wang SchoolofChemistry and Chemical Engineering Jiangsu University ofTechnology Changzhou,Jiangsu 2130o1,P.R.China E-mail: Ibwang@jsut.edu.cn \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[84,951,216,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Adv. Sci. 2023,10,2303830 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[84,34,297,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RESEARCH ARTICLE \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[443,950,919,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2303830 (1 of 10) $\\\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[746,61,919,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"www.advancedscience.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[84,855,487,901]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution License,which permits use,distribution and reproduction in any medium,provided the original work is properly cited. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[84,828,489,852]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The ORClD identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/advs.202303830 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[786,0,998,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates ADVANCED SCIENCE OpenAccess \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[84,909,275,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DOI: 10.1002/advs.202303830 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[78,94,485,276]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"presented unique $\\\\mathrm { V O } _ { \\\\mathrm { x } }$ hollow spheres,as representative metal oxide,for chemically adsorbing LiPS following a“Goldilocks\\\" principle to resist the shutling behavior[9] Cui's group systematically studied the catalytic oxidation of $\\\\mathrm { L i } _ { 2 } \\\\mathrm { S }$ on different polar sulfide surfaces, demonstrating that $\\\\mathrm { V S } _ { 2 }$ ， $\\\\mathrm { T i } \\\\mathsf { S } _ { 2 }$ ,and ${ \\\\mathrm { C o } } { \\\\mathrm { S } } _ { 2 }$ deliver relatively superior catalytic activity due to their strong interaction with LiPS and the great facilitation of Li-ion transport.10l Apart from that,metal-organic frameworks (MOFs) are also potential host matrixes for improving sulfur electrochemistry. Huang's group recently reported an ultralight electroconductive Ni-TABQ, which efficiently promotes the multistep sulfur conversions benefiting from the dual catalytic centers within the framework,i.e., Ni-N4 and quinone.[11] \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,276,485,753]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Despite the encouraging progress, common metal compounds encounter some substantial shortcomings,represented by their poor electrical conductivity and limited access to the active sites, struggling to boost battery performance further. Given this, metal borides (MBs) recently become a rising star in this field due to their intrinsically high conductivity and unique chemical structures.[12] Metal-rich MB with alloy lattice structure inherits good metallicity, endowing high electrical conductivity up to $1 0 ^ { 7 } \\\\mathrm { ~ S ~ m ^ { - 1 } }$ .Additionally,MBs deliver a controllable transition from superconductor to semiconductor and further into insulator along with the increase of metal-to-boron ratio in stoichiometry,enabling tailorable chemical and electronic structures that afford attractive opportunities to meet specific demands in extensive areas.[13] Given the challenges in the Li-S system, MBs have shown considerable promise in regulating sulfur electrochemistry. Qian's group pioneered a layer-structure and highly conductive titanium diboride $\\\\left( \\\\mathrm { T i B } _ { 2 } \\\\right)$ as the polar sulfur host ma-terial in Li-S batteries.[14] A surface passivation mechanism was proposed for enhancing the polysulfide anchoring effect of $\\\\mathrm { T i B } _ { 2 }$ ， thus contributing to a strong sulfur immobilization with stable cell cycling.Li et al.compared $\\\\mathrm { T i B } _ { 2 }$ with other Ti-based compounds, i.e., $\\\\mathrm { T i O } _ { 2 }$ and TiC,and revealed the highest LiPS adsorption strength by boride that mostly favors the catalytic sulfur conversions.[15] Recently, MoB was also explored as an electrocatalyst for sulfur electrochemistry.16] Combining high conductivity,rich catalytic sites,and good hydrophilicity,the MoB catalyst enables a capacity of $1 2 5 3 \\\\mathrm { \\\\ m A h \\\\ g ^ { - 1 } }$ ,a long lifespan of over 1000 cycles,and reliable sulfur redox under high sulfur loading and limited electrolyte.These advances have demonstrated the great potential of MBs for catalyzing sulfur conversions and improving Li-S battery performance.However, this family of electrocatalysts is still in its infancy,with their structural potentialities to be further exploited,nanostructure to be rationally designed,and underlying functioning mechanisms to be deeply probed into. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,754,485,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this contribution,chromium boride (CrB) nanocatalyst was developed for the first time to power superior Li-S battery chemistry. The $\\\\mathrm { C r B }$ catalyst with unique laminated nanostructure was prepared via a well-confined,mild-temperature,and solid-state thermal method.Benefiting from its 2D interstitial-alloy latice structure,CrB inherits good metallicity with a high conductivity of $3 5 7 \\\\mathrm { S } \\\\mathrm { m } ^ { - 1 }$ .More importantly, the unique dual-sulphophilic character of $\\\\mathrm { C r B }$ imposes simultaneous p-d and p-p hybridizations with LiPS,which is distinct from the Lewis acid-base mode in conventional catalyst scenarios,and enables strong sulfur immobilization against the shuttle effect and lowered energy barriers for multistep sulfur conversions.As a result, fast,efficient, and durable sulfur redox electrochemistry was realized,instantiated by the excellent cyclability over 20oo cycles,superb rate capability up to 7 C,and decent energy output even under the highloading (sulfur loading $: = 5 . 0 \\\\mathrm { m g } \\\\mathrm { c m } ^ { - 2 }$ ),and lean-electrolyte (E/S ratio $= 5 . 5 \\\\mathrm { m L g ^ { - 1 } }$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,94,914,164]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,181,714,198]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. Results and Discussion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,207,914,332]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The $\\\\mathrm { C r B }$ nanocatalyst was prepared via a well-confined,mildtemperature，solid-state method. $\\\\mathrm { C r } { \\\\mathrm { O } } _ { 3 }$ was utilized as the chromium source,while $\\\\mathrm { N a B H } _ { 4 }$ was simultaneously the boron source and reductive agent.The as-builthigh-pressure and highly reductive environment enables the mild-temperature boronization to yield the well-dispersed nanometric $\\\\mathrm { C r B }$ .It should be noted that the one-pot synthesis with the simple setup and mild conditions endows this method with good scalability for potential practical implementation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,326,914,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The morphology and microstructure of the as-prepared $\\\\mathrm { C r B }$ were first investigated by scanning electron microscopy (SEM). It is observed that the $\\\\mathrm { C r B }$ delivers a nanometric lamination structure with a lateral size of ${ \\\\approx } 2 0 0 { \\\\mathrm { - } } 5 0 0 \\\\ \\\\mathrm { n m }$ (Figure 1A,B). The welldefined edges and flat surfaces can be noticed, signifying a layered lattice structure. Transmission electron microscope (TEM) characterization was carried out for a more profound structural study as shown in Figure 1C.As expected,stacked layers can be noted with some exfoliated nanosheets scattering around, likely due to their peeled-off upon the ultrasonic process, which strongly indicates the 2D lattice structure. The corresponding selected area electron diffraction (SAED) pattern is depicted in Figure 1D.The as-developed CrB exhibits a somewhat singlecrystalline character, showing bright spots in an ordered arangement. The marked spots can be assigned to the (111),(110), and (130) crystal faces of the orthorhombic CrB,respectively. Meanwhile,the high-resolution TEM (HRTEM) and inverse fast Fourier transform (IFFT) reveal an explicit interplanar space of $0 . 2 7 7 \\\\mathrm { n m }$ assigned to the (110) facet of CrB (Figure 1E; Figure S1, Supporting Information),while the high-angle annular dark field (HAADF) scanning transmission electron microscopy (STEM) image and energy dispersive X-ray spectroscopy (EDS) mapping in Figure 1F confirm the uniform distribution of Cr and B in the layered product. In addition, the crystallographic structure of the as-developed CrB was further confirmed by X-Ray Diffrac-tion (XRD, Figure 1G). The resulting pattern shows sharp peaks at 32.3,38.3,45.0,and 46.2 degrees corresponding to the (110), (021),(111),and (130) facets,respectively,of the orthorhombic CrB in Cmcm space group (PDF# 32-0277),which is well con-sistent with the SAED and HRTEM results. Therefore,it is rea-sonable to conclude that well-crystallized $\\\\mathrm { C r B }$ was successfully obtained in a typical 2D lattice structure,where the slabs of face-sharing $\\\\mathrm { C r } _ { 6 } \\\\mathrm { B }$ trigonal prisms extend along the (010) plane and stack in the b-direction,and the boron atoms are covalently bonded with each other to form zigzag chains parallel to the (001) direction.What should be further noted is that the interstitially alloyed structure with well-inherited metal bonds in the matrix endows $\\\\mathrm { C r B }$ with good metallicity and high electrical conductiv-ity that significantly outperforms its oxide counterpart (357 vs $4 . 5 \\\\times 1 0 ^ { - 3 } \\\\ \\\\mathrm { S \\\\ m ^ { - 1 } }$ ,Figure S2, Supporting Information). Beyond that, the coexistence of ${ \\\\mathrm { C r } } { - } { \\\\mathrm { C r } }$ (metallic)，B-B (covalent)，and $\\\\mathrm { C r - B }$ (ionic) bonds could trigger unique chemical features for potential host-guest interactions with LiPS and thus regulates the sulfur electrochemical behaviors. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,26,282,74]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADVANCED SCIENCENEWS www.advancedsciencenews.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[78,951,211,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Adv.Sci. 2023,10,2303830 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[440,950,914,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2303830 (2 of 10) $\\\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[783,20,912,58]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADVANCED SCIENCE OpenAccess \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[742,61,914,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"www.advancedscience.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[206,94,806,514]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure1.A,B)Saes,C)age,D)paernE)Hge,F)HDD-STgeamentalapg,G) projected crystalline structure (inset),high-resolution XPS H) $C r 2 \\\\\\\\mathsf { p }$ and I) B 1s spectra of CrB. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0034/2d182cb2010eac968873925ce0b30500d876a7c1e40f33981fa0517bd71a9f38.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[84,571,489,598]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,599,489,892]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Given this,the surface chemistry of the obtained $\\\\mathrm { C r B }$ was studied by X-ray photoelectron spectroscopy (XPS).All the spectra were calibrated according to the $\\\\mathrm { C - C }$ peak position of $2 8 4 . 8 \\\\ \\\\mathrm { e V }$ (Figure S3,Supporting Information).Figure 1H depicts the Cr $2 \\\\mathrm { p }$ spectrum of $\\\\mathrm { C r B }$ ，which witnesses three subpeaks at 574.1, 575.6,and $5 7 7 . 1 \\\\ \\\\mathrm { e V }$ ascribed to the $\\\\mathrm { C r } ^ { 0 }$ ， $\\\\mathrm { C r } ^ { 3 + }$ ,and Cr hydroxide species,respectively[17] The strong $\\\\mathrm { C r } ^ { 0 }$ signal confirms the metallic character of $\\\\mathrm { C r B }$ ,while the rest species suggest the ionic feature as well as the potential partial oxidization/hydroxidization at the surface.Meanwhile,the B 1s spectrum exhibits characteristic peaks at 187.5 and $1 8 8 . 5 \\\\ \\\\mathrm { e V }$ assigned to the $\\\\mathrm { B - C r }$ and $\\\\mathrm { B - B }$ ,respectively (Figure 1I)[18] manifesting the coexistence of local ionic and covalent features in the $\\\\mathrm { C r B }$ lattice.Oxidized B species such as ${ \\\\bf B } _ { 2 } \\\\mathrm { O } _ { 3 }$ also exist,which is commonly observed in the literature.[19] These results confirm the successful acquisition of $\\\\mathrm { C r B }$ based on mild-temperature thermal synthesis.The alloy structure well inherits the metallic character that would greatly benefit the electron conduction,while the as-established local chemical polarity is expected to impose favorable chemical affinity to LiPS for achieving reliable sulfur immobilization and catalyzation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,893,489,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In view of these potential benefits,the Li-S battery performance based on the CrB nanocatalyst was evaluated in coin cells (see details in Experimental Section).The $\\\\mathrm { C r B }$ dosage in the electrodes was first optimized to $1 0 \\\\mathrm { w t . } \\\\%$ as shown in Figure S4 (Supporting Information).Skimpy addition seems hardly fulfill the catalytic advantages of $\\\\mathrm { C r B }$ ,while excessive $\\\\mathrm { C r B }$ likely densifies the electrode with potential structural nonuniformity and thus deteriorates the electrochemical performance.Figure 2A shows the CV curves of the cell based on the $\\\\mathrm { C r B }$ catalyst (denoted as w CrB),while the one based on super $\\\\mathrm { P }$ (SP) was employed as a contrast sample (denoted as w/o CrB).Distinctly, the $\\\\mathrm { C r B }$ cell exhibits a much higher current response with a much sharper peak shape and smaller redox overpotential gap compared with those of the SP cell (61 vs $9 9 ~ \\\\mathrm { m V }$ for Gap I,248 vs $2 9 3 ~ \\\\mathrm { m V }$ for Gap II),suggesting the considerably enhanced conversion kinetics by the CrB catalyst. This advance can also be confirmed by the Tafel plots as shown in Figure S5 (Supporting Information), the redox peaks referring to the $\\\\operatorname { L i } _ { 2 } \\\\mathrm { S }$ conversion,which forms the Gap II,are specifically studied as it is generally considered the most kinetic-limiting process in sulfur electrochemistry. During both the cathodic and anodic scanning,the CrB cell shows a much smaller Tafel slope $( - 3 3 . 4 $ and $6 5 . 1 \\\\mathrm { m V } \\\\mathrm { d e c ^ { - 1 } }$ ) compared with those without the CrB catalyst $( - 6 7 . 3$ and $7 1 . 3 ~ \\\\mathrm { m V ~ d e c ^ { - 1 } }$ ）， indicating the lowered energy barrier and facilitated $\\\\operatorname { L i } _ { 2 } \\\\mathrm { S }$ redox by the CrB catalyst. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,571,919,892]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,893,919,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Correspondingly, the galvanostatic charge-discharge profiles (Figure 2B)of the CrB cell demonstrate a much higher capacity (1260.9 vs $1 0 0 2 . 7 \\\\mathrm { \\\\ m A h \\\\ g ^ { - 1 } } .$ ）and smaller charge-discharge potential hysteresis (denoted as $\\\\Delta \\\\mathrm { E }$ ）compared with its SP counterpart (121 vs $1 6 5 \\\\ \\\\mathrm { m V } ,$ Figure S6, Supporting Information). Apart from that, the capacity ratio of the low-potential plateau to the high-potential plateau in the discharge curve $( \\\\mathrm { Q _ { L } } / \\\\mathrm { Q _ { H } } )$ is another critical indicator of the sulfur redox efficiency.The much higher $\\\\mathrm { Q _ { L } } / \\\\mathrm { Q _ { H } }$ (1.975) of the $\\\\mathrm { C r B }$ cell over its w/o CrB counterpart (1.594) illustrates that the sulfur species dissolved during the high-plateau discharging can be reutilized and precipitated back on the cathode surface upon the low-plateau stage.These results consistently and collectively demonstrate the much faster and more efficient sulfur redox behaviors on the $\\\\mathrm { C r B }$ surface. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,26,287,74]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADVANCED SCIENCENEWS www.advancedsciencenews.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[84,951,216,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Adv. Sci. 2023,10,2303830 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[443,950,917,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2303830 (3 of 10) $\\\\circledcirc$ 2023 The Authors. Advanced Science published by Wiley-VCH GmbH \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[788,20,919,58]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADVANCED SCIENCE Open Access \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[746,61,919,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"www.advancedscience.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[189,94,794,457]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figurte ofLiSestal; H) long-term cycling performance of CrB cell at $_ { 2 \\\\\\\\subset }$ \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0975/de1c08473c1cc1b6c6b24cf3543259b71db0e90115340768493e9110e3a88fa4.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[78,529,485,695]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,698,485,864]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The kinetic improvement can also be perceived from the EIS analysis.As shown in Figure 2C,the Nyquist plots of both cells consist of a suppressed semicircle at the high-medium frequency range,which is assigned to the charge-transfer resistance $\\\\mathrm { ( R _ { c t } ) }$ ,as well as an oblique line at the low-frequency range referring to the Warburg resistance.The inset depicts the corresponding equivalent circuit. It can be noticed that the $\\\\mathrm { C r B }$ cell delivers a much lower $\\\\mathrm { R _ { c t } }$ compared with the w/o CrB cell, while the much higher slope of the oblique line suggests the faster $\\\\mathrm { L i ^ { + } }$ diffusion on the $\\\\mathrm { C r B }$ surface.These results confirm the more facile charge supplies in the CrB-based configuration and support its faster kinetic behaviors. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,865,485,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The galvanostatic cycling behaviors of different cells at $0 . 2 ~ \\\\mathrm { { C } }$ were compared in Figure 2D.The CrB cell retains a capacity of $9 5 4 . 7 \\\\mathrm { ~ m A h ~ g ^ { - 1 } }$ after 50 cycles, corresponding to a capacity retention of $7 5 . 7 \\\\%$ .By contrast, the SP cell experiences much faster capacity decay, showing a capacity of $6 4 4 . 4 \\\\ \\\\mathrm { m A h \\\\ g ^ { - 1 } }$ at the $5 0 ^ { \\\\mathrm { t h } }$ cycle with alower retention of $6 4 . 2 \\\\%$ .Apart from that,the rate capability of different cells was also compared by cycling at varied current rates,as shown in Figure 2E.The $\\\\mathrm { C r B }$ electrode retains a favorably high capacity of $7 7 5 \\\\mathrm { m A h g ^ { - 1 } }$ even under a high rate up to $7 \\\\mathrm { C }$ ,which recovers to $1 1 1 2 . 4 \\\\mathrm { ~ m A h ~ g ^ { - 1 } }$ as the applied current returns to $0 . 2 \\\\ : \\\\mathrm { C }$ .In comparison, the control cell undergoes fast capacity decaying along with the increase of current rate and retains only a capacity of $2 2 8 . 4 \\\\ : \\\\mathrm { m A h \\\\ : g ^ { - 1 } }$ at7C.Sucha distinctive comparison demonstrates the excellent reaction kinetic realized by the $\\\\mathrm { C r B }$ electrocatalyst, which significantly promotes the sulfur conversions toward a superior rate capability. This improvement can be more vividly evidenced by the evolution of chargedischarge curves upon the multi-rate cycling. Figure 2F shows the voltage profiles of the $\\\\mathrm { C r B }$ cell at different current rates.It is noticed that the potential gap between the charge and discharge plateaus is gradually expanded along the rate increase from 0.2 to 7 C,yet the two-platform discharge profile can be maintained. Whereas for the control cell without CrB,the severe polarization deforms the charge-discharge profiles significantly, failing to maintain the two-plateau profile at 7 C (Figure S7, Supporting Information).The potential gap evolutions along the current rate increase are summarized in Figure 2G.It reveals the persistently lower overpotential of the CrB cellat the whole rate range compared with its w/o CrB counterpart,strongly confirming its much superior kinetic properties. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,530,914,892]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,893,915,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Additionally, the cycling behaviors of cells based on $\\\\mathrm { C r B }$ and $\\\\mathrm { C r } _ { 2 } \\\\mathrm { O } _ { 3 }$ were compared (Figure S8, Supporting Information). The $\\\\mathrm { C r } _ { 2 } \\\\mathrm { O } _ { 3 }$ cell shows higher capacity，stability，and rate capability than the w/o $\\\\mathrm { C r B }$ cell but is inferior to the $\\\\mathrm { C r B }$ cell. The electrochemical improvement by $\\\\mathrm { C r } _ { 2 } \\\\mathrm { O } _ { 3 }$ could be attributed to the decent sulfur confinement as illustrated by the static adsorption test (Figure S9, Supporting Information),whereas the lack of conductivity and catalytic activity limit the sulfur redox effciency[20] Such a comparison further highlights the balanced adsorption and catalyzation by the $\\\\mathrm { C r B }$ design,which enables fast and reversible sulfur electrochemistry. Given these advantages,a longterm cycling evaluation was performed for the $\\\\mathrm { C r B }$ cell. As depicted in Figure 2H,a stable cycling behavior can be achieved with a minimum capacity fading rate of $0 . 0 1 7 6 \\\\%$ per cycle over a prolonged 2000 cycles at $2 \\\\mathrm { C }$ Meanwhile,a high coulombic effciency close to unity can also be sustained, indicating excellent sulfur redox reversibility. These electrochemical performances are highly competitive among those reported in recent literature (Table S1, Supporting Information),further illustrating the superiorities of the $\\\\mathrm { C r B }$ catalyst design. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,26,282,74]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADVANCED SCIENCENEWS www.advancedsciencenews.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[78,951,211,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Adv.Sci. 2023,10,2303830 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[440,950,914,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2303830 (4 of 10) $\\\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[783,20,914,58]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADVANCED SCIENCE Open Access \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[742,61,914,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"www.advancedscience.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[193,94,805,517]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure3A)UVisectadticalagsfluldsobdreles;Botefsly) surface; C) geometrical configuration,D) bond lengths, XPS $S 2 \\\\\\\\mathsf { p }$ spectra of ${ \\\\\\\\dot { \\\\\\\\mathsf { L i } } } _ { 2 } { \\\\\\\\mathsf { S } } _ { 4 }$ E) before and F)after adsorption on CrB; G) energy profiles of sulfur conversion with or without CrB catalyst; H) DOS pattern of CrB; I) DOS pattern of sulfur in ${ \\\\\\\\dot { \\\\\\\\mathsf { L i } } } _ { 2 } { \\\\\\\\mathsf { S } } _ { 4 }$ before and after adsorption. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0532/9e608d47a6b599da6684ee311bf747f74fe83d1bdb8fe93695b35eebf1354f8e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[84,584,489,836]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,837,489,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To understand these electrochemical improvements,the chemical interaction between the $\\\\mathrm { C r B }$ catalyst and LiPS was studied first by static adsorption test.As presented in the inset of Figure 3A,the blank $\\\\mathrm { L i } _ { 2 } \\\\mathrm { S } _ { 4 } /$ /tetrahydrofuran (THF) solution shows a typical brownish color. When immersed with the same amount of $\\\\mathrm { S P }$ and $\\\\mathrm { C r B }$ and let stand for $^ { 1 2 \\\\mathrm { ~ h ~ } }$ ，the solution with SP turns slightly lighter, while the one with $\\\\mathrm { C r B }$ undergoes a strong decoloration,resulting in an almost transparent and colorless supernatant. Such a contrastive comparison strongly confirms the much superior LiPS adsorbability of $\\\\mathrm { C r B }$ ,which can be further verified by the lowest UV-vis absorbance of its corresponding supernatant (Figure 3A).Further probe into the interactive mechanism was carried out by DFT calculation.The $\\\\mathrm { C r B }$ (111) facet was selected as a typical platform for the DFT study as it is the strongest exposed facet in the XRD result.A variety of sulfur species were simulated anchoring on the $\\\\mathrm { C r B }$ (111) surface,contributing to adsorption energies of $- 2 . 7 1$ ,-5.37,-5.19, $- 5 . 0 2$ ， $- 5 . 1 6$ ，and $- 5 . 1 9 \\\\ \\\\mathrm { e V } ,$ respectively (Figure 3B). The corresponding geometrically stable configurations are presented in Figure S10 (Supporting Information). These values are among the relatively high level in literature based on conventional carbon or metal compound substrates,21] confirming the strong chemical interactions between $\\\\mathrm { C r B }$ and LiPS.Figure 3C shows specifically the geometry of $\\\\mathrm { L i } _ { 2 } \\\\mathrm { S } _ { 4 }$ adsorption on $\\\\mathrm { C r B }$ .Interestingly,both chromium and boron are available and preferential to bond with terminal sulfur, forming a $\\\\mathrm { C r - } { \\\\mathsf { S } } { \\\\mathrm { - } } { \\\\mathsf { B } }$ bridging structure.This distinguishes from what is commonly observed in conventional metal compounds scenarios,such as oxides and sulfide,where the Lewis acid-base principle governs the interactions with one end (e.g.,metal sites) sulphophilic and the other (e.g.,oxygen or sulfur sites) lithiophilic.[22] The bond lengths of ${ \\\\mathrm { C r } } { - } S$ are 2.333 and $2 . 4 0 9 \\\\mathrm { ~ \\\\AA ~ }$ ,while the B-S lengths are 1.819 and $1 . 9 5 0 \\\\mathring { \\\\mathrm { A } }$ , suggesting a strong bonding strength. Such a dual sulfur-bonding mode contributes to favorably high adsorption energy of $- 5 . 0 2 \\\\mathrm { e V } ,$ while the lessened bondage with lithium may facilitate the ion transfer, as demonstrated above.Beyond that,a considerable distortion of polysulfide geometry can be noticed upon the interaction.As shown in Figure 3D and Figure S11 (Supporting Information),the bond lengths of S1-S2 and S2-S3 decrease significantly,accompanied by the drastic increase of the S3-S4 bond,referring to the compression of the $\\\\mathrm { L i } _ { 2 } \\\\mathrm { S } _ { 4 }$ molecule at one side and the stretch at the other. Such a distortion is also supported by the consistent variations of the Li-S bonds (Figure S12,Supporting Information).As a result, the lengthened S-S bond tends to be more easily cleaved,contributing to the facilitated conversion kinetics. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,585,919,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,26,287,74]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADVANCED SCIENCENEWS www.advancedsciencenews.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[84,951,216,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Adv. Sci. 2023,10,2303830 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[443,950,919,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2303830 (5 of 10) $\\\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[788,20,919,58]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADVANCED SCIENCE Open Access \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[746,61,919,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"www.advancedscience.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[78,94,485,318]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,319,485,765]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Furthermore,the interactive behaviors between $\\\\mathrm { C r B }$ and polysulfide were analyzed by XPS.Figure 3E,F shows the S 2p XPS spectra of $\\\\mathrm { L i } _ { 2 } \\\\mathsf { S } _ { 4 }$ before and after the adsorption.Typically,the bare $\\\\mathrm { L i } _ { 2 } \\\\mathrm { S } _ { 4 }$ exhibits two pairs of peaks assigned to the terminal $ { ( \\\\mathrm { S _ { T } } ^ { - 1 } }$ ， $1 6 1 . 6 \\\\ \\\\mathrm { e V } )$ and bridging sulfur $( \\\\mathrm { S _ { B } } ^ { 0 } , 1 6 3 . 5 \\\\ : \\\\mathrm { e V } )$ , respectively.[23] A small peak can be detected at a relatively higher binding energy (BE),which could be ascribed to the slight oxidization upon the sampling.By contrast, the adsorption of $\\\\mathrm { L i } _ { 2 } \\\\mathrm { S } _ { 4 }$ on the $\\\\mathrm { C r B }$ surface witnesses the peak shift to lower BE,particularly for the ${ \\\\mathrm { S } _ { \\\\mathrm { B } } } ^ { 0 }$ species, suggesting the electron-accepting behavior of sulfur that increases the electron density and lowers the valence state.This well accords with the lengthened S-S bond as well as the formation of ${ \\\\mathrm { C r } } { - } S$ bonds in the DFT result. Accordingly, the $\\\\mathrm { C r ~ } 2 \\\\mathrm { p }$ and B 1s XPS spectra were also compared. As shown in Figure S13A,B (Supporting Information), the Cr 2p XPS signal becomes drastically weaker after the adsorption, likely due to the strong polysulfide adsorption and coverage thereon.More importantly, the relative content of $\\\\mathrm { C r }$ (III) species significantly increases after the adsorption,confirming the establishment of the ${ \\\\mathrm { C r } } { - } S$ bonding that oxidizes $\\\\mathrm { C r }$ and reduces S.The decrease of Cr hydroxide is likely due to its consumption by LiPS to form the sulfate-related species,which is consistent with the S 2p spectrum.Meanwhile,a considerable peak shift to lower BE can be noticed in B 1s spectra (Figure S13C,D,Supporting Information),reflecting the electron transfer from sulfur to the electrondeficient boron upon the $\\\\mathtt { B - S }$ bonding. This behavior seems mainly occur at the terminal sulfur with higher electron density and partially offset the electron gained from Cr, thus resulting in the limited peak shift for ${ \\\\mathrm { S _ { T } } } ^ { - 1 }$ . Overall,the XPS results are highly consistent with the DFT calculation,collectively confirming the strong and unique chemical interactions between $\\\\mathrm { C r B }$ and polysulfides. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,767,485,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Given these interesting interactions,the kinetic behaviors of sulfur conversions on the $\\\\mathrm { C r B }$ surface were studied as depicted in Figure 3G.The energy profile for bare sulfur conversion was also calculated as a control. It can be recognized that the ratedetermining step for sulfur conversions is the redox of $\\\\operatorname { L i } _ { 2 } \\\\mathrm { S }$ in both cases,which is consistent with that commonly and experimentally observed.Notably, the $\\\\mathrm { C r B }$ catalyst effectively lowers the energy barrier for $\\\\operatorname { L i } _ { 2 } \\\\mathrm { S }$ redox from 0.936 to $0 . 8 9 9 \\\\ \\\\mathrm { e V } .$ Meanwhile,a significantly lower energy barrier of $0 . 7 2 9 \\\\mathrm { e V }$ (vs 0.849 for the control) can be also achieved for another kinetic-limiting step from $\\\\mathrm { L i } _ { 2 } \\\\mathrm { S } _ { 4 }$ to $\\\\mathrm { L i } _ { 2 } \\\\mathrm { S } _ { 2 }$ . These results strongly illustrate the substantial promotion of sulfur conversion kinetics by the highly catalytic $\\\\mathrm { C r B }$ .The density of states (DOS)of CrB was further calculated to unveil the underlying mechanism (Figure 3H). Strong and broad DOS can be observed around the Femilevel, which is mainly contributed by the 3d electrons of Cr, indicating its excellent metallic character for rapid electron conduction.By contrast,the S pDOs of $\\\\mathrm { L i } _ { 2 } \\\\mathrm { S } _ { 4 }$ shows a discrete and cleaved pattern with several isolated bands along the energy axis and no distribution at the Fermi level (Figure 3I; Figure S14,Supporting Information),which well accords with its insulating nature.Interestingly,after the adsorption,the S pDOS converts into a continuous pattern with consideration distribution near the Femi level (Figure 3I).Such a drastic change in the S pDOS pattern strongly indicates that the adsorption on $\\\\mathrm { C r B }$ provides sulfur with significantly enhanced electron transfer for fast reaction kinetics.Apart from that,strong overlapping between S and Cr as well as between S and B pDOS patterns can be witnessed (Figure S15,Supporting Information),further testifying the dual-site interaction via strong S-Cr (p-d) and S-B (p-p)hybridizations that expedite the charge transfer for fast sulfur electrochemistry. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,94,914,373]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,377,914,751]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Inspired by the above computational results,the practical catalytic effect of $\\\\mathrm { C r B }$ was further investigated by symmetric cell characterizations.Figure 4A shows the CV profiles of polysulfide conversions at a scanning rate of $5 0 ~ \\\\mathrm { m V ~ s ^ { - 1 } }$ based on symmetric electrodes with or without the $\\\\mathrm { C r B }$ catalyst.As expected, the electrode shows undetectable redox peaks without LiPS as the active species.Notably, the $\\\\mathrm { C r B }$ catalyst enables a much higher current response with narrowed redox polarization compared with that without $\\\\mathrm { C r B }$ (0.18 vs 0.45 V),manifesting the promoted polysulfide conversion on the $\\\\mathrm { C r B }$ surface.In addition, CV curves at varied scanning rates were also recorded.The profle is well maintained upon the increase of scanning rate for the $\\\\mathrm { C r B }$ cell, while serious profile deformation can be observed for that without $\\\\mathrm { C r B }$ (Figure S16, Supporting Information).This can be more quantitively comprehended via the overpotential comparison as depicted in Figure 4B,where $\\\\mathrm { C r B }$ enables much lower overpotential as well as its slower growth along the increase of scanning rate. Such a distinct comparison further confirms the significant kinetic promotion by the highly catalytic $\\\\mathrm { C r B }$ ,which can also be supported by the much smaller internal resistance of the CrB-based symmetric cell (Figure 4C). Beyond that, the $\\\\mathrm { L i ^ { + } }$ diffusion property is also a critical factor governing kinetic behaviors.Based on the Randles-Sevcik equation, the symmetric cells' peak currents versus the scanning rate's square root were fitted, as shown in Figure 4D.It is noticed that the $\\\\mathrm { C r B }$ cell delivers a more significant slope $( - 3 . 9 9 1$ VS $- 2 . 0 6 6 ]$ ,indicating its higher $\\\\mathrm { L i ^ { + } }$ diffusion coefficient that facilitates sulfur conversions. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,754,914,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In addition to the polysulfide redox,the liquid-solid $\\\\operatorname { L i } _ { 2 } \\\\mathrm { S }$ precipitation, which is considered the most kinetic-limiting process, was specifically studied via the potentiostatic measurement.As shown in Figure 4E,F,the potentiostatic LiPS reduction initiates a current downslope followed by an upward peak,and gradually approaches the current limit after $^ { 8 \\\\mathrm { ~ h ~ } }$ .The downslope and level-off current (dark region) correspond to the reduction of residue $\\\\mathrm { L i } _ { 2 } \\\\mathrm S _ { 8 }$ and $\\\\mathrm { L i } _ { 2 } \\\\mathrm { S } _ { 6 } ,$ respectively, while the left-over upward peak part (light region) can be assigned to the $\\\\operatorname { L i } _ { 2 } \\\\mathrm { S }$ formation and deposition.[24] By comparison, the $\\\\mathrm { C r B }$ catalyst enables a much earlier peak position (2800 vs 7240 s) and sharper peak shape compared with its counterpart, suggesting a lower energy barrier for $\\\\operatorname { L i } _ { 2 } \\\\mathrm { S }$ nucleation and accumulation growth.As such, the CrB electrode obtains a higher precipitation capacity than that without $\\\\mathrm { C r B }$ (303.8 vs $2 5 7 . 1 \\\\mathrm { m A h } \\\\ \\\\mathrm { g } ^ { - 1 }$ ). Such an improvement can be also verified by the more uniform and smoother $\\\\operatorname { L i } _ { 2 } \\\\mathrm { S }$ precipitation on the CrB-loaded carbon paper compared with that without CrB (Figure S17, Supporting Information).Based on these specific studies,Galvanostatic Intermittent Titration Technique (GITT)and operando impedance tests further probed the overall sulfur kinetic behaviors.Figure S18 (Supporting Information) shows the GITT profles of cells with and without the CrB catalyst. The corresponding IR drop evolutions upon the chargedischarge process were compared in Figure 4G.The IR drop refers to the gap between working potential and equilibrium potential at certain reaction status,which reflects the dynamic polarization upon the cell charge and discharge process.It is worth noting that the CrB cell shows persistently lower IR drop along the whole discharge-charge process, strongly confirming the alleviated polarization and the enhanced reaction kinetics by the $\\\\mathrm { C r B }$ catalyst. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,26,282,74]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADVANCED SCIENCENEWS www.advancedsciencenews.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[78,951,211,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Adv. Sci. 2023,10,2303830 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[440,950,914,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2303830 (6 of 10) $\\\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[783,20,912,58]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADVANCED SCIENCE OpenAccess \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[742,61,914,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"www.advancedscience.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[194,95,805,462]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure4)CfpoeldserilititCrB; peak current versus square root of scanning rate; $\\\\\\\\mathsf { L i } _ { 2 } \\\\\\\\mathsf { S }$ precipitation profiles on electrode E) without and F) with CrB; G) IR drops in GITT profiles,H operando resistances and their first-derivative of Li-S cells with or without CrB. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0293/5c9e2714ca446885230bf39e9ee4cd9581e36ecb009656930a896777bc73e7ef.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[84,530,489,794]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,795,489,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Furthermore,the operando resistance profles were recorded for different cells as shown in Figure 4H.Upon the voltammetric scanning from 2.6 to $1 . 8 \\\\mathrm { V }$ at $0 . 2 \\\\mathrm { m V s ^ { - 1 } }$ ,both the cells begin with a fairly steady resistance and encounter a sudden drop at ${ \\\\approx } 2 . 3 \\\\mathrm { V } ,$ which results from the initiation of polysulfide dissolution that expose the conductive surfaces.After that,a steep resistance rise can be noted at ${ \\\\approx } 2 . 2 \\\\ \\\\mathrm { V } ,$ owing to the polysulfide accumulation that increases the viscosity and deteriorates the ion transportation.Further cathodic scanning witnesses the drastic decline of resistance at ${ \\\\approx } 2 . 1 \\\\ \\\\mathrm { V } ,$ where $\\\\operatorname { L i } _ { 2 } \\\\mathrm { S }$ starts to precipitate,accompanied by the recovery of electrolyte conductivity. Interestingly,at the end section of discharge, the $\\\\mathrm { C r B }$ cell shows a continuous decrease of resistance,while the control cell undergoes a re-rise of resistance,likely due to the electrode surface passivation by $\\\\operatorname { L i } _ { 2 } \\\\mathbf { S } .$ This contrast further suggests that the $\\\\mathrm { C r B }$ could regulate the $\\\\operatorname { L i } _ { 2 } \\\\mathrm { S }$ precipitation behavior that facilitates the reaction kinetics.Apart from that, the operando profile of the CrB cell exhibits persistently lower resistance upon the discharging process compared with the w/o CrB counterpart.The first derivation of the profile more intuitively distinguishes the hysteretic and steadier resistance fluctuation for the CrB-based configuration.Based on these results,it is reasonable to conclude that the unique CrB catalyst delivers a great capability of lowering the electrochemical resistance and promoting sulfur conversion,which contributes to excellent cycling and rate performance. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,530,919,751]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,754,919,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Based on these encouraging superiorities,Li-S cells based on the $\\\\mathrm { C r B }$ catalyst were further examined under high sulfur loading and limited electrolyte for potential practical application. Figure 5A shows the rate performance of the $\\\\mathrm { C r B }$ cell at a sulfur loading of $5 . 0 \\\\mathrm { m g } \\\\mathrm { c m } ^ { - 2 }$ and an electrolyte-to-sulfur (E/S) ratio of $5 . 5 \\\\mathrm { m L g ^ { - 1 } }$ .An areal capacity of $5 . 2 7 \\\\mathrm { m A h } \\\\mathrm { c m } ^ { - 2 }$ can be obtained at $0 . 1 \\\\mathrm { C }$ ,which retains at $4 . 5 3 \\\\mathrm { \\\\ m A h \\\\ c m ^ { - 2 } }$ even under a high rate of $1 \\\\mathrm { { C } }$ .What should also be noted is that the high-loading CrB cell can basically maintain the two-plateau discharging profile even under $1 . 0 ~ \\\\mathrm { { C } }$ rate despite certain deformation (Figure 5B). These results strongly demonstrate the excellent reaction effciency and kinetics realized by the $\\\\mathrm { C r B }$ catalyst even under the high-loading configuration.Moreover, the cycling performance was also evaluated.As shown in Figure 5C,a stable cycling behavior can be achieved by the $\\\\mathrm { C r B }$ cell with a highly reversible capacity of $3 . 9 8 \\\\mathrm { \\\\ m A h \\\\ c m ^ { - 2 } }$ after 50 cycles at $0 . 2 \\\\textrm { C }$ Meanwhile, high coulombic efficiency close to unity can be obtained.These favorable results should be attributed to the strong sulfur adsorption and effective catalyzation of $\\\\mathrm { C r B }$ ,which inhibits the shuttle effect and promote the conversion reaction toward fast and durable sulfur electrochemistry. On this basis,the Li-S pouch cell with $\\\\mathrm { C r B }$ electrode was further fabricated (Figure S19, Supporting Information),which is capable of successfully charging a smartphone,as well as powering electric clock and electric fan as depicted in Figure 5D and Figure S20 (Supporting Information), demonstrating an intriguing potential for practical application. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,26,285,74]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADVANCED SCIENCENEWS www.advancedsciencenews.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[84,951,216,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Adv. Sci. 2023,10,2303830 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[443,950,919,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2303830 (7 of 10) $\\\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[788,20,919,58]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADVANCED SCIENCE Open Access \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[746,61,919,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"www.advancedscience.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[191,95,801,488]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 5.A) Rate performance, B) voltage profiles at diferent curent rates,and C) cycling performance at $_ { 0 . 2 \\\\\\\\textsf { C } }$ of CrB cells under sulfur loading ol $5 ~ \\\\\\\\mathrm { { m g } } \\\\\\\\thinspace \\\\\\\\mathrm { { c m } } ^ { - 2 }$ and E/S ratio of $5 . 5 ~ \\\\\\\\mathsf { m L } ~ \\\\\\\\mathsf { g } ^ { - 1 }$ ; D) Optical image of CrB-based Li-S pouch cell charging a cellphone. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0293/87e0d37c711af554cc2d47d445644a9816115d28ee6d0358ef070006841b5f82.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[78,543,485,739]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,757,188,772]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3. Conclusion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,782,485,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In summary, this work developed a unique CrB nanocatalyst for boosting sulfur electrochemistry.A well-confined,mild-temperature,solid-state thermal method was designed to prepare $\\\\mathrm { C r B }$ with lamination nanostructure.The highly conductive nature ensures a fast charge supply for electrochemical reactions, while more importantly, the dual-sulphophilic feature of $\\\\mathrm { C r B }$ establishes strong interactions with LiPS via simultaneous p-d and p-p hybridizations.Such a unique interaction imposes strong sulfur adsorption against the shuttle effect,and lowers energy barriers for multistep sulfur conversions,thus contributing to fast and efficient sulfur electrochemistry.As a result,Li-S cells based on $\\\\mathrm { C r B }$ catalyst realize excellent cyclability over 2ooo cycles, rate capability up to 7 C,and decent performance under high sulfur loading and limited electrolyte,demonstratinga good promise for the development of high-performance and practically viable Li-S batteries. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,543,914,625]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,667,697,682]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4. Experimental Section \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,690,914,800]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Preparation ofCrB:CrB was prepared via a confined solid thermal reaction.All the chemicals were used directly without further purification.Typically, $0 . 3 \\\\mathrm { g }$ chromium oxide $\\\\left( \\\\mathsf { C r O } _ { 3 } \\\\right)$ and $2 . 0 { \\\\ g }$ sodium borohydride $( N a B H _ { 4 } )$ were mixed and sealed in a $2 0 ~ \\\\mathsf { m L }$ stainless high-pressure reactor under an Aratmosphere,and then subjected to the thermal treatment at $5 0 0 ^ { \\\\circ } \\\\mathsf { C }$ for $\\\\mathsf { \\\\Delta } \\\\rceil 0 \\\\mathrm { ~ h ~ }$ .After cooling,the product was washed with diluted hydrochloric acid to remove the by-product and $\\\\mathsf { N a B H } _ { 4 }$ residue,and then washed repeatedly with deionized water and ethanol.The final CrB product was collected after vacuum drying for over $1 2 \\\\mathrm { ~ h ~ }$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,800,914,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Preparation ofSulfur Electrodes:Sulfur electrodes were prepared by the conventional casting method.Typically, the slurry was prepared by dispersing elemental sulfur, SP,CrB,and PVDF with a weight ratio of 5:3:i:1 in NMP solvent.The homogeneous slurry was then cast onto the carboncoated aluminum foil and vacuum dried at $6 0 ~ ^ { \\\\circ } \\\\mathsf { C }$ for over $^ { 1 2 \\\\mathrm { ~ h ~ } }$ .Thedried electrode was then cut into discs with a diameter of $1 2 \\\\mathsf { m m }$ for use.Sulfur loadings of 1.5 and $5 . 0 \\\\mathsf { m g } \\\\mathsf { c m } ^ { - 2 }$ were applied for regular and high-loading electrode evaluations with the electrode thickness of 33 and $9 6 ~ { \\\\mu \\\\mathrm { m } }$ respectively.Sulfur electrode containing elemental sulfur,SP,and PVDF in a weight ratio of 5:4:1 was also prepared for comparison. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,26,282,74]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADVANCED SCIENCENEWS www.advancedsciencenews.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[78,951,211,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Adv.Sci. 2023,10,2303830 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[440,950,914,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2303830 (8 of 10) $\\\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[783,20,914,58]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADVANCED SCIENCE Open Access \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[742,61,914,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"www.advancedscience.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[84,95,489,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Electrochemical Evaluation: The electrochemical performance was evaluated in coin-cell configuration. The cells were assembled using the prepared sulfur electrode as the cathode,Li foil as the anode, and Celgard 240O membrane as the separator in an Ar-filled glove box $( \\\\mathrm { O } _ { 2 } { < } 0 . 1$ ppm, $H _ { 2 } O < 0 . 1$ ppm).The electrolyte contains 1 M lithium bis(trifluoromethane)sulfonimide (LiTFSl) in the mixed solvent of 1,2- dimethoxyethane (DME) and 1,3-dioxolane (DOL) (1:l in volume ratio) with 2 wt. $\\\\%$ lithium nitrate $( \\\\mathsf { L N O } _ { 3 } )$ additive.The electrolyte-to-sulfur (E/S) ratio was controlled at 12 and $5 . 5 ~ \\\\mathsf { m L } ~ \\\\mathsf { g } ^ { - 1 }$ ,respectively，for regular and high-loading configurations.The pouch cell was assembled using the CrBbased sulfur cathode with a sulfur loading of $4 . 4 \\\\ : \\\\mathsf { m g c m } ^ { - 2 }$ and a thickness of $8 5 ~ { \\\\mu \\\\mathrm { m } }$ ,lithium foil anode with a thickness of $5 0 ~ { \\\\mu \\\\mathrm { m } }$ ,and a controlled E/S ratio of $4 . 0 \\\\mathsf { m L } \\\\mathsf { g } ^ { - 1 }$ .Galvanostatic charge/discharge was performed by LAND testers (CT3002A).The current was calculated based on the mass of sulfur ( $\\\\mathsf { I } ~ { \\\\mathsf { C } } = 1 6 7 5 ~ \\\\mathsf { m A } ~ \\\\mathsf { g } ^ { - 1 } ,$ ).The cyclic voltammetry of Li-S cells was recorded by CHl604E electrochemical workstation with a scanning rate of $0 . 1 \\\\mathrm { \\\\ m V \\\\ } \\\\mathsf { s } ^ { - 1 }$ ,while the electrochemical impedance spectra were collected within the frequency range of $0 . 7 { - } 7 0 0 \\\\mathrm { ~ k ~ } \\\\mathsf { H z }$ under an amplitude of $5 \\\\mathsf { m V } .$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,313,489,373]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Polysulfide Adsorption Test:The ${ \\\\dot { \\\\mathsf { L i } } } _ { 2 } { \\\\mathsf { S } } _ { 4 }$ solution $( 5 0 \\\\ : \\\\mathrm { m M } )$ was prepared by dissolving elemental sulfur and $\\\\mathsf { L i } _ { 2 } \\\\mathsf { S }$ in a molar ratio of3:1in THF under vigorous stirring.After that,the same amount of absorbent,i.e.,SP or CrB, was added to the as-prepared homogenous ${ \\\\dot { \\\\mathsf { L i } } } _ { 2 } { \\\\mathsf { S } } _ { 4 }$ solution and let stand for $1 2 \\\\mathrm { ~ h ~ }$ to observe the color variation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,374,489,554]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Kinetic Characterizations:The symmetric cells were fabricated by using identical electrodes as cathode and anode with an electrolyte containing 0.1 M ${ \\\\dot { \\\\mathsf { L } } } { \\\\dot { \\\\mathsf { I } } } _ { 2 } { \\\\mathsf { S } } _ { 6 }$ .The electrodes were prepared by loading the same amount $( \\\\approx 1 ~ \\\\mathsf { m g } ~ \\\\mathsf { c m } ^ { - 2 } )$ of SP or CrB on carbon paper. The corresponding CV measurement was conducted at varied scanning rates from 10 to $5 0 \\\\mathsf { m } \\\\mathsf { V } \\\\mathsf { s } ^ { - 1 }$ ,while the ElS spectra were recorded as described above.Moreover, the $\\\\mathsf { L i } _ { 2 } \\\\mathsf { S }$ deposition profiles were collected in asymmetric cells with CrB (or SP) cathode and Li foil anode.The $\\\\mathsf { L i } _ { 2 } \\\\mathsf { S } _ { 8 }$ /tetraglyme $( 0 . 2 5 ~ \\\\mathsf { M } )$ solution was specifically employed as the electrolyte according to the previous report.[24a,25] The cells were galvanostatically discharged to $2 . 0 6 \\\\mathrm { ~ V ~ }$ and then subjected to potentiostatically discharge at $2 . 0 5 \\\\lor$ for $\\\\mathsf { L i } _ { 2 } \\\\mathsf { S }$ nucleation and precipitation until the current was lower than ${ 7 0 ^ { - 5 } } \\\\mathsf { A }$ In addition, operando resistance was recorded upon the voltammetric scanning from 2.6 to $\\\\phantom { - } 7 . 8 \\\\ : \\\\mathsf { V }$ at a rate of $0 . 1 \\\\ : \\\\mathrm { m V } \\\\ : \\\\mathsf { s } ^ { - 1 }$ .The applied amplitude was $5 \\\\mathsf { m V } ,$ and the constant frequency was set at $1 5 \\\\mathsf { H z }$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,556,489,713]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Material Characterizations:The phase and crystalline characters were studied by X-ray diffraction (XRD, Bruker-AXS D8 Advances).Scanning electron microscopy (SEM,FEl Quanta 250F) and transmission electron microscopy (TEM,FEl Talos F200X G2) were applied to collect the morphological and microstructure information of the product.The chemical status was measured by X-ray photoelectron spectroscopy (XPS,Thermo Scientific K-Alpha).The UV-vis spectra were recorded on a UV-3600 SHl-MADZU spectrophotometer.The conductivities were obtained via the twoelectrode measurement.The CrB powder was sealed and densified in a plastic tube with tabs at each end connecting to the workstation.The conductivity (σ) was calculated based on the equation $\\\\sigma = \\\\mathsf { L } / ( \\\\mathsf { A } { \\\\cdot } \\\\mathsf { R } )$ ,where L, A,and R refer to the length,sectional area,and measured resistance,respectively,of the sample cylinder. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,713,489,858]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theoretical Calculations:The Vienna Ab-lnitio Simulation Package (VASP) was employed for density functional theory (DFT) calculations.[26] The Perdew-Burke-Ernzerhof (PBE) formulation was used within the generalized gradient approximation (GGA),[27] while the projected augmented wave (PAW) potentials were applied to describe the ionic cores,[28] The self-consistency of electronic energy was determined by the energy change smaller than $1 0 ^ { - 5 } \\\\thinspace \\\\thinspace \\\\mathrm { e V } .$ The geometryoptimization was converged to the energy change smaller than $0 . \\\\mathsf { \\\\bar { O } } 2 \\\\mathsf { e } \\\\mathsf { V } \\\\mathring { \\\\mathsf { A } } ^ { - 1 }$ .The vacuum spacing perpendicular to the plane was set to be $1 8 { \\\\mathring { \\\\mathsf { A } } } .$ The weak interaction was described by $\\\\mathsf { D F T } + \\\\mathsf { D } 3$ method using empirical correction in Grimme's scheme,[29] Additionally, the Gibbs free energy was calculated based on the equation below: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,870,223,886]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0670/83129f5f913ff145e71c3584e1842a1bb3ac1f345862f79a103b1967278f7063.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathsf { C } = E _ { e l e c } + E _ { Z P E } - \\\\mathsf { \\\\Delta } T S\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[84,896,489,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\mathsf { E } _ { \\\\mathrm { e } | \\\\mathrm { e } \\\\mathrm { c } }$ refers to the electronic energy at $0 \\\\mathrm { ~ K ~ }$ calculated by DFT, $\\\\mathsf { E } _ { \\\\mathsf { Z P E } }$ is assigned to the zero-point energy term. S and $\\\\intercal$ are the entropy and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,95,919,131]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"the absolute temperature (i.e.,298.15 K in this case),respectively.The adsorption energies were determined as the Gibbs energy differences of the projected system before and after relaxation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,157,700,173]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Supporting Information \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,181,919,205]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Supporting Information is available from the Wiley Online Library or from the author. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,231,668,248]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acknowledgements \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,255,919,364]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The authors thank the financial support from the National Natural Science Foundation of China (22109072,52372286),Natural Science Foundation of Jiangsu Province (BK20210349),Fundamental Research Funds for the Central Universities (30922010304),Foundation of Key Laboratory for Palygorskite Science and Applied Technology of Jiangsu Province (HPK202102),Open Project Foundation of National & Local Joint Engineering Research Center for Mineral Salt Deep Utilization (SF202105),the 2021 Qinlan Project of Jiangsu Province,the Jiangsu University of Technology Zhongwu Youth Innovative Talent Support Program. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,390,662,406]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conflict of Interest \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,414,747,425]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The authors declare no conflict of interest. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,451,729,468]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data Availability Statement \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,475,919,512]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The data that support the findings of this study are available on request from the corresponding author.The data are not publicly available due to privacy or ethical restrictions. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,538,591,553]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,562,917,586]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"chromium boride,dual sulphophilicity,electrocatalysis,lithium-sulfur batteries \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[707,602,919,638]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Received: June 12,2023 \\nRevised: August 5,2023 \\nPublished online: September 25,2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,951,216,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Adv. Sci. 2023,10,2303830 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[788,20,917,58]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADVANCED SCIENCE OpenAccess \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[443,950,919,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2303830 (9 of 10) $\\\\circledcirc$ 2023 The Authors.Advanced Science published by Wiley-VCH GmbH \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[84,26,282,74]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADVANCED SCIENCENEWS www.advancedsciencenews.con \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[517,679,919,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"[1] C. Steven,A. Majumdar,Nature 20l2,488, 294. [2] a) M.Zhao,B.Q.Li, H.J.Peng,H.Yuan,J.Y.Wei,J.Q.Huang, Angew. Chem.,Int.Ed.2020,59,12636;b) M.Armand,J.M.Tarascon,Nature 2008,451, 652. [3] a)S.Chu,Y. Cui,N. Liu,Nat.Mater. 2017,16,16; b)J. M.Tarascon, M.Armand,Nature 200l,414,359. [4] a) Y.Chen,T. Y.Wang,H.J.Tian, D.W.Su,Q.Zhang, G.X.Wang, Adv. Mater.2021,33,2003666; b) S.Feng,Z. H.Fu,X.Chen,Q.Zhang, Infomat 2022,4, e12304. [5] a) G.R.Li,S.Wang,Y.N.Zhang,M.Li,Z.W.Chen,J. Lu,Adv. Mater. 2018,30,1705590; b) C.Y.Li, Y.Y. Li,Y.X. Fan,F.Wang,T.Y. Lei,W. Chen,A.J.Hu,L.J. Xue,J. W. Huang,C.T. Yang,Y. Hu,Y.T.Yan,Small 2022,18,2106657. [6] a) M.Jana, R.Xu,X. B.Cheng,J.S.Yeon,J. M.Park,J. Q. Huang, Q. Zhang,H.S.Park,Energy Environ.Sci.2020,13,1049;b) Z.Zhuang, Q. Kang,D.Wang,Y.LiNano Res.2020,13,1856;c)J.Pu,W.B.Gong, Z.X. Shen,L.T. Wang,Y. G.Yao,G. Hong,Adv.Sci. 2022,9,2104375; d) N.Sapkota,S.Chiluwal,P.Parajuli,A. Rowland,R.Podila,Adv. Sci. 2023,10,2206901. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[747,61,917,72]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"www.advancedscience.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[78,95,485,478]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"[7]a)F.Y. Shi,J.Y.Yu,C.H.Chen, S.P. Lau,W. Lv,Z.L. Xu,J.Mater. Chem.A 2022,10,19412; b) H.Tang,W.L.Li,L.M.Pan, C.P.Cullen, Y. Liu,A. Pakdel,D.H. Long,J. Yang,N. McEvoy, G.S. Duesberg, V. Nicolosi, C.F. Zhang,Adv.Sci.2018,5,1800502. \\n[8] a) M.L. Wang, Z.T. Sun,H.N.Ci,Z.X. Shi,L.Shen,C. H.Wei,Y.D. Yi,X.Z.Yang,J.Y. Sun,Angew. Chem.,Int.Ed.2021,60,24558; b) D. Q.Cai,J.L.Yang,T. Liu,S.X. Zhao,G. Z. Cao,Nano Energy 2021,86, 106452; c) K.Y. Zou,X.X. Chen,W.T. Jing,X. Dai,P.F.Wang,Y. Liu, R. Qiao,M.Shi,Y.Z.Chen,J.J.Sun,Y.N. Liu,Energy Storage Mater. 2022,48,133;d) K. Xi,W.Wang,S.Liu,G.R.Li, X.P.Gao,Energy Storage Mater 2022,45,1229. \\n[9] X.Liang,C.Y. Kwok,L.M.Fernanda,Q. Pang,C. Marine,H. Huang, C.J.Hart,D.Houtarde, K. Kaup,H.Sommer,Adv.Energy Mater.2016, 6,1501636. \\n[10] G. M. Zhou,H.Z. Tian,Y. Jin,X.Y. Tao,B.F. Liu,R.F. Zhang,W.S. Zhi, D. Zhuo,Y.Y.Liu,J.Sun,j.Zhao,C.X.Zu,D.S.Wu,Q.F. Zhang, Y. Cui, PANS 2017,114,840. \\n[11] Y. B. Xiao, Y. C. Xiang,S.J. Guo,J. Wang, Y. Ouyang, D. X. Li, Q. H. Zeng,W. Gong,L.Y. Gan,Q. Zhang, Energy Storage Mater.2022,51, 882. \\n[12] a) G.R.Li, H.Y.Li, H. B. Zeng,J. Inorg. Mater. 2021,32,152; b) B. Wang,L.Wang,B. Zhang,Z. Kong,S.Y. Zeng,M.W. Zhao,Y.T. Qian, L. Q.Xu,Energy Storage Mater.202,45,130;c) B.Wang,L.Wang, Y. Y. Kong,F. B. Wang, Z.X.Jing, X. F. Yang,Y.T. Qian,M. Chen,L. Q.Xu,Adv. Energy Mater. 2023,13,2300590;d) B.Wang,L.Wang, B. Zhang,S.Y. Zeng,F.Tian,J. M. Dou,Y.T. Qian,L. Q. Xu,ACS Nano 2022, 16,4947. \\n[13] a) S.Carenco,D. Portehault,C. Boissiere,N.Mezailles,C.Sanchez, Chem.Rev.2013,113,7981; b)J.P.Scheifers,Y.M. Zhang,B.P.T. Fokwa,Acc.Chem.Res.2017,50,2317. \\n[14] C. C. Li,X.B. Liu,L. Zhu,R.Z. Huang,M.W. Zhao,L. Q.Xu,Y.T. Qian,Chem.Mater 2018,30,6969. \\n[15] Z.J.Li, H.R.Jiang,N.C.Lai,T.S. Zhao,Y.C.Lu,Chem. Mater. 2019, 31,10186. \\n[16] J. R. He,A. Bhargav,A. Manthiram,Adv. Mater. 2020,32,2004741. \\n[17] V.Natu,S.S.Kota,M.W. Barsoum,J. Eur. Ceram. Soc.2020,40,305. \\n[18] S.L. Ma, K. Bao,Q.Tao,P.W. Zhu,T. Ma,B. Liu,Y.Z. Liu,T.Cui, Sci. Rep.2017,7,43759. \\n[19] X.Y. Liu,Y. Gong,ACS Appl. Nano Mater. 2022,5,10183. \\n[20] S.Yu,W.Cai,L.Chen,L.Song,Y. Song,J.Energy Chem.2021,55, 533. \\n[21] X. R. Zhang,Z. H.Shen,Y. Wen,Q.Y. He,J.Yao, H.Y. Cheng,T. Gao, X.M.Wang,H.G. Zhang,H. Jiao,ACS Appl. Mater. Interfaces 2023, 15,21040. \\n[22] a) X. Z. Zhang,C.Q. Shang,E.M.Akinoglu, X.Wang, G.F. Zhou, Adv. Sci. 2020,7,2002037; b) J.Y. Zhang,G.B.Xu,Q. Zhang,X. Li,Y. Yang,L.W. Yang,J.Y. Huang,G.M. Zhou,Adv. Sci.2022,9,2201579. \\n[23] a) G. R. Li, W. Lei, D. Luo,Y.P.Deng,Z. P. Deng,D.L. Wang,A. P. Yu, Z.W. Chen, Energy Environ. Sci.2018,11,2372; b) W.L.Qiu,G.R.Li, D. Luo,Y.G. Zhang, Y. Zhao,G.F. Zhou,L. L. Shui, X. Wang, Z. W. Chen,Adv. Sci.2021,8,2003400. \\n[24] a) F.Y. Fan,W.C. Carter,Y. M. Chiang,Adv. Mater. 2015,27,5203;b) K. Kim,J. Kim,J.H.Moon,Adv.Sci.2023,10,2206057. \\n[25] Z.J. Li,Y. C. Zhou,Y. Wang,Y. C. Lu,Adv. Energy Mater. 2019,9, 1802207. \\n[26] G. Kresse,J.Furthmuller, Comput. Mater. Sci.1996,6,15. \\n[27] J.P.Perdew,K. Burke,M.Ernzerhof,Phys.Rev. Lett.1996,77,3865. \\n[28] P.E. Blochl,Phys.Rev. B1994,50,17953. \\n[29] S.Grimme,J.Antony,S.Ehrlich,H. Krieg,J. 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We also administered a questionnaire to assess the risk factors associated with the occurrence of resistance to commonly used antibiotics. Materials and Methods: We collected milk samples from 110 lactating dairy goats in Thika subcounty to screen for subclinical mastitis using the California mastitis test. Bacterial isolation and identification were performed according to colony morphology, the hemolytic pattern on sheep blood agar, lactose fermentation on MacConkey plates, Gram staining, and standard biochemical tests. The antibiotic susceptibility of the isolates was determined by the agar disk diffusion method using penicillin G, cephalexin, cefoxitin, and cefotaxime antibiotic disks. The double-disk synergy test using amoxicillin-clavulanic acid was employed as a confirmatory test for extended-spectrum beta-lactamase (ESBL) production. Fisher's exact test was used to determine the risk factors associated with the occurrence of antibiotic resistance (p <= 0.05 was considered significant). Results: Of the 110 dairy goats sampled, 72.7% (80) were positive for subclinical mastitis. Isolation and identification of the bacteria from the positive samples yielded 149 bacteria isolates, including Staphylococcus aureus, Klebsiella pneumoniae, Acinetobacter spp., Yersinia spp., coagulase-negative staphylococci, and Escherichia coli. A high percentage (76.5%, 114/149) of the bacterial isolates was resistant to at least one of the tested antibiotics. At least 56/106 isolates (52.8%) showing cross-resistance to the beta-lactam antibiotics were resistant to all four of the tested antibiotics, while only one isolate was resistant to three antibiotics (penicillin G, cephalexin, and cefoxitin). The double-disk synergy test confirmed that none of the isolates possessed ESBLs. Pre- and post-milking practices (p=0.0336) were found to be significantly associated with the occurrence of antibiotic resistance. Conclusion: A large proportion of the goats in our study cohort were infected with beta-lactam-resistant bacteria associated with subclinical mastitis. Because the identified bacteria are of zoonotic importance, further studies should be undertaken to determine the transmission dynamics between humans and livestock and to identify novel intervention strategies."},"author":{"kind":"string","value":"Okoko, Irene Mkavi|Maina, Naomi|Kiboi, Daniel|Kagira, John"},"content_list":{"kind":"list like","value":[{"bbox":"[102,87,894,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\beta$ -lactam resistance in bacteria associated with subclinical mastitis in goats in Thika Subcounty, Kenya ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[245,131,749,147]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Irene Mkavi Okoko<sup>1</sup>, Naomi Maina<sup>1,2</sup>, Daniel Kiboi<sup>2</sup> and John Kagira<sup>3</sup> ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,156,902,204]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Department of Molecular Biology and Biotechnology, Pan-African University, Institute of Basic Sciences, Technology and Innovation, P.O. Box 62000-00200, Nairobi, Kenya; 2. Department of Biochemistry, Jomo Kenyatta University of Agriculture and Technology, P.O. Box 62000-00200, Nairobi, Kenya; 3. Department of Animal Sciences, Jomo Kenyatta University of Agriculture and Technology, P.O. Box 62000-00200, Nairobi, Kenya. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[164,205,833,239]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corresponding author: John Kagira, email: jkagira@gmail.com \nCo-authors: IMO: okokomkavi@gmail.com, NM: nmaina@jkuat.ac.ke, DK: dmuthui@jkuat.ac.ke \nReceived: 01-02-2020, Accepted: 08-06-2020, Published online: 25-07-2020 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,244,915,282]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"doi: www.doi.org/10.14202/vetworld.2020.1448-1456 How to cite this article: Okoko IM, Maina N, Kiboi D, Kagira J (2020) β-lactam resistance in bacteria associated with subclinical mastitis in goats in Thika Subcounty, Kenya, Veterinary World, 13(7): 1448-1456. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[455,291,542,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,312,914,354]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Aim: This study determined the resistance pattern to $\\beta$ -lactam antibiotics of bacteria isolated from goats with subclinical mastitis in Thika subcounty, Kenya. We also administered a questionnaire to assess the risk factors associated with the occurrence of resistance to commonly used antibiotics. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,359,914,464]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Materials and Methods: We collected milk samples from 110 lactating dairy goats in Thika subcounty to screen for subclinical mastitis using the California mastitis test. Bacterial isolation and identification were performed according to colony morphology, the hemolytic pattern on sheep blood agar, lactose fermentation on MacConkey plates, Gram staining, and standard biochemical tests. The antibiotic susceptibility of the isolates was determined by the agar disk diffusion method using penicillin G, cephalexin, cefoxitin, and cefotaxime antibiotic disks. The double-disk synergy test using amoxicillin-clavulanic acid was employed as a confirmatory test for extended-spectrum $\\beta$ -lactamase (ESBL) production. Fisher's exact test was used to determine the risk factors associated with the occurrence of antibiotic resistance $(p \\leq 0.05$ was considered significant). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,470,912,576]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Results: Of the 110 dairy goats sampled, $72.7\\%$ (80) were positive for subclinical mastitis. Isolation and identification of the bacteria from the positive samples yielded 149 bacteria isolates, including Staphylococcus aureus, Klebsiella pneumoniae, Acinetobacter spp., Yersinia spp., coagulase-negative staphylococci, and Escherichia coli. A high percentage $(76.5\\%, 114/149)$ of the bacterial isolates was resistant to at least one of the tested antibiotics. At least 56/106 isolates $(52.8\\%)$ showing cross-resistance to the $\\beta$ -lactam antibiotics were resistant to all four of the tested antibiotics, while only one isolate was resistant to three antibiotics (penicillin G, cephalexin, and cefoxitin). The double-disk synergy test confirmed that none of the isolates possessed ESBLs. Pre- and post-milking practices $(p = 0.0336)$ were found to be significantly associated with the occurrence of antibiotic resistance. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,582,914,624]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conclusion: A large proportion of the goats in our study cohort were infected with $\\beta$ -lactam-resistant bacteria associated with subclinical mastitis. Because the identified bacteria are of zoonotic importance, further studies should be undertaken to determine the transmission dynamics between humans and livestock and to identify novel intervention strategies. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[82,627,640,643]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords: bacteria, dairy goats, Kenya, subclinical mastitis, $\\beta$ -lactam resistance. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[82,661,194,674]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,680,487,813]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The dairy goat agricultural industry is rapidly expanding. In 2017, the global dairy goat population was estimated at 218 million, producing approximately 18.7 million tons of milk [1]. Sub-Saharan African countries account for a goat population of 372,716,040, with $16\\%$ farmed as dairy goats [2]. The dairy goat population in Kenya was estimated as $>251,100$ in 2011, with the milk used as an alternative to cow's milk. At present, approximately $4.7\\%$ of ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,661,914,931]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"the total milk consumed in Kenya is obtained from goats [3]. The emerging interest in goats and their derivative products, including milk, is due to the associated health and nutritional benefits, low production cost, ease of management, and goat dairy farming being a source of livelihood, among other factors [1,3,4]. Despite the vast interest in dairy goat production, the enterprise faces a myriad of abiotic and biotic challenges. Of major concern is mastitis, a disease resulting from the infection of mammary glands by viruses, fungi, or bacteria, with the latter being the major etiological agents [5]. Following mammary gland infection, white blood cells and epithelial (somatic) cells are produced by the goat as a defense mechanism [6]. A relatively small number of epithelial cells are present in milk from healthy milk-secreting tissue (1,000,000 cells/ml), and somatic cell counts (SCC) $>1,500,000$ cells/ml indicate intramammary ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,48,515,74]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Veterinary World, EISSN: 2231-0916 \nAvailable at www.veterinaryworld.org/Vol.13/July-2020/27.pdf ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[759,48,914,74]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RESEARCH ARTICLE Open Access ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[80,826,485,929]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Copyright: Okoko, et al. Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[82,936,339,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Veterinary World, EISSN: 2231-0916 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[875,938,912,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1448 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[85,71,484,290]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"infection [6]. Mastitis is associated with symptoms such as udder inflammation and tissue damage, leading to reduced milk quality and quantity [7]. Mastitis manifests either clinically (symptomatic) or subclinically (asymptomatic), the latter being more prevalent in goats [8]. Coupled with the treatment cost, disease management has led to significant economic losses [9]. The mastitis management strategies in Kenya entail the use of antibiotics, including tetracyclines, $\\beta$ -lactams, trimethoprim/sulfamethoxazole, nitrofurans, aminoglycosides, and quinolones [10], the majority of which belong to the $\\beta$ -lactam family, such as penicillin and cephalosporins [11]. $\\beta$ -lactam antibiotics share a common structural feature, the $\\beta$ -lactam ring, that inhibits bacterial cell wall synthesis [12]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[85,290,485,580]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Reports of bacterial resistance to $\\beta$ -lactam antibiotics in cases of mastitis have recently emerged [13-15] and have been attributed to the ability of the bacteria causing mastitis to produce $\\beta$ -lactamase enzymes and modified penicillin-binding proteins that hydrolyze or inhibit the binding of $\\beta$ -lactam antibiotics, respectively [12,16]. Antibiotic resistance in pathogens that cause subclinical mastitis in animals poses a risk of the transfer of antibiotic resistance to humans, especially where raw milk and raw milk products are consumed, such as in Kenya [17,18]. Antibiotic-resistant bacteria cause severe infections with limited treatment options and fatalities, leading to a high cost of treatment, loss of productivity, and adverse economic consequences in the affected country [19,20]. Although a Kenyan study reported resistance to penicillin G in $100\\%$ of the bacterial isolates causing subclinical mastitis in goats [21], the resistance of mastitis-causing pathogens to other $\\beta$ -lactam antibiotics remains undocumented in Kenya. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[85,581,484,696]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This study determined the prevalence of bacterial resistance to selected $\\beta$ -lactam antibiotics in cases of subclinical mastitis using the agar disk diffusion method. We also report the risk factor associated with the occurrence of resistance to these antibiotics. The study highlighted the consumption of raw milk as a potential vehicle for the transfer of bacterial resistance to humans. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[85,705,278,717]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Materials and Methods ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[85,724,208,734]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ethical approval ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[85,737,484,794]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since no invasive procedure was performed on the animals, approval from the Institutional Animal Ethics Committee to conduct the study was not required. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[85,801,253,812]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Study area and period ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[85,813,485,929]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The study was conducted in Thika subcounty of Kiambu County, located in Central Kenya, approximately $40\\mathrm{km}$ from Nairobi $(1^{\\circ}1^{\\prime}\\mathrm{S}37^{\\circ}5^{\\prime}\\mathrm{E})$ . Rainfall is bimodal, ranging from 500 to $1300~\\mathrm{mm}$ , and the average temperature is $18.7^{\\circ}\\mathrm{C}$ . The study was conducted for a period of 10 months (January 2019 to October 2019). Although no formal census has been undertaken, farmers have taken up dairy goat farming ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,72,910,117]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"in this region [21]. Farmers practice zero-grazing, open-grazing, and tethering-grazing systems [Personal communication, Veterinary officer, Thika subcounty]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,123,910,145]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Study design, sample size calculation, and questionnaire administration ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,148,910,483]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A cross-sectional field- and laboratory-based study design was used. The sample size of 110 lactating dairy goats was calculated using an adjusted formula for small populations [22], whereby the sample size for an infinite population with an estimated prevalence can be calculated for a $95\\%$ confidence interval as $\\mathrm{n = (1.96^2[\\mathrm{P_{exp}}] / [1 - \\mathrm{P_{exp}}]) / d^2}$ where $n$ is the required sample size, $\\mathrm{P_{exp}}$ is the expected prevalence, and $d$ is the desired absolute precision. We used a prevalence of $50.9\\%$ subclinical mastitis, which was determined in a study of goats in Thika East subcounty [21], and a desired absolute precision of $5\\%$ , resulting in a required sample size of 384 goats. In relatively small populations, such as that of dairy goats in Thika subcounty, it is possible to select a smaller sample size than one from a theoretically infinite population to achieve the same degree of precision using the adjusted formula $\\mathrm{N_{adj} = N\\times n / N + n}$ , where $n$ is the sample size as calculated above and $N$ is the size of the study population [22]. Given that the size of the study population of lactating dairy goats was approximately 150 [21], a sample size of 108 does was calculated. We sampled 110 does. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,485,910,703]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We targeted small-scale farmers with a herd size of $\\leq 15$ goats who consented to participate. Sampling was performed using the snowball technique and sampling to redundancy due to the absence of a formal list of dairy farmers in the area [23]. The local agriculture extension officer assisted in identifying the first farmers, who later recruited other farmers until all the farmers had been included. The final cohort comprised 110 lactating goats with no clinical signs of mastitis, sampled from 41 farms. We administered a questionnaire to the farmers to collect data pertaining to age, parity, mastitis history, mastitis management, cleaning schedules of the housing structure, pre- and post-milking practices, and other variables. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,711,877,722]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Milk sampling and California mastitis test (CMT) ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,725,910,929]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The milk samples were aseptically collected, as previously described [21]. CMTs (ImmuCell, ME, USA) were performed to screen for subclinical mastitis, according to the manufacturer's instructions with minor modifications. Briefly, $5\\mathrm{ml}$ of milk from different teats were milked into the CMT paddle, and an equal volume of the CMT reagent was added. A positive CMT test was scored as 2 or 3, indicating an SCC of 2,560,000 and $\\geq 10,000,000$ respectively [6]. In addition, $5\\mathrm{ml}$ of milk from the samples with CMT scores of 2 and 3 were collected into sterile universal bottles, placed in a cooler box containing ice packs, and transported within $12\\mathrm{h}$ of collection to the microbiology laboratory at Jomo Kenyatta University of ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[284,49,712,61]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Available at www.veterinaryworld.org/Vol.13/July-2020/27.pdf ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[85,938,337,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Veterinary World, EISSN: 2231-0916 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[877,938,910,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1449 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[85,72,482,101]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Agriculture and Technology, Juja, Kenya, for bacteriological culture and isolation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[85,108,349,118]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bacterial culture and identification ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[85,121,482,514]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sheep blood agar (HiMedia, Mumbai, India) and MacConkey agar (Oxoid, Basingstoke, UK) were freshly prepared as per the manufacturer's guidelines and inoculated with approximately $100\\mu \\mathrm{l}$ of each milk sample per plate, respectively. In routine microbiological diagnosis, a sample of $10 - 100\\mu \\mathrm{l}$ of milk is used for culture. However, an inoculum of $100\\mu \\mathrm{l}$ has a higher sensitivity when isolating bacteria causing intramammary infection [24]. Hence, we used $100\\mu \\mathrm{l}$ for inoculation. The milk was evenly spread onto the surface of the agar plate using a sterile glass spreader and then incubated at $37^{\\circ}\\mathrm{C}$ for up to $48\\mathrm{h}$ . The morphological characteristics of the resulting bacterial colonies were assessed for shape, color, size, and texture as well as lactose fermentation ability on MacConkey plates and hemolysis patterns on sheep blood agar plates. Hemolysis of the blood cells in the sheep blood agar was evaluated by noting any color change in the media, particularly around and under the colonies. A plate was said to have mixed growth when the colonies on that plate showed distinct morphological characteristics, and each of the distinguishable colonies was then subcultured in nutrient agar to obtain pure colonies. The pure cultures were examined for colony morphology, Gram staining characteristics (to distinguish between Gram-positive and Gram-negative isolates), and biochemical reactions. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[85,516,482,646]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A catalase test using $3\\%$ hydrogen peroxide was performed on all Gram-positive isolates to identify Staphylococcus spp. Staphylococcus spp. were also identified based on their growth and fermentative attributes on mannitol salt agar (HiMedia) and the results of an oxidase test (HiMedia). Staphylococcus aureus was differentiated from coagulase-negative staphylococci (CNS) using slide and tube coagulase tests that employed rabbit plasma. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[85,648,482,851]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The oxidase test was performed on all Gram-negative isolates to identify Enterobacteriaceae, and those isolates that were oxidase negative were subsequently subcultured on MacConkey agar to assess lactose fermentation. Then, the triple sugar iron (Oxoid), methyl red/Voges-Proskauer (HiMedia), urease (urea broth base; HiMedia), indole (tryptone broth; HiMedia), citrate utilization (Simmons citrate agar; HiMedia), motility (sulfur indole motility media; HiMedia), catalase, and mannitol salt agar (HiMedia) tests were employed to classify the Gram-negative bacteria. The results of culture, Gram staining, and biochemical tests were used for final bacterial identification [25-27]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[85,859,433,870]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Antibiotic susceptibility testing of the isolates ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[85,872,482,929]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Antibiotic susceptibility testing was performed using the Kirby-Bauer disk diffusion method [28], and the results were interpreted following the Clinical Laboratory Standard Institute guidelines (https:// ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,72,912,449]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"clsi.org/). Briefly, the bacterial suspension was prepared in sterile distilled water and standardized to $0.5\\mathrm{McFarland}$ 's standard (corresponding to approximately $1.5\\times 10^{8}$ colony-forming units/ml). Mueller-Hinton agar was prepared following the manufacturer's instructions, poured into inoculation plates to solidify, and inoculated with the standardized inoculums. Escherichia coli ATCC 25922 (American Type Culture Collection, Rockville, MA) was used for quality control. Bacterial resistance was tested using commercially available antibiotic disks of penicillin G $(10~\\mu \\mathrm{g})$ , cephalexin $(30~\\mu \\mathrm{g})$ , cefoxitin $(30~\\mu \\mathrm{g})$ , and cefotaxime $(30~\\mu \\mathrm{g})$ . These were randomly selected from $\\beta$ -lactam antibiotics commonly used in Kenya to treat bacterial infections in both animals and humans and included different generations and classes of $\\beta$ -lactam antibiotics, with penicillin G as a representative of the penicillin class of $\\beta$ -lactams. Cephalexin, cefoxitin, and cefotaxime were representative of the first-, second- and third-generation cephalosporins, respectively. The disks were placed on the agar surface $24\\mathrm{mm}$ from center to center, and the plates were incubated at $37^{\\circ}C$ for $18\\mathrm{h}$ . The zones of growth inhibition, including the disk diameters, were measured in millimeters and recorded as resistant, intermediate, or susceptible to the specific antibiotics [29]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,456,910,480]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Determining the proportion of extended-spectrum $\\beta$ -lactamase (ESBL) producers ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,481,912,784]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The double-disk synergy test was used for phenotypic confirmation of ESBL producers, as previously described [30]. E. coli ATCC 25922 and E. coli ATCC 35218 (American Type Culture Collection) were used as negative and positive controls, respectively. Commercially available disks (HiMedia) of cefotaxime $(30\\mu \\mathrm{g})$ and ceftazidime $(30\\mu \\mathrm{g})$ together with amoxicillin-clavulanic acid $(30\\mu \\mathrm{g})$ were placed $25\\mathrm{mm}$ apart from each other on the surface of inoculated Mueller-Hinton agar plates and incubated at $37^{\\circ}\\mathrm{C}$ for $24\\mathrm{h}$ . The test was regarded as positive when the decreased susceptibility to cefotaxime and/or ceftazidime was joined with a clear-cut augmentation of the inhibition zone of cefotaxime and/or ceftazidime in front of the amoxicillin-clavulanic acid antibiotic disk, forming a characteristic \"champagne-cork\" or \"keyhole\" shape [30]; otherwise, the results were regarded as negative. The bacterial samples were stored in tryptic soy broth (Scharlab SL, Barcelona, Spain) containing $20\\%$ glycerol at $-70^{\\circ}\\mathrm{C}$ for long-term preservation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,791,910,826]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Determining the relationship between the occurrence of resistance against the selected $\\beta$ -lactam antibiotics and doe- and farm-level variables ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,829,910,929]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A well-structured questionnaire was developed based on that of Mahlangu et al. [21]. The questionnaire was used to collect data regarding farm-level and doe-level variables, such as mastitis history, previous antibiotic treatment of mastitis, pre- and post-milk-ing practices, and the cleaning schedule of the housing structure as well as the doe name, age, parity, and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[284,49,712,61]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Available at www.veterinaryworld.org/Vol.13/July-2020/27.pdf ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[85,938,337,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Veterinary World, EISSN: 2231-0916 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[877,938,910,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1450 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[80,71,485,159]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"lactation stage. It was written in English and administered to the farmers in English before sample collection. However, based on the farmers' level of understanding, the agriculture extension officer helped with the translation to Swahili or Kikuyu (the vernacular language used by the local farmers in the study area). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,159,485,233]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The results of the agar disk diffusion test were used to determine the relationship between the occurrence of resistance against the $\\beta$ -lactam antibiotics and doe- and farm-level variables obtained from the questionnaire. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[82,237,230,250]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Statistical analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,250,485,500]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Data were entered into a Microsoft Office Excel spreadsheet Version 16.0.12827.20336 (Microsoft Corp., Redmond, WA, USA) and then analyzed using StatXact 11 (Cytel Inc., Cambridge, MA, USA) and GraphPad Prism software version 7.0.0 (GraphPad Software, CA, USA). Descriptive statistics were expressed as percentages and frequencies, and data were presented in tables and histograms. The prevalence of subclinical mastitis was determined as the proportion of CMT-positive goats out of the total number of sampled goats. The proportion of the resistant bacterial isolates was determined as the number of all the isolates resistant to at least one of the antibiotics out of the total number of bacterial isolates. The association of the doe- and farm-level variables with the occurrence of resistance was determined by Fisher's exact test. $p \\leq 0.05$ was considered statistically significant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[82,506,152,519]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Results ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[82,524,485,549]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Prevalence of subclinical mastitis in goats from Thika subcounty ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,549,485,624]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We sampled 110 lactating dairy goats from 41 farms in Thika subcounty with no more than 15 lactating dairy goats per farm (average, three lactating goats). Subclinical mastitis was present in 80 of the dairy goats sampled $(72.7\\%)$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[82,627,485,653]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bacteria causing subclinical mastitis in goats from Thika subcounty ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,653,485,843]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bacteria were cultured in $97.5\\%$ (78/80) of the CMT-positive milk samples. The volume of $2.5\\%$ (2/80) of the samples was insufficient for microbiology culture and analysis after performing the CMT test. We cultured 149 isolates from the raw milk samples. In descending order of occurrence, the isolates included S. aureus, Klebsiella pneumoniae, Acinetobacter spp., Yersinia spp., CNS, E. coli, Enterobacter intermedius, Klebsiella oxytoca, Proteus vulgaris, Citrobacter freundii, Citrobacter diversus, Morganella morganii, Proteus mirabilis, Serratia fonticola, Providencia stuartii, Serratia marcescens, and Staphylococcus epidermidis (Table-1). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,848,485,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Resistance profiles of bacteria isolated from CMT-positive milk samples from goats in Thika subcounty to the selected $\\beta$ -lactam antibiotics ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,885,485,929]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The disk diffusion tests showed that $23.5\\%$ (35/149) of the bacterial isolates were susceptible to all the antibiotics tested, and $76.5\\%$ (114/149) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,111,907,376]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0994/81cb1fbcfc8ea541b87411756b35132f95c6c5b9f9667d294d6c5dbe07805e11.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":"<table><tr><td>Bacterial isolate</td><td>No. of isolates n=149</td><td>Proportion (%)</td></tr><tr><td>Staphylococcus aureus</td><td>26</td><td>17.5</td></tr><tr><td>Klebsiella pneumoniae</td><td>26</td><td>17.5</td></tr><tr><td>Acinetobacter spp.</td><td>24</td><td>16.1</td></tr><tr><td>Yersinia spp.</td><td>21</td><td>14.1</td></tr><tr><td>CNS</td><td>15</td><td>10.1</td></tr><tr><td>Escherichia coli</td><td>6</td><td>4.0</td></tr><tr><td>Enterobacter intermedius</td><td>5</td><td>3.4</td></tr><tr><td>Klebsiella oxytoca</td><td>4</td><td>2.7</td></tr><tr><td>Proteus vulgaris</td><td>4</td><td>2.7</td></tr><tr><td>Citrobacter freundii</td><td>3</td><td>2.0</td></tr><tr><td>Citrobacter diversus</td><td>3</td><td>2.0</td></tr><tr><td>Morganella morganii</td><td>3</td><td>2.0</td></tr><tr><td>Proteus mirabilis</td><td>2</td><td>1.3</td></tr><tr><td>Serratia fonticola</td><td>2</td><td>1.3</td></tr><tr><td>Providencia stuartii</td><td>2</td><td>1.3</td></tr><tr><td>Serratia marcescens</td><td>2</td><td>1.3</td></tr><tr><td>Staphylococcus epidermidis</td><td>1</td><td>0.7</td></tr><tr><td>Total</td><td>149</td><td></td></tr></table>","table_caption":"[\"Table-1: The proportion of bacteria isolated from CMT-positive milk samples obtained from goats in Thika subcounty. \"]","table_footnote":"[\"CNS=Coagulase-negative staphylococcus, CMT=California mastitis test \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[507,418,912,564]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"showed intermediate or complete resistance to at least one $\\beta$ -lactam. The susceptibility of the isolates to penicillin G, cephalexin, cefoxitin, and cefotaxime was $23.5\\%$ , $32.9\\%$ , $61.7\\%$ , and $36.2\\%$ , respectively. Intermediate resistance of the bacterial isolates to penicillin G, cephalexin, cefoxitin, and cefotaxime was $0\\%$ , $4.0\\%$ , $4.0\\%$ , and $10.7\\%$ , respectively. Complete resistance to penicillin G, cephalexin, cefoxitin, and cefotaxime was $76.5\\%$ , $63.1\\%$ , $34.2\\%$ , and $53.0\\%$ , respectively (Figure-1a). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,564,914,753]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cross-resistance to the $\\beta$ -lactam antibiotics was observed in $93.0\\%$ (106/114) of the resistant bacterial isolates, with $10.4\\%$ (11/106) resistant to penicillin G and cephalexin; $0.9\\%$ (1/106) resistant to penicillin G, cephalexin, and cefoxitin; and the majority $(52.8\\%, 56/106)$ resistant to penicillin G, cephalexin, cefoxitin, and cefotaxime. Resistance to penicillin G, cephalexin, and cefotaxime was observed in $30.2\\%$ (32/106) of the isolates, while $5.7\\%$ (6/106) showed resistance to penicillin G and cefotaxime (Figure-1b). In general, a larger proportion of the bacteria resistant to penicillin G showed cross-resistance to other $\\beta$ -lactam antibiotics (Table-2). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,759,914,784]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Non-ESBL-producing bacterial isolates from CMT-positive milk samples from goats in Thika subcounty ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[507,784,912,858]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The double-disk synergy test did not show any characteristic \"champagne-cork\" or \"keyhole\" inhibition zones. Thus, this phenotypic method confirmed that none of the bacterial isolates were ESBL producers. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,863,912,898]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Relationship between the occurrence of resistance to the selected $\\beta$ -lactam antibiotics and doe- and farm-level variables ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[507,900,912,929]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In terms of a history of mastitis, higher resistance to the selected $\\beta$ -lactam antibiotics occurred in the ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[282,49,712,61]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Available at www.veterinaryworld.org/Vol.13/July-2020/27.pdf ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[82,936,339,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Veterinary World, EISSN: 2231-0916 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[875,936,912,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1451 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[191,72,500,286]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0656/3a52f561c82e2cfc7e3f87c73caacbea7cc62e6aeccec5fabdb1225b768066cd.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[509,74,806,286]","code_body":null,"code_caption":null,"image_caption":"[\"Figure-1: Resistance pattern of the bacterial isolates to beta-lactam antibiotics. (a) Overall susceptibility profiles of the bacterial isolates from CMT-positive milk samples to penicillin G, cephalexin, cefoxitin, and cefotaxime. (b) Cross-resistance of the bacterial isolates from CMT-positive milk samples to the selected beta-lactam antibiotics. P=Penicillin G, CN=Cephalexin, FOX=Cefoxitin, CTX=Cefotaxime. \"]","image_footnote":"[]","img_path":"images/0969/252a72095c974736529657f3415d0906a13142502a90a8f5f5d0ea31509864d6.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[80,382,915,521]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0380/b42ec535406c7e9db2594ea4f2564bb2dba9f7f20476da25bf40ea10c589a480.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":"<table><tr><td rowspan=\"2\">Antibiotic</td><td colspan=\"6\">% proportion of resistant bacterial isolates</td></tr><tr><td>Staphylococcus aureus (n=26)</td><td>Klebsiella pneumoniae (n=26)</td><td>Acinetobacter spp. (n=24)</td><td>Yersinia spp. (n=21)</td><td>CNS (n=15)</td><td>Escherichia coli (n=6)</td></tr><tr><td>P</td><td>26.9</td><td>0</td><td>0</td><td>4.8</td><td>0</td><td>0</td></tr><tr><td>P and CN</td><td>0</td><td>3.8</td><td>8.3</td><td>4.8</td><td>6.7</td><td>33.3</td></tr><tr><td>P, CN, and FOX</td><td>3.8</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>P, CN, FOX, and CTX</td><td>38.5</td><td>34.6</td><td>54.2</td><td>33.3</td><td>40</td><td>16.7</td></tr><tr><td>P, CN, and CTX</td><td>3.8</td><td>42.3</td><td>4.2</td><td>19</td><td>20</td><td>33.3</td></tr><tr><td>P and CTX</td><td>0</td><td>3.8</td><td>8.3</td><td>0</td><td>20</td><td>0</td></tr></table>","table_caption":"[\"Table-2: Resistance to penicillin G, cephalexin, cefoxitin, and cefotaxime of the predominant bacterial isolates in CMT-positive milk samples from goats in Thika subcounty. \"]","table_footnote":"[\"CNS=Coagulase-negative staphylococcus, P=Penicillin G, CN=Cephalexin, FOX=Cefoxitin, CTX=Cefotaxime \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[80,552,485,687]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"does that had previous cases of mastitis compared with no mastitis history, but this relationship was not statistically significant $(p = 0.4242)$ . A higher prevalence of $\\beta$ -lactam resistance occurred in does that were administered antibiotics when they had mastitis compared with those where no treatment was received, but there was no significant association $(p = 0.5781)$ between antibiotic treatment and the occurrence of resistance (Table-3). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,688,485,929]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The occurrence of resistance to $\\beta$ -lactams was significantly affected by the pre- and post-milk-ing practices adopted by the farmers in the region $(p = 0.0336)$ . The highest prevalence $(94.1\\%)$ of resistance was observed in bacteria isolated from does where the farmers only used water and a reusable towel for cleaning before and after milking. There was a lower prevalence $(77.8\\%)$ of resistance in does where the farmers used water, disinfectant, and a reusable towel, while no resistance $(0.0\\%)$ was observed where the farmers used water and a disposable towel during milking. The highest resistance $(100.0\\%)$ occurred in does whose housing structures were cleaned every 2 days, after 2 weeks, 3 times a week, or irregularly compared with those that were cleaned once a week $(81.8\\%)$ , twice a week $(81.8\\%)$ , and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,604,915,750]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0260/1ab47889cf78de8d62a00a2e4387b9a9dd1cd3f6618adb4af6dbdb0b72918d5d.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":"<table><tr><td>Risk factor</td><td>Frequency of positive resistant</td><td>% proportion of positive resistant</td><td>p-value</td></tr><tr><td>History of mastitis</td><td></td><td></td><td></td></tr><tr><td>Yes</td><td>21/23</td><td>91.3</td><td>0.4242</td></tr><tr><td>No</td><td>46/55</td><td>83.6</td><td></td></tr><tr><td>Antibiotic treatment</td><td></td><td></td><td></td></tr><tr><td>Treatment administered</td><td>17/19</td><td>89.5</td><td>0.5781</td></tr><tr><td>No treatment administered</td><td>3/3</td><td>100</td><td></td></tr></table>","table_caption":"[\"Table-3: Effects of history of mastitis and antibiotic treatment on occurrence of resistance to the selected beta-lactam antibiotics in dairy goats with subclinical mastitis in Thika subcounty. \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[507,766,914,931]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"daily (87.5%). No resistance was seen in those that were cleaned every 3 days (0.0%). From the data in the questionnaire on goats whose house were cleaned every 3 days, the farmers used water alone and udder and teats were dried with a disposable towel during milking. From the data analysis on milking practices used by the farmers, which was found to be significantly associated with the occurrence of resistance, the goats whose farmers used water alone and udder and teats dried with disposable a towel showed no resistance to beta-lactam antibiotics. Therefore, despite the ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[282,49,715,61]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Available at www.veterinaryworld.org/Vol.13/July-2020/27.pdf ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[82,936,341,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Veterinary World, EISSN: 2231-0916 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[875,936,912,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1452 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[80,71,485,130]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"goats' housing structure being cleaned every 3 days, the milking practices used on these goats might be a contributing factor in preventing the occurrence of resistance in these goats. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,130,485,261]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On the other hand, the goats whose houses were cleaned every 2 days had their udder and teats dried using a reusable towel during milking. Therefore, despite the goats' housing structure being cleaned every 2 days, the milking practices used on these goats might be a factor contributing to the occurrence of resistance in these goats. However, resistance was not significantly affected by the cleaning schedule of the housing structure $(p = 0.2141)$ (Table-4). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,261,485,363]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Resistance was highest among does $>4$ years old $(88.9\\%)$ and does with a parity $\\leq 4$ $(85.7\\%)$ . Resistance was lowest in does $\\leq 4$ years old $(86.7\\%)$ and does with a parity $>4$ parity $(80.0\\%)$ . However, there was no significant association between age and parity, respectively, and the occurrence of resistance $(p = 0.6203$ and $p = 0.333$ , respectively) (Table-5). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[82,370,179,384]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Discussion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,388,487,667]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Mastitis is an economically important disease of dairy goats and cattle. This infection is mainly subclinical in goats, making it impossible to diagnose by physical examination [8]. Losses resulting from subclinical mastitis include decreased milk production and quality and increased veterinary expenses as well as progression to the clinical form [31]. This study determined resistance against penicillin G, cephalexin, cefoxitin, and cefotaxime in bacterial isolates from milk of goats with subclinical mastitis in Thika subcounty. We found a higher prevalence of subclinical mastitis than that reported in Northern region, Bangladesh [32], Tando Jam, Pakistan [33] and Nigeria [34]. However, our result was in agreement with that recorded in the Mount Kenya region, Kenya [35], the state of Minas Gerais, Brazil [36], and Thika East subcounty [21]. Fluctuations in the prevalence of mastitis in different geographical regions are attributed to the complexity of the disease resulting from the interaction of several ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,71,912,145]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"factors, including management and husbandry strategies, geographical distribution, the herd's health condition, biological agents causing mastitis, and environmental and nutritional conditions as well as the sample size used in the studies [34]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,145,914,479]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our results showed that the highest proportion $(28.18\\%)$ of the bacterial isolates belonged to the Staphylococcus genus, particularly S. aureus $(17.45\\%)$ , which was similar to other studies reporting that the highest proportion of the isolates from goats with subclinical mastitis was S. aureus, ranging from $23.4\\%$ to $61.64\\%$ [15,33,37,38]. Staphylococci are listed as the major causative agents in the development of contagious mastitis [5]. This is attributed to the ability of these bacteria to produce highly toxic metabolites that damage the membranes of mammary glands [39]. In addition, Staphylococcus is effectively transmitted through several routes [39]. Therefore, in regions where hand milking is mostly practiced, such as Thika subcounty, Staphylococcus can be spread during milking operations, which could be associated with its high prevalence. An intramammary infection of S. aureus is a public health concern since the bacteria produce thermostable exotoxins that cause toxic shock syndrome and staphylococcal food poisoning. Furthermore, S. aureus can also infect humans and animals when unpasteurized milk and milk products are consumed [34,40]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,479,914,668]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Enterobacteriaceae are also associated with mastitis [41]. We isolated a relatively high proportion of members of the Enterobacteriaceae family $(55.5\\%)$ in our study, which is comparable to a previous report of the isolation of $64.5\\%$ Enterobacteriaceae from milk samples of goats with mastitis in Thika East subcounty [21]. Coliforms, represented by Klebsiella spp. $(20.1\\%)$ , Escherichia spp. $(4.0\\%)$ , Citrobacter spp. $(4.0\\%)$ , and Enterobacter spp. $(3.3\\%)$ , have been identified as the main cause of environmental mastitis in small ruminants. Coliforms are normally associated with environmental contamination of the udder and the establishment of intramammary infection [31]. The ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,708,914,932]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0613/4a9d25b16cba314384e324aef5d31391b3bd2cc0f62bbde47364b05259195d2f.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":"<table><tr><td>Risk factor</td><td>Frequency of positive resistant</td><td>% proportion of positive resistant</td><td>p-value</td></tr><tr><td>Hygiene during milking</td><td></td><td></td><td></td></tr><tr><td>Use of water alone and udder and teat dried with disposable towel</td><td>0/2</td><td>0</td><td>0.0336</td></tr><tr><td>Use of water, disinfectant, and udder and teat dried with reusable towel</td><td>14/18</td><td>77.8</td><td></td></tr><tr><td>Use of water alone and udder and teat dried with reusable towel</td><td>16/17</td><td>94.1</td><td></td></tr><tr><td>Use of water alone and udder and teat are not dried</td><td>23/26</td><td>88.5</td><td></td></tr><tr><td>Goat house cleaning schedule</td><td></td><td></td><td></td></tr><tr><td>Once a week</td><td>18/22</td><td>81.8</td><td>0.2141</td></tr><tr><td>Daily</td><td>21/24</td><td>87.5</td><td></td></tr><tr><td>After 3 days</td><td>0/2</td><td>0</td><td></td></tr><tr><td>After 2 days</td><td>4/4</td><td>100</td><td></td></tr><tr><td>After 2 weeks</td><td>4/4</td><td>100</td><td></td></tr><tr><td>Twice a week</td><td>9/11</td><td>81.8</td><td></td></tr><tr><td>Thrice a week</td><td>2/2</td><td>100</td><td></td></tr><tr><td>Irregularly</td><td>9/9</td><td>100</td><td></td></tr></table>","table_caption":"[\"Table-4: Effects of pre- and post-milking practices and cleaning schedule on occurrence of resistance to the selected beta-lactam antibiotics in dairy goats with subclinical mastitis in Thika subcounty. \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[282,49,712,61]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Available at www.veterinaryworld.org/Vol.13/July-2020/27.pdf ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[82,936,337,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Veterinary World, EISSN: 2231-0916 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[875,936,912,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1453 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[82,111,485,233]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0930/575641aa0de8957ff4465a242f1183bfda49d9b0fa2c833342c5e222d42e4857.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":"<table><tr><td>Risk factor</td><td>Frequency of positive resistant</td><td>% proportion of positive resistant</td><td>p-value</td></tr><tr><td>Age</td><td></td><td></td><td></td></tr><tr><td>≤4 years</td><td>39/45</td><td>86.7</td><td>0.6203</td></tr><tr><td>>4 years</td><td>24/27</td><td>88.9</td><td></td></tr><tr><td>Parity</td><td></td><td></td><td></td></tr><tr><td>≤4</td><td>54/63</td><td>85.7</td><td>0.333</td></tr><tr><td>>4</td><td>4/5</td><td>80.0</td><td></td></tr></table>","table_caption":"[\"Table-5: Effects of age and parity on occurrence of resistance to the selected beta-lactam antibiotics in dairy goats with subclinical mastitis in Thika subcounty. \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[80,250,485,382]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"presence of coliform bacteria could be linked to poor hygiene practices in dairy goat farms. Poor hygiene is also reflected by an increase in the prevalence of subclinical mastitis (72.7%) recorded in this study, compared with 50.9% previously reported in Thika East subcounty. In addition, the frequency of isolation of different bacteria from goat milk is strongly associated with the overall hygiene status as well as the management strategy of the farms [42]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,384,485,501]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Although not listed as a major etiological agent, Acinetobacter spp. have also been reported in cases of subclinical mastitis in goats. We isolated a higher proportion $(16.1\\%)$ of Acinetobacter spp. in our study compared with that reported in goats in Nigeria $(3\\%)$ [34]. Yersinia spp. have not been associated with subclinical mastitis in goats; however, they have been widely isolated from raw goat milk [43,44]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,501,485,929]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The high resistance to penicillin G, cephalosporins, and cephamycin observed in our study was unforeseen. However, our findings corroborate with several other studies of dairy goats that reported high resistance to penicillin and cephalosporins of the bacterial isolates causing mastitis [13,36,37]. In addition, we observed cross-resistance among the bacterial isolates causing subclinical mastitis in goats. Similar results showing cross-resistance to penicillin and other $\\beta$ -lactams have been observed in goats [45] and cows [46]. The bacterial resistance mechanism to $\\beta$ -lactam antibiotics involves the production of $\\beta$ -lactamases and the synthesis of modified penicillin-binding proteins that inactivate $\\beta$ -lactam antibiotics by hydrolyzing the amide bond of the four-membered $\\beta$ -lactam ring, preventing drug binding [12,16]. The primary mechanism of bacterial resistance to $\\beta$ -lactam antibiotics is the production of $\\beta$ -lactamases, such as penicillinases, cephalosporinases, broad-spectrum $\\beta$ -lactamases, ESBLs, and carbapenemases [16]. These enzymes hydrolyze penicillins (including benzylpenicillin, also known as penicillin G), third-generation cephalosporins (including cefotaxime and ceftazidime), aztreonam, cephamycins (including cefoxitin), and carbapenems, but are inhibited by $\\beta$ -lactamase inhibitors, such as clavulanic acid [16]. At present, antibiotic therapy is the recommended treatment option for clinical and subclinical mastitis, and $\\beta$ -lactams are among the antibiotics used for mastitis treatment in ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,71,914,204]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Kenya. However, antibiotic resistance has spread over the years, mostly due to farmers initiating antibiotic therapy for mastitis treatment before conducting an antibiotic susceptibility test [47]. The high resistance to $\\beta$ -lactam antibiotics is concerning because limited therapeutic options are available for the treatment of such infections compounded with difficulty in controlling these bacteria due to their ability to rapidly spread and transfer the resistance phenotype. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,204,914,639]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hygiene is a major contributing factor to the spread of resistance in farms [48,49]. In our study, hygiene was defined by pre- and post-milking practices and the cleaning schedule of the goats' housing structure. We isolated a higher proportion of antibiotic-resistant bacteria isolated from goats whose farmers used water only and reusable towels during pre- and post-milking compared with those who used water, disinfectant, and reusable towels. This observation is probably due to the bactericidal effect of disinfectant [50]. The use of disinfectant during milking has been recommended as a control strategy to eradicate and prevent the spread of antibiotic-resistant bacteria in dairy farms [50]. However, the occurrence of antibiotic-resistant bacteria among goats whose farmers used disinfectant and reusable towels during milking was probably due to the use of reusable towels that might have been contaminated or not cleaned properly or due to other unknown factors. Furthermore, we found that hygiene practices during milking were significantly associated with the occurrence of resistance. However, hygiene in the goat housing structures was not significantly associated with the occurrence of resistance, which was probably due to the small sample size used in our study or other unknown factors. In addition, mastitis history, previous antibiotic treatment, parity, and age did not significantly impact the occurrence of resistance, and the observed variability can be attributed to different management practices in the farms or other unknown factors. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,640,915,800]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We did not phenotypically detect any ESBL producers. However, a negative synergy test for the phenotypic detection of ESBLs does not necessarily indicate that the bacteria are negative for ESBL production. False-negative ESBL results can be attributed to the overproduction of AmpC $\\beta$ -lactamases, inhibitor-resistant TEM $\\beta$ -lactamases, or TEM and SHV $\\beta$ -lactamases in ESBL-producing bacteria [30,51]. Therefore, there is a need for genotypic detection of ESBLs in bacterial isolates that show resistance to $\\beta$ -lactam antibiotics. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,808,608,821]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conclusion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[507,827,914,931]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We found a high prevalence of subclinical mastitis in the sampled dairy goats in Thika subcounty. The etiological agents causing subclinical mastitis are of zoonotic importance and are resistant to $\\beta$ -lactam antibiotics. This poses a health risk to humans, especially when raw milk and raw milk products are consumed. Hygiene during milking contributes to the occurrence ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[284,49,712,61]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Available at www.veterinaryworld.org/Vol.13/July-2020/27.pdf ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[84,936,339,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Veterinary World, EISSN: 2231-0916 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[875,936,912,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1454 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[80,72,487,175]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"of resistance to $\\beta$ -lactam antibiotics. More stringent measures, such as good animal husbandry and milking practices, must be adopted to control mastitis and prevent the spread of resistance to $\\beta$ -lactam antibiotics. Further studies should be undertaken to determine transmission dynamics between humans and livestock and identify novel intervention strategies. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[82,181,228,197]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Data Availability ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,200,485,247]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The raw data used to support the findings of this study are available from the corresponding author on request. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[82,252,280,266]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Authors' Contributions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,272,485,404]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IMO, JK, and NM involved in the conception of the research idea. IMO and JK planned the study design. IMO performed data collection, sample collection, and laboratory work. IMO and DK contributed in statistical analysis and interpretation of the data results. JK and DK provided laboratory guidance. IMO drafted the manuscript. JK, DK, and NM corrected the manuscript. All the authors read and approved the final manuscript. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[82,411,243,425]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acknowledgments ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,430,485,752]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This research was financially supported by Institute of Basic Sciences, Technology and Innovation, Pan-African University (PAUSTI) and Jomo Kenyatta University of Agriculture and Technology (JKUAT). The Grand Challenges Africa program (GCA/AMR/rnd2/079) also financially supported the work. Grand Challenges Africa is a program of the African Academy of Sciences (AAS) implemented through the Alliance for Accelerating Excellence in Science in Africa (AESA) platform, an initiative of the AAS and the African Union Development Agency (AUDA-NEPAD). GC Africa is supported by the Bill and Melinda Gates Foundation (BMGF) and The African Academy of Sciences and partners. The views expressed herein are those of the author(s) and not necessarily those of the AAS and her partners. The authors acknowledge the technical assistance provided by JKUAT staff, including Dr. Macharia Lewis and Mr. Muthanga Josephat, the extension officer from the veterinary office Thika subcounty, Mr. Gachina as well as PAUSTI students Armel Tangomo and Ronald Tanui. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[82,759,263,772]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Competing Interests ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[82,778,485,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The authors declare that they have no competing interests. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[82,814,226,828]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Publisher's Note ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,834,485,878]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Veterinary World remains neutral with regard to jurisdictional claims in published institutional affiliation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,885,181,898]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[84,903,485,929]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Miller, B.A. and Lu, C.D. (2019) Current status of global dairy goat production: An overview. Asian Australas. J. ","text_format":null,"text_level":null,"type":"ref_text"},{"bbox":"[512,72,912,929]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"Anim. Sci., 32(8): 1219-1232. \",\"2. Kahi, A.K. and Wasike, C.B. (2019) Dairy goat production in sub-Saharan Africa: Current status, constraints and prospects for research and development. Asian Australas. J. Anim. Sci., 32(8): 1266-1274. \",\"3. Makini, F. and Mose, L. (2019) Innovation Dairy Livestock in Kenya. Global Trend Report. \",\"4. Turkmen, N. (2017) The nutritional value and health benefits of goat milk components. In: Watson, R.R., Collier, R.J. and Preedy, V.R., editors. Nutrients in Dairy and their Implications on Health and Disease. Ch. 35. Academic Press, Cambridge, Massachusetts. p441-449. \",\"5. Contreras, A., Sierra, D., Sánchez, A., Corrales, J.C., Marco, J.C., Paape, M.J. and Gonzalo, C. (2007) Mastitis in small ruminants. Small Rumin. Res., 68(1): 145-153. \",\"6. Shearer, J.K. and Harris, B. (1992) DS 85 Mastitis in Dairy Goats. Vol. 1. University of Florida, Florida. p1-7. \",\"7. Abebe, R., Hatiya, H., Abera, M., Megersa, B. and Asmare, K. (2016) Bovine mastitis: Prevalence, risk factors and isolation of Staphylococcus aureus in dairy herds at Hawassa milk shed, South Ethiopia. BMC Vet. Res., 12(1): 270. \",\"8. Mishra, A.K., Sharma, N., Singh, D.D., Gururaj, K., Abhishek, Kumar, V. and Sharma, D.K. (2018) Prevalence and bacterial etiology of subclinical mastitis in goats reared in organized farms. Vet. World, 11(1): 20-24. \",\"9. Abdalhamed, A.M., Zeedan, G.S.G. and Abou Zeina, H.A.A. (2018) Isolation and identification of bacteria causing mastitis in small ruminants and their susceptibility to antibiotics, honey, essential oils, and plant extracts. Vet. World, 11(3): 355-362. \",\"10. Mitema, E.S., Kikuvi, G.M., Wegener, H.C. and Stohr, K. (2001) An assessment of antimicrobial consumption in food producing animals in Kenya. J. Vet. Pharmacol. Ther., 24(6): 385-390. \",\"11. Oliver, S.P. and Murinda, S.E. (2012) Antimicrobial resistance of mastitis pathogens. Vet. Clin. Food Anim. Pract., 28(2): 165-185. \",\"12. Bush, K. and Bradford, P.A. (2016) $\\\\beta$ -Lactams and $\\\\beta$ -lactamase inhibitors: An overview. Cold Spring Harb. Perspect. Med., 6(8): a025247. \",\"13. Scaccabarozzi, L., Leoni, L., Ballarini, A., Barberio, A., Locatelli, C., Casula, A. and Moroni, P. (2015) Pseudomonas aeruginosa in dairy goats: Genotypic and phenotypic comparison of intramammary and environmental isolates. PLoS One, 10(11): e0142973. \",\"14. Mello, P.L., Pinheiro, L., Martins, L., Brito, M.A.V. and da Cunha, M.L. (2017) Short communication: $\\\\beta$ -Lactam resistance and vancomycin heteroresistance in Staphylococcus spp. isolated from bovine subclinical mastitis. J. Dairy Sci., 100(8): 6567-6571. \",\"15. Aqib, A.I., Nighat, S., Ahmed, R., Sana, S., Jamal, M.A., Kulyar, M.F.A. and Rahman, S. (2018) Drug susceptibility profile of Staphylococcus aureus isolated from mastitic milk of goats and risk factors associated with goat mastitis in Pakistan. Pak. J. Zool., 51(1): 307-315. \",\"16. Santajit, S. and Indrawattana, N. (2016) Mechanisms of antimicrobial resistance in ESKAPE pathogens. *BioMed Res. Int*, 2016: Article ID 2475067. \",\"17. Muriuki, H.G. (2003) A Review of the Small Scale Dairy Sector-Kenya, No. 60. FAO Prevention of Food Losses Programme. \",\"18. Njarui, D.M.G., Gatheru, M., Wambua, J.M., Nguluu, S.N., Mwangi, D.M. and Keya, G.A. (2011) Consumption patterns and preference of milk and milk products among rural and urban consumers in semi-arid Kenya. Ecol. Food Nutr., 50(3): 240-262. \",\"19. Hwang, A.Y. and Gums, J.G. (2016) The emergence and evolution of antimicrobial resistance: Impact on a global scale. Bioorg. Med. Chem., 24(24): 6440-6445. \",\"20. Scarafile, G. (2016) Antibiotic resistance: Current issues and future strategies. Rev. Health Care, 7(1): 3-16. \",\"21. Mahlangu, P., Maina, N. and Kagira, J. (2018) Prevalence, \"]","page_idx":"7","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[282,49,712,61]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Available at www.veterinaryworld.org/Vol.13/July-2020/27.pdf ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[82,936,337,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Veterinary World, EISSN: 2231-0916 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[875,936,912,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1455 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[84,71,485,760]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"risk factors, and antibiogram of bacteria isolated from milk of goats with subclinical mastitis in Thika East Subcounty, Kenya. J. Vet. Med., 2018: Article ID 3801479. \",\"22. Thrusfield, M.V. (2007) Veterinary Epidemiology. $3^{\\\\mathrm{rd}}$ ed. Blackwell Science, Oxford. \",\"23. Handcock, M.S. and Gile, K.J. (2011) On the Concept of Snowball Sampling. Available from: http://www.arxiv.org/abs/1108.0301. Retrieved on 28-08-2019. \",\"24. Walker, J. B., Rajala-Schultz, P. J., and DeGraves, F. J. (2010) The effect of inoculum volume on the microbiologic detection of naturally occurring Staphylococcus aureus intramammary infections. J. Vet. Diagn. Invest., 22(5): 720-724. \",\"25. Garrity, G., Brenner, D.J., Krieg, N.R. and Staley, J.R., editors. (2005) The proteobacteria, Part B: The gammaproteobacteria. In: Bergey's Manual® of Systematic Bacteriology. 2nd ed., Vol. 2. Springer, United States. \",\"26. Cheesbrough, M. (2006) District Laboratory Practice in Tropical Countries. $2^{\\\\mathrm{nd}}$ ed. Cambridge University Press, Cambridge, New York. \",\"27. Quinn, P.J., Markey, B.K., Leonard, F.C., Hartigan, P., Fanning, S. and Fitzpatrick, E.S. (2011) Veterinary Microbiology and Microbial Disease. John Wiley and Sons, Hoboken, New Jersey. \",\"28. Bauer, A. W., Kirby, W. M., Sherris, J. C., and Turck, M. (1966) Antibiotic susceptibility testing by a standardized single disk method. Am. J. Clin. Pathol., 45(4), 493-496. \",\"29. Clinical and Laboratory Standards Institute. (2016) M100S Performance Standards for Antimicrobial Susceptibility Testing. $26^{\\\\mathrm{th}}$ ed. Clinical and Laboratory Standards Institute, Wayne, PA. \",\"30. Drieux, L., Brossier, F., Sougakoff, W. and Jarlier, V. (2008) Phenotypic detection of extended-spectrum $\\\\beta$ -lactamase production in Enterobacteriaceae: Review and bench guide. Clin. Microbiol. Infect., 14(Suppl 1): 90-103. \",\"31. Gelasakis, A.I., Angelidis, A.S., Giannakou, R., Filioussis, G., Kalamaki, M.S. and Arsenos, G. (2016) Bacterial subclinical mastitis and its effect on milk yield in low-input dairy goat herds. J. Dairy Sci., 99(5): 3698-3708. \",\"32. Ferdous, J., Rahman, M., Khan, M., Khan, M., & Rima, U. (2018). Prevalence of clinical and subclinical caprine mastitis of northern region in Bangladesh. Progress. Agric., 29(2): 127-138. \",\"33. Pirzada, M. and Ali, A. (2016) Prevalence of subclinical mastitis in dairy goats caused by bacterial species. J. Anim. Health Prod., 4(2): 55-59. \",\"34. Danmallam, F.A. and Pimenov, N.V. (2019) Study on prevalence, clinical presentation, and associated bacterial pathogens of goat mastitis in Bauchi, Plateau, and Edo states, Nigeria. Vet. World, 12(5): 638-645. \",\"35. Mbindyo, C.M., Gitao, C.G. and Bebora, L. (2014) A cross-sectional study on the prevalence of subclinical mastitis and antimicrobial susceptibility patterns of the bacterial isolates in milk samples of smallholder dairy goats in Kenya. Am. J. Res. Commun., 2(8): 22-50. \",\"36. Lima, M.C., Souza, M.C.C., Espeschit, I.F., Maciel, P.A.C., Sousa, J.E., Moraes, G. and Moreira, M.A.S. (2018) Mastitis in dairy goats from the state of Minas Gerais, Brazil: Profiles of farms, risk factors and characterization \"]","page_idx":"8","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[512,72,912,762]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"of bacteria. Pesqui. Vet. Bras., 38(9): 1742-1751. \",\"37. Najeeb, M.F., Anjum, A.A., Ahmad, M.U.D., Khan, H.M., Ali, M.A. and Sattar, M.M.K. (2013) Bacterial etiology of subclinical mastitis in dairy goats and multiple drug resistance of the isolates. J. Anim. Plant Sci., 23(6): 1541-1544. \",\"38. Haftay, A., Habtamu, T.M. and Abebe, M.S. (2016) Susceptibility of subclinical mastitis causing bacteria from goats in Aba'lla district, Afar, North-Eastern Ethiopia. Rev. Med. Vet., 6(1): 1-6. \",\"39. Petersson-Wolfe, C.S., Mullarky, I.K. and Jones, G.M. (2010) Staphylococcus aureus Mastitis: Cause, Detection, and Control. Vol. 7. VCE Publications, VA, USA. \",\"40. Merz, A., Stephan, R. and Johler, S. (2016) Staphylococcus aureus isolates from goat and sheep milk seem to be closely related and differ from isolates detected from bovine milk. Front. Microbiol., 7: 319. \",\"41. Machado, G.P., Silva, R.C., Guimarães, F.F., Salina, A., Langoni, H., Machado, G. and Langoni, H. (2018) Detection of Staphylococcus aureus, Streptococcus agalactiae and Escherichia coli in Brazilian mastitic milk goats by multiplex-PCR. Pesqui. Vet. Bras., 38(7): 1358-1364. \",\"42. McDougall, S., Pankey, W., Delaney, C., Barlow, J., Murdough, P.A. and Scruton, D. (2002) Prevalence and incidence of subclinical mastitis in goats and dairy ewes in Vermont, USA. Small Rumin. Res., 46(2): 115-121. \",\"43. Rahimi, E., Sepehri, S., Dehkordi, F.S., Shaygan, S. and Momtaz, H. (2014) Prevalence of Yersinia species in traditional and commercial dairy products in Isfahan Province, Iran. *Jundishapur J. Microbiol.*, 7(4): e9249. \",\"44. Jamali, H., Paydar, M., Radmehr, B. and Ismail, S. (2015) Prevalence, characterization, and antimicrobial resistance of Yersinia species and Yersinia enterocolitica isolated from raw milk in farm bulk tanks. J. Dairy Sci., 98(2): 798-803. \",\"45. Bochev, I.V. and Russenova, N. (2005) Resistance of Staphylococcus spp. Strains isolated from goats with subclinical mastitis. *Bulg J. Vet. Med.*, 8(8): 109-118. \",\"46. Berghash, S.R., Davidson, J.N., Armstrong, J.C. and Dunny, G.M. (1983) Effects of antibiotic treatment of non-lactating dairy cows on antibiotic resistance patterns of bovine mastitis pathogens. Antimicrob. Agents Chemother., 24(5): 771-776. \",\"47. Ceniti, C., Britti, D., Santoro, A.M.L., Musarella, R., Ciambrone, L., Casalinuovo, F. and Costanzo, N. (2017) Phenotypic antimicrobial resistance profile of isolates causing clinical mastitis in dairy animals. Ital. J. Food Saf., 6(2): 6612. \",\"48. Lee, C.R., Cho, I.H., Jeong, B.C. and Lee, S.H. (2013) Strategies to minimize antibiotic resistance. Int. J. Environ. Res. Public Health, 10(9): 4274-4305. \",\"49. Adnan, M., Khan, H., Kashif, J., Ahmad, S., Gohar, A., Ali, A. and Khan, N.A. (2017) Clonal expansion of sulfonamide resistant Escherichia coli isolates recovered from diarrheic calves. *Pak. Vet.* J., 37(2): 253-8318. \",\"50. Gohary, A., Gohary, F., Elsayed, M. and ElFateh, M. (2019) In-vitro investigation on the antiseptic efficacy of commonly used disinfectants in dairy farms against methicillin-resistant Staphylococcus aureus. Alex. J. Vet. Sci., 60(1): 86. \",\"51. Lalitha, D.M.K. (2004) Manual on antimicrobial susceptibility testing. Indian Assoc. Med. Microbiol., 47: 34. \"]","page_idx":"8","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[436,768,559,780]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"\\*\\*\\*\\*\\*\\*\\*\\* ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[282,49,712,61]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Available at www.veterinaryworld.org/Vol.13/July-2020/27.pdf ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[84,936,337,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Veterinary World, EISSN: 2231-0916 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[875,936,912,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1456 ","text_format":null,"text_level":null,"type":"page_number"}],"string":"[\n {\n \"bbox\": \"[102,87,894,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\beta$ -lactam resistance in bacteria associated with subclinical mastitis in goats in Thika Subcounty, Kenya \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[245,131,749,147]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Irene Mkavi Okoko<sup>1</sup>, Naomi Maina<sup>1,2</sup>, Daniel Kiboi<sup>2</sup> and John Kagira<sup>3</sup> \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,156,902,204]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Department of Molecular Biology and Biotechnology, Pan-African University, Institute of Basic Sciences, Technology and Innovation, P.O. Box 62000-00200, Nairobi, Kenya; 2. Department of Biochemistry, Jomo Kenyatta University of Agriculture and Technology, P.O. Box 62000-00200, Nairobi, Kenya; 3. Department of Animal Sciences, Jomo Kenyatta University of Agriculture and Technology, P.O. Box 62000-00200, Nairobi, Kenya. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[164,205,833,239]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corresponding author: John Kagira, email: jkagira@gmail.com \\nCo-authors: IMO: okokomkavi@gmail.com, NM: nmaina@jkuat.ac.ke, DK: dmuthui@jkuat.ac.ke \\nReceived: 01-02-2020, Accepted: 08-06-2020, Published online: 25-07-2020 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,244,915,282]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"doi: www.doi.org/10.14202/vetworld.2020.1448-1456 How to cite this article: Okoko IM, Maina N, Kiboi D, Kagira J (2020) β-lactam resistance in bacteria associated with subclinical mastitis in goats in Thika Subcounty, Kenya, Veterinary World, 13(7): 1448-1456. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[455,291,542,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,312,914,354]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Aim: This study determined the resistance pattern to $\\\\beta$ -lactam antibiotics of bacteria isolated from goats with subclinical mastitis in Thika subcounty, Kenya. We also administered a questionnaire to assess the risk factors associated with the occurrence of resistance to commonly used antibiotics. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,359,914,464]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Materials and Methods: We collected milk samples from 110 lactating dairy goats in Thika subcounty to screen for subclinical mastitis using the California mastitis test. Bacterial isolation and identification were performed according to colony morphology, the hemolytic pattern on sheep blood agar, lactose fermentation on MacConkey plates, Gram staining, and standard biochemical tests. The antibiotic susceptibility of the isolates was determined by the agar disk diffusion method using penicillin G, cephalexin, cefoxitin, and cefotaxime antibiotic disks. The double-disk synergy test using amoxicillin-clavulanic acid was employed as a confirmatory test for extended-spectrum $\\\\beta$ -lactamase (ESBL) production. Fisher's exact test was used to determine the risk factors associated with the occurrence of antibiotic resistance $(p \\\\leq 0.05$ was considered significant). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,470,912,576]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Results: Of the 110 dairy goats sampled, $72.7\\\\%$ (80) were positive for subclinical mastitis. Isolation and identification of the bacteria from the positive samples yielded 149 bacteria isolates, including Staphylococcus aureus, Klebsiella pneumoniae, Acinetobacter spp., Yersinia spp., coagulase-negative staphylococci, and Escherichia coli. A high percentage $(76.5\\\\%, 114/149)$ of the bacterial isolates was resistant to at least one of the tested antibiotics. At least 56/106 isolates $(52.8\\\\%)$ showing cross-resistance to the $\\\\beta$ -lactam antibiotics were resistant to all four of the tested antibiotics, while only one isolate was resistant to three antibiotics (penicillin G, cephalexin, and cefoxitin). The double-disk synergy test confirmed that none of the isolates possessed ESBLs. Pre- and post-milking practices $(p = 0.0336)$ were found to be significantly associated with the occurrence of antibiotic resistance. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,582,914,624]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conclusion: A large proportion of the goats in our study cohort were infected with $\\\\beta$ -lactam-resistant bacteria associated with subclinical mastitis. Because the identified bacteria are of zoonotic importance, further studies should be undertaken to determine the transmission dynamics between humans and livestock and to identify novel intervention strategies. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,627,640,643]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords: bacteria, dairy goats, Kenya, subclinical mastitis, $\\\\beta$ -lactam resistance. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,661,194,674]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,680,487,813]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The dairy goat agricultural industry is rapidly expanding. In 2017, the global dairy goat population was estimated at 218 million, producing approximately 18.7 million tons of milk [1]. Sub-Saharan African countries account for a goat population of 372,716,040, with $16\\\\%$ farmed as dairy goats [2]. The dairy goat population in Kenya was estimated as $>251,100$ in 2011, with the milk used as an alternative to cow's milk. At present, approximately $4.7\\\\%$ of \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,661,914,931]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"the total milk consumed in Kenya is obtained from goats [3]. The emerging interest in goats and their derivative products, including milk, is due to the associated health and nutritional benefits, low production cost, ease of management, and goat dairy farming being a source of livelihood, among other factors [1,3,4]. Despite the vast interest in dairy goat production, the enterprise faces a myriad of abiotic and biotic challenges. Of major concern is mastitis, a disease resulting from the infection of mammary glands by viruses, fungi, or bacteria, with the latter being the major etiological agents [5]. Following mammary gland infection, white blood cells and epithelial (somatic) cells are produced by the goat as a defense mechanism [6]. A relatively small number of epithelial cells are present in milk from healthy milk-secreting tissue (1,000,000 cells/ml), and somatic cell counts (SCC) $>1,500,000$ cells/ml indicate intramammary \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,48,515,74]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Veterinary World, EISSN: 2231-0916 \\nAvailable at www.veterinaryworld.org/Vol.13/July-2020/27.pdf \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[759,48,914,74]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RESEARCH ARTICLE Open Access \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[80,826,485,929]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Copyright: Okoko, et al. Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[82,936,339,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Veterinary World, EISSN: 2231-0916 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[875,938,912,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1448 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[85,71,484,290]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"infection [6]. Mastitis is associated with symptoms such as udder inflammation and tissue damage, leading to reduced milk quality and quantity [7]. Mastitis manifests either clinically (symptomatic) or subclinically (asymptomatic), the latter being more prevalent in goats [8]. Coupled with the treatment cost, disease management has led to significant economic losses [9]. The mastitis management strategies in Kenya entail the use of antibiotics, including tetracyclines, $\\\\beta$ -lactams, trimethoprim/sulfamethoxazole, nitrofurans, aminoglycosides, and quinolones [10], the majority of which belong to the $\\\\beta$ -lactam family, such as penicillin and cephalosporins [11]. $\\\\beta$ -lactam antibiotics share a common structural feature, the $\\\\beta$ -lactam ring, that inhibits bacterial cell wall synthesis [12]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[85,290,485,580]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Reports of bacterial resistance to $\\\\beta$ -lactam antibiotics in cases of mastitis have recently emerged [13-15] and have been attributed to the ability of the bacteria causing mastitis to produce $\\\\beta$ -lactamase enzymes and modified penicillin-binding proteins that hydrolyze or inhibit the binding of $\\\\beta$ -lactam antibiotics, respectively [12,16]. Antibiotic resistance in pathogens that cause subclinical mastitis in animals poses a risk of the transfer of antibiotic resistance to humans, especially where raw milk and raw milk products are consumed, such as in Kenya [17,18]. Antibiotic-resistant bacteria cause severe infections with limited treatment options and fatalities, leading to a high cost of treatment, loss of productivity, and adverse economic consequences in the affected country [19,20]. Although a Kenyan study reported resistance to penicillin G in $100\\\\%$ of the bacterial isolates causing subclinical mastitis in goats [21], the resistance of mastitis-causing pathogens to other $\\\\beta$ -lactam antibiotics remains undocumented in Kenya. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[85,581,484,696]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This study determined the prevalence of bacterial resistance to selected $\\\\beta$ -lactam antibiotics in cases of subclinical mastitis using the agar disk diffusion method. We also report the risk factor associated with the occurrence of resistance to these antibiotics. The study highlighted the consumption of raw milk as a potential vehicle for the transfer of bacterial resistance to humans. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[85,705,278,717]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Materials and Methods \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[85,724,208,734]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ethical approval \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[85,737,484,794]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since no invasive procedure was performed on the animals, approval from the Institutional Animal Ethics Committee to conduct the study was not required. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[85,801,253,812]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Study area and period \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[85,813,485,929]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The study was conducted in Thika subcounty of Kiambu County, located in Central Kenya, approximately $40\\\\mathrm{km}$ from Nairobi $(1^{\\\\circ}1^{\\\\prime}\\\\mathrm{S}37^{\\\\circ}5^{\\\\prime}\\\\mathrm{E})$ . Rainfall is bimodal, ranging from 500 to $1300~\\\\mathrm{mm}$ , and the average temperature is $18.7^{\\\\circ}\\\\mathrm{C}$ . The study was conducted for a period of 10 months (January 2019 to October 2019). Although no formal census has been undertaken, farmers have taken up dairy goat farming \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,72,910,117]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"in this region [21]. Farmers practice zero-grazing, open-grazing, and tethering-grazing systems [Personal communication, Veterinary officer, Thika subcounty]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,123,910,145]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Study design, sample size calculation, and questionnaire administration \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,148,910,483]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A cross-sectional field- and laboratory-based study design was used. The sample size of 110 lactating dairy goats was calculated using an adjusted formula for small populations [22], whereby the sample size for an infinite population with an estimated prevalence can be calculated for a $95\\\\%$ confidence interval as $\\\\mathrm{n = (1.96^2[\\\\mathrm{P_{exp}}] / [1 - \\\\mathrm{P_{exp}}]) / d^2}$ where $n$ is the required sample size, $\\\\mathrm{P_{exp}}$ is the expected prevalence, and $d$ is the desired absolute precision. We used a prevalence of $50.9\\\\%$ subclinical mastitis, which was determined in a study of goats in Thika East subcounty [21], and a desired absolute precision of $5\\\\%$ , resulting in a required sample size of 384 goats. In relatively small populations, such as that of dairy goats in Thika subcounty, it is possible to select a smaller sample size than one from a theoretically infinite population to achieve the same degree of precision using the adjusted formula $\\\\mathrm{N_{adj} = N\\\\times n / N + n}$ , where $n$ is the sample size as calculated above and $N$ is the size of the study population [22]. Given that the size of the study population of lactating dairy goats was approximately 150 [21], a sample size of 108 does was calculated. We sampled 110 does. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,485,910,703]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We targeted small-scale farmers with a herd size of $\\\\leq 15$ goats who consented to participate. Sampling was performed using the snowball technique and sampling to redundancy due to the absence of a formal list of dairy farmers in the area [23]. The local agriculture extension officer assisted in identifying the first farmers, who later recruited other farmers until all the farmers had been included. The final cohort comprised 110 lactating goats with no clinical signs of mastitis, sampled from 41 farms. We administered a questionnaire to the farmers to collect data pertaining to age, parity, mastitis history, mastitis management, cleaning schedules of the housing structure, pre- and post-milking practices, and other variables. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,711,877,722]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Milk sampling and California mastitis test (CMT) \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,725,910,929]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The milk samples were aseptically collected, as previously described [21]. CMTs (ImmuCell, ME, USA) were performed to screen for subclinical mastitis, according to the manufacturer's instructions with minor modifications. Briefly, $5\\\\mathrm{ml}$ of milk from different teats were milked into the CMT paddle, and an equal volume of the CMT reagent was added. A positive CMT test was scored as 2 or 3, indicating an SCC of 2,560,000 and $\\\\geq 10,000,000$ respectively [6]. In addition, $5\\\\mathrm{ml}$ of milk from the samples with CMT scores of 2 and 3 were collected into sterile universal bottles, placed in a cooler box containing ice packs, and transported within $12\\\\mathrm{h}$ of collection to the microbiology laboratory at Jomo Kenyatta University of \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[284,49,712,61]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Available at www.veterinaryworld.org/Vol.13/July-2020/27.pdf \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[85,938,337,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Veterinary World, EISSN: 2231-0916 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[877,938,910,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1449 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[85,72,482,101]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Agriculture and Technology, Juja, Kenya, for bacteriological culture and isolation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[85,108,349,118]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bacterial culture and identification \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[85,121,482,514]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sheep blood agar (HiMedia, Mumbai, India) and MacConkey agar (Oxoid, Basingstoke, UK) were freshly prepared as per the manufacturer's guidelines and inoculated with approximately $100\\\\mu \\\\mathrm{l}$ of each milk sample per plate, respectively. In routine microbiological diagnosis, a sample of $10 - 100\\\\mu \\\\mathrm{l}$ of milk is used for culture. However, an inoculum of $100\\\\mu \\\\mathrm{l}$ has a higher sensitivity when isolating bacteria causing intramammary infection [24]. Hence, we used $100\\\\mu \\\\mathrm{l}$ for inoculation. The milk was evenly spread onto the surface of the agar plate using a sterile glass spreader and then incubated at $37^{\\\\circ}\\\\mathrm{C}$ for up to $48\\\\mathrm{h}$ . The morphological characteristics of the resulting bacterial colonies were assessed for shape, color, size, and texture as well as lactose fermentation ability on MacConkey plates and hemolysis patterns on sheep blood agar plates. Hemolysis of the blood cells in the sheep blood agar was evaluated by noting any color change in the media, particularly around and under the colonies. A plate was said to have mixed growth when the colonies on that plate showed distinct morphological characteristics, and each of the distinguishable colonies was then subcultured in nutrient agar to obtain pure colonies. The pure cultures were examined for colony morphology, Gram staining characteristics (to distinguish between Gram-positive and Gram-negative isolates), and biochemical reactions. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[85,516,482,646]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A catalase test using $3\\\\%$ hydrogen peroxide was performed on all Gram-positive isolates to identify Staphylococcus spp. Staphylococcus spp. were also identified based on their growth and fermentative attributes on mannitol salt agar (HiMedia) and the results of an oxidase test (HiMedia). Staphylococcus aureus was differentiated from coagulase-negative staphylococci (CNS) using slide and tube coagulase tests that employed rabbit plasma. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[85,648,482,851]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The oxidase test was performed on all Gram-negative isolates to identify Enterobacteriaceae, and those isolates that were oxidase negative were subsequently subcultured on MacConkey agar to assess lactose fermentation. Then, the triple sugar iron (Oxoid), methyl red/Voges-Proskauer (HiMedia), urease (urea broth base; HiMedia), indole (tryptone broth; HiMedia), citrate utilization (Simmons citrate agar; HiMedia), motility (sulfur indole motility media; HiMedia), catalase, and mannitol salt agar (HiMedia) tests were employed to classify the Gram-negative bacteria. The results of culture, Gram staining, and biochemical tests were used for final bacterial identification [25-27]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[85,859,433,870]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Antibiotic susceptibility testing of the isolates \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[85,872,482,929]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Antibiotic susceptibility testing was performed using the Kirby-Bauer disk diffusion method [28], and the results were interpreted following the Clinical Laboratory Standard Institute guidelines (https:// \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,72,912,449]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"clsi.org/). Briefly, the bacterial suspension was prepared in sterile distilled water and standardized to $0.5\\\\mathrm{McFarland}$ 's standard (corresponding to approximately $1.5\\\\times 10^{8}$ colony-forming units/ml). Mueller-Hinton agar was prepared following the manufacturer's instructions, poured into inoculation plates to solidify, and inoculated with the standardized inoculums. Escherichia coli ATCC 25922 (American Type Culture Collection, Rockville, MA) was used for quality control. Bacterial resistance was tested using commercially available antibiotic disks of penicillin G $(10~\\\\mu \\\\mathrm{g})$ , cephalexin $(30~\\\\mu \\\\mathrm{g})$ , cefoxitin $(30~\\\\mu \\\\mathrm{g})$ , and cefotaxime $(30~\\\\mu \\\\mathrm{g})$ . These were randomly selected from $\\\\beta$ -lactam antibiotics commonly used in Kenya to treat bacterial infections in both animals and humans and included different generations and classes of $\\\\beta$ -lactam antibiotics, with penicillin G as a representative of the penicillin class of $\\\\beta$ -lactams. Cephalexin, cefoxitin, and cefotaxime were representative of the first-, second- and third-generation cephalosporins, respectively. The disks were placed on the agar surface $24\\\\mathrm{mm}$ from center to center, and the plates were incubated at $37^{\\\\circ}C$ for $18\\\\mathrm{h}$ . The zones of growth inhibition, including the disk diameters, were measured in millimeters and recorded as resistant, intermediate, or susceptible to the specific antibiotics [29]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,456,910,480]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Determining the proportion of extended-spectrum $\\\\beta$ -lactamase (ESBL) producers \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,481,912,784]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The double-disk synergy test was used for phenotypic confirmation of ESBL producers, as previously described [30]. E. coli ATCC 25922 and E. coli ATCC 35218 (American Type Culture Collection) were used as negative and positive controls, respectively. Commercially available disks (HiMedia) of cefotaxime $(30\\\\mu \\\\mathrm{g})$ and ceftazidime $(30\\\\mu \\\\mathrm{g})$ together with amoxicillin-clavulanic acid $(30\\\\mu \\\\mathrm{g})$ were placed $25\\\\mathrm{mm}$ apart from each other on the surface of inoculated Mueller-Hinton agar plates and incubated at $37^{\\\\circ}\\\\mathrm{C}$ for $24\\\\mathrm{h}$ . The test was regarded as positive when the decreased susceptibility to cefotaxime and/or ceftazidime was joined with a clear-cut augmentation of the inhibition zone of cefotaxime and/or ceftazidime in front of the amoxicillin-clavulanic acid antibiotic disk, forming a characteristic \\\"champagne-cork\\\" or \\\"keyhole\\\" shape [30]; otherwise, the results were regarded as negative. The bacterial samples were stored in tryptic soy broth (Scharlab SL, Barcelona, Spain) containing $20\\\\%$ glycerol at $-70^{\\\\circ}\\\\mathrm{C}$ for long-term preservation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,791,910,826]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Determining the relationship between the occurrence of resistance against the selected $\\\\beta$ -lactam antibiotics and doe- and farm-level variables \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,829,910,929]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A well-structured questionnaire was developed based on that of Mahlangu et al. [21]. The questionnaire was used to collect data regarding farm-level and doe-level variables, such as mastitis history, previous antibiotic treatment of mastitis, pre- and post-milk-ing practices, and the cleaning schedule of the housing structure as well as the doe name, age, parity, and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[284,49,712,61]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Available at www.veterinaryworld.org/Vol.13/July-2020/27.pdf \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[85,938,337,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Veterinary World, EISSN: 2231-0916 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[877,938,910,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1450 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[80,71,485,159]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"lactation stage. It was written in English and administered to the farmers in English before sample collection. However, based on the farmers' level of understanding, the agriculture extension officer helped with the translation to Swahili or Kikuyu (the vernacular language used by the local farmers in the study area). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,159,485,233]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The results of the agar disk diffusion test were used to determine the relationship between the occurrence of resistance against the $\\\\beta$ -lactam antibiotics and doe- and farm-level variables obtained from the questionnaire. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,237,230,250]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Statistical analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,250,485,500]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data were entered into a Microsoft Office Excel spreadsheet Version 16.0.12827.20336 (Microsoft Corp., Redmond, WA, USA) and then analyzed using StatXact 11 (Cytel Inc., Cambridge, MA, USA) and GraphPad Prism software version 7.0.0 (GraphPad Software, CA, USA). Descriptive statistics were expressed as percentages and frequencies, and data were presented in tables and histograms. The prevalence of subclinical mastitis was determined as the proportion of CMT-positive goats out of the total number of sampled goats. The proportion of the resistant bacterial isolates was determined as the number of all the isolates resistant to at least one of the antibiotics out of the total number of bacterial isolates. The association of the doe- and farm-level variables with the occurrence of resistance was determined by Fisher's exact test. $p \\\\leq 0.05$ was considered statistically significant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,506,152,519]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Results \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,524,485,549]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Prevalence of subclinical mastitis in goats from Thika subcounty \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,549,485,624]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We sampled 110 lactating dairy goats from 41 farms in Thika subcounty with no more than 15 lactating dairy goats per farm (average, three lactating goats). Subclinical mastitis was present in 80 of the dairy goats sampled $(72.7\\\\%)$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,627,485,653]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bacteria causing subclinical mastitis in goats from Thika subcounty \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,653,485,843]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bacteria were cultured in $97.5\\\\%$ (78/80) of the CMT-positive milk samples. The volume of $2.5\\\\%$ (2/80) of the samples was insufficient for microbiology culture and analysis after performing the CMT test. We cultured 149 isolates from the raw milk samples. In descending order of occurrence, the isolates included S. aureus, Klebsiella pneumoniae, Acinetobacter spp., Yersinia spp., CNS, E. coli, Enterobacter intermedius, Klebsiella oxytoca, Proteus vulgaris, Citrobacter freundii, Citrobacter diversus, Morganella morganii, Proteus mirabilis, Serratia fonticola, Providencia stuartii, Serratia marcescens, and Staphylococcus epidermidis (Table-1). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,848,485,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Resistance profiles of bacteria isolated from CMT-positive milk samples from goats in Thika subcounty to the selected $\\\\beta$ -lactam antibiotics \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,885,485,929]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The disk diffusion tests showed that $23.5\\\\%$ (35/149) of the bacterial isolates were susceptible to all the antibiotics tested, and $76.5\\\\%$ (114/149) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,111,907,376]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0994/81cb1fbcfc8ea541b87411756b35132f95c6c5b9f9667d294d6c5dbe07805e11.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Bacterial isolate</td><td>No. of isolates n=149</td><td>Proportion (%)</td></tr><tr><td>Staphylococcus aureus</td><td>26</td><td>17.5</td></tr><tr><td>Klebsiella pneumoniae</td><td>26</td><td>17.5</td></tr><tr><td>Acinetobacter spp.</td><td>24</td><td>16.1</td></tr><tr><td>Yersinia spp.</td><td>21</td><td>14.1</td></tr><tr><td>CNS</td><td>15</td><td>10.1</td></tr><tr><td>Escherichia coli</td><td>6</td><td>4.0</td></tr><tr><td>Enterobacter intermedius</td><td>5</td><td>3.4</td></tr><tr><td>Klebsiella oxytoca</td><td>4</td><td>2.7</td></tr><tr><td>Proteus vulgaris</td><td>4</td><td>2.7</td></tr><tr><td>Citrobacter freundii</td><td>3</td><td>2.0</td></tr><tr><td>Citrobacter diversus</td><td>3</td><td>2.0</td></tr><tr><td>Morganella morganii</td><td>3</td><td>2.0</td></tr><tr><td>Proteus mirabilis</td><td>2</td><td>1.3</td></tr><tr><td>Serratia fonticola</td><td>2</td><td>1.3</td></tr><tr><td>Providencia stuartii</td><td>2</td><td>1.3</td></tr><tr><td>Serratia marcescens</td><td>2</td><td>1.3</td></tr><tr><td>Staphylococcus epidermidis</td><td>1</td><td>0.7</td></tr><tr><td>Total</td><td>149</td><td></td></tr></table>\",\n \"table_caption\": \"[\\\"Table-1: The proportion of bacteria isolated from CMT-positive milk samples obtained from goats in Thika subcounty. \\\"]\",\n \"table_footnote\": \"[\\\"CNS=Coagulase-negative staphylococcus, CMT=California mastitis test \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[507,418,912,564]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"showed intermediate or complete resistance to at least one $\\\\beta$ -lactam. The susceptibility of the isolates to penicillin G, cephalexin, cefoxitin, and cefotaxime was $23.5\\\\%$ , $32.9\\\\%$ , $61.7\\\\%$ , and $36.2\\\\%$ , respectively. Intermediate resistance of the bacterial isolates to penicillin G, cephalexin, cefoxitin, and cefotaxime was $0\\\\%$ , $4.0\\\\%$ , $4.0\\\\%$ , and $10.7\\\\%$ , respectively. Complete resistance to penicillin G, cephalexin, cefoxitin, and cefotaxime was $76.5\\\\%$ , $63.1\\\\%$ , $34.2\\\\%$ , and $53.0\\\\%$ , respectively (Figure-1a). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,564,914,753]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cross-resistance to the $\\\\beta$ -lactam antibiotics was observed in $93.0\\\\%$ (106/114) of the resistant bacterial isolates, with $10.4\\\\%$ (11/106) resistant to penicillin G and cephalexin; $0.9\\\\%$ (1/106) resistant to penicillin G, cephalexin, and cefoxitin; and the majority $(52.8\\\\%, 56/106)$ resistant to penicillin G, cephalexin, cefoxitin, and cefotaxime. Resistance to penicillin G, cephalexin, and cefotaxime was observed in $30.2\\\\%$ (32/106) of the isolates, while $5.7\\\\%$ (6/106) showed resistance to penicillin G and cefotaxime (Figure-1b). In general, a larger proportion of the bacteria resistant to penicillin G showed cross-resistance to other $\\\\beta$ -lactam antibiotics (Table-2). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,759,914,784]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Non-ESBL-producing bacterial isolates from CMT-positive milk samples from goats in Thika subcounty \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,784,912,858]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The double-disk synergy test did not show any characteristic \\\"champagne-cork\\\" or \\\"keyhole\\\" inhibition zones. Thus, this phenotypic method confirmed that none of the bacterial isolates were ESBL producers. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,863,912,898]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Relationship between the occurrence of resistance to the selected $\\\\beta$ -lactam antibiotics and doe- and farm-level variables \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,900,912,929]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In terms of a history of mastitis, higher resistance to the selected $\\\\beta$ -lactam antibiotics occurred in the \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[282,49,712,61]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Available at www.veterinaryworld.org/Vol.13/July-2020/27.pdf \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[82,936,339,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Veterinary World, EISSN: 2231-0916 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[875,936,912,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1451 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[191,72,500,286]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0656/3a52f561c82e2cfc7e3f87c73caacbea7cc62e6aeccec5fabdb1225b768066cd.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[509,74,806,286]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure-1: Resistance pattern of the bacterial isolates to beta-lactam antibiotics. (a) Overall susceptibility profiles of the bacterial isolates from CMT-positive milk samples to penicillin G, cephalexin, cefoxitin, and cefotaxime. (b) Cross-resistance of the bacterial isolates from CMT-positive milk samples to the selected beta-lactam antibiotics. P=Penicillin G, CN=Cephalexin, FOX=Cefoxitin, CTX=Cefotaxime. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0969/252a72095c974736529657f3415d0906a13142502a90a8f5f5d0ea31509864d6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[80,382,915,521]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0380/b42ec535406c7e9db2594ea4f2564bb2dba9f7f20476da25bf40ea10c589a480.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=\\\"2\\\">Antibiotic</td><td colspan=\\\"6\\\">% proportion of resistant bacterial isolates</td></tr><tr><td>Staphylococcus aureus (n=26)</td><td>Klebsiella pneumoniae (n=26)</td><td>Acinetobacter spp. (n=24)</td><td>Yersinia spp. (n=21)</td><td>CNS (n=15)</td><td>Escherichia coli (n=6)</td></tr><tr><td>P</td><td>26.9</td><td>0</td><td>0</td><td>4.8</td><td>0</td><td>0</td></tr><tr><td>P and CN</td><td>0</td><td>3.8</td><td>8.3</td><td>4.8</td><td>6.7</td><td>33.3</td></tr><tr><td>P, CN, and FOX</td><td>3.8</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>P, CN, FOX, and CTX</td><td>38.5</td><td>34.6</td><td>54.2</td><td>33.3</td><td>40</td><td>16.7</td></tr><tr><td>P, CN, and CTX</td><td>3.8</td><td>42.3</td><td>4.2</td><td>19</td><td>20</td><td>33.3</td></tr><tr><td>P and CTX</td><td>0</td><td>3.8</td><td>8.3</td><td>0</td><td>20</td><td>0</td></tr></table>\",\n \"table_caption\": \"[\\\"Table-2: Resistance to penicillin G, cephalexin, cefoxitin, and cefotaxime of the predominant bacterial isolates in CMT-positive milk samples from goats in Thika subcounty. \\\"]\",\n \"table_footnote\": \"[\\\"CNS=Coagulase-negative staphylococcus, P=Penicillin G, CN=Cephalexin, FOX=Cefoxitin, CTX=Cefotaxime \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[80,552,485,687]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"does that had previous cases of mastitis compared with no mastitis history, but this relationship was not statistically significant $(p = 0.4242)$ . A higher prevalence of $\\\\beta$ -lactam resistance occurred in does that were administered antibiotics when they had mastitis compared with those where no treatment was received, but there was no significant association $(p = 0.5781)$ between antibiotic treatment and the occurrence of resistance (Table-3). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,688,485,929]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The occurrence of resistance to $\\\\beta$ -lactams was significantly affected by the pre- and post-milk-ing practices adopted by the farmers in the region $(p = 0.0336)$ . The highest prevalence $(94.1\\\\%)$ of resistance was observed in bacteria isolated from does where the farmers only used water and a reusable towel for cleaning before and after milking. There was a lower prevalence $(77.8\\\\%)$ of resistance in does where the farmers used water, disinfectant, and a reusable towel, while no resistance $(0.0\\\\%)$ was observed where the farmers used water and a disposable towel during milking. The highest resistance $(100.0\\\\%)$ occurred in does whose housing structures were cleaned every 2 days, after 2 weeks, 3 times a week, or irregularly compared with those that were cleaned once a week $(81.8\\\\%)$ , twice a week $(81.8\\\\%)$ , and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,604,915,750]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0260/1ab47889cf78de8d62a00a2e4387b9a9dd1cd3f6618adb4af6dbdb0b72918d5d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Risk factor</td><td>Frequency of positive resistant</td><td>% proportion of positive resistant</td><td>p-value</td></tr><tr><td>History of mastitis</td><td></td><td></td><td></td></tr><tr><td>Yes</td><td>21/23</td><td>91.3</td><td>0.4242</td></tr><tr><td>No</td><td>46/55</td><td>83.6</td><td></td></tr><tr><td>Antibiotic treatment</td><td></td><td></td><td></td></tr><tr><td>Treatment administered</td><td>17/19</td><td>89.5</td><td>0.5781</td></tr><tr><td>No treatment administered</td><td>3/3</td><td>100</td><td></td></tr></table>\",\n \"table_caption\": \"[\\\"Table-3: Effects of history of mastitis and antibiotic treatment on occurrence of resistance to the selected beta-lactam antibiotics in dairy goats with subclinical mastitis in Thika subcounty. \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[507,766,914,931]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"daily (87.5%). No resistance was seen in those that were cleaned every 3 days (0.0%). From the data in the questionnaire on goats whose house were cleaned every 3 days, the farmers used water alone and udder and teats were dried with a disposable towel during milking. From the data analysis on milking practices used by the farmers, which was found to be significantly associated with the occurrence of resistance, the goats whose farmers used water alone and udder and teats dried with disposable a towel showed no resistance to beta-lactam antibiotics. Therefore, despite the \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[282,49,715,61]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Available at www.veterinaryworld.org/Vol.13/July-2020/27.pdf \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[82,936,341,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Veterinary World, EISSN: 2231-0916 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[875,936,912,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1452 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[80,71,485,130]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"goats' housing structure being cleaned every 3 days, the milking practices used on these goats might be a contributing factor in preventing the occurrence of resistance in these goats. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,130,485,261]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On the other hand, the goats whose houses were cleaned every 2 days had their udder and teats dried using a reusable towel during milking. Therefore, despite the goats' housing structure being cleaned every 2 days, the milking practices used on these goats might be a factor contributing to the occurrence of resistance in these goats. However, resistance was not significantly affected by the cleaning schedule of the housing structure $(p = 0.2141)$ (Table-4). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,261,485,363]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Resistance was highest among does $>4$ years old $(88.9\\\\%)$ and does with a parity $\\\\leq 4$ $(85.7\\\\%)$ . Resistance was lowest in does $\\\\leq 4$ years old $(86.7\\\\%)$ and does with a parity $>4$ parity $(80.0\\\\%)$ . However, there was no significant association between age and parity, respectively, and the occurrence of resistance $(p = 0.6203$ and $p = 0.333$ , respectively) (Table-5). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,370,179,384]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Discussion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,388,487,667]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Mastitis is an economically important disease of dairy goats and cattle. This infection is mainly subclinical in goats, making it impossible to diagnose by physical examination [8]. Losses resulting from subclinical mastitis include decreased milk production and quality and increased veterinary expenses as well as progression to the clinical form [31]. This study determined resistance against penicillin G, cephalexin, cefoxitin, and cefotaxime in bacterial isolates from milk of goats with subclinical mastitis in Thika subcounty. We found a higher prevalence of subclinical mastitis than that reported in Northern region, Bangladesh [32], Tando Jam, Pakistan [33] and Nigeria [34]. However, our result was in agreement with that recorded in the Mount Kenya region, Kenya [35], the state of Minas Gerais, Brazil [36], and Thika East subcounty [21]. Fluctuations in the prevalence of mastitis in different geographical regions are attributed to the complexity of the disease resulting from the interaction of several \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,71,912,145]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"factors, including management and husbandry strategies, geographical distribution, the herd's health condition, biological agents causing mastitis, and environmental and nutritional conditions as well as the sample size used in the studies [34]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,145,914,479]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our results showed that the highest proportion $(28.18\\\\%)$ of the bacterial isolates belonged to the Staphylococcus genus, particularly S. aureus $(17.45\\\\%)$ , which was similar to other studies reporting that the highest proportion of the isolates from goats with subclinical mastitis was S. aureus, ranging from $23.4\\\\%$ to $61.64\\\\%$ [15,33,37,38]. Staphylococci are listed as the major causative agents in the development of contagious mastitis [5]. This is attributed to the ability of these bacteria to produce highly toxic metabolites that damage the membranes of mammary glands [39]. In addition, Staphylococcus is effectively transmitted through several routes [39]. Therefore, in regions where hand milking is mostly practiced, such as Thika subcounty, Staphylococcus can be spread during milking operations, which could be associated with its high prevalence. An intramammary infection of S. aureus is a public health concern since the bacteria produce thermostable exotoxins that cause toxic shock syndrome and staphylococcal food poisoning. Furthermore, S. aureus can also infect humans and animals when unpasteurized milk and milk products are consumed [34,40]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,479,914,668]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Enterobacteriaceae are also associated with mastitis [41]. We isolated a relatively high proportion of members of the Enterobacteriaceae family $(55.5\\\\%)$ in our study, which is comparable to a previous report of the isolation of $64.5\\\\%$ Enterobacteriaceae from milk samples of goats with mastitis in Thika East subcounty [21]. Coliforms, represented by Klebsiella spp. $(20.1\\\\%)$ , Escherichia spp. $(4.0\\\\%)$ , Citrobacter spp. $(4.0\\\\%)$ , and Enterobacter spp. $(3.3\\\\%)$ , have been identified as the main cause of environmental mastitis in small ruminants. Coliforms are normally associated with environmental contamination of the udder and the establishment of intramammary infection [31]. The \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,708,914,932]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0613/4a9d25b16cba314384e324aef5d31391b3bd2cc0f62bbde47364b05259195d2f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Risk factor</td><td>Frequency of positive resistant</td><td>% proportion of positive resistant</td><td>p-value</td></tr><tr><td>Hygiene during milking</td><td></td><td></td><td></td></tr><tr><td>Use of water alone and udder and teat dried with disposable towel</td><td>0/2</td><td>0</td><td>0.0336</td></tr><tr><td>Use of water, disinfectant, and udder and teat dried with reusable towel</td><td>14/18</td><td>77.8</td><td></td></tr><tr><td>Use of water alone and udder and teat dried with reusable towel</td><td>16/17</td><td>94.1</td><td></td></tr><tr><td>Use of water alone and udder and teat are not dried</td><td>23/26</td><td>88.5</td><td></td></tr><tr><td>Goat house cleaning schedule</td><td></td><td></td><td></td></tr><tr><td>Once a week</td><td>18/22</td><td>81.8</td><td>0.2141</td></tr><tr><td>Daily</td><td>21/24</td><td>87.5</td><td></td></tr><tr><td>After 3 days</td><td>0/2</td><td>0</td><td></td></tr><tr><td>After 2 days</td><td>4/4</td><td>100</td><td></td></tr><tr><td>After 2 weeks</td><td>4/4</td><td>100</td><td></td></tr><tr><td>Twice a week</td><td>9/11</td><td>81.8</td><td></td></tr><tr><td>Thrice a week</td><td>2/2</td><td>100</td><td></td></tr><tr><td>Irregularly</td><td>9/9</td><td>100</td><td></td></tr></table>\",\n \"table_caption\": \"[\\\"Table-4: Effects of pre- and post-milking practices and cleaning schedule on occurrence of resistance to the selected beta-lactam antibiotics in dairy goats with subclinical mastitis in Thika subcounty. \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[282,49,712,61]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Available at www.veterinaryworld.org/Vol.13/July-2020/27.pdf \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[82,936,337,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Veterinary World, EISSN: 2231-0916 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[875,936,912,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1453 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[82,111,485,233]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0930/575641aa0de8957ff4465a242f1183bfda49d9b0fa2c833342c5e222d42e4857.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Risk factor</td><td>Frequency of positive resistant</td><td>% proportion of positive resistant</td><td>p-value</td></tr><tr><td>Age</td><td></td><td></td><td></td></tr><tr><td>≤4 years</td><td>39/45</td><td>86.7</td><td>0.6203</td></tr><tr><td>>4 years</td><td>24/27</td><td>88.9</td><td></td></tr><tr><td>Parity</td><td></td><td></td><td></td></tr><tr><td>≤4</td><td>54/63</td><td>85.7</td><td>0.333</td></tr><tr><td>>4</td><td>4/5</td><td>80.0</td><td></td></tr></table>\",\n \"table_caption\": \"[\\\"Table-5: Effects of age and parity on occurrence of resistance to the selected beta-lactam antibiotics in dairy goats with subclinical mastitis in Thika subcounty. \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[80,250,485,382]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"presence of coliform bacteria could be linked to poor hygiene practices in dairy goat farms. Poor hygiene is also reflected by an increase in the prevalence of subclinical mastitis (72.7%) recorded in this study, compared with 50.9% previously reported in Thika East subcounty. In addition, the frequency of isolation of different bacteria from goat milk is strongly associated with the overall hygiene status as well as the management strategy of the farms [42]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,384,485,501]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Although not listed as a major etiological agent, Acinetobacter spp. have also been reported in cases of subclinical mastitis in goats. We isolated a higher proportion $(16.1\\\\%)$ of Acinetobacter spp. in our study compared with that reported in goats in Nigeria $(3\\\\%)$ [34]. Yersinia spp. have not been associated with subclinical mastitis in goats; however, they have been widely isolated from raw goat milk [43,44]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,501,485,929]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The high resistance to penicillin G, cephalosporins, and cephamycin observed in our study was unforeseen. However, our findings corroborate with several other studies of dairy goats that reported high resistance to penicillin and cephalosporins of the bacterial isolates causing mastitis [13,36,37]. In addition, we observed cross-resistance among the bacterial isolates causing subclinical mastitis in goats. Similar results showing cross-resistance to penicillin and other $\\\\beta$ -lactams have been observed in goats [45] and cows [46]. The bacterial resistance mechanism to $\\\\beta$ -lactam antibiotics involves the production of $\\\\beta$ -lactamases and the synthesis of modified penicillin-binding proteins that inactivate $\\\\beta$ -lactam antibiotics by hydrolyzing the amide bond of the four-membered $\\\\beta$ -lactam ring, preventing drug binding [12,16]. The primary mechanism of bacterial resistance to $\\\\beta$ -lactam antibiotics is the production of $\\\\beta$ -lactamases, such as penicillinases, cephalosporinases, broad-spectrum $\\\\beta$ -lactamases, ESBLs, and carbapenemases [16]. These enzymes hydrolyze penicillins (including benzylpenicillin, also known as penicillin G), third-generation cephalosporins (including cefotaxime and ceftazidime), aztreonam, cephamycins (including cefoxitin), and carbapenems, but are inhibited by $\\\\beta$ -lactamase inhibitors, such as clavulanic acid [16]. At present, antibiotic therapy is the recommended treatment option for clinical and subclinical mastitis, and $\\\\beta$ -lactams are among the antibiotics used for mastitis treatment in \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,71,914,204]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Kenya. However, antibiotic resistance has spread over the years, mostly due to farmers initiating antibiotic therapy for mastitis treatment before conducting an antibiotic susceptibility test [47]. The high resistance to $\\\\beta$ -lactam antibiotics is concerning because limited therapeutic options are available for the treatment of such infections compounded with difficulty in controlling these bacteria due to their ability to rapidly spread and transfer the resistance phenotype. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,204,914,639]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hygiene is a major contributing factor to the spread of resistance in farms [48,49]. In our study, hygiene was defined by pre- and post-milking practices and the cleaning schedule of the goats' housing structure. We isolated a higher proportion of antibiotic-resistant bacteria isolated from goats whose farmers used water only and reusable towels during pre- and post-milking compared with those who used water, disinfectant, and reusable towels. This observation is probably due to the bactericidal effect of disinfectant [50]. The use of disinfectant during milking has been recommended as a control strategy to eradicate and prevent the spread of antibiotic-resistant bacteria in dairy farms [50]. However, the occurrence of antibiotic-resistant bacteria among goats whose farmers used disinfectant and reusable towels during milking was probably due to the use of reusable towels that might have been contaminated or not cleaned properly or due to other unknown factors. Furthermore, we found that hygiene practices during milking were significantly associated with the occurrence of resistance. However, hygiene in the goat housing structures was not significantly associated with the occurrence of resistance, which was probably due to the small sample size used in our study or other unknown factors. In addition, mastitis history, previous antibiotic treatment, parity, and age did not significantly impact the occurrence of resistance, and the observed variability can be attributed to different management practices in the farms or other unknown factors. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,640,915,800]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We did not phenotypically detect any ESBL producers. However, a negative synergy test for the phenotypic detection of ESBLs does not necessarily indicate that the bacteria are negative for ESBL production. False-negative ESBL results can be attributed to the overproduction of AmpC $\\\\beta$ -lactamases, inhibitor-resistant TEM $\\\\beta$ -lactamases, or TEM and SHV $\\\\beta$ -lactamases in ESBL-producing bacteria [30,51]. Therefore, there is a need for genotypic detection of ESBLs in bacterial isolates that show resistance to $\\\\beta$ -lactam antibiotics. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,808,608,821]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conclusion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,827,914,931]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We found a high prevalence of subclinical mastitis in the sampled dairy goats in Thika subcounty. The etiological agents causing subclinical mastitis are of zoonotic importance and are resistant to $\\\\beta$ -lactam antibiotics. This poses a health risk to humans, especially when raw milk and raw milk products are consumed. Hygiene during milking contributes to the occurrence \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[284,49,712,61]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Available at www.veterinaryworld.org/Vol.13/July-2020/27.pdf \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[84,936,339,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Veterinary World, EISSN: 2231-0916 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[875,936,912,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1454 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[80,72,487,175]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"of resistance to $\\\\beta$ -lactam antibiotics. More stringent measures, such as good animal husbandry and milking practices, must be adopted to control mastitis and prevent the spread of resistance to $\\\\beta$ -lactam antibiotics. Further studies should be undertaken to determine transmission dynamics between humans and livestock and identify novel intervention strategies. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,181,228,197]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data Availability \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,200,485,247]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The raw data used to support the findings of this study are available from the corresponding author on request. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,252,280,266]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Authors' Contributions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,272,485,404]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IMO, JK, and NM involved in the conception of the research idea. IMO and JK planned the study design. IMO performed data collection, sample collection, and laboratory work. IMO and DK contributed in statistical analysis and interpretation of the data results. JK and DK provided laboratory guidance. IMO drafted the manuscript. JK, DK, and NM corrected the manuscript. All the authors read and approved the final manuscript. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,411,243,425]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acknowledgments \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,430,485,752]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This research was financially supported by Institute of Basic Sciences, Technology and Innovation, Pan-African University (PAUSTI) and Jomo Kenyatta University of Agriculture and Technology (JKUAT). The Grand Challenges Africa program (GCA/AMR/rnd2/079) also financially supported the work. Grand Challenges Africa is a program of the African Academy of Sciences (AAS) implemented through the Alliance for Accelerating Excellence in Science in Africa (AESA) platform, an initiative of the AAS and the African Union Development Agency (AUDA-NEPAD). GC Africa is supported by the Bill and Melinda Gates Foundation (BMGF) and The African Academy of Sciences and partners. The views expressed herein are those of the author(s) and not necessarily those of the AAS and her partners. The authors acknowledge the technical assistance provided by JKUAT staff, including Dr. Macharia Lewis and Mr. Muthanga Josephat, the extension officer from the veterinary office Thika subcounty, Mr. Gachina as well as PAUSTI students Armel Tangomo and Ronald Tanui. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,759,263,772]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Competing Interests \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,778,485,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The authors declare that they have no competing interests. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,814,226,828]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Publisher's Note \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,834,485,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Veterinary World remains neutral with regard to jurisdictional claims in published institutional affiliation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,885,181,898]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,903,485,929]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Miller, B.A. and Lu, C.D. (2019) Current status of global dairy goat production: An overview. Asian Australas. J. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"ref_text\"\n },\n {\n \"bbox\": \"[512,72,912,929]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"Anim. Sci., 32(8): 1219-1232. \\\",\\\"2. Kahi, A.K. and Wasike, C.B. (2019) Dairy goat production in sub-Saharan Africa: Current status, constraints and prospects for research and development. Asian Australas. J. Anim. Sci., 32(8): 1266-1274. \\\",\\\"3. Makini, F. and Mose, L. (2019) Innovation Dairy Livestock in Kenya. Global Trend Report. \\\",\\\"4. Turkmen, N. (2017) The nutritional value and health benefits of goat milk components. In: Watson, R.R., Collier, R.J. and Preedy, V.R., editors. Nutrients in Dairy and their Implications on Health and Disease. Ch. 35. Academic Press, Cambridge, Massachusetts. p441-449. \\\",\\\"5. Contreras, A., Sierra, D., Sánchez, A., Corrales, J.C., Marco, J.C., Paape, M.J. and Gonzalo, C. (2007) Mastitis in small ruminants. Small Rumin. Res., 68(1): 145-153. \\\",\\\"6. Shearer, J.K. and Harris, B. (1992) DS 85 Mastitis in Dairy Goats. Vol. 1. University of Florida, Florida. p1-7. \\\",\\\"7. Abebe, R., Hatiya, H., Abera, M., Megersa, B. and Asmare, K. (2016) Bovine mastitis: Prevalence, risk factors and isolation of Staphylococcus aureus in dairy herds at Hawassa milk shed, South Ethiopia. BMC Vet. Res., 12(1): 270. \\\",\\\"8. Mishra, A.K., Sharma, N., Singh, D.D., Gururaj, K., Abhishek, Kumar, V. and Sharma, D.K. (2018) Prevalence and bacterial etiology of subclinical mastitis in goats reared in organized farms. Vet. World, 11(1): 20-24. \\\",\\\"9. Abdalhamed, A.M., Zeedan, G.S.G. and Abou Zeina, H.A.A. (2018) Isolation and identification of bacteria causing mastitis in small ruminants and their susceptibility to antibiotics, honey, essential oils, and plant extracts. Vet. World, 11(3): 355-362. \\\",\\\"10. Mitema, E.S., Kikuvi, G.M., Wegener, H.C. and Stohr, K. (2001) An assessment of antimicrobial consumption in food producing animals in Kenya. J. Vet. Pharmacol. Ther., 24(6): 385-390. \\\",\\\"11. Oliver, S.P. and Murinda, S.E. (2012) Antimicrobial resistance of mastitis pathogens. Vet. Clin. Food Anim. Pract., 28(2): 165-185. \\\",\\\"12. Bush, K. and Bradford, P.A. (2016) $\\\\\\\\beta$ -Lactams and $\\\\\\\\beta$ -lactamase inhibitors: An overview. Cold Spring Harb. Perspect. Med., 6(8): a025247. \\\",\\\"13. Scaccabarozzi, L., Leoni, L., Ballarini, A., Barberio, A., Locatelli, C., Casula, A. and Moroni, P. (2015) Pseudomonas aeruginosa in dairy goats: Genotypic and phenotypic comparison of intramammary and environmental isolates. PLoS One, 10(11): e0142973. \\\",\\\"14. Mello, P.L., Pinheiro, L., Martins, L., Brito, M.A.V. and da Cunha, M.L. (2017) Short communication: $\\\\\\\\beta$ -Lactam resistance and vancomycin heteroresistance in Staphylococcus spp. isolated from bovine subclinical mastitis. J. Dairy Sci., 100(8): 6567-6571. \\\",\\\"15. Aqib, A.I., Nighat, S., Ahmed, R., Sana, S., Jamal, M.A., Kulyar, M.F.A. and Rahman, S. (2018) Drug susceptibility profile of Staphylococcus aureus isolated from mastitic milk of goats and risk factors associated with goat mastitis in Pakistan. Pak. J. Zool., 51(1): 307-315. \\\",\\\"16. Santajit, S. and Indrawattana, N. (2016) Mechanisms of antimicrobial resistance in ESKAPE pathogens. *BioMed Res. Int*, 2016: Article ID 2475067. \\\",\\\"17. Muriuki, H.G. (2003) A Review of the Small Scale Dairy Sector-Kenya, No. 60. FAO Prevention of Food Losses Programme. \\\",\\\"18. Njarui, D.M.G., Gatheru, M., Wambua, J.M., Nguluu, S.N., Mwangi, D.M. and Keya, G.A. (2011) Consumption patterns and preference of milk and milk products among rural and urban consumers in semi-arid Kenya. Ecol. Food Nutr., 50(3): 240-262. \\\",\\\"19. Hwang, A.Y. and Gums, J.G. (2016) The emergence and evolution of antimicrobial resistance: Impact on a global scale. Bioorg. Med. Chem., 24(24): 6440-6445. \\\",\\\"20. Scarafile, G. (2016) Antibiotic resistance: Current issues and future strategies. Rev. Health Care, 7(1): 3-16. \\\",\\\"21. Mahlangu, P., Maina, N. and Kagira, J. (2018) Prevalence, \\\"]\",\n \"page_idx\": \"7\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[282,49,712,61]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Available at www.veterinaryworld.org/Vol.13/July-2020/27.pdf \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[82,936,337,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Veterinary World, EISSN: 2231-0916 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[875,936,912,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1455 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[84,71,485,760]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"risk factors, and antibiogram of bacteria isolated from milk of goats with subclinical mastitis in Thika East Subcounty, Kenya. J. Vet. Med., 2018: Article ID 3801479. \\\",\\\"22. Thrusfield, M.V. (2007) Veterinary Epidemiology. $3^{\\\\\\\\mathrm{rd}}$ ed. Blackwell Science, Oxford. \\\",\\\"23. Handcock, M.S. and Gile, K.J. (2011) On the Concept of Snowball Sampling. Available from: http://www.arxiv.org/abs/1108.0301. Retrieved on 28-08-2019. \\\",\\\"24. Walker, J. B., Rajala-Schultz, P. J., and DeGraves, F. J. (2010) The effect of inoculum volume on the microbiologic detection of naturally occurring Staphylococcus aureus intramammary infections. J. Vet. Diagn. Invest., 22(5): 720-724. \\\",\\\"25. Garrity, G., Brenner, D.J., Krieg, N.R. and Staley, J.R., editors. (2005) The proteobacteria, Part B: The gammaproteobacteria. In: Bergey's Manual® of Systematic Bacteriology. 2nd ed., Vol. 2. Springer, United States. \\\",\\\"26. Cheesbrough, M. (2006) District Laboratory Practice in Tropical Countries. $2^{\\\\\\\\mathrm{nd}}$ ed. Cambridge University Press, Cambridge, New York. \\\",\\\"27. Quinn, P.J., Markey, B.K., Leonard, F.C., Hartigan, P., Fanning, S. and Fitzpatrick, E.S. (2011) Veterinary Microbiology and Microbial Disease. John Wiley and Sons, Hoboken, New Jersey. \\\",\\\"28. Bauer, A. W., Kirby, W. M., Sherris, J. C., and Turck, M. (1966) Antibiotic susceptibility testing by a standardized single disk method. Am. J. Clin. Pathol., 45(4), 493-496. \\\",\\\"29. Clinical and Laboratory Standards Institute. (2016) M100S Performance Standards for Antimicrobial Susceptibility Testing. $26^{\\\\\\\\mathrm{th}}$ ed. Clinical and Laboratory Standards Institute, Wayne, PA. \\\",\\\"30. Drieux, L., Brossier, F., Sougakoff, W. and Jarlier, V. (2008) Phenotypic detection of extended-spectrum $\\\\\\\\beta$ -lactamase production in Enterobacteriaceae: Review and bench guide. Clin. Microbiol. Infect., 14(Suppl 1): 90-103. \\\",\\\"31. Gelasakis, A.I., Angelidis, A.S., Giannakou, R., Filioussis, G., Kalamaki, M.S. and Arsenos, G. (2016) Bacterial subclinical mastitis and its effect on milk yield in low-input dairy goat herds. J. Dairy Sci., 99(5): 3698-3708. \\\",\\\"32. Ferdous, J., Rahman, M., Khan, M., Khan, M., & Rima, U. (2018). Prevalence of clinical and subclinical caprine mastitis of northern region in Bangladesh. Progress. Agric., 29(2): 127-138. \\\",\\\"33. Pirzada, M. and Ali, A. (2016) Prevalence of subclinical mastitis in dairy goats caused by bacterial species. J. Anim. Health Prod., 4(2): 55-59. \\\",\\\"34. Danmallam, F.A. and Pimenov, N.V. (2019) Study on prevalence, clinical presentation, and associated bacterial pathogens of goat mastitis in Bauchi, Plateau, and Edo states, Nigeria. Vet. World, 12(5): 638-645. \\\",\\\"35. Mbindyo, C.M., Gitao, C.G. and Bebora, L. (2014) A cross-sectional study on the prevalence of subclinical mastitis and antimicrobial susceptibility patterns of the bacterial isolates in milk samples of smallholder dairy goats in Kenya. Am. J. Res. Commun., 2(8): 22-50. \\\",\\\"36. Lima, M.C., Souza, M.C.C., Espeschit, I.F., Maciel, P.A.C., Sousa, J.E., Moraes, G. and Moreira, M.A.S. (2018) Mastitis in dairy goats from the state of Minas Gerais, Brazil: Profiles of farms, risk factors and characterization \\\"]\",\n \"page_idx\": \"8\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[512,72,912,762]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"of bacteria. Pesqui. Vet. Bras., 38(9): 1742-1751. \\\",\\\"37. Najeeb, M.F., Anjum, A.A., Ahmad, M.U.D., Khan, H.M., Ali, M.A. and Sattar, M.M.K. (2013) Bacterial etiology of subclinical mastitis in dairy goats and multiple drug resistance of the isolates. J. Anim. Plant Sci., 23(6): 1541-1544. \\\",\\\"38. Haftay, A., Habtamu, T.M. and Abebe, M.S. (2016) Susceptibility of subclinical mastitis causing bacteria from goats in Aba'lla district, Afar, North-Eastern Ethiopia. Rev. Med. Vet., 6(1): 1-6. \\\",\\\"39. Petersson-Wolfe, C.S., Mullarky, I.K. and Jones, G.M. (2010) Staphylococcus aureus Mastitis: Cause, Detection, and Control. Vol. 7. VCE Publications, VA, USA. \\\",\\\"40. Merz, A., Stephan, R. and Johler, S. (2016) Staphylococcus aureus isolates from goat and sheep milk seem to be closely related and differ from isolates detected from bovine milk. Front. Microbiol., 7: 319. \\\",\\\"41. Machado, G.P., Silva, R.C., Guimarães, F.F., Salina, A., Langoni, H., Machado, G. and Langoni, H. (2018) Detection of Staphylococcus aureus, Streptococcus agalactiae and Escherichia coli in Brazilian mastitic milk goats by multiplex-PCR. Pesqui. Vet. Bras., 38(7): 1358-1364. \\\",\\\"42. McDougall, S., Pankey, W., Delaney, C., Barlow, J., Murdough, P.A. and Scruton, D. (2002) Prevalence and incidence of subclinical mastitis in goats and dairy ewes in Vermont, USA. Small Rumin. Res., 46(2): 115-121. \\\",\\\"43. Rahimi, E., Sepehri, S., Dehkordi, F.S., Shaygan, S. and Momtaz, H. (2014) Prevalence of Yersinia species in traditional and commercial dairy products in Isfahan Province, Iran. *Jundishapur J. Microbiol.*, 7(4): e9249. \\\",\\\"44. Jamali, H., Paydar, M., Radmehr, B. and Ismail, S. (2015) Prevalence, characterization, and antimicrobial resistance of Yersinia species and Yersinia enterocolitica isolated from raw milk in farm bulk tanks. J. Dairy Sci., 98(2): 798-803. \\\",\\\"45. Bochev, I.V. and Russenova, N. (2005) Resistance of Staphylococcus spp. Strains isolated from goats with subclinical mastitis. *Bulg J. Vet. Med.*, 8(8): 109-118. \\\",\\\"46. Berghash, S.R., Davidson, J.N., Armstrong, J.C. and Dunny, G.M. (1983) Effects of antibiotic treatment of non-lactating dairy cows on antibiotic resistance patterns of bovine mastitis pathogens. Antimicrob. Agents Chemother., 24(5): 771-776. \\\",\\\"47. Ceniti, C., Britti, D., Santoro, A.M.L., Musarella, R., Ciambrone, L., Casalinuovo, F. and Costanzo, N. (2017) Phenotypic antimicrobial resistance profile of isolates causing clinical mastitis in dairy animals. Ital. J. Food Saf., 6(2): 6612. \\\",\\\"48. Lee, C.R., Cho, I.H., Jeong, B.C. and Lee, S.H. (2013) Strategies to minimize antibiotic resistance. Int. J. Environ. Res. Public Health, 10(9): 4274-4305. \\\",\\\"49. Adnan, M., Khan, H., Kashif, J., Ahmad, S., Gohar, A., Ali, A. and Khan, N.A. (2017) Clonal expansion of sulfonamide resistant Escherichia coli isolates recovered from diarrheic calves. *Pak. Vet.* J., 37(2): 253-8318. \\\",\\\"50. Gohary, A., Gohary, F., Elsayed, M. and ElFateh, M. (2019) In-vitro investigation on the antiseptic efficacy of commonly used disinfectants in dairy farms against methicillin-resistant Staphylococcus aureus. Alex. J. Vet. Sci., 60(1): 86. \\\",\\\"51. Lalitha, D.M.K. (2004) Manual on antimicrobial susceptibility testing. Indian Assoc. Med. 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Rice|Kigocha Okeng'o|Emmanuel Massawe|Seif Ismail|Notburga A. Mworia|Faraja Chiwanga|Boniface Kapina|Michael Wasserman|Farrah J. Mateen"},"content_list":{"kind":"list like","value":[{"bbox":"[116,95,881,146]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Efficacy of Fluoxetine for Post-Ischemic Stroke Depression in Tanzania ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[150,174,844,244]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Dylan R. Rice, $^{a}$ Kigocha Okeng'o, $^{b}$ Emmanuel Massawe, $^{b}$ Seif Ismail, $^{b}$ Notburga A. Mworia, $^{b}$ Faraja Chiwanga, $^{b}$ Boniface Kapina, $^{b}$ Michael Wasserman, $^{a}$ and Farrah J. Mateen, $^{a}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[234,295,763,627]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Objective: Post-stroke fluoxetine trials are primarily conducted in high-income countries. We characterize post-ischemic stroke depression in fluoxetine-treated and -untreated study participants in urban Tanzania. Methods: Adults $(>18$ years old) within 14 days of CT-confirmed acute ischemic stroke onset were enrolled at Muhimbili National Hospital, Tanzania. The fluoxetine-treated group took $20\\mathrm{mg}$ fluoxetine daily for 90 days in a phase II trial and were compared to fluoxetine-untreated historical controls. The primary outcome was depression at 90 days, measured by the Patient Health Questionnaire-9 (PHQ-9). PHQ-9 scores were compared between fluoxetine-treated and -untreated groups. A score $>=9$ points was considered to reflect depression. A multivariable linear regression model assessed associations with post-stroke PHQ-9 scores. Results: Of the fluoxetine-treated $(n = 27)$ and -untreated $(n = 32)$ participants, the average age was 56.8 years old $(39\\%$ women, $100\\%$ Black/African). The average presentation NIHSS score was 12.1 points and modified Rankin Scale (mRS) score was 3.5. The average mRS score at 90-day follow-up was 2.3. There was no significant difference between 90-day PHQ-9 scores in the fluoxetine-treated (mean=4.1 points, standard deviation=3.2; $11\\%$ depression) and untreated (mean=4.4, standard deviation=4.8; $19\\%$ depression) groups, $p = .69$ . In the multivariable analysis, older age $(\\beta = 0.08, p = .03)$ and higher NIHSS score $(\\beta = 0.15, p = .04)$ , but neither fluoxetine $(\\beta = 0.57, p = .59)$ nor sex $(\\beta = -0.51, p = .63)$ , were significantly associated with more depressive symptoms. Conclusions: Our findings parallel results from trials from higher income settings that fluoxetine does not significantly improve post-ischemic stroke depression, although our sample size was small. More work is needed to depict the longitudinal nature and treatment of post-stroke depression in Sub-Saharan Africa. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[234,628,664,642]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Key Words: Depression—Stroke—Fluoxetine—Treatment—Africa ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[234,642,483,655]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"© 2021 Elsevier Inc. All rights reserved. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[107,688,212,704]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[91,712,480,758]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A diagnosis of post-stroke depression is defined as a period of depressed mood, lasting two or more weeks after the occurrence of stroke, and the presence of at least ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,689,903,750]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"four other symptoms of depression as characterized by the Diagnostic and Statistical Manual of Mental Disorders-5. Post-stroke depression affects nearly one-third of patients who have experienced stroke.2 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,750,903,902]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A meta-analysis exploring depression after stroke in Sub-Saharan Africa found a post-stroke depression prevalence of $31\\%$ in approximately 1500 stroke patients across 17 studies. The distinction between ischemic and hemorrhagic stroke was not made. Lower education level, poststroke cognitive impairment, physical disability, lower quality of life, and divorced marital status have each been associated with post-stroke depression in Sub-Saharan Africa. However, the associations with post-stroke depression in this context are not consistently measured ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,786,482,811]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From the aMassachusetts General Hospital, Boston, MA, United States; and bMuhimbili National Hospital, Dar es Salaam, Tanzania. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[91,811,480,837]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Received September 1, 2021; revision received October 14, 2021; accepted October 17, 2021. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[91,837,480,875]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Address correspondence to Massachusetts General Hospital, Department of Neurology, 165 Cambridge Street, #627, Boston, MA, United States. E-mail: fmateen@mgh.harvard.edu. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[104,875,274,887]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1052-3057/$ - see front matter ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[104,888,325,900]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"© 2021 Elsevier Inc. All rights reserved. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[104,900,438,912]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.1016/j.jstrokerebprovdis.2021.106181 ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[91,930,602,945]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Journal of Stroke and Cerebrovascular Diseases, Vol. 31, No. 1 (January), 2022: 106181 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[890,931,902,941]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[94,70,478,143]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"or identified across studies. Of all studies in this meta-analysis from the African continent, none explored the impact of fluoxetine, a low-cost selective serotonin reuptake inhibitor (SSRI) typically indicated for depression and readily available on most Essential Medicines Lists. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[94,146,478,294]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Researchers from the AFFINITY trial, originally reporting on the lack of impact of fluoxetine on post-stroke motor recovery, recently reported on the post-depression outcomes in their participants. They concluded that daily fluoxetine, administered for 26 weeks post-stroke, did not significantly reduce depressive symptoms compared to placebo. Notably, AFFINITY enrolled $< 2\\%$ of participants of Black or African ancestry. To date, no trials have reported on post-stroke fluoxetine outcomes from Sub-Saharan Africa. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[94,296,478,415]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We recently completed a phase II trial - MAMBO (Measuring Ambulation, Motor, and Behavioral Outcomes) - of fluoxetine $20\\mathrm{mg}$ , administered for 90 days post-ischemic stroke to adults in urban Tanzania. Here, we aim to report on the secondary outcome of post-stroke depression in MAMBO's fluoxetine-treated participants compared to historical ischemic stroke fluoxetine-untreated controls enrolled from the same Tanzanian hospital. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,431,180,445]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Methods ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[112,455,215,469]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ethics approvals ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[94,478,478,580]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ethics boards approvals were obtained from the Muhimbili National Hospital, the National Institute of Medical Research of Tanzania, the Tanzania Medicine and Medical Devices Authority, and the Partners Healthcare Institutional Review Board. Each participant or when necessary, a next of kin proxy, provided his or her own individual consent. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,599,156,613]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Setting ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[94,621,478,830]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Tanzania has a gross national income per capita (Atlas method, 2020) of 1080 USD. All participants were prospectively recruited from the Muhimbili National Hospital in Dar es Salaam. This is the largest academic referral hospital in Tanzania with an estimated 1000-1200 patients admitted per week. The hospital has four neurologists. Stroke services available through the neurology department include basic care and thrombolysis for those who can afford it. The yearly age-standardized incidence rate of stroke in Dar es Salaam, Tanzania is presumed to be approximately 316 per 100,000 persons. Not all patients have health insurance and most have government funded support or family support for hospital expenses. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,848,188,861]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Participants ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[94,869,478,914]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A flow diagram of participants is provided in Figure 1. Participants were (1) fluoxetine-treated participants in a phase II trial of fluoxetine (MAMBO; 11/2019 to 10/2020) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,70,900,112]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and (2) prospectively enrolled participants from an observational cohort study (07/2016 to 03/2017) who served as historical controls. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,114,900,340]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The participants were age $>18$ years old; admitted within 14 days of stroke symptom onset, meeting the WHO definition of acute stroke; and had a CT-confirmed ischemic stroke at the time of onset. The fluoxetine-treated group had additional exclusion criteria: dysphagia preventing pill swallowing, hyponatremia at stroke onset $(<125\\mathrm{mmol / L})$ , and/or hepatic impairment (serum ALT) $>120$ units/L). In this group, participants were administered open-label $20\\mathrm{mg}$ oral fluoxetine daily for 90 days. Fluoxetine for this trial was procured through the University of Iowa Pharmacy laboratory. For complete details on the inclusion and exclusion criteria for this group, see the published MAMBO protocol. Enrollment in the MAMBO trial stopped early due to COVID-19, leading to a lower than intended sample size. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[532,359,603,372]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Procedures ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,381,900,547]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Both groups had baseline and 90-day follow up visits conducted by physicians and study staff at the Muhimbili National Hospital. NIH Stroke Scale (NIHSS) scores, sociodemographic variables, and modified Rankin Scale (mRS) scores were assessed at baseline. Adherence in the fluoxetine-treated group was assessed using a medication possession ratio, assessed in person by the investigators every 30 days. At 90-days, mRS scores were reassessed, and the Patient Health Questionnaire-9 (PHQ-9) was administered in Kiswahili or English, as preferred by the participant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[532,566,591,578]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Outcome ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,588,900,752]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The primary outcome reported here is the burden of depressive symptoms at the 90-day follow up visit, as measured by the PHQ-9.[12] This measure has previously been translated and validated in Kiswahili in a Sub-Saharan African setting.[13] The PHQ-9 assesses depressive symptoms within the last two weeks with a series of 9 items, each using scores from 0 (not at all) to 3 (nearly every day) points. Overall scores range from 0–27 points. A score of $>=9$ points was used to classify significant depressive symptom burden, consistent with a recent large trial definition.[5] ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[532,772,620,786]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Data analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,794,900,914]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Variables were described by their means and standard deviations (SDs). Baseline characteristics between the fluoxetine-treated and -untreated groups were compared using $t$ -tests, tests of two proportions, or chi-square tests, as appropriate. A Mann-Whitney $U$ -test was used to compare PHQ-9 scores between fluoxetine-treated and -untreated participant groups, and a Fisher's exact test was used to compare the proportions of participants ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[94,47,104,56]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[790,46,900,56]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"D.R. RICE ET AL. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[129,65,866,489]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 1. Flow of participants. \"]","image_footnote":"[]","img_path":"images/0528/90a297e98390b30893f407d7a420562b6912fd4e3068bebe9a8dc688a2aa3d94.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[89,537,482,750]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"meeting criteria for depressive symptom burden between these two groups. Ninety-five percent confidence intervals (CIs) were calculated for the proportion of participants in each group who met criteria for depressive symptom burden. A multivariable linear regression model was used to assess associations between PHQ-9 scores at 90 days as the continuous outcome variable and participant sex (binary: male or female), age (continuous), fluoxetine administration (binary: treated or untreated), and NIHSS score (continuous) as explanatory variables. No significant deviations from standard statistical assumptions were found for this model. All analyses were conducted in R (Version 1.0.153). A p-value $< .05$ was considered statistically significant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[107,770,170,784]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Results ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[91,794,480,883]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Participant characteristics at baseline, comparing fluoxetine-treated $(n = 27)$ and -untreated $(n = 32)$ groups, are reported in Table 1. Overall, the groups were balanced on the clinical and sociodemographic variables assessed with the exception that more fluoxetine-treated participants were employed. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,885,480,915]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Participants were on average 56.8 years old (SD=14.1 years). Thirty-nine percent were women, $100\\%$ were ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,537,903,686]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"African Black, and most had a highest educational attainment level of primary school (58%, $n = 34$ ). At enrollment, the average NIHSS score was 12.1 points (SD=7.4 points), and the average mRS was 3.5 points (SD=1.0 points), indicating that participants had mild to moderately severe strokes and experienced moderate to moderately severe disability. At 90 days, the average mRS score was 2.3 (SD=1.4), indicating mild disability. In the fluoxetine-treated group, adherence to fluoxetine was 96% at 90 days. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,688,903,883]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The distribution of PHQ-9 scores for both groups is shown in Figure 2. The average PHQ-9 score at 90-day follow up was 4.1 points $(\\mathrm{SD} = 3.2)$ for fluoxetine-treated patients and 4.4 points $(\\mathrm{SD} = 4.8)$ for fluoxetine-untreated patients. Eleven percent of fluoxetine-treated participants met criteria for significant depressive symptom burden $(n = 3; 95\\% \\text{ CI}[0,24])$ , compared to $19\\%$ $(n = 6; 95\\% \\text{ CI}[4,33])$ in untreated participants. We did not observe a statistically significant difference in the mean PHQ-9 scores between the two groups, $p = .69$ . Similarly, we did not observe a statistically significant difference in the proportion of participants who met criteria for depression by fluoxetine treatment versus not, $p = .49$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,885,903,915]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the multivariable linear regression model (Table 2) with PHQ-9 score as the outcome variable and age, sex, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,46,409,56]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"FLUOXETINE FOR POST-STROKE DEPRESSION ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[892,47,902,56]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[92,89,903,547]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0460/3be1182391992ca414351b4c19ee8d0ae89f6faa993c88e518999936f7f34779.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":"<table><tr><td></td><td>Fluoxetine-treated (MAMBO participants)</td><td>Fluoxetine-untreated (Tanzanian controls)</td><td>Comparison between groups, p</td></tr><tr><td>n</td><td>27</td><td>32</td><td>-</td></tr><tr><td>Age in years, mean (standard deviation; SD)</td><td>53.7 (13.0)</td><td>59.5 (14.6)</td><td>.11</td></tr><tr><td>Sex, n (%)</td><td></td><td></td><td>.28</td></tr><tr><td>Female</td><td>8 (30)</td><td>15 (47)</td><td></td></tr><tr><td>Male</td><td>19 (70)</td><td>17 (53)</td><td></td></tr><tr><td>NIH Stroke Scale score, mean (SD)</td><td>10.5 (4.4)</td><td>13.5 (9.0)</td><td>.12</td></tr><tr><td>Modified Rankin Scale score, mean (SD)</td><td></td><td></td><td></td></tr><tr><td>At baseline</td><td>3.7 (0.9)</td><td>3.3 (1.1)</td><td>.10</td></tr><tr><td>At 90-day follow up</td><td>2.1 (1.0)</td><td>2.4 (1.6)</td><td>.32</td></tr><tr><td>Highest level of education, n (%)</td><td></td><td></td><td>.28</td></tr><tr><td>None</td><td>0</td><td>4 (13)</td><td></td></tr><tr><td>Primary school</td><td>16 (59)</td><td>18 (56)</td><td></td></tr><tr><td>Some secondary school</td><td>2 (7)</td><td>4 (13)</td><td></td></tr><tr><td>Complete secondary school</td><td>6 (22)</td><td>4 (13)</td><td></td></tr><tr><td>University</td><td>2 (7)</td><td>1 (3)</td><td></td></tr><tr><td>Employment status, n (%)</td><td></td><td></td><td><.01</td></tr><tr><td>Employed</td><td>25 (93)</td><td>19 (59)</td><td></td></tr><tr><td>Unemployed</td><td>2 (7)</td><td>13 (41)</td><td></td></tr><tr><td>Comorbidities,† n yes (%)</td><td></td><td></td><td></td></tr><tr><td>BMI >30 kg/m2</td><td>1 (4)</td><td>6 (19)</td><td>.17</td></tr><tr><td>Cigarette smoking</td><td>6 (22)</td><td>3 (9)</td><td>.32</td></tr><tr><td>Diabetes</td><td>6 (22)</td><td>5 (16)</td><td>.75</td></tr><tr><td>History of depression</td><td>1 (4)</td><td>2 (6)</td><td>1</td></tr><tr><td>History of stroke</td><td>0</td><td>2 (6)</td><td>.12</td></tr><tr><td>HIV/AIDS</td><td>0</td><td>0</td><td>-</td></tr><tr><td>Hypertension</td><td>25 (93)</td><td>23 (72)</td><td>.09</td></tr><tr><td>Substance use, including alcohol</td><td>5 (19)</td><td>5 (16)</td><td>1</td></tr><tr><td>90-day Patient Health Questionnaire-9 (PHQ-9) score</td><td></td><td></td><td></td></tr><tr><td>Mean (SD)</td><td>4.1 (3.2)</td><td>4.4 (4.8)</td><td>.81 .42</td></tr><tr><td>PHQ-9 score >=9, n (%)</td><td>3 (11)</td><td>6 (19)</td><td></td></tr></table>","table_caption":"[\"Table 1. Participant characteristics at enrollment. \"]","table_footnote":"[\"+Comorbidities were primarily self-reported by participants. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[91,598,480,688]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"fluoxetine treatment, and NIHSS score as explanatory variables, only older age $(\\beta = 0.08, p = .03)$ and higher baseline NIHSS score $(\\beta = 0.15, p = .04)$ were statistically significantly associated with depressive symptom burden. There were no significant association with depressive symptom burden by sex or fluoxetine administration. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[109,710,200,724]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Discussion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[91,733,482,914]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We develop an evidence base on the limited efficacy of fluoxetine for post-ischemic stroke depression in Sub-Saharan Africa. In our fluoxetine-treated MAMBO trial participants from Tanzania, $11\\%$ met criteria for significant depressive symptom burden at 90-day follow up. Although we did not have a placebo arm in the MAMBO trial, using historical controls from the same Tanzanian hospital we found $19\\%$ of this fluoxetine-untreated sample met criteria for significant depressive symptom burden at 90 days. However, the difference in depressive symptom burden between these groups was not statistically significant. In both groups, higher stroke severity at ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,598,903,627]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"onset and older age are associated with a higher burden of depressive symptoms by 90 days. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,628,903,915]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A network of clinical trials examined the role of poststroke fluoxetine on post-stroke recovery and depression. The FLAME (France), EFFECTS (Sweden), and FOCUS (United Kingdom) trials[14-16] utilized different measures for depression, but the AFFINITY trial (Australasia) used the PHQ-9. We found a remarkably similar prevalence of depressive symptom burden in our fluoxetine-treated sample $(11\\%)$ at 90 days compared to the $10.8\\%$ reported in the fluoxetine-treated arm of AFFINTY at 84 days. While $19\\%$ of our fluoxetine-untreated sample met criteria for depressive symptom burden at 90 days, only $9.7\\%$ of the placebo arm patients at 84 days in the AFFINITY trial met such criteria. This may relate to milder strokes in the AFFINITY trial, better stroke services, or other factors. Notably, our fluoxetine-treated and control participants were younger at enrollment (mean age 56.8 years in Tanzania versus 63.8 years in AFFINTY, 64.7 in FLAME; $>56\\%$ of participants in both FOCUS and EFFECTS were $>70$ years of age) and experienced more severe stroke ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[94,47,104,56]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[790,46,900,56]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"D.R. RICE ET AL. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[131,71,866,364]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0587/01d6458ea083e3d56bb1ca7b3677d541b33cfb5c6f9f97acfac606936a66eb1b.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[133,375,831,681]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 2. Comparison of 90-day Patient Health Questionnaire-9 scores between fluoxetine-treated and -untreated participants depicted by A) box plots and B) proportional bar charts. \"]","image_footnote":"[]","img_path":"images/0299/4be000b6ae623be92022ffcbe854772572690d0ab5a72d4788b084c559ee9c6a.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[92,768,903,854]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0938/7c588dea5f77680d4ce578db434ed903f41f5267873d1f4879d255dc3af5722b.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":"<table><tr><td></td><td>β</td><td>Standard error</td><td>95% confidence interval of β</td><td>p</td></tr><tr><td>Fluoxetine treated</td><td>0.57</td><td>1.06</td><td>[-1.55, 2.69]</td><td>.59</td></tr><tr><td>Age (continuous)</td><td>0.08</td><td>0.04</td><td>[0.01, 0.16]</td><td>.03*</td></tr><tr><td>Female sex</td><td>-0.51</td><td>1.05</td><td>[-2.61, 1.60]</td><td>.63</td></tr><tr><td>NIH Stroke Scale score (continuous)</td><td>0.15</td><td>0.07</td><td>[0.01, 0.29]</td><td>.04*</td></tr></table>","table_caption":"[\"Table 2. Multivariable linear regression statistics evaluating variables associated with post-stroke depression in Tanzania. \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[109,857,161,869]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n^ {*} \\mathrm {p} < . 0 5\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[92,46,409,56]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"FLUOXETINE FOR POST-STROKE DEPRESSION ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[893,47,903,56]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[91,69,480,130]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"presentations as measured by the median NIHSS score (11 points in Tanzanian participants versus 6 points in AFFINITY, 6 points in FOCUS, and 3 points in EFFECTS; the mean NIHSS score was 13 points in FLAME). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,130,480,234]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"However, like the AFFINITY trial, we also found no statistically significant difference between fluoxetine-treated and untreated post-stroke depression in our study participants in Tanzania. Although our sample size was much smaller, we observed that fluoxetine has no overall measurable benefit on post-stroke depression in unselected patients at 90 days in this Sub-Saharan African setting. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[94,237,483,566]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The prevalence of post-stroke depression reported in our participants study is also notably less than the approximately one-third of post-stroke patients commonly reported in the literature.[2,3] There are several possible explanations for this. Our participants were well-evaluated and received better than usual care because of study enrollment, including access to free neuroimaging as well as 30-, 60-, and 90-day follow up visits by a physician. Participants may have had response bias, wanting to report fewer depressive symptoms to an investigator. In addition, there is a very high post-stroke mortality at the Muhimbili National Hospital, approximating $50\\%$ .[17] Participants with the most severe strokes were more likely to die and could not be included in 90-day analyses. Another source of differences across post-stroke depression studies globally may be due to the lack of consensus on which scales should be used to accurately measure post-stroke depression.[2] The PHQ-9 is the most frequently used scale to assess post-stroke depression in Sub-Saharan Africa,[3] and a cutoff value of $>=9$ points indicating depressive symptom burden has been established prior to our report.[5] ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,568,480,748]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our data provide class IV evidence regarding the efficacy of fluoxetine for post-stroke depression in Tanzanian adults. Our findings are limited in that they use historical controls and derive from relatively small numbers of participants, subjecting our findings to a type II error. Another limitation of the current research is the greater loss to follow up and death in the historical fluoxetine-untreated controls compared to the fluoxetine-treated MAMBO group: it is possible that certain factors related to the death and loss to follow-up in the historical controls could be related to the depressive outcomes of participants. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,750,480,915]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our results suggest that fluoxetine, a low-cost SSRI, may not be efficacious for treating post-stroke depressive symptoms in this urban Tanzanian setting. Fluoxetine may still be efficacious in selected patients or those with a clinical history of pre-stroke depression which we were unable to adequately measure in our study. Especially given the lower access in such settings to psychiatric and rehabilitative services and the low-cost and accessibility of fluoxetine, future large, multi-site randomized controlled trials should incorporate the Sub-Saharan African region to evaluate post-stroke depression more thoroughly. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[529,69,807,85]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Declaration of Competing Interest ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[529,92,816,106]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The authors declare no conflicts of interest. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[514,125,903,154]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Funding source: U.S. National Institutes of Health (R21TW011051) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[529,167,623,181]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,196,903,915]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. American Psychiatric Association. Diagnostic and statistical manual of mental disorders. United States: DSM-5; 2013. \",\"2. Towfighi A, Ovbiagele B, El Husseini N, Hackett ML, Jorge RE, Kissela BM, Mitchell PH, Skolarus LE, Whooley MA, Williams LS. Poststroke depression: a scientific statement for healthcare professionals from the American Heart Association/American Stroke Association. Stroke 2017;48(2):e30-e43. \",\"3. Ojagbemi A, Akpa O, Elugbadebo F, Owolabi M, Ovbiagele B. Depression after Stroke in Sub-Saharan Africa: A Systematic Review and Meta-Analysis. Behav Neurol 2017;4160259. \",\"4. Hankey GJ, Hackett ML, Almeida OP, Flicker L, Mead GE, Dennis MS, Etherton-Beer C, Ford AH, Billot L, Jan S, Lung T. Safety and efficacy of fluoxetine on functional outcome after acute stroke (AFFINITY): a randomised, double-blind, placebo-controlled trial. The Lancet Neurol 2020;19(8):651-660. \",\"5. Almeida OP, Hankey GJ, Ford A, et al. Depression Outcomes Among Patients Treated With Fluoxetine for Stroke Recovery: The AFFINITY Randomized Clinical Trial. JAMA Neurol 2021;78(9):1072-1079. https://doi.org/10.1001/jamaneurol.2021.2418. PMID: 34338714. Published online August. \",\"6. Vogel AC, Okeng'o K, Chiwanga F, Ismail SS, Buma D, Pothier L, Mateen FJ. MAMBO: Measuring ambulation, motor, and behavioral outcomes with post-stroke fluoxetine in Tanzania: Protocol of a phase II clinical trial. J Neurol Sci 2020;408:116563. \",\"7. Saadi A, Okeng'o K, Biseko MR, Shayo AF, Mmbando TN, Grundy SJ, Xu A, Parker RA, Wibecan L, Iyer G, Onesmo PM, Kapina BN, Regenhardt RW, Mateen FJ. Post-stroke social networks, depressive symptoms, and disability in Tanzania: A prospective study. Int J Stroke 2018;13(8):840-848. \",\"8. World Bank. GNI per capita, Atlas method (current US). Available from https://data.worldbank.org/indicator/NY.GNP.PCAP.CD Accessed 23 Aug. \",\"9. Muhimbili National Hospital Profile. Muhimbili National Hospital. www.mnh.or.tz/index.php/our-profile Accessed 23 Aug. \",\"10. Walker R, Whiting D, Unwin N, Mugusi F, Swai M, Aris E, Jusabani A, Kabadi G, Gray WK, Lewanga M, Alberti G. Stroke incidence in rural and urban Tanzania: a prospective, community-based study. The Lancet Neurol 2010;9(8):786-792. \",\"11. Pharmaccess Group. A Closer Look at the Healthcare System in Tanzania. Available from: https://www.pharmaccess.org/wp-content/uploads/2018/01/The-healthcare-system-in-Tanzania.pdf Accessed 27 Aug. \",\"12. Kroenke K, Spitzer RL, Williams JB. The PHQ-9: validity of a brief depression severity measure. J Gen Intern Med 2001;16(9):606-613. \",\"13. Omoro SA, Fann JR, Weymuller EA, Macharia IM, Yueh B. Swahili translation and validation of the Patient Health Questionnaire-9 depression scale in the Kenyan head and \"]","page_idx":"5","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[94,47,104,56]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[789,45,900,56]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"D.R. RICE ET AL. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[94,71,480,234]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"neck cancer patient population. Int J Psychiat Med 2006;36(3):367-381. \",\"14. Chollet F, Tardy J, Albucher JF, Thalamas C, Berard E, Lamy C, Bejot Y, Deltour S, Jaillard A, Niclot P, Guillon B. Fluoxetine for motor recovery after acute ischaemic stroke (FLAME): a randomised placebo-controlled trial. The Lancet Neurol 2011;10(2):123-130. \",\"15. Lundström E, Isaksson E, Näsmann P, Wester P, Martensson B, Norrving B, Wallén H, Borg J, Dennis M, Mead G, Hankey GJ. Safety and efficacy of fluoxetine on functional recovery after acute stroke (EFFECTS): a randomised, double-blind, placebo-controlled trial. The Lancet Neurol 2020;19(8):661-669. \"]","page_idx":"6","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[517,70,903,210]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"16. Dennis M, Mead G, Forbes J, Graham C, Hackett M, Hankey GJ, House A, Lewis S, Lundström E, Sandercock P, Innes K. Effects of fluoxetine on functional outcomes after acute stroke (FOCUS): a pragmatic, double-blind, randomised, controlled trial. Lancet North Am Ed 2019;393 (10168):265-274. \",\"17. Regenhardt RW, Biseko MR, Shayo AF, Mmbando TN, Grundy SJ, Xu AI, Saadi A, Wibecan L, Khalal GA, Parker R, Klein JP. Opportunities for intervention: stroke treatments, disability and mortality in urban Tanzania. Int J Qual Health Care 2019;31(5):385-392. \"]","page_idx":"6","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[92,46,409,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"FLUOXETINE FOR POST-STROKE DEPRESSION ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[892,47,903,56]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7 ","text_format":null,"text_level":null,"type":"page_number"}],"string":"[\n {\n \"bbox\": \"[116,95,881,146]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Efficacy of Fluoxetine for Post-Ischemic Stroke Depression in Tanzania \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[150,174,844,244]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Dylan R. Rice, $^{a}$ Kigocha Okeng'o, $^{b}$ Emmanuel Massawe, $^{b}$ Seif Ismail, $^{b}$ Notburga A. Mworia, $^{b}$ Faraja Chiwanga, $^{b}$ Boniface Kapina, $^{b}$ Michael Wasserman, $^{a}$ and Farrah J. Mateen, $^{a}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[234,295,763,627]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Objective: Post-stroke fluoxetine trials are primarily conducted in high-income countries. We characterize post-ischemic stroke depression in fluoxetine-treated and -untreated study participants in urban Tanzania. Methods: Adults $(>18$ years old) within 14 days of CT-confirmed acute ischemic stroke onset were enrolled at Muhimbili National Hospital, Tanzania. The fluoxetine-treated group took $20\\\\mathrm{mg}$ fluoxetine daily for 90 days in a phase II trial and were compared to fluoxetine-untreated historical controls. The primary outcome was depression at 90 days, measured by the Patient Health Questionnaire-9 (PHQ-9). PHQ-9 scores were compared between fluoxetine-treated and -untreated groups. A score $>=9$ points was considered to reflect depression. A multivariable linear regression model assessed associations with post-stroke PHQ-9 scores. Results: Of the fluoxetine-treated $(n = 27)$ and -untreated $(n = 32)$ participants, the average age was 56.8 years old $(39\\\\%$ women, $100\\\\%$ Black/African). The average presentation NIHSS score was 12.1 points and modified Rankin Scale (mRS) score was 3.5. The average mRS score at 90-day follow-up was 2.3. There was no significant difference between 90-day PHQ-9 scores in the fluoxetine-treated (mean=4.1 points, standard deviation=3.2; $11\\\\%$ depression) and untreated (mean=4.4, standard deviation=4.8; $19\\\\%$ depression) groups, $p = .69$ . In the multivariable analysis, older age $(\\\\beta = 0.08, p = .03)$ and higher NIHSS score $(\\\\beta = 0.15, p = .04)$ , but neither fluoxetine $(\\\\beta = 0.57, p = .59)$ nor sex $(\\\\beta = -0.51, p = .63)$ , were significantly associated with more depressive symptoms. Conclusions: Our findings parallel results from trials from higher income settings that fluoxetine does not significantly improve post-ischemic stroke depression, although our sample size was small. More work is needed to depict the longitudinal nature and treatment of post-stroke depression in Sub-Saharan Africa. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[234,628,664,642]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Key Words: Depression—Stroke—Fluoxetine—Treatment—Africa \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[234,642,483,655]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"© 2021 Elsevier Inc. All rights reserved. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[107,688,212,704]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,712,480,758]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A diagnosis of post-stroke depression is defined as a period of depressed mood, lasting two or more weeks after the occurrence of stroke, and the presence of at least \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,689,903,750]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"four other symptoms of depression as characterized by the Diagnostic and Statistical Manual of Mental Disorders-5. Post-stroke depression affects nearly one-third of patients who have experienced stroke.2 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,750,903,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A meta-analysis exploring depression after stroke in Sub-Saharan Africa found a post-stroke depression prevalence of $31\\\\%$ in approximately 1500 stroke patients across 17 studies. The distinction between ischemic and hemorrhagic stroke was not made. Lower education level, poststroke cognitive impairment, physical disability, lower quality of life, and divorced marital status have each been associated with post-stroke depression in Sub-Saharan Africa. However, the associations with post-stroke depression in this context are not consistently measured \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,786,482,811]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From the aMassachusetts General Hospital, Boston, MA, United States; and bMuhimbili National Hospital, Dar es Salaam, Tanzania. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[91,811,480,837]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Received September 1, 2021; revision received October 14, 2021; accepted October 17, 2021. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[91,837,480,875]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Address correspondence to Massachusetts General Hospital, Department of Neurology, 165 Cambridge Street, #627, Boston, MA, United States. E-mail: fmateen@mgh.harvard.edu. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[104,875,274,887]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1052-3057/$ - see front matter \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[104,888,325,900]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"© 2021 Elsevier Inc. All rights reserved. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[104,900,438,912]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.1016/j.jstrokerebprovdis.2021.106181 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[91,930,602,945]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Journal of Stroke and Cerebrovascular Diseases, Vol. 31, No. 1 (January), 2022: 106181 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[890,931,902,941]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[94,70,478,143]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"or identified across studies. Of all studies in this meta-analysis from the African continent, none explored the impact of fluoxetine, a low-cost selective serotonin reuptake inhibitor (SSRI) typically indicated for depression and readily available on most Essential Medicines Lists. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[94,146,478,294]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Researchers from the AFFINITY trial, originally reporting on the lack of impact of fluoxetine on post-stroke motor recovery, recently reported on the post-depression outcomes in their participants. They concluded that daily fluoxetine, administered for 26 weeks post-stroke, did not significantly reduce depressive symptoms compared to placebo. Notably, AFFINITY enrolled $< 2\\\\%$ of participants of Black or African ancestry. To date, no trials have reported on post-stroke fluoxetine outcomes from Sub-Saharan Africa. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[94,296,478,415]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We recently completed a phase II trial - MAMBO (Measuring Ambulation, Motor, and Behavioral Outcomes) - of fluoxetine $20\\\\mathrm{mg}$ , administered for 90 days post-ischemic stroke to adults in urban Tanzania. Here, we aim to report on the secondary outcome of post-stroke depression in MAMBO's fluoxetine-treated participants compared to historical ischemic stroke fluoxetine-untreated controls enrolled from the same Tanzanian hospital. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,431,180,445]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Methods \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,455,215,469]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ethics approvals \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[94,478,478,580]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ethics boards approvals were obtained from the Muhimbili National Hospital, the National Institute of Medical Research of Tanzania, the Tanzania Medicine and Medical Devices Authority, and the Partners Healthcare Institutional Review Board. Each participant or when necessary, a next of kin proxy, provided his or her own individual consent. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,599,156,613]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Setting \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[94,621,478,830]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Tanzania has a gross national income per capita (Atlas method, 2020) of 1080 USD. All participants were prospectively recruited from the Muhimbili National Hospital in Dar es Salaam. This is the largest academic referral hospital in Tanzania with an estimated 1000-1200 patients admitted per week. The hospital has four neurologists. Stroke services available through the neurology department include basic care and thrombolysis for those who can afford it. The yearly age-standardized incidence rate of stroke in Dar es Salaam, Tanzania is presumed to be approximately 316 per 100,000 persons. Not all patients have health insurance and most have government funded support or family support for hospital expenses. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,848,188,861]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Participants \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[94,869,478,914]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A flow diagram of participants is provided in Figure 1. Participants were (1) fluoxetine-treated participants in a phase II trial of fluoxetine (MAMBO; 11/2019 to 10/2020) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,70,900,112]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and (2) prospectively enrolled participants from an observational cohort study (07/2016 to 03/2017) who served as historical controls. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,114,900,340]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The participants were age $>18$ years old; admitted within 14 days of stroke symptom onset, meeting the WHO definition of acute stroke; and had a CT-confirmed ischemic stroke at the time of onset. The fluoxetine-treated group had additional exclusion criteria: dysphagia preventing pill swallowing, hyponatremia at stroke onset $(<125\\\\mathrm{mmol / L})$ , and/or hepatic impairment (serum ALT) $>120$ units/L). In this group, participants were administered open-label $20\\\\mathrm{mg}$ oral fluoxetine daily for 90 days. Fluoxetine for this trial was procured through the University of Iowa Pharmacy laboratory. For complete details on the inclusion and exclusion criteria for this group, see the published MAMBO protocol. Enrollment in the MAMBO trial stopped early due to COVID-19, leading to a lower than intended sample size. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[532,359,603,372]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Procedures \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,381,900,547]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Both groups had baseline and 90-day follow up visits conducted by physicians and study staff at the Muhimbili National Hospital. NIH Stroke Scale (NIHSS) scores, sociodemographic variables, and modified Rankin Scale (mRS) scores were assessed at baseline. Adherence in the fluoxetine-treated group was assessed using a medication possession ratio, assessed in person by the investigators every 30 days. At 90-days, mRS scores were reassessed, and the Patient Health Questionnaire-9 (PHQ-9) was administered in Kiswahili or English, as preferred by the participant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[532,566,591,578]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Outcome \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,588,900,752]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The primary outcome reported here is the burden of depressive symptoms at the 90-day follow up visit, as measured by the PHQ-9.[12] This measure has previously been translated and validated in Kiswahili in a Sub-Saharan African setting.[13] The PHQ-9 assesses depressive symptoms within the last two weeks with a series of 9 items, each using scores from 0 (not at all) to 3 (nearly every day) points. Overall scores range from 0–27 points. A score of $>=9$ points was used to classify significant depressive symptom burden, consistent with a recent large trial definition.[5] \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[532,772,620,786]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,794,900,914]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Variables were described by their means and standard deviations (SDs). Baseline characteristics between the fluoxetine-treated and -untreated groups were compared using $t$ -tests, tests of two proportions, or chi-square tests, as appropriate. A Mann-Whitney $U$ -test was used to compare PHQ-9 scores between fluoxetine-treated and -untreated participant groups, and a Fisher's exact test was used to compare the proportions of participants \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[94,47,104,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[790,46,900,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"D.R. RICE ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[129,65,866,489]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 1. Flow of participants. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0528/90a297e98390b30893f407d7a420562b6912fd4e3068bebe9a8dc688a2aa3d94.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[89,537,482,750]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"meeting criteria for depressive symptom burden between these two groups. Ninety-five percent confidence intervals (CIs) were calculated for the proportion of participants in each group who met criteria for depressive symptom burden. A multivariable linear regression model was used to assess associations between PHQ-9 scores at 90 days as the continuous outcome variable and participant sex (binary: male or female), age (continuous), fluoxetine administration (binary: treated or untreated), and NIHSS score (continuous) as explanatory variables. No significant deviations from standard statistical assumptions were found for this model. All analyses were conducted in R (Version 1.0.153). A p-value $< .05$ was considered statistically significant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[107,770,170,784]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Results \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,794,480,883]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Participant characteristics at baseline, comparing fluoxetine-treated $(n = 27)$ and -untreated $(n = 32)$ groups, are reported in Table 1. Overall, the groups were balanced on the clinical and sociodemographic variables assessed with the exception that more fluoxetine-treated participants were employed. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,885,480,915]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Participants were on average 56.8 years old (SD=14.1 years). Thirty-nine percent were women, $100\\\\%$ were \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,537,903,686]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"African Black, and most had a highest educational attainment level of primary school (58%, $n = 34$ ). At enrollment, the average NIHSS score was 12.1 points (SD=7.4 points), and the average mRS was 3.5 points (SD=1.0 points), indicating that participants had mild to moderately severe strokes and experienced moderate to moderately severe disability. At 90 days, the average mRS score was 2.3 (SD=1.4), indicating mild disability. In the fluoxetine-treated group, adherence to fluoxetine was 96% at 90 days. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,688,903,883]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The distribution of PHQ-9 scores for both groups is shown in Figure 2. The average PHQ-9 score at 90-day follow up was 4.1 points $(\\\\mathrm{SD} = 3.2)$ for fluoxetine-treated patients and 4.4 points $(\\\\mathrm{SD} = 4.8)$ for fluoxetine-untreated patients. Eleven percent of fluoxetine-treated participants met criteria for significant depressive symptom burden $(n = 3; 95\\\\% \\\\text{ CI}[0,24])$ , compared to $19\\\\%$ $(n = 6; 95\\\\% \\\\text{ CI}[4,33])$ in untreated participants. We did not observe a statistically significant difference in the mean PHQ-9 scores between the two groups, $p = .69$ . Similarly, we did not observe a statistically significant difference in the proportion of participants who met criteria for depression by fluoxetine treatment versus not, $p = .49$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,885,903,915]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the multivariable linear regression model (Table 2) with PHQ-9 score as the outcome variable and age, sex, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,46,409,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"FLUOXETINE FOR POST-STROKE DEPRESSION \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[892,47,902,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[92,89,903,547]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0460/3be1182391992ca414351b4c19ee8d0ae89f6faa993c88e518999936f7f34779.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td></td><td>Fluoxetine-treated (MAMBO participants)</td><td>Fluoxetine-untreated (Tanzanian controls)</td><td>Comparison between groups, p</td></tr><tr><td>n</td><td>27</td><td>32</td><td>-</td></tr><tr><td>Age in years, mean (standard deviation; SD)</td><td>53.7 (13.0)</td><td>59.5 (14.6)</td><td>.11</td></tr><tr><td>Sex, n (%)</td><td></td><td></td><td>.28</td></tr><tr><td>Female</td><td>8 (30)</td><td>15 (47)</td><td></td></tr><tr><td>Male</td><td>19 (70)</td><td>17 (53)</td><td></td></tr><tr><td>NIH Stroke Scale score, mean (SD)</td><td>10.5 (4.4)</td><td>13.5 (9.0)</td><td>.12</td></tr><tr><td>Modified Rankin Scale score, mean (SD)</td><td></td><td></td><td></td></tr><tr><td>At baseline</td><td>3.7 (0.9)</td><td>3.3 (1.1)</td><td>.10</td></tr><tr><td>At 90-day follow up</td><td>2.1 (1.0)</td><td>2.4 (1.6)</td><td>.32</td></tr><tr><td>Highest level of education, n (%)</td><td></td><td></td><td>.28</td></tr><tr><td>None</td><td>0</td><td>4 (13)</td><td></td></tr><tr><td>Primary school</td><td>16 (59)</td><td>18 (56)</td><td></td></tr><tr><td>Some secondary school</td><td>2 (7)</td><td>4 (13)</td><td></td></tr><tr><td>Complete secondary school</td><td>6 (22)</td><td>4 (13)</td><td></td></tr><tr><td>University</td><td>2 (7)</td><td>1 (3)</td><td></td></tr><tr><td>Employment status, n (%)</td><td></td><td></td><td><.01</td></tr><tr><td>Employed</td><td>25 (93)</td><td>19 (59)</td><td></td></tr><tr><td>Unemployed</td><td>2 (7)</td><td>13 (41)</td><td></td></tr><tr><td>Comorbidities,† n yes (%)</td><td></td><td></td><td></td></tr><tr><td>BMI >30 kg/m2</td><td>1 (4)</td><td>6 (19)</td><td>.17</td></tr><tr><td>Cigarette smoking</td><td>6 (22)</td><td>3 (9)</td><td>.32</td></tr><tr><td>Diabetes</td><td>6 (22)</td><td>5 (16)</td><td>.75</td></tr><tr><td>History of depression</td><td>1 (4)</td><td>2 (6)</td><td>1</td></tr><tr><td>History of stroke</td><td>0</td><td>2 (6)</td><td>.12</td></tr><tr><td>HIV/AIDS</td><td>0</td><td>0</td><td>-</td></tr><tr><td>Hypertension</td><td>25 (93)</td><td>23 (72)</td><td>.09</td></tr><tr><td>Substance use, including alcohol</td><td>5 (19)</td><td>5 (16)</td><td>1</td></tr><tr><td>90-day Patient Health Questionnaire-9 (PHQ-9) score</td><td></td><td></td><td></td></tr><tr><td>Mean (SD)</td><td>4.1 (3.2)</td><td>4.4 (4.8)</td><td>.81 .42</td></tr><tr><td>PHQ-9 score >=9, n (%)</td><td>3 (11)</td><td>6 (19)</td><td></td></tr></table>\",\n \"table_caption\": \"[\\\"Table 1. Participant characteristics at enrollment. \\\"]\",\n \"table_footnote\": \"[\\\"+Comorbidities were primarily self-reported by participants. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[91,598,480,688]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"fluoxetine treatment, and NIHSS score as explanatory variables, only older age $(\\\\beta = 0.08, p = .03)$ and higher baseline NIHSS score $(\\\\beta = 0.15, p = .04)$ were statistically significantly associated with depressive symptom burden. There were no significant association with depressive symptom burden by sex or fluoxetine administration. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[109,710,200,724]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Discussion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,733,482,914]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We develop an evidence base on the limited efficacy of fluoxetine for post-ischemic stroke depression in Sub-Saharan Africa. In our fluoxetine-treated MAMBO trial participants from Tanzania, $11\\\\%$ met criteria for significant depressive symptom burden at 90-day follow up. Although we did not have a placebo arm in the MAMBO trial, using historical controls from the same Tanzanian hospital we found $19\\\\%$ of this fluoxetine-untreated sample met criteria for significant depressive symptom burden at 90 days. However, the difference in depressive symptom burden between these groups was not statistically significant. In both groups, higher stroke severity at \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,598,903,627]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"onset and older age are associated with a higher burden of depressive symptoms by 90 days. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,628,903,915]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A network of clinical trials examined the role of poststroke fluoxetine on post-stroke recovery and depression. The FLAME (France), EFFECTS (Sweden), and FOCUS (United Kingdom) trials[14-16] utilized different measures for depression, but the AFFINITY trial (Australasia) used the PHQ-9. We found a remarkably similar prevalence of depressive symptom burden in our fluoxetine-treated sample $(11\\\\%)$ at 90 days compared to the $10.8\\\\%$ reported in the fluoxetine-treated arm of AFFINTY at 84 days. While $19\\\\%$ of our fluoxetine-untreated sample met criteria for depressive symptom burden at 90 days, only $9.7\\\\%$ of the placebo arm patients at 84 days in the AFFINITY trial met such criteria. This may relate to milder strokes in the AFFINITY trial, better stroke services, or other factors. Notably, our fluoxetine-treated and control participants were younger at enrollment (mean age 56.8 years in Tanzania versus 63.8 years in AFFINTY, 64.7 in FLAME; $>56\\\\%$ of participants in both FOCUS and EFFECTS were $>70$ years of age) and experienced more severe stroke \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[94,47,104,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[790,46,900,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"D.R. RICE ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[131,71,866,364]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0587/01d6458ea083e3d56bb1ca7b3677d541b33cfb5c6f9f97acfac606936a66eb1b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[133,375,831,681]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 2. Comparison of 90-day Patient Health Questionnaire-9 scores between fluoxetine-treated and -untreated participants depicted by A) box plots and B) proportional bar charts. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0299/4be000b6ae623be92022ffcbe854772572690d0ab5a72d4788b084c559ee9c6a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[92,768,903,854]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0938/7c588dea5f77680d4ce578db434ed903f41f5267873d1f4879d255dc3af5722b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td></td><td>β</td><td>Standard error</td><td>95% confidence interval of β</td><td>p</td></tr><tr><td>Fluoxetine treated</td><td>0.57</td><td>1.06</td><td>[-1.55, 2.69]</td><td>.59</td></tr><tr><td>Age (continuous)</td><td>0.08</td><td>0.04</td><td>[0.01, 0.16]</td><td>.03*</td></tr><tr><td>Female sex</td><td>-0.51</td><td>1.05</td><td>[-2.61, 1.60]</td><td>.63</td></tr><tr><td>NIH Stroke Scale score (continuous)</td><td>0.15</td><td>0.07</td><td>[0.01, 0.29]</td><td>.04*</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 2. Multivariable linear regression statistics evaluating variables associated with post-stroke depression in Tanzania. \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[109,857,161,869]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n^ {*} \\\\mathrm {p} < . 0 5\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[92,46,409,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"FLUOXETINE FOR POST-STROKE DEPRESSION \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[893,47,903,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[91,69,480,130]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"presentations as measured by the median NIHSS score (11 points in Tanzanian participants versus 6 points in AFFINITY, 6 points in FOCUS, and 3 points in EFFECTS; the mean NIHSS score was 13 points in FLAME). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,130,480,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"However, like the AFFINITY trial, we also found no statistically significant difference between fluoxetine-treated and untreated post-stroke depression in our study participants in Tanzania. Although our sample size was much smaller, we observed that fluoxetine has no overall measurable benefit on post-stroke depression in unselected patients at 90 days in this Sub-Saharan African setting. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[94,237,483,566]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The prevalence of post-stroke depression reported in our participants study is also notably less than the approximately one-third of post-stroke patients commonly reported in the literature.[2,3] There are several possible explanations for this. Our participants were well-evaluated and received better than usual care because of study enrollment, including access to free neuroimaging as well as 30-, 60-, and 90-day follow up visits by a physician. Participants may have had response bias, wanting to report fewer depressive symptoms to an investigator. In addition, there is a very high post-stroke mortality at the Muhimbili National Hospital, approximating $50\\\\%$ .[17] Participants with the most severe strokes were more likely to die and could not be included in 90-day analyses. Another source of differences across post-stroke depression studies globally may be due to the lack of consensus on which scales should be used to accurately measure post-stroke depression.[2] The PHQ-9 is the most frequently used scale to assess post-stroke depression in Sub-Saharan Africa,[3] and a cutoff value of $>=9$ points indicating depressive symptom burden has been established prior to our report.[5] \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,568,480,748]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our data provide class IV evidence regarding the efficacy of fluoxetine for post-stroke depression in Tanzanian adults. Our findings are limited in that they use historical controls and derive from relatively small numbers of participants, subjecting our findings to a type II error. Another limitation of the current research is the greater loss to follow up and death in the historical fluoxetine-untreated controls compared to the fluoxetine-treated MAMBO group: it is possible that certain factors related to the death and loss to follow-up in the historical controls could be related to the depressive outcomes of participants. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,750,480,915]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our results suggest that fluoxetine, a low-cost SSRI, may not be efficacious for treating post-stroke depressive symptoms in this urban Tanzanian setting. Fluoxetine may still be efficacious in selected patients or those with a clinical history of pre-stroke depression which we were unable to adequately measure in our study. Especially given the lower access in such settings to psychiatric and rehabilitative services and the low-cost and accessibility of fluoxetine, future large, multi-site randomized controlled trials should incorporate the Sub-Saharan African region to evaluate post-stroke depression more thoroughly. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[529,69,807,85]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Declaration of Competing Interest \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[529,92,816,106]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The authors declare no conflicts of interest. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,125,903,154]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Funding source: U.S. National Institutes of Health (R21TW011051) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[529,167,623,181]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,196,903,915]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. American Psychiatric Association. Diagnostic and statistical manual of mental disorders. United States: DSM-5; 2013. \\\",\\\"2. Towfighi A, Ovbiagele B, El Husseini N, Hackett ML, Jorge RE, Kissela BM, Mitchell PH, Skolarus LE, Whooley MA, Williams LS. Poststroke depression: a scientific statement for healthcare professionals from the American Heart Association/American Stroke Association. Stroke 2017;48(2):e30-e43. \\\",\\\"3. Ojagbemi A, Akpa O, Elugbadebo F, Owolabi M, Ovbiagele B. Depression after Stroke in Sub-Saharan Africa: A Systematic Review and Meta-Analysis. Behav Neurol 2017;4160259. \\\",\\\"4. Hankey GJ, Hackett ML, Almeida OP, Flicker L, Mead GE, Dennis MS, Etherton-Beer C, Ford AH, Billot L, Jan S, Lung T. Safety and efficacy of fluoxetine on functional outcome after acute stroke (AFFINITY): a randomised, double-blind, placebo-controlled trial. The Lancet Neurol 2020;19(8):651-660. \\\",\\\"5. Almeida OP, Hankey GJ, Ford A, et al. Depression Outcomes Among Patients Treated With Fluoxetine for Stroke Recovery: The AFFINITY Randomized Clinical Trial. JAMA Neurol 2021;78(9):1072-1079. https://doi.org/10.1001/jamaneurol.2021.2418. PMID: 34338714. Published online August. \\\",\\\"6. Vogel AC, Okeng'o K, Chiwanga F, Ismail SS, Buma D, Pothier L, Mateen FJ. MAMBO: Measuring ambulation, motor, and behavioral outcomes with post-stroke fluoxetine in Tanzania: Protocol of a phase II clinical trial. J Neurol Sci 2020;408:116563. \\\",\\\"7. Saadi A, Okeng'o K, Biseko MR, Shayo AF, Mmbando TN, Grundy SJ, Xu A, Parker RA, Wibecan L, Iyer G, Onesmo PM, Kapina BN, Regenhardt RW, Mateen FJ. Post-stroke social networks, depressive symptoms, and disability in Tanzania: A prospective study. Int J Stroke 2018;13(8):840-848. \\\",\\\"8. World Bank. GNI per capita, Atlas method (current US). Available from https://data.worldbank.org/indicator/NY.GNP.PCAP.CD Accessed 23 Aug. \\\",\\\"9. Muhimbili National Hospital Profile. Muhimbili National Hospital. www.mnh.or.tz/index.php/our-profile Accessed 23 Aug. \\\",\\\"10. Walker R, Whiting D, Unwin N, Mugusi F, Swai M, Aris E, Jusabani A, Kabadi G, Gray WK, Lewanga M, Alberti G. Stroke incidence in rural and urban Tanzania: a prospective, community-based study. The Lancet Neurol 2010;9(8):786-792. \\\",\\\"11. Pharmaccess Group. A Closer Look at the Healthcare System in Tanzania. Available from: https://www.pharmaccess.org/wp-content/uploads/2018/01/The-healthcare-system-in-Tanzania.pdf Accessed 27 Aug. \\\",\\\"12. Kroenke K, Spitzer RL, Williams JB. The PHQ-9: validity of a brief depression severity measure. J Gen Intern Med 2001;16(9):606-613. \\\",\\\"13. Omoro SA, Fann JR, Weymuller EA, Macharia IM, Yueh B. Swahili translation and validation of the Patient Health Questionnaire-9 depression scale in the Kenyan head and \\\"]\",\n \"page_idx\": \"5\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[94,47,104,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[789,45,900,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"D.R. RICE ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[94,71,480,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"neck cancer patient population. Int J Psychiat Med 2006;36(3):367-381. \\\",\\\"14. Chollet F, Tardy J, Albucher JF, Thalamas C, Berard E, Lamy C, Bejot Y, Deltour S, Jaillard A, Niclot P, Guillon B. Fluoxetine for motor recovery after acute ischaemic stroke (FLAME): a randomised placebo-controlled trial. The Lancet Neurol 2011;10(2):123-130. \\\",\\\"15. Lundström E, Isaksson E, Näsmann P, Wester P, Martensson B, Norrving B, Wallén H, Borg J, Dennis M, Mead G, Hankey GJ. Safety and efficacy of fluoxetine on functional recovery after acute stroke (EFFECTS): a randomised, double-blind, placebo-controlled trial. The Lancet Neurol 2020;19(8):661-669. \\\"]\",\n \"page_idx\": \"6\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[517,70,903,210]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"16. Dennis M, Mead G, Forbes J, Graham C, Hackett M, Hankey GJ, House A, Lewis S, Lundström E, Sandercock P, Innes K. Effects of fluoxetine on functional outcomes after acute stroke (FOCUS): a pragmatic, double-blind, randomised, controlled trial. Lancet North Am Ed 2019;393 (10168):265-274. \\\",\\\"17. Regenhardt RW, Biseko MR, Shayo AF, Mmbando TN, Grundy SJ, Xu AI, Saadi A, Wibecan L, Khalal GA, Parker R, Klein JP. Opportunities for intervention: stroke treatments, disability and mortality in urban Tanzania. Int J Qual Health Care 2019;31(5):385-392. \\\"]\",\n \"page_idx\": \"6\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[92,46,409,58]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"FLUOXETINE FOR POST-STROKE DEPRESSION \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[892,47,903,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.1016/j.jstrokecerebrovasdis.2021.106181"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Medicine and Health Sciences"},"sha256":{"kind":"string","value":"709b369b341bc41342d66bf0e05481cc2eaf8e0cefee2b643e05a4d585fa96ca"},"title":{"kind":"string","value":"Efficacy of Fluoxetine for Post-Ischemic Stroke Depression in Tanzania"}}},{"rowIdx":47,"cells":{"abstract":{"kind":"string","value":"The study of shared variation in gray matter morphology may define neurodegenerative diseases beyond what can be detected from the isolated assessment of regional brain volumes. We, therefore, aimed to (1) identify SCNs (structural covariance networks) that discriminate between Alzheimer's disease (AD) patients and healthy controls (HC), (2) investigate their diagnostic accuracy in comparison and above established markers, and (3) determine if they are associated with cognitive abilities. We applied a random forest algorithm to identify discriminating networks from a set of 20 SCNs. The algorithm was trained on a main sample of 104 AD patients and 104 age-matched HC and was then validated in an independent sample of 28 AD patients and 28 controls from another center. Only two of the 20 SCNs contributed significantly to the discrimination between AD and controls. These were a temporal and a secondary somatosensory SCN. Their diagnostic accuracy was 74% in the original cohort and 80% in the independent samples. The diagnostic accuracy of SCNs was comparable with that of conventional volumetric MRI markers including whole brain volume and hippocampal volume. SCN did not significantly increase diagnostic accuracy beyond that of conventional MRI markers. We found the temporal SCN to be associated with verbal memory at baseline. No other associations with cognitive functions were seen. SCNs failed to predict the course of cognitive decline over an average of 18 months. We conclude that SCNs have diagnostic potential, but the diagnostic information gain beyond conventional MRI markers is limited."},"author":{"kind":"string","value":"Benno G Gesierich|Florian Mayer|Peter Dal-Bianco|Christian Enzinger|Stefan Ropele|Reinhold Schmidt|Fabian Wagner|Marisa Koini|Marco Duering"},"content_list":{"kind":"list like","value":[{"bbox":"[290,211,867,327]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Gray Matter Covariance Networks as Classifiers and Predictors of Cognitive Function in Alzheimer's Disease ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[289,344,924,374]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Fabian Wagner<sup>1</sup>, Marco Duering<sup>2</sup>, Benno G. Gesierich<sup>2</sup>, Christian Enzinger<sup>1</sup>, Stefan Ropele<sup>1</sup>, Peter Dal-Bianco<sup>3</sup>, Florian Mayer<sup>3</sup>, Reinhold Schmidt<sup>1*</sup> and Marisa Koini<sup>1</sup> ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[289,386,915,410]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$^{1}$ Department of Neurology, Medical University of Graz, Graz, Austria, $^{2}$ Institute for Stroke and Dementia Research (ISD), University Hospital, LMU Munich, Munich, Germany, $^{3}$ Department of Neurology, Medical University of Vienna, Vienna, Austria ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[142,442,268,456]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"OPEN ACCESS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[206,469,267,481]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Edited by: ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,483,267,494]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Francesco Ernesto Pontieri, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,496,267,505]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sapienza University of Rome, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[243,509,267,519]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Italy ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[189,526,267,537]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Reviewed by: ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[157,540,265,550]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sara L. 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","text_format":null,"text_level":null,"type":"text"},{"bbox":"[134,771,267,783]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Published: 05 May 2020 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[216,789,267,799]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Citation: ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,802,267,813]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Wagner F, Duering M, Gesierich BG 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","text_format":null,"text_level":null,"type":"text"},{"bbox":"[137,892,267,903]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Front. Psychiatry 11:360. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[105,905,267,916]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"doi: 10.3389/fpsyt.2020.00360 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[289,432,927,772]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The study of shared variation in gray matter morphology may define neurodegenerative diseases beyond what can be detected from the isolated assessment of regional brain volumes. We, therefore, aimed to (1) identify SCNs (structural covariance networks) that discriminate between Alzheimer's disease (AD) patients and healthy controls (HC), (2) investigate their diagnostic accuracy in comparison and above established markers, and (3) determine if they are associated with cognitive abilities. We applied a random forest algorithm to identify discriminating networks from a set of 20 SCNs. The algorithm was trained on a main sample of 104 AD patients and 104 age-matched HC and was then validated in an independent sample of 28 AD patients and 28 controls from another center. Only two of the 20 SCNs contributed significantly to the discrimination between AD and controls. These were a temporal and a secondary somatosensory SCN. Their diagnostic accuracy was $74\\%$ in the original cohort and $80\\%$ in the independent samples. The diagnostic accuracy of SCNs was comparable with that of conventional volumetric MRI markers including whole brain volume and hippocampal volume. SCN did not significantly increase diagnostic accuracy beyond that of conventional MRI markers. We found the temporal SCN to be associated with verbal memory at baseline. No other associations with cognitive functions were seen. SCNs failed to predict the course of cognitive decline over an average of 18 months. We conclude that SCNs have diagnostic potential, but the diagnostic information gain beyond conventional MRI markers is limited. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[290,781,803,795]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords: structural covariance network, longitudinal, Alzheimer, cognition, random forest ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[290,829,452,845]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"INTRODUCTION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[289,857,926,917]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Alzheimer's Disease (AD) has been recognized as a disconnection syndrome (1, 2) leading to increasing cognitive deficits as the disease progresses. Resting state functional magnetic resonance imaging (rs-fMRI) has been used to study functional connectivity at rest and showed decreased connectivity in brain networks of AD patients (3-5). The default mode network (DMN) (3, 6) is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[72,30,105,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"# ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[107,43,203,57]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"frontiers ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[107,60,242,75]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"in Psychiatry ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[794,43,922,53]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ORIGINAL RESEARCH ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[800,55,922,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"published: 05 May 2020 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[764,65,922,75]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"doi: 10.3389/fpsyt.2020.00360 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[873,92,922,129]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[72,949,299,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Psychiatry | www.frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[492,951,504,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[732,949,924,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"May 2020 | Volume 11 | Article 360 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[73,77,485,444]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"most affected (7-9). Alterations in connectivity have also been found in the salience and frontal executive networks (6, 8). In healthy subjects, DMN connectivity is inhibited while other networks increase their connectivity. AD patients lose the ability to suppress the DMN network during cognitive activity, (8) and a task-based fMRI study showed an association between de-synchronized hippocampus and DMN activity and impaired memory (10). In line with functional disconnection in AD, multi-level deficiency of white matter (WM) connectivity has been shown in DTI studies (11, 12). The authors reported a decreased fiber-count of WM and loss of connective efficiency within and between AD relevant brain areas like the hippocampus, precuneus, and temporal gyrus (11). Another study showed degeneration in 18 major WM tracts, including the fornix, cingulum, and corpus callosum (12). Connectivity between gray matter regions can also be assessed by a method called structural covariance networks (SCNs) (13). SCNs are networks found on the basis of distinct gray matter (GM) covariation patterns in remote cortical areas. In healthy subjects, SCN integrity decreases with age (14, 15), and relates to impairment of cognitive and motor functions (16, 17). In patients with mild cognitive impairment and AD, SCNs containing temporal and limbic regions as well as the precuneus were found to predict the rate of decline in memory over time (13, 18, 19). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,446,485,622]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the present study we aimed to identify those SCNs that best discriminate between AD patients and healthy controls by utilizing twenty SCNs obtained in a group of 257 healthy aging subjects (16). We hypothesized, that networks including the temporal lobes as well as the DMN discriminate best between patients and controls. To validate the diagnostic accuracy of potentially significant identified networks, we applied them on an independent sample. The diagnostic value of SCNs was then compared to conventional MRI markers of AD including the medial temporal lobe atrophy score, total brain volume, and hippocampal volume. We also tested whether SCNs relate to cognitive functioning cross-sectionally and longitudinally. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,657,176,672]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"METHODS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[73,684,147,700]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sample ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[73,702,485,892]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The data from AD patients are from the Prospective Registry on Dementia (PRODEM), a longitudinal multi-center study on disease-progression of dementia patients in Austria. The inclusion criteria were as follows: (a) dementia diagnosis according to DSM-IV criteria, (b) non-institutionalization and no need for 24-h care and (c) availability of a caregiver who is able to provide additional information on the patient. We excluded patients with co-morbidities that were likely to cause early termination from the study such as cancer. Subjects who were not able to sign an informed consent form were also excluded. For further detail see (20). The data from the controls are from the Austrian Stroke Prevention Study-Family (ASPS-Fam). The ASPS-Fam is a single-center, prospective study ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,77,922,134]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"on risk factors and their effects on the brain in the normal elderly. None of our control subjects had a history of neuropsychiatric disease, including cerebrovascular attacks and dementia. All had a normal neurologic examination (21). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,136,922,400]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For the present study all PRODEM patients with (1) either possible or probable AD defined by the NINCDS-ADRDA Alzheimer's criteria (22), (2) a 3-Tesla T1-weighted 3D MRI scan, and (3) a cognitive assessment within 3 months of the imaging examination were selected. The cognitive assessment included the Mini Mental State examination (MMSE) and the \"Consortium to Establish a Registry for Alzheimer's Disease\"—Test battery—plus version (CERAD-Plus). In total 104 AD patients (mean age $= 71.45 \\pm 7.97$ years, range: 51–87 years, 59 females) from Graz and 28 AD patients (mean age $= 73.79 \\pm 6.17$ , range: 58–82, 14 females) from Vienna were included. The Graz sample was used for statistical modeling, while the Viennese sample served as the independent validation cohort. In each center a cohort of controls was matched to each AD patient. Matching was done for age $(+/- 3$ years) and sex (Graz: mean age $= 71.09 \\pm 7.38$ years; range: 53–86, 59 females; Vienna: $n = 28$ , mean age $= 72.44 \\pm 7.18$ years; range: 56–85 years, 14 female). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,401,922,487]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Eighty-two AD patients from the Graz set had a follow-up clinical and cognitive assessment that ranged between 6 and 37 months with a mean of 18 months. Patients who dropped out from the study did not differ from subjects with follow-up in terms of and the Mini Mental State Examination scores at baseline. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,490,922,548]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The study was approved by the ethics committees of the Medical Universities of Graz, Austria and Vienna, Austria. Informed consent was obtained from all subjects and/or caregivers included in the study. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,560,638,576]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"MRI Protocol ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,578,922,738]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Structural data were assessed using a three-dimensional, T1-weighted, magnetization prepared rapid gradient echo sequence (MPRAGE) in Graz and Vienna (Trio Tim 3.0T, manufactured by Siemens Healthcare, Erlangen, Germany) with whole brain coverage. The selected sequence parameters for PRODEM were as follows: Repetition time: $1,900 / 2,000\\mathrm{ms}$ , inversion time: 900 ms, echo time: $2.19\\mathrm{ms}$ , flip angle: $9^{\\circ}$ and a resolution of: $1\\mathrm{mm} \\times 1\\mathrm{mm} \\times 1\\mathrm{mm} / 1.2\\mathrm{mm}$ . For ASPS-Fam, the selected sequence parameters were as follows: Repetition time: 1,900, inversion time: 900 ms, echo time: $2.19\\mathrm{ms}$ , flip angle: $9^{\\circ}$ an isotropic resolution of $1\\mathrm{mm}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,758,657,774]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Image Analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,775,757,788]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Structural Covariance Networks ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,790,922,890]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To prevent biasing data, all image processing steps were done independently for the Graz and Vienna cohorts of this sample. The T1 weighted images were visually checked to ensure consistent image quality for analysis. Pre-processing and SCN analyses were performed using FMRIB's Software Library (FSL) Version 5.0.9 (23). A detailed version of processing steps can be found in (14, 16, 18, 24) In brief, the brain was extracted from ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,35,142,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Wagner et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[732,35,924,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Structural Covariance Networks in AD ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[73,949,297,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Psychiatry | www.frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[495,951,502,960]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[734,949,922,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"May 2020 | Volume 11 | Article 360 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[73,77,485,518]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"non-brain tissue in the images with a semiautomatic brain extraction tool (BET) implemented in FSL (25). The extracted brains were visually checked for artefacts (remaining skull or missing part of sulci). In case of missing part of sulci, parameters for BET were changed. Remaining skull was removed by hand. Then followed tissue-type segmented into gray matter (GM), white matter (WM), and cerebrospinal fluid (CSF) using a voxel-based morphometric analysis (VBM) within the FSL framework (26). After visual inspection, the individual GM images were non-linear registered (27) to the GM MNI152 standard space (Montreal Neurological Institute, Montreal, QC, Canada) (23). By averaging the resulting images, a study-specific GM template was created to which the native GM images were non-linearly re-registered. Following, the images were modulated to correct for distortions due to non-linear transformation (26, 28). Finally, the GM images were then spatially smoothed with an isotropic Gaussian kernel with a sigma of $3\\mathrm{mm}$ and concatenated to a 4D-data set. Thereafter, this 4D-data set was spatially regressed onto 20 networks obtained by Koini and colleagues (16) in a healthy sample, using spatial regression (Step 1 of the dual regression script provided by FSL) (29) to calculate individual network integrity scores. These networks included a temporal, a temporo-cerebellar, a superior temporal gyrus, a supplementary motor, a secondary somatosensory, a limbic, a fronto-parietal, a cuneal, a precuneus, a fronto-occipital, an insular including the gyrus cinguli, an occipital including the posterior cingulate cortex, six cerebellar, and an amygdala SCN. The outcome of the spatial regression consists of individual beta scores (positive and negative) per person and per SCN, representing network integrity values, which were used in the statistical analyses. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,537,369,551]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Visual and Automated Atrophy Marker ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[73,553,485,698]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The native T1-weighted images of all subjects were visually rated by trained specialists using the medial temporal lobe atrophy (MTA)-score scale, a five-point rating scale, ranging from no atrophy (0) to maximum atrophy (4) (30). Ratings were made for both hemispheres, with a mean value calculated for statistical analyses. Automatic brain volume measurements from the native T1-weighted images of all subjects were done by using FreeSurfer (Version 5.3) (31). The brain volume and the hippocampus volumes, averaged across hemispheres, were normalized for intracranial volume. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,716,287,732]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cognitive Assessment ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[73,732,485,908]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The PRODEM study protocol includes the CERAD-Plus (Consortium to Establish a Registry for Alzheimer's Disease) test battery, which among others includes subtests for verbal memory and figural memory (32). Verbal memory is tested by the recall of a wordlist and figural memory is assessed by drawing geometric figures from a template and later from memory. For both subtests \"savings\" scores, the ratio of correctly recalled items to learned items (times 100), were available as measurement for memory retention. We also determined a composite score of CERAD-Plus subtest, the Chandler-score (33). Verbal and figural memory savings and the Chandler-score were selected as outcome variables in the cross sectional and longitudinal analyses. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,77,692,93]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Statistical Analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,94,922,372]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The statistical software R (version 3.5.3 on Windows 10) was used (34). Training of the classification algorithms were done solely on the cohort from Graz. For discrimination between AD patients and HC the value of the integrity score of each of the 20 SCNs was used in a random forest analysis (35) using the \"cforest\" function within the \"party\" package (36-38). 1001 condition inference trees and the associated conditional permutation importance for the SCN integrity scores were calculated (39). Numbers of random variables at the nodes (\"mtry\" parameter) were left at the standard of five variables. The conditional permutation importance is a measure of variable importance and is defined as a score for mean loss of accuracy in classification, if a given SCN is excluded from the random forest model. SCNs with a variable importance score of less than one were not regarded as crucial for classification. Since the random forest estimations and variable importance calculations contain steps based on randomness, the results typically vary in small ranges. To compensate for this, the calculations were repeated 100 times to build mean values and standard deviation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,374,922,578]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Next, random forest models containing distinct combinations of crucial SCNs were compared to check if comparable classification accuracy can be achieved with sparser models. Crucial SCNs were identified and selected by variable importance as mentioned above. Accuracy, sensitivity, and specificity calculations were done with the \"caret\"-package (40). These models were again calculated 100 times to obtain mean accuracy, sensitivity, and specificity values. The models were then tested on the independent data set containing AD patients from Vienna and HC from the ASPS-Fam. Thereafter, a model containing the crucial SCNs was extended by the established markers MTA-score, total brain volume, and averaged hippocampus volume to measure their joint discriminative ability. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,581,922,698]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To examine the association between the SCNs integrity and cognitive ability, six multiple linear regression models were computed using the longitudinal cohort from Graz. Outcome variables were baseline (BL) values (cross-sectional) and annualized change values from BL to follow-up (FU, longitudinal) of (a) the Chandler-score, (b) the verbal memory savings, and (c) the figural memory savings. Predictors were the crucial SCNs evaluated by variable importance. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,732,605,748]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RESULTS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,761,710,777]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Diagnostic Accuracy ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,777,922,849]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From the 20 template networks included in our random forest analysis (Model A), the temporal network (Figure 1A, Table 1) and the secondary somatosensory (S2) network (Figure 1B, Table 1) reached a variable importance score above one (Figure 2), and discriminative value. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,851,922,908]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As can be seen from Table 2, sensitivity, specificity, and diagnostic accuracy of the temporal SCN is 72, 77, and $74\\%$ , respectively. These values were replicated in the validation cohort. It is also shown in this table that the S2 SCN has lower ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,35,142,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Wagner et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[732,35,924,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Structural Covariance Networks in AD ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[73,949,297,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Psychiatry | www.frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[495,951,502,960]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[734,949,922,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"May 2020 | Volume 11 | Article 360 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[211,100,786,288]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0471/5ccef15251674497d758135d5f38bb85326697976c168711105ee365f02ce6c2.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[211,300,786,487]","code_body":null,"code_caption":null,"image_caption":"[\"FIGURE 1 | Sagittal, coronal, and axial slices of the temporal SCN (MNI coordinates: $x = -50$ , $y = -20$ , $z = -32$ ) and the secondary somatosensory SCN (S2; MNI coordinates: $x = -50$ , $y = -26$ , $z = 18$ ). The images are taken from the Koini masks (16). \"]","image_footnote":"[]","img_path":"images/0850/27f9940346cccde11a9a70d40c964d345a221f49c73225781344fcca01a7317b.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[70,555,487,614]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"diagnostic accuracy. A combination of the S2 and temporal SCNs and even a combination of all 20 SCNs increased the diagnostic accuracy only marginally when compared to the temporal SCN alone. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,616,485,779]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In Table 3 and Figure 3 we show the random forest models assessing diagnostic accuracy measure for volumetric MRI markers of AD alone and for the full model including all volumetric measures and the temporal and S2 SCN. In both the Graz cohort and the Vienna validation cohort single volumetric MRI markers performed similar to SCNs. Joint use of volumetric markers and SCNs discriminated best between AD and controls with a diagnostic accuracy of 82 and $86\\%$ in the founder and validation cohort, respectively. The increase in accuracy when adding SCNs to conventional MRI volumetric measures was near zero and non-significant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,789,169,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cognition ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,806,485,894]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cross-sectionally, in AD, the temporal SCN showed a significant association with verbal memory savings (Table 4), explaining nine percent of the variance. No association between SCN integrity and any other test scores were found. For change in cognitive measures over time, no associations with SCN integrity were found either (Table 4). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,555,640,572]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DISCUSSION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[507,584,924,790]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Among 20 covariance networks, a temporal gray matter network and a secondary somatosensory gray matter network showed a diagnostic accuracy of 74 and $80\\%$ in two independent cohorts. AD patients were correctly identified in 76 and $83\\%$ of cases, controls were correctly identified in 78 and $75\\%$ of cases. The gray matter networks did not significantly surpass the diagnostic accuracy of traditional volumetric markers including brain volume, hippocampal volume, and MTA score. When adding the covariance networks to a model of traditional volumetric MR markers, the diagnostic accuracy remained virtually unchanged. This contrasts the view that the assessment of networks, which degenerate in synchrony, provides more diagnostic information than a simple volumetric assessment of selected brain regions known to be involved in AD (1, 41, 42). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,790,924,849]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Loss of connectivity within the temporal network was related to lower verbal memory, but neither the temporal network nor other SCNs were able to predict the slope of cognitive decline over the observational period. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,849,924,894]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The covariance networks in our study are partly overlapping with those described by Hafkemeijer and co-workers (18). The temporal gray matter network, which related to verbal memory in ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,35,142,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Wagner et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[729,35,924,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Structural Covariance Networks in AD ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[72,949,299,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Psychiatry | www.frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[492,949,504,960]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[732,949,924,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"May 2020 | Volume 11 | Article 360 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[72,91,924,243]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0148/8c3b21a187be7f66f32532922334b663847bacdc3bf9a247e723f33acedc2a21.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":"<table><tr><td rowspan=\"2\">SCN</td><td rowspan=\"2\">Voxels</td><td colspan=\"3\">MNI coordinates</td><td rowspan=\"2\">Region</td><td rowspan=\"2\">Hemisphere</td></tr><tr><td>x</td><td>y</td><td>z</td></tr><tr><td rowspan=\"5\">Temporal</td><td>10,442</td><td>40</td><td>-4</td><td>-40</td><td>Inferior temporal gyrus, anterior division</td><td>R</td></tr><tr><td>5,355</td><td>-38</td><td>8</td><td>-38</td><td>Temporal pole</td><td>L</td></tr><tr><td>1,466</td><td>0</td><td>20</td><td>40</td><td>Paracingulate gyrus</td><td>R</td></tr><tr><td>1,313</td><td>10</td><td>38</td><td>-4</td><td>Cingulate gyrus, anterior division</td><td>R</td></tr><tr><td>1,106</td><td>-32</td><td>22</td><td>-4</td><td>Insular Cortex</td><td>L</td></tr><tr><td rowspan=\"3\">Secondary Somatosensory (S2)</td><td>9,650</td><td>-48</td><td>-26</td><td>16</td><td>Parietal operculum cortex</td><td>L</td></tr><tr><td>7,478</td><td>50</td><td>-28</td><td>16</td><td>Parietal operculum cortex</td><td>R</td></tr><tr><td>1,028</td><td>10</td><td>-26</td><td>34</td><td>Cingulate gyrus, posterior division</td><td>R</td></tr></table>","table_caption":"[\"TABLE 1 | Regions included in the Temporal SCN and Secondary Somatosensory SCN (16). \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[231,282,764,587]","code_body":null,"code_caption":null,"image_caption":"[\"FIGURE 2 | Variable importance (mean accuracy loss) of 20 structural covariance networks (SCNs) in a random forest classification model. Exclusion of the temporal and the secondary somatosensory network show a variable importance score above one. The error bars show one standard deviation. \"]","image_footnote":"[]","img_path":"images/0237/73ca8144e946cc0c6f32ecf59a27c53a9c40a50e6f2da9a85312482344e51c1c.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[70,679,924,785]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0914/02e0aeef1c1e29d6f2007c15f4d698ec953760e0d8d2fd3a6c882071002a85ad.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":"<table><tr><td rowspan=\"2\">Model</td><td rowspan=\"2\">Predictors</td><td colspan=\"3\">Main sample (Graz)</td><td colspan=\"3\">Independent sample (Vienna)</td></tr><tr><td>Accuracy</td><td>Sensitivity</td><td>Specificity</td><td>Accuracy</td><td>Sensitivity</td><td>Specificity</td></tr><tr><td>A</td><td>All SCNs</td><td>77</td><td>76</td><td>78</td><td>79</td><td>83</td><td>75</td></tr><tr><td>B</td><td>Temporal + S2</td><td>74</td><td>73</td><td>75</td><td>80</td><td>85</td><td>75</td></tr><tr><td>C</td><td>Temporal SCN</td><td>74</td><td>72</td><td>77</td><td>74</td><td>75</td><td>73</td></tr><tr><td>D</td><td>S2 SCN</td><td>67</td><td>67</td><td>67</td><td>54</td><td>54</td><td>53</td></tr></table>","table_caption":"[\"TABLE 2 | Accuracy, sensitivity, and specificity measures for the classification models in the original Graz sample and the independent Viennese sample. \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[70,808,487,912]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"our study, contains connections between the anterior part of the right inferior temporal gyrus, the left temporal pole, the right paracingulate gyrus, the anterior part of the right cingulate gyrus, and the left insular cortex. These areas have been reported to be particularly affected by AD pathology (43-46). However, the inclusion of the anterior cingulate and paracingulate gyri in the temporal network is somehow unusual, but in case of the former a ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,808,924,913]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"comparable network was also found in a study by Hafkemeijer and co-workers (14). Since there is quite some randomness in network creation depending on both sample data and parameter choice, we are unable to determine whether the inclusion of the anterior division of the cingulate gyrus and the paracingulate gyrus in the temporal SCN occurred on true biological grounds. A statistical change finding cannot be excluded with certainty. The ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,35,142,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Wagner et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[731,35,924,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Structural Covariance Networks in AD ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[72,949,299,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Psychiatry | www.frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[492,949,504,960]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[732,949,924,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"May 2020 | Volume 11 | Article 360 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[70,91,926,233]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0827/2e1e512ee75f298ee5df996bf1b69af3d9346bf53db8e716a22d531bbb11f916.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":"<table><tr><td rowspan=\"2\">Model</td><td colspan=\"3\">Graz AD-HC</td><td colspan=\"3\">Vienna AD-HC</td></tr><tr><td>Accuracy</td><td>Sensitivity</td><td>Specificity</td><td>Accuracy</td><td>Sensitivity</td><td>Specificity</td></tr><tr><td>MTA</td><td>77</td><td>74</td><td>81</td><td>82</td><td>82</td><td>82</td></tr><tr><td>BV</td><td>74</td><td>72</td><td>75</td><td>73</td><td>75</td><td>72</td></tr><tr><td>Hc</td><td>76</td><td>73</td><td>79</td><td>79</td><td>88</td><td>74</td></tr><tr><td>MTA + Hc + BV</td><td>82</td><td>80</td><td>84</td><td>88</td><td>92</td><td>84</td></tr><tr><td>MTA + Hc + BV + Temporal + S2</td><td>82</td><td>81</td><td>82</td><td>86</td><td>92</td><td>81</td></tr></table>","table_caption":"[\"TABLE 3 | Accuracy, sensitivity, and specificity measures for MTA-score, normalized brain volume (BV), normalized hippocampus volume (Hc), and combined models. \"]","table_footnote":"[\"Calculations were done in the original Graz sample and the independent Viennese sample. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[231,308,764,611]","code_body":null,"code_caption":null,"image_caption":"[\"FIGURE 3 | Variable importance (mean accuracy loss) of the temporal and the S2 SCNs and three alternative markers (MTA-score, normalized brain volume, normalized hippocampus volume) in a random forest classification model. The error bars depict one standard deviation. \"]","image_footnote":"[]","img_path":"images/0718/34b2acd16b59c654c821c772a06286ba4165c111f228fb5ab28cbdf7f3a4d5aa.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[70,704,927,872]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0001/c5fe60cdc40e334137f20061437b2086639d809bec400e726ca374d4b9ab66ea.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":"<table><tr><td>N = 82</td><td colspan=\"3\">Chandler-score</td><td colspan=\"3\">Verbal memory savings</td><td colspan=\"3\">Figural memory savings</td></tr><tr><td>Cross-sectional</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Variable</td><td>B</td><td>SE</td><td>beta</td><td>B</td><td>SE</td><td>beta</td><td>B</td><td>SE</td><td>Beta</td></tr><tr><td>Temporal SCN</td><td>276.34</td><td>200.33</td><td>.15</td><td>1,060.42*</td><td>508.55</td><td>.23*</td><td>-168.11</td><td>508.11</td><td>-.03</td></tr><tr><td>Secondary Somatosensory SCN</td><td>204.857</td><td>225.80</td><td>.10</td><td>805.79</td><td>573.23</td><td>.15</td><td>569.17</td><td>572.73</td><td>.11</td></tr><tr><td>R2(p-value)</td><td></td><td>.04 (.17)</td><td></td><td></td><td>.09 (.01)*</td><td></td><td></td><td>.01 (.61)</td><td></td></tr><tr><td colspan=\"10\">Longitudinal</td></tr><tr><td>Variable</td><td>B</td><td>SE</td><td>beta</td><td>B</td><td>SE</td><td>beta</td><td>B</td><td>SE</td><td>Beta</td></tr><tr><td>Temporal SCN</td><td>-7.26</td><td>126.24</td><td><-0.00</td><td>155.77</td><td>484.45</td><td>.03</td><td>-182.99</td><td>391.47</td><td>< .00</td></tr><tr><td>Secondary Somatosensory SCN</td><td>-71.78</td><td>143.45</td><td>-0.06</td><td>-902.78</td><td>570.06</td><td>-.19</td><td>87.47</td><td>441.26</td><td>< .00</td></tr><tr><td>R2(p-value)</td><td></td><td>< .00 (.86)</td><td></td><td></td><td>.03 (.29)</td><td></td><td></td><td>< .00 (.89)</td><td></td></tr></table>","table_caption":"[\"TABLE 4 | Multiple linear regression models to evaluate the predictive value of the SCNs for cognition. \"]","table_footnote":"[\"*significance at.05 level. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[73,35,142,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Wagner et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[729,35,924,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Structural Covariance Networks in AD ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[72,949,299,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Psychiatry | www.frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[492,951,504,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[732,949,924,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"May 2020 | Volume 11 | Article 360 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[73,77,485,902]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"secondary somatosensory SCN, which was the second gray matter network that discriminated AD from controls in the current study, contains connections between the parietal operculum and the posterior part of the cingulate gyrus, which is also involved in the AD pathophysiologic processes (44). Assessment of SCNs as potential MRI markers for the differential diagnosis of dementia follows the network degeneration hypothesis of Seeley and coworkers (13). It suggests disease-dependent simultaneous network degeneration of gray matter rather than isolated and successive involvement of brain areas to by typical for primary degenerative dementias including AD, behavioral variant fronto-temporal dementia, semantic dementia, progressive nonfluent aphasia, and cortico-basal syndrome. Our results are somewhat contrasting this assumption in that the potential of SCNs to discriminate between AD and controls was almost identical to that of isolated volumetric measures. So far, five studies have used SCNs as a diagnostic tool to separate AD cases from controls (18, 19, 47-49). They identified multiple SCNs that differ between AD and HC, such as the default mode network, a hippocampal network, the salience network, and the executive control network. We also found two networks, the temporal and the somatosensory SCN to contribute to the classification between patients and controls. Our study extends previous investigations by not only defining SCNs that may assist in AD diagnosis but by also assessing their diagnostic potential relative to established MR-based biomarkers. In our study SCNs did not increase the diagnostic accuracy beyond that of whole brain volume, hippocampal volume, and MTA rating. Similar assessments were not done in previous studies but are needed to better define the diagnostic potential of SCNs in clinical settings. Another reason for insignificant results may be the use of MNI152 for co-registration standard space during processing. MNI152 may not be ideal, since the brains of the elderly might significantly differ from the brains used for the MNI152 template. However, Fillmore et al. (50) and Huang et al. (51) discussed this problem and indicated that these effects are diminished in case of VBM because it results in a study specific template. This was the case in our study. As Ashburner and Friston discussed in their technical paper on VBM, a version of standard space template is needed to create this study specific template, and it is merely used for correction of global brain shape difference (26). However, we cannot completely rule out that the use of the default workflow might have had an impact on our analyses. Possibly another crucial factor for our results may be the use of SCN templates from Koini et al. (52). To our knowledge, there exist two commonly used options for template selection. One is using templates from prior studies, the other is to generate study specific templates. Bijsterbosch et al. (53) discuss these choices of network templates and associated advantages and disadvantages. They point out that the first choice is typically to create study specific network templates on the study sample. This approach increases the likelihood for finding statistical differences between investigational groups, because it represents the best fit for the sample under investigation and provides a way to control for noise in the data. This advantage may come at a price of interpretability. If we had created study specific templates, they ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,77,922,355]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"would include both experimental groups (controls and patients) and therefore represent a form of hybrid networks rather than being specific for each comparative sample. Using data of only one of the experimental groups in template creation (controls for example) would introduce a strong bias for later statistical analyses (53). Hence, we decided to use templates from an independent healthy sample with adequate sample size, since we expected sufficient differences and a high potential for classification accuracy between patients and controls when using networks derived from healthy individuals. Another factor regarding interpretation of our results is the inclusion of imaging data from two different centers. While from the same vendor, slightly different image acquisition parameters were used at the two centers which could have affected the study results (54). It is important to note, however, that both cohorts were preprocessed independently, and although some center effects cannot be excluded with certainty, the results on the diagnostic accuracy of SCNs were almost identical at both sites. This is a clear indication for the robustness of the study's finding across centers. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,356,922,562]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our study has several strengths. The size of the cohort is considerable, and we used an independent sample for validation of our results. The diagnostic accuracy was almost identical in both samples which underscores the robustness of our findings. A limitation of our study was the absence of imaging data at follow-up. Longitudinal change in network integrity over time may be a better diagnostic marker of AD. Moreover, although we did not find an association between SCNs at baseline and future cognitive functioning, we cannot rule out that loss of SCN integrity over time parallels cognitive decline in AD patients. In this study, we used subtests of the CERAD-Plus for assessment of verbal and figural memory. While CERAD is a well-established test battery in AD research, more extensive memory tests to reduce the possibility of measurement error may be considered. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,564,922,680]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In conclusion, we found that established volumetric markers and the visual MTA score perform similarly well in differentiating AD from healthy controls than SCNs which need extensive image postprocessing. As to whether SCNs are helpful in discriminating between different forms of dementia syndromes needs to be determined and it is likely that longitudinal assessment of SCNs can increase our understanding of the spatial and timely evolution of neurodegenerative processes. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,716,836,731]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DATA AVAILABILITY STATEMENT ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,745,922,816]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The datasets for this article are not publicly available due to ethical and privacy restrictions. Information contained in the data may compromise the privacy of research participants. Requests to access the datasets should be directed to RS (reinhold.schmidt@medunigraz.at). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,845,715,860]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ETHICS STATEMENT ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,874,922,902]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The studies involving human participants were reviewed and approved by Ethics committees of the Medical Universities of ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,35,142,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Wagner et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[732,35,924,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Structural Covariance Networks in AD ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[73,949,297,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Psychiatry | www.frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[495,951,502,960]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[734,949,922,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"May 2020 | Volume 11 | Article 360 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[73,77,484,106]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Graz, Austria and Vienna, Austria. The patients/participants provided their written informed consent to participate in this study. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,139,334,156]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"AUTHOR CONTRIBUTIONS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[73,168,485,227]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RS, MK, MD, SR, CE, and FW contributed to the conception and design of the study. FM, SR, PD-B, and RS implemented MRI as well as clinical data collection. FW, RS, MK, and BG performed the statistical analysis. FW wrote the first draft of the manuscript. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,77,922,120]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RS, MK, MD, and CE wrote sections of the manuscript. All authors contributed to the manuscript revision, read and approved the submitted version. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,156,605,170]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"FUNDING ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,184,922,227]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This study was funded by a D-A-CH grand (German Research DF DU1626/1-1, and an Austrian Fund FWF 12889-B31) and the Austrian Science Fund (FWF, KLI 546-BBL). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,288,211,304]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"REFERENCES ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[73,315,485,913]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. Brier MR, Thomas JB, Ances BM. Network dysfunction in Alzheimer's disease: refining the disconnection hypothesis. Brain Connect (2014) 4 (5):299-311. doi: 10.1089/brain.2014.0236 \",\"2. Palop JJ, Mucke L. Network abnormalities and interneuron dysfunction in Alzheimer disease. Nat Rev Neurosci (2016) 17(12):777-92. doi: 10.1038/nrn.2016.141 \",\"3. Greicius MD, Srivastava G, Reiss AL, Menon V. Default-mode network activity distinguishes Alzheimer's disease from healthy aging: evidence from functional MRI. Proc Natl Acad Sci U S A (2004) 101(13):4637-42. doi: 10.1073/pnas.0308627101 \",\"4. Buckner RL, Sepulcre J, Talukdar T, Krienen FM, Liu H, Hedden T, et al. Cortical Hubs Revealed by Intrinsic Functional Connectivity: Mapping, Assessment of Stability, and Relation to Alzheimer's Disease. J Neurosci (2009) 29(6):1860-73. doi: 10.1523/JNEUROSCI.5062-08.2009 \",\"5. Wang K, Liang M, Wang L, Tian L, Zhang X, Li K, et al. Altered functional connectivity in early Alzheimer's disease: A resting-state fMRI study. Hum Brain Mapp (2007) 28(10):967-78. doi: 10.1002/hbm.20324 \",\"6. Agosta F, Pievani M, Geroldi C, Copetti M, Frisoni GB, Filippi M. Resting state fMRI in Alzheimer's disease: Beyond the default mode network. Neurobiol Aging (2012) 33(8):1564-78. doi: 10.1016/j.neurobiolaging.2011.06.007 \",\"7. Hafkemeijer A, Moller C, Dopper EGP, Jiskoot LC, Schouten TM, van Swieten JC, et al. Resting state functional connectivity differences between behavioral variant frontotemporal dementia and Alzheimer's disease. Front Hum Neurosci (2015) 9(September):1-12. doi: 10.3389/fnhum.2015.00474 \",\"8. Brier M, Thomas JB, Snyder AZ, Benzinger TL, Zhang D. Loss of Intra- and Inter-Network Resting State Functional Connections with Alzheimer's Disease Progression. J Neurosci (2012) 32(26):8890-9. doi: 10.1523/JNEUROSCI.5698-11.2012.Loss \",\"9. Klaassens BL, van Gerven JMA, van der Grond J, de Vos F, Moller C, Rombouts SARB. Diminished posterior precuneus connectivity with the default mode network differentiates normal aging from Alzheimer's Disease. Front Aging Neurosci (2017) 9(APR):1-13. doi: 10.3389/fnagi.2017.00097 \",\"10. Celone KA, Calhoun VD, Dickerson BC, Atri A, Chua EF, Miller SL, et al. Alterations in Memory Networks in Mild Cognitive Impairment and Alzheimer's Disease: An Independent Component Analysis. J Neurosci (2006) 26(40):10222-31. doi: 10.1523/JNEUROSCI.2250-06.2006 \",\"11. Wang T, Shi F, Jin Y, Yap PT, Wee CY, Zhang J, et al. Multilevel Deficiency of White Matter Connectivity Networks in Alzheimer's Disease: A Diffusion MRI Study with DTI and HARDI Models. *Neural Plast* (2016). doi: 10.1155/2016/2947136 \",\"12. Jin Y, Huang C, Daianu M, Zhan L, Dennis EL, Reid RI, et al. 3D Tract-Specific Local and Global Analysis of White Matter Integrity in Alzheimer's Disease. Hum Brain Mapp (2017) 38(3):1191-207. doi: 10.1002/hbm.23448 \",\"13. Seeley WW, Crawford RK, Zhou J, Miller BL, Greicius MD. Neurodegenerative Diseases Target Large-Scale Human Brain Networks. Neuron (2009) 62(1):42-52. doi: 10.1016/j.neuron.2009.03.024 \",\"14. Hafkemeijer A, Altmann-Schneider I, de Craen AJM, Slagboom PE, van der Grond J, Rombouts SARB. Associations between age and gray matter volume \"]","page_idx":"7","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[512,292,924,913]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"in anatomical brain networks in middle-aged to older adults. Aging Cell (2014) 13(6):1068-74. doi: 10.1111/accel.12271 \",\"15. Montembeault M, Joubert S, Doyon J, Carrier J, Gagnon JF, Monchi O, et al. The impact of aging on gray matter structural covariance networks. Neuroimage (2012) 63(2):754-9. doi: 10.1016/j.neuroimage.2012.06.052 \",\"16. Koini M, Duering M, Gesierich BG, Rombouts SARB, Ropele S, Wagner F, et al. Grey-matter network disintegration as predictor of cognitive and motor function with aging. Brain Struct Funct (2018) 0(0):0. doi: 10.1007/s00429-018-1642-0 \",\"17. Foster-Dingley JC, Hafkemeijer A, van den Berg-Huysmans AA, Moonen JEF, de Ruijter W, de Craen AJM, et al. Structural Covariance Networks and Their Association with Age, Features of Cerebral Small-Vessel Disease, and Cognitive Functioning in Older Persons. Brain Connect (2016) 6(9):681-90. doi: 10.1089/brain.2016.0434 \",\"18. Hafkemeijer A, Möller C, Dopper EGP, Jiskoot LC, van den Berg-Huysmans AA, van Swieten JC, et al. Differences in structural covariance brain networks between behavioral variant frontotemporal dementia and Alzheimer's disease. Hum Brain Mapp (2016) 37(3):978-88. doi: 10.1002/hbm.23081 \",\"19. Li K, Luo X, Zeng Q, Huang P, Shen Z, Xu X, et al. Gray matter structural covariance networks changes along the Alzheimer's disease continuum. NeuroImage Clin (2019) 23(October 2018):101828. doi: 10.1016/j.nicl.2019.101828 \",\"20. Seiler S, Schmidt H, Lechner A, Benke T, Sanin G, Ransmayr G, et al. Driving Cessation and Dementia: Results of the Prospective Registry on Dementia in Austria (PRODEM). *PloS One* (2012) 7(12):e52710. doi: 10.1371/journal.pone.0052710 \",\"21. Schmidt R, Ropele S, Enzinger C, Petrovic K, Smith S, Schmidt H, et al. White matter lesion progression, brain atrophy, and cognitive decline: The Austrian stroke prevention study. Ann Neurol (2005) 58(4):610-6. doi: 10.1002/ana.20630 \",\"22. McKhann G, Drachman D, Folstein M, Katzman R, Price D, Stadlan EM. Clinical diagnosis of Alzheimer's disease: Report of the NINCDS-ADRDA Work Group* under the auspices of Department of Health and Human Services Task Force on Alzheimer's Disease. *Neurology* (1984) 34(7):939. doi: 10.1212/WNL.34.7.939 \",\"23. Jenkinson M, Beckmann CF, Behrens TEJ, Woolrich MW, Smith SM. Fsl. Neuroimage (2012) 62(2):782-90. doi: 10.1016/j.neuroimage.2011.09.015 \",\"24. Douaud G, MacKay C, Andersson J, James S, Quested D, Ray MK, et al. Schizophrenia delays and alters maturation of the brain in adolescence. Brain (2009) 132(9):2437-48. doi: 10.1093/brain/awp126 \",\"25. Smith SM. Fast robust automated brain extraction. Hum Brain Mapp (2002) 17(3):143-55. doi: 10.1002/hbm.10062 \",\"26. Ashburner J, Friston KJ. Voxel-based morphometry - The methods. Neuroimage (2000) 11(6 I):805-21. doi: 10.1006/nimg.2000.0582 \",\"27. Andersson JLR, Jenkinson M, Smith S. Non-linear registration, aka spatial normalisation. In: . FMRIB Technical Report TR07JA2. UK: Oxford Cent Funct Magn Reson Imaging Brain, Dep Clin Neurol Oxford Univ Oxford (2007). p. 22. \",\"28. Good CD, Johnsrude IS, Ashburner J, Henson RNA, Friston KJ. Frackowiak RSJ. A voxel-based morphometric study of ageing in 465 normal adult human brains. Neuroimage (2001) 14(1 Pt 1):21-36. doi: 10.1006/nimg.2001.0786 \"]","page_idx":"7","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[73,35,142,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Wagner et al. 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Filippini N, MacIntosh BJ, Hough MG, Goodwin GM, Frisoni GB, Smith SM, et al. Distinct patterns of brain activity in young carriers of the APOE-epsilon4 allele. PNAS (2009) 106(17):7209-14. doi: 10.1073/pnas.0811879106 \",\"30. Scheltens P, Leys D, Barkhof F, Hugo D, Weinstein HC, Vermersch P, et al. Atrophy of medial temporal lobes on MRI in \\\"probable\\\" Alzheimer's disease and normal ageing: diagnostic value and neuropsychological correlates. J Neurol Neurosurg Psychiatry (1992) 55:967-72. doi: 10.1136/jnnp.55.10.967 \",\"31. Fischl B. Freesurfer. Neuroimage (2012) 62(2):774-81. doi: 10.1016/j.neuroimage.2012.01.021 \",\"32. Morris JC, Heyman A, Mohs RC, Hughes JP, van Belle G, Fillenbaum G, et al. The Consortium to Establish a Registry for Alzheimer's Disease (CERAD). Neurology (1989) 39(9):1159-65. doi: 10.1212/WNL.39.9.1159 \",\"33. Chandler MJ, Lacritz LH, Hynan LS, Barnard HD, Allen G, Deschner M, et al. A total score for the CERAD neuropsychological battery. *Neurology* (2005) 65 (1):102-6. doi: 10.1212/01.wnl.0000167607.63000.38 \",\"34. R-Core-Team. A language and environment for statistical computing. R Found Stat Comput (2016). \",\"35. Breiman L. Random forests. Mach Learn (2001) 45(1):5-32. doi: 10.1023/A:1010933404324 \",\"36. Hothorn T, Buhlmann P, Dudoit S, Molinaro A, Van Der Laan MJ. Survival ensembles. Biostatistics (2006) 7(3):355-73. doi: 10.1093/biostatistics/kxj011 \",\"37. Strobl C, Malley J, Tutz G. An Introduction to Recursive Partitioning: An Introduction to Recursive Partitioning: Rationale. Psychol Methods (2009) 14 (4):323-38. doi: 10.1037/a0016973. Technical Report Number 55. \",\"38. Strobl C, Boulesteix AL, Zeileis A, Hothorn T. Bias in random forest variable importance measures: Illustrations, sources and a solution. BMC Bioinf (2007) 8:25. doi: 10.1186/1471-2105-8-25 \",\"39. Strobl C, Boulesteix AL, Kneib T, Augustin T, Zeileis A. Conditional variable importance for random forests. BMC Bioinf (2008) 9:1-11. doi: 10.1186/1471-2105-9-307 \",\"40. Kuhn M, Wing J, Weston S, Williams A, Keefer C, Engelhardt A, et al. caret: Classification and Regression Training. (2018). \",\"41. Zhou L, Wang Y, Li Y, Yap PT, Shen D. Alzheimer's Disease Neuroimaging Initiative (ADNI). Hierarchical anatomical brain networks for MCI prediction: Revisiting volumetric measures. PloS One (2011) 6(7):1-14. doi: 10.1371/journal.pone.0021935 \",\"42. Reid AT, Evans AC. Structural networks in Alzheimer's disease. Eur Neuropsychopharmacol. (2013) 23(1):63-77. doi: 10.1016/j.euroneuro.2012.11.010 \",\"43. Scheff SW, Price DA, Schmitt FA, Scheff MA, Mufson EJ. Synaptic Loss in the Inferior Temporal Gyrus in Mild Cognitive Impairment and Alzheimer Disease. J Alzheimer's Dis (2011) 24(3):547-57. doi: 10.3233/JAD-2011-101782 \",\"44. Jones BF, Barnes J, Uylings HBM, Fox NC, Frost C, Witter MP, et al. Differential regional atrophy of the cingulate gyrus in Alzheimer disease: A volumetric MRI study. Cereb Cortex (2006) 16(12):1701-8. doi: 10.1093/cercor/bhj105 \"]","page_idx":"8","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[510,75,924,490]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"45. Frisoni GB, Testa C, Zorzan A, Sabattoli F, Beltramello A, Soininen H, et al. Detection of grey matter loss in mild Alzheimer's disease with voxel based morphometry. J Neurol Neurosurg Psychiatry (2002) 73(6):657-64. doi: 10.1136/jnnp.73.6.657 \",\"46. Steven AE, Hyman BT, Van Hoesen GW. Neuropathologic Changes of the Temporal Pole in Alzheimer's Disease and Pick's Disease. Arch Neurol (1994) 51(2):145-50. doi: 10.1001/archneur.1994.00540140051014 \",\"47. Seeley WW, Crawford RK, Zhou J, Miller BL, Greicius MD. Neurodegenerative Diseases Target Large-Scale Human Brain Networks. Neuron (2009) 62(1):42-52. doi: 10.1016/j.neuron.2009.03.024 \",\"48. Montembeault M, Rouleau I, Provost JS, Brambati SM. Altered Gray Matter Structural Covariance Networks in Early Stages of Alzheimer's Disease. Cereb Cortex (2016) 26(6):2650-62. doi: 10.1093/cercor/bhv105 \",\"49. Spreng RN, Turner GR. Structural Covariance of the Default Network in Healthy and Pathological Aging. J Neurosci (2013) 33(38):15226-34. doi: 10.1523/JNEUROSCI.2261-13.2013 \",\"50. Fillmore PT, Phillips-Meek MC, Richards JE. Age-specific MRI brain and head templates for healthy adults from 20 through 89 years of age. Front Aging Neurosci (2015) 7(MAR):1-14. doi: 10.3389/fnagi.2015.00044 \",\"51. Huang CM, Lee SH, Hsiao IT, Kuan WC, Wai YY, Ko HJ, et al. Study-specific EPI template improves group analysis in functional MRI of young and older adults. J Neurosci Methods (2010) 189(2):257-66. doi: 10.1016/j.jneumeth.2010.03.021 \",\"52. Koini M, Duering M, Gesierich BG, Rombouts SARB, Ropele S, Wagner F, et al. Grey-matter network disintegration as predictor of cognitive and motor function with aging. Brain Struct Funct (2018) 0(0):0. doi: 10.1007/s00429-018-1642-0 \",\"53. Bijsterbosch J, Smith SM, Beckmann CF. Introduction to resting state FMRI functional connectivity. Oxford: Oxford University Press (2017). \",\"54. Jovicich J, Czanner S, Han X, Salat D, van der Kouwe A, Quinn B, et al. MRI-derived measurements of human subcortical, ventricular and intracranial brain volumes: Reliability effects of scan sessions, acquisition sequences, data analyses, scanner upgrade, scanner vendors and field strengths. NeuroImage [Internet] (2009) 46(1):177-92. doi: 10.1016/j.neuroimage.2009.02.010 \"]","page_idx":"8","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[510,503,922,540]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conflict of Interest: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,551,924,637]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Copyright © 2020 Wagner, Duering, Gesierich, Enzinger, Ropele, Dal-Bianco, Mayer, Schmidt and Koini. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. 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\"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"a section of the journal \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[152,727,267,739]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Psychiatry \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,745,267,756]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Received: 21 October 2019 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,758,267,770]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Accepted: 09 April 2020 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,771,267,783]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Published: 05 May 2020 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[216,789,267,799]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Citation: \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,802,267,813]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Wagner F, Duering M, Gesierich BG \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,815,267,826]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Enzinger C, Ropele S, Dal-Bianco P, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[99,827,267,839]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Mayer F, Schmidt R and Koini M \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,840,267,852]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(2020) Gray Matter Covariance \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[124,853,267,863]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Networks as Classifiers and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[147,866,267,876]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Predictors of Cognitive \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,879,267,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Function in Alzheimer's Disease. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[137,892,267,903]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Front. Psychiatry 11:360. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[105,905,267,916]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"doi: 10.3389/fpsyt.2020.00360 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[289,432,927,772]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The study of shared variation in gray matter morphology may define neurodegenerative diseases beyond what can be detected from the isolated assessment of regional brain volumes. We, therefore, aimed to (1) identify SCNs (structural covariance networks) that discriminate between Alzheimer's disease (AD) patients and healthy controls (HC), (2) investigate their diagnostic accuracy in comparison and above established markers, and (3) determine if they are associated with cognitive abilities. We applied a random forest algorithm to identify discriminating networks from a set of 20 SCNs. The algorithm was trained on a main sample of 104 AD patients and 104 age-matched HC and was then validated in an independent sample of 28 AD patients and 28 controls from another center. Only two of the 20 SCNs contributed significantly to the discrimination between AD and controls. These were a temporal and a secondary somatosensory SCN. Their diagnostic accuracy was $74\\\\%$ in the original cohort and $80\\\\%$ in the independent samples. The diagnostic accuracy of SCNs was comparable with that of conventional volumetric MRI markers including whole brain volume and hippocampal volume. SCN did not significantly increase diagnostic accuracy beyond that of conventional MRI markers. We found the temporal SCN to be associated with verbal memory at baseline. No other associations with cognitive functions were seen. SCNs failed to predict the course of cognitive decline over an average of 18 months. We conclude that SCNs have diagnostic potential, but the diagnostic information gain beyond conventional MRI markers is limited. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[290,781,803,795]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords: structural covariance network, longitudinal, Alzheimer, cognition, random forest \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[290,829,452,845]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"INTRODUCTION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[289,857,926,917]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Alzheimer's Disease (AD) has been recognized as a disconnection syndrome (1, 2) leading to increasing cognitive deficits as the disease progresses. Resting state functional magnetic resonance imaging (rs-fMRI) has been used to study functional connectivity at rest and showed decreased connectivity in brain networks of AD patients (3-5). The default mode network (DMN) (3, 6) is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[72,30,105,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"# \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[107,43,203,57]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"frontiers \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[107,60,242,75]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"in Psychiatry \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[794,43,922,53]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ORIGINAL RESEARCH \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[800,55,922,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"published: 05 May 2020 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[764,65,922,75]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"doi: 10.3389/fpsyt.2020.00360 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[873,92,922,129]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[72,949,299,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Psychiatry | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[492,951,504,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[732,949,924,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"May 2020 | Volume 11 | Article 360 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[73,77,485,444]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"most affected (7-9). Alterations in connectivity have also been found in the salience and frontal executive networks (6, 8). In healthy subjects, DMN connectivity is inhibited while other networks increase their connectivity. AD patients lose the ability to suppress the DMN network during cognitive activity, (8) and a task-based fMRI study showed an association between de-synchronized hippocampus and DMN activity and impaired memory (10). In line with functional disconnection in AD, multi-level deficiency of white matter (WM) connectivity has been shown in DTI studies (11, 12). The authors reported a decreased fiber-count of WM and loss of connective efficiency within and between AD relevant brain areas like the hippocampus, precuneus, and temporal gyrus (11). Another study showed degeneration in 18 major WM tracts, including the fornix, cingulum, and corpus callosum (12). Connectivity between gray matter regions can also be assessed by a method called structural covariance networks (SCNs) (13). SCNs are networks found on the basis of distinct gray matter (GM) covariation patterns in remote cortical areas. In healthy subjects, SCN integrity decreases with age (14, 15), and relates to impairment of cognitive and motor functions (16, 17). In patients with mild cognitive impairment and AD, SCNs containing temporal and limbic regions as well as the precuneus were found to predict the rate of decline in memory over time (13, 18, 19). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,446,485,622]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the present study we aimed to identify those SCNs that best discriminate between AD patients and healthy controls by utilizing twenty SCNs obtained in a group of 257 healthy aging subjects (16). We hypothesized, that networks including the temporal lobes as well as the DMN discriminate best between patients and controls. To validate the diagnostic accuracy of potentially significant identified networks, we applied them on an independent sample. The diagnostic value of SCNs was then compared to conventional MRI markers of AD including the medial temporal lobe atrophy score, total brain volume, and hippocampal volume. We also tested whether SCNs relate to cognitive functioning cross-sectionally and longitudinally. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,657,176,672]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"METHODS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,684,147,700]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sample \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,702,485,892]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The data from AD patients are from the Prospective Registry on Dementia (PRODEM), a longitudinal multi-center study on disease-progression of dementia patients in Austria. The inclusion criteria were as follows: (a) dementia diagnosis according to DSM-IV criteria, (b) non-institutionalization and no need for 24-h care and (c) availability of a caregiver who is able to provide additional information on the patient. We excluded patients with co-morbidities that were likely to cause early termination from the study such as cancer. Subjects who were not able to sign an informed consent form were also excluded. For further detail see (20). The data from the controls are from the Austrian Stroke Prevention Study-Family (ASPS-Fam). The ASPS-Fam is a single-center, prospective study \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,77,922,134]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"on risk factors and their effects on the brain in the normal elderly. None of our control subjects had a history of neuropsychiatric disease, including cerebrovascular attacks and dementia. All had a normal neurologic examination (21). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,136,922,400]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For the present study all PRODEM patients with (1) either possible or probable AD defined by the NINCDS-ADRDA Alzheimer's criteria (22), (2) a 3-Tesla T1-weighted 3D MRI scan, and (3) a cognitive assessment within 3 months of the imaging examination were selected. The cognitive assessment included the Mini Mental State examination (MMSE) and the \\\"Consortium to Establish a Registry for Alzheimer's Disease\\\"—Test battery—plus version (CERAD-Plus). In total 104 AD patients (mean age $= 71.45 \\\\pm 7.97$ years, range: 51–87 years, 59 females) from Graz and 28 AD patients (mean age $= 73.79 \\\\pm 6.17$ , range: 58–82, 14 females) from Vienna were included. The Graz sample was used for statistical modeling, while the Viennese sample served as the independent validation cohort. In each center a cohort of controls was matched to each AD patient. Matching was done for age $(+/- 3$ years) and sex (Graz: mean age $= 71.09 \\\\pm 7.38$ years; range: 53–86, 59 females; Vienna: $n = 28$ , mean age $= 72.44 \\\\pm 7.18$ years; range: 56–85 years, 14 female). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,401,922,487]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Eighty-two AD patients from the Graz set had a follow-up clinical and cognitive assessment that ranged between 6 and 37 months with a mean of 18 months. Patients who dropped out from the study did not differ from subjects with follow-up in terms of and the Mini Mental State Examination scores at baseline. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,490,922,548]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The study was approved by the ethics committees of the Medical Universities of Graz, Austria and Vienna, Austria. Informed consent was obtained from all subjects and/or caregivers included in the study. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,560,638,576]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"MRI Protocol \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,578,922,738]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Structural data were assessed using a three-dimensional, T1-weighted, magnetization prepared rapid gradient echo sequence (MPRAGE) in Graz and Vienna (Trio Tim 3.0T, manufactured by Siemens Healthcare, Erlangen, Germany) with whole brain coverage. The selected sequence parameters for PRODEM were as follows: Repetition time: $1,900 / 2,000\\\\mathrm{ms}$ , inversion time: 900 ms, echo time: $2.19\\\\mathrm{ms}$ , flip angle: $9^{\\\\circ}$ and a resolution of: $1\\\\mathrm{mm} \\\\times 1\\\\mathrm{mm} \\\\times 1\\\\mathrm{mm} / 1.2\\\\mathrm{mm}$ . For ASPS-Fam, the selected sequence parameters were as follows: Repetition time: 1,900, inversion time: 900 ms, echo time: $2.19\\\\mathrm{ms}$ , flip angle: $9^{\\\\circ}$ an isotropic resolution of $1\\\\mathrm{mm}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,758,657,774]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Image Analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,775,757,788]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Structural Covariance Networks \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,790,922,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To prevent biasing data, all image processing steps were done independently for the Graz and Vienna cohorts of this sample. The T1 weighted images were visually checked to ensure consistent image quality for analysis. Pre-processing and SCN analyses were performed using FMRIB's Software Library (FSL) Version 5.0.9 (23). A detailed version of processing steps can be found in (14, 16, 18, 24) In brief, the brain was extracted from \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,35,142,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Wagner et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[732,35,924,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Structural Covariance Networks in AD \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[73,949,297,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Psychiatry | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[495,951,502,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[734,949,922,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"May 2020 | Volume 11 | Article 360 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[73,77,485,518]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"non-brain tissue in the images with a semiautomatic brain extraction tool (BET) implemented in FSL (25). The extracted brains were visually checked for artefacts (remaining skull or missing part of sulci). In case of missing part of sulci, parameters for BET were changed. Remaining skull was removed by hand. Then followed tissue-type segmented into gray matter (GM), white matter (WM), and cerebrospinal fluid (CSF) using a voxel-based morphometric analysis (VBM) within the FSL framework (26). After visual inspection, the individual GM images were non-linear registered (27) to the GM MNI152 standard space (Montreal Neurological Institute, Montreal, QC, Canada) (23). By averaging the resulting images, a study-specific GM template was created to which the native GM images were non-linearly re-registered. Following, the images were modulated to correct for distortions due to non-linear transformation (26, 28). Finally, the GM images were then spatially smoothed with an isotropic Gaussian kernel with a sigma of $3\\\\mathrm{mm}$ and concatenated to a 4D-data set. Thereafter, this 4D-data set was spatially regressed onto 20 networks obtained by Koini and colleagues (16) in a healthy sample, using spatial regression (Step 1 of the dual regression script provided by FSL) (29) to calculate individual network integrity scores. These networks included a temporal, a temporo-cerebellar, a superior temporal gyrus, a supplementary motor, a secondary somatosensory, a limbic, a fronto-parietal, a cuneal, a precuneus, a fronto-occipital, an insular including the gyrus cinguli, an occipital including the posterior cingulate cortex, six cerebellar, and an amygdala SCN. The outcome of the spatial regression consists of individual beta scores (positive and negative) per person and per SCN, representing network integrity values, which were used in the statistical analyses. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,537,369,551]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Visual and Automated Atrophy Marker \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,553,485,698]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The native T1-weighted images of all subjects were visually rated by trained specialists using the medial temporal lobe atrophy (MTA)-score scale, a five-point rating scale, ranging from no atrophy (0) to maximum atrophy (4) (30). Ratings were made for both hemispheres, with a mean value calculated for statistical analyses. Automatic brain volume measurements from the native T1-weighted images of all subjects were done by using FreeSurfer (Version 5.3) (31). The brain volume and the hippocampus volumes, averaged across hemispheres, were normalized for intracranial volume. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,716,287,732]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cognitive Assessment \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,732,485,908]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The PRODEM study protocol includes the CERAD-Plus (Consortium to Establish a Registry for Alzheimer's Disease) test battery, which among others includes subtests for verbal memory and figural memory (32). Verbal memory is tested by the recall of a wordlist and figural memory is assessed by drawing geometric figures from a template and later from memory. For both subtests \\\"savings\\\" scores, the ratio of correctly recalled items to learned items (times 100), were available as measurement for memory retention. We also determined a composite score of CERAD-Plus subtest, the Chandler-score (33). Verbal and figural memory savings and the Chandler-score were selected as outcome variables in the cross sectional and longitudinal analyses. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,77,692,93]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Statistical Analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,94,922,372]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The statistical software R (version 3.5.3 on Windows 10) was used (34). Training of the classification algorithms were done solely on the cohort from Graz. For discrimination between AD patients and HC the value of the integrity score of each of the 20 SCNs was used in a random forest analysis (35) using the \\\"cforest\\\" function within the \\\"party\\\" package (36-38). 1001 condition inference trees and the associated conditional permutation importance for the SCN integrity scores were calculated (39). Numbers of random variables at the nodes (\\\"mtry\\\" parameter) were left at the standard of five variables. The conditional permutation importance is a measure of variable importance and is defined as a score for mean loss of accuracy in classification, if a given SCN is excluded from the random forest model. SCNs with a variable importance score of less than one were not regarded as crucial for classification. Since the random forest estimations and variable importance calculations contain steps based on randomness, the results typically vary in small ranges. To compensate for this, the calculations were repeated 100 times to build mean values and standard deviation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,374,922,578]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Next, random forest models containing distinct combinations of crucial SCNs were compared to check if comparable classification accuracy can be achieved with sparser models. Crucial SCNs were identified and selected by variable importance as mentioned above. Accuracy, sensitivity, and specificity calculations were done with the \\\"caret\\\"-package (40). These models were again calculated 100 times to obtain mean accuracy, sensitivity, and specificity values. The models were then tested on the independent data set containing AD patients from Vienna and HC from the ASPS-Fam. Thereafter, a model containing the crucial SCNs was extended by the established markers MTA-score, total brain volume, and averaged hippocampus volume to measure their joint discriminative ability. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,581,922,698]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To examine the association between the SCNs integrity and cognitive ability, six multiple linear regression models were computed using the longitudinal cohort from Graz. Outcome variables were baseline (BL) values (cross-sectional) and annualized change values from BL to follow-up (FU, longitudinal) of (a) the Chandler-score, (b) the verbal memory savings, and (c) the figural memory savings. Predictors were the crucial SCNs evaluated by variable importance. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,732,605,748]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RESULTS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,761,710,777]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Diagnostic Accuracy \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,777,922,849]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From the 20 template networks included in our random forest analysis (Model A), the temporal network (Figure 1A, Table 1) and the secondary somatosensory (S2) network (Figure 1B, Table 1) reached a variable importance score above one (Figure 2), and discriminative value. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,851,922,908]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As can be seen from Table 2, sensitivity, specificity, and diagnostic accuracy of the temporal SCN is 72, 77, and $74\\\\%$ , respectively. These values were replicated in the validation cohort. It is also shown in this table that the S2 SCN has lower \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,35,142,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Wagner et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[732,35,924,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Structural Covariance Networks in AD \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[73,949,297,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Psychiatry | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[495,951,502,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[734,949,922,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"May 2020 | Volume 11 | Article 360 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[211,100,786,288]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0471/5ccef15251674497d758135d5f38bb85326697976c168711105ee365f02ce6c2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[211,300,786,487]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIGURE 1 | Sagittal, coronal, and axial slices of the temporal SCN (MNI coordinates: $x = -50$ , $y = -20$ , $z = -32$ ) and the secondary somatosensory SCN (S2; MNI coordinates: $x = -50$ , $y = -26$ , $z = 18$ ). The images are taken from the Koini masks (16). \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0850/27f9940346cccde11a9a70d40c964d345a221f49c73225781344fcca01a7317b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[70,555,487,614]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"diagnostic accuracy. A combination of the S2 and temporal SCNs and even a combination of all 20 SCNs increased the diagnostic accuracy only marginally when compared to the temporal SCN alone. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,616,485,779]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In Table 3 and Figure 3 we show the random forest models assessing diagnostic accuracy measure for volumetric MRI markers of AD alone and for the full model including all volumetric measures and the temporal and S2 SCN. In both the Graz cohort and the Vienna validation cohort single volumetric MRI markers performed similar to SCNs. Joint use of volumetric markers and SCNs discriminated best between AD and controls with a diagnostic accuracy of 82 and $86\\\\%$ in the founder and validation cohort, respectively. The increase in accuracy when adding SCNs to conventional MRI volumetric measures was near zero and non-significant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,789,169,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cognition \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,806,485,894]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cross-sectionally, in AD, the temporal SCN showed a significant association with verbal memory savings (Table 4), explaining nine percent of the variance. No association between SCN integrity and any other test scores were found. For change in cognitive measures over time, no associations with SCN integrity were found either (Table 4). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,555,640,572]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DISCUSSION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,584,924,790]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Among 20 covariance networks, a temporal gray matter network and a secondary somatosensory gray matter network showed a diagnostic accuracy of 74 and $80\\\\%$ in two independent cohorts. AD patients were correctly identified in 76 and $83\\\\%$ of cases, controls were correctly identified in 78 and $75\\\\%$ of cases. The gray matter networks did not significantly surpass the diagnostic accuracy of traditional volumetric markers including brain volume, hippocampal volume, and MTA score. When adding the covariance networks to a model of traditional volumetric MR markers, the diagnostic accuracy remained virtually unchanged. This contrasts the view that the assessment of networks, which degenerate in synchrony, provides more diagnostic information than a simple volumetric assessment of selected brain regions known to be involved in AD (1, 41, 42). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,790,924,849]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Loss of connectivity within the temporal network was related to lower verbal memory, but neither the temporal network nor other SCNs were able to predict the slope of cognitive decline over the observational period. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,849,924,894]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The covariance networks in our study are partly overlapping with those described by Hafkemeijer and co-workers (18). The temporal gray matter network, which related to verbal memory in \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,35,142,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Wagner et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[729,35,924,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Structural Covariance Networks in AD \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[72,949,299,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Psychiatry | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[492,949,504,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[732,949,924,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"May 2020 | Volume 11 | Article 360 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[72,91,924,243]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0148/8c3b21a187be7f66f32532922334b663847bacdc3bf9a247e723f33acedc2a21.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=\\\"2\\\">SCN</td><td rowspan=\\\"2\\\">Voxels</td><td colspan=\\\"3\\\">MNI coordinates</td><td rowspan=\\\"2\\\">Region</td><td rowspan=\\\"2\\\">Hemisphere</td></tr><tr><td>x</td><td>y</td><td>z</td></tr><tr><td rowspan=\\\"5\\\">Temporal</td><td>10,442</td><td>40</td><td>-4</td><td>-40</td><td>Inferior temporal gyrus, anterior division</td><td>R</td></tr><tr><td>5,355</td><td>-38</td><td>8</td><td>-38</td><td>Temporal pole</td><td>L</td></tr><tr><td>1,466</td><td>0</td><td>20</td><td>40</td><td>Paracingulate gyrus</td><td>R</td></tr><tr><td>1,313</td><td>10</td><td>38</td><td>-4</td><td>Cingulate gyrus, anterior division</td><td>R</td></tr><tr><td>1,106</td><td>-32</td><td>22</td><td>-4</td><td>Insular Cortex</td><td>L</td></tr><tr><td rowspan=\\\"3\\\">Secondary Somatosensory (S2)</td><td>9,650</td><td>-48</td><td>-26</td><td>16</td><td>Parietal operculum cortex</td><td>L</td></tr><tr><td>7,478</td><td>50</td><td>-28</td><td>16</td><td>Parietal operculum cortex</td><td>R</td></tr><tr><td>1,028</td><td>10</td><td>-26</td><td>34</td><td>Cingulate gyrus, posterior division</td><td>R</td></tr></table>\",\n \"table_caption\": \"[\\\"TABLE 1 | Regions included in the Temporal SCN and Secondary Somatosensory SCN (16). \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[231,282,764,587]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIGURE 2 | Variable importance (mean accuracy loss) of 20 structural covariance networks (SCNs) in a random forest classification model. Exclusion of the temporal and the secondary somatosensory network show a variable importance score above one. The error bars show one standard deviation. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0237/73ca8144e946cc0c6f32ecf59a27c53a9c40a50e6f2da9a85312482344e51c1c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[70,679,924,785]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0914/02e0aeef1c1e29d6f2007c15f4d698ec953760e0d8d2fd3a6c882071002a85ad.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=\\\"2\\\">Model</td><td rowspan=\\\"2\\\">Predictors</td><td colspan=\\\"3\\\">Main sample (Graz)</td><td colspan=\\\"3\\\">Independent sample (Vienna)</td></tr><tr><td>Accuracy</td><td>Sensitivity</td><td>Specificity</td><td>Accuracy</td><td>Sensitivity</td><td>Specificity</td></tr><tr><td>A</td><td>All SCNs</td><td>77</td><td>76</td><td>78</td><td>79</td><td>83</td><td>75</td></tr><tr><td>B</td><td>Temporal + S2</td><td>74</td><td>73</td><td>75</td><td>80</td><td>85</td><td>75</td></tr><tr><td>C</td><td>Temporal SCN</td><td>74</td><td>72</td><td>77</td><td>74</td><td>75</td><td>73</td></tr><tr><td>D</td><td>S2 SCN</td><td>67</td><td>67</td><td>67</td><td>54</td><td>54</td><td>53</td></tr></table>\",\n \"table_caption\": \"[\\\"TABLE 2 | Accuracy, sensitivity, and specificity measures for the classification models in the original Graz sample and the independent Viennese sample. \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[70,808,487,912]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"our study, contains connections between the anterior part of the right inferior temporal gyrus, the left temporal pole, the right paracingulate gyrus, the anterior part of the right cingulate gyrus, and the left insular cortex. These areas have been reported to be particularly affected by AD pathology (43-46). However, the inclusion of the anterior cingulate and paracingulate gyri in the temporal network is somehow unusual, but in case of the former a \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,808,924,913]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"comparable network was also found in a study by Hafkemeijer and co-workers (14). Since there is quite some randomness in network creation depending on both sample data and parameter choice, we are unable to determine whether the inclusion of the anterior division of the cingulate gyrus and the paracingulate gyrus in the temporal SCN occurred on true biological grounds. A statistical change finding cannot be excluded with certainty. The \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,35,142,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Wagner et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[731,35,924,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Structural Covariance Networks in AD \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[72,949,299,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Psychiatry | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[492,949,504,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[732,949,924,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"May 2020 | Volume 11 | Article 360 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[70,91,926,233]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0827/2e1e512ee75f298ee5df996bf1b69af3d9346bf53db8e716a22d531bbb11f916.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=\\\"2\\\">Model</td><td colspan=\\\"3\\\">Graz AD-HC</td><td colspan=\\\"3\\\">Vienna AD-HC</td></tr><tr><td>Accuracy</td><td>Sensitivity</td><td>Specificity</td><td>Accuracy</td><td>Sensitivity</td><td>Specificity</td></tr><tr><td>MTA</td><td>77</td><td>74</td><td>81</td><td>82</td><td>82</td><td>82</td></tr><tr><td>BV</td><td>74</td><td>72</td><td>75</td><td>73</td><td>75</td><td>72</td></tr><tr><td>Hc</td><td>76</td><td>73</td><td>79</td><td>79</td><td>88</td><td>74</td></tr><tr><td>MTA + Hc + BV</td><td>82</td><td>80</td><td>84</td><td>88</td><td>92</td><td>84</td></tr><tr><td>MTA + Hc + BV + Temporal + S2</td><td>82</td><td>81</td><td>82</td><td>86</td><td>92</td><td>81</td></tr></table>\",\n \"table_caption\": \"[\\\"TABLE 3 | Accuracy, sensitivity, and specificity measures for MTA-score, normalized brain volume (BV), normalized hippocampus volume (Hc), and combined models. \\\"]\",\n \"table_footnote\": \"[\\\"Calculations were done in the original Graz sample and the independent Viennese sample. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[231,308,764,611]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIGURE 3 | Variable importance (mean accuracy loss) of the temporal and the S2 SCNs and three alternative markers (MTA-score, normalized brain volume, normalized hippocampus volume) in a random forest classification model. The error bars depict one standard deviation. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0718/34b2acd16b59c654c821c772a06286ba4165c111f228fb5ab28cbdf7f3a4d5aa.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[70,704,927,872]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0001/c5fe60cdc40e334137f20061437b2086639d809bec400e726ca374d4b9ab66ea.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>N = 82</td><td colspan=\\\"3\\\">Chandler-score</td><td colspan=\\\"3\\\">Verbal memory savings</td><td colspan=\\\"3\\\">Figural memory savings</td></tr><tr><td>Cross-sectional</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Variable</td><td>B</td><td>SE</td><td>beta</td><td>B</td><td>SE</td><td>beta</td><td>B</td><td>SE</td><td>Beta</td></tr><tr><td>Temporal SCN</td><td>276.34</td><td>200.33</td><td>.15</td><td>1,060.42*</td><td>508.55</td><td>.23*</td><td>-168.11</td><td>508.11</td><td>-.03</td></tr><tr><td>Secondary Somatosensory SCN</td><td>204.857</td><td>225.80</td><td>.10</td><td>805.79</td><td>573.23</td><td>.15</td><td>569.17</td><td>572.73</td><td>.11</td></tr><tr><td>R2(p-value)</td><td></td><td>.04 (.17)</td><td></td><td></td><td>.09 (.01)*</td><td></td><td></td><td>.01 (.61)</td><td></td></tr><tr><td colspan=\\\"10\\\">Longitudinal</td></tr><tr><td>Variable</td><td>B</td><td>SE</td><td>beta</td><td>B</td><td>SE</td><td>beta</td><td>B</td><td>SE</td><td>Beta</td></tr><tr><td>Temporal SCN</td><td>-7.26</td><td>126.24</td><td><-0.00</td><td>155.77</td><td>484.45</td><td>.03</td><td>-182.99</td><td>391.47</td><td>< .00</td></tr><tr><td>Secondary Somatosensory SCN</td><td>-71.78</td><td>143.45</td><td>-0.06</td><td>-902.78</td><td>570.06</td><td>-.19</td><td>87.47</td><td>441.26</td><td>< .00</td></tr><tr><td>R2(p-value)</td><td></td><td>< .00 (.86)</td><td></td><td></td><td>.03 (.29)</td><td></td><td></td><td>< .00 (.89)</td><td></td></tr></table>\",\n \"table_caption\": \"[\\\"TABLE 4 | Multiple linear regression models to evaluate the predictive value of the SCNs for cognition. \\\"]\",\n \"table_footnote\": \"[\\\"*significance at.05 level. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[73,35,142,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Wagner et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[729,35,924,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Structural Covariance Networks in AD \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[72,949,299,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Psychiatry | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[492,951,504,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[732,949,924,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"May 2020 | Volume 11 | Article 360 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[73,77,485,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"secondary somatosensory SCN, which was the second gray matter network that discriminated AD from controls in the current study, contains connections between the parietal operculum and the posterior part of the cingulate gyrus, which is also involved in the AD pathophysiologic processes (44). Assessment of SCNs as potential MRI markers for the differential diagnosis of dementia follows the network degeneration hypothesis of Seeley and coworkers (13). It suggests disease-dependent simultaneous network degeneration of gray matter rather than isolated and successive involvement of brain areas to by typical for primary degenerative dementias including AD, behavioral variant fronto-temporal dementia, semantic dementia, progressive nonfluent aphasia, and cortico-basal syndrome. Our results are somewhat contrasting this assumption in that the potential of SCNs to discriminate between AD and controls was almost identical to that of isolated volumetric measures. So far, five studies have used SCNs as a diagnostic tool to separate AD cases from controls (18, 19, 47-49). They identified multiple SCNs that differ between AD and HC, such as the default mode network, a hippocampal network, the salience network, and the executive control network. We also found two networks, the temporal and the somatosensory SCN to contribute to the classification between patients and controls. Our study extends previous investigations by not only defining SCNs that may assist in AD diagnosis but by also assessing their diagnostic potential relative to established MR-based biomarkers. In our study SCNs did not increase the diagnostic accuracy beyond that of whole brain volume, hippocampal volume, and MTA rating. Similar assessments were not done in previous studies but are needed to better define the diagnostic potential of SCNs in clinical settings. Another reason for insignificant results may be the use of MNI152 for co-registration standard space during processing. MNI152 may not be ideal, since the brains of the elderly might significantly differ from the brains used for the MNI152 template. However, Fillmore et al. (50) and Huang et al. (51) discussed this problem and indicated that these effects are diminished in case of VBM because it results in a study specific template. This was the case in our study. As Ashburner and Friston discussed in their technical paper on VBM, a version of standard space template is needed to create this study specific template, and it is merely used for correction of global brain shape difference (26). However, we cannot completely rule out that the use of the default workflow might have had an impact on our analyses. Possibly another crucial factor for our results may be the use of SCN templates from Koini et al. (52). To our knowledge, there exist two commonly used options for template selection. One is using templates from prior studies, the other is to generate study specific templates. Bijsterbosch et al. (53) discuss these choices of network templates and associated advantages and disadvantages. They point out that the first choice is typically to create study specific network templates on the study sample. This approach increases the likelihood for finding statistical differences between investigational groups, because it represents the best fit for the sample under investigation and provides a way to control for noise in the data. This advantage may come at a price of interpretability. If we had created study specific templates, they \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,77,922,355]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"would include both experimental groups (controls and patients) and therefore represent a form of hybrid networks rather than being specific for each comparative sample. Using data of only one of the experimental groups in template creation (controls for example) would introduce a strong bias for later statistical analyses (53). Hence, we decided to use templates from an independent healthy sample with adequate sample size, since we expected sufficient differences and a high potential for classification accuracy between patients and controls when using networks derived from healthy individuals. Another factor regarding interpretation of our results is the inclusion of imaging data from two different centers. While from the same vendor, slightly different image acquisition parameters were used at the two centers which could have affected the study results (54). It is important to note, however, that both cohorts were preprocessed independently, and although some center effects cannot be excluded with certainty, the results on the diagnostic accuracy of SCNs were almost identical at both sites. This is a clear indication for the robustness of the study's finding across centers. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,356,922,562]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our study has several strengths. The size of the cohort is considerable, and we used an independent sample for validation of our results. The diagnostic accuracy was almost identical in both samples which underscores the robustness of our findings. A limitation of our study was the absence of imaging data at follow-up. Longitudinal change in network integrity over time may be a better diagnostic marker of AD. Moreover, although we did not find an association between SCNs at baseline and future cognitive functioning, we cannot rule out that loss of SCN integrity over time parallels cognitive decline in AD patients. In this study, we used subtests of the CERAD-Plus for assessment of verbal and figural memory. While CERAD is a well-established test battery in AD research, more extensive memory tests to reduce the possibility of measurement error may be considered. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,564,922,680]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In conclusion, we found that established volumetric markers and the visual MTA score perform similarly well in differentiating AD from healthy controls than SCNs which need extensive image postprocessing. As to whether SCNs are helpful in discriminating between different forms of dementia syndromes needs to be determined and it is likely that longitudinal assessment of SCNs can increase our understanding of the spatial and timely evolution of neurodegenerative processes. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,716,836,731]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DATA AVAILABILITY STATEMENT \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,745,922,816]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The datasets for this article are not publicly available due to ethical and privacy restrictions. Information contained in the data may compromise the privacy of research participants. Requests to access the datasets should be directed to RS (reinhold.schmidt@medunigraz.at). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,845,715,860]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ETHICS STATEMENT \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,874,922,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The studies involving human participants were reviewed and approved by Ethics committees of the Medical Universities of \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,35,142,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Wagner et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[732,35,924,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Structural Covariance Networks in AD \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[73,949,297,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Psychiatry | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[495,951,502,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[734,949,922,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"May 2020 | Volume 11 | Article 360 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[73,77,484,106]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Graz, Austria and Vienna, Austria. The patients/participants provided their written informed consent to participate in this study. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,139,334,156]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"AUTHOR CONTRIBUTIONS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,168,485,227]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RS, MK, MD, SR, CE, and FW contributed to the conception and design of the study. FM, SR, PD-B, and RS implemented MRI as well as clinical data collection. FW, RS, MK, and BG performed the statistical analysis. FW wrote the first draft of the manuscript. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,77,922,120]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RS, MK, MD, and CE wrote sections of the manuscript. All authors contributed to the manuscript revision, read and approved the submitted version. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,156,605,170]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"FUNDING \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,184,922,227]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This study was funded by a D-A-CH grand (German Research DF DU1626/1-1, and an Austrian Fund FWF 12889-B31) and the Austrian Science Fund (FWF, KLI 546-BBL). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,288,211,304]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"REFERENCES \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,315,485,913]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. Brier MR, Thomas JB, Ances BM. Network dysfunction in Alzheimer's disease: refining the disconnection hypothesis. Brain Connect (2014) 4 (5):299-311. doi: 10.1089/brain.2014.0236 \\\",\\\"2. Palop JJ, Mucke L. Network abnormalities and interneuron dysfunction in Alzheimer disease. Nat Rev Neurosci (2016) 17(12):777-92. doi: 10.1038/nrn.2016.141 \\\",\\\"3. Greicius MD, Srivastava G, Reiss AL, Menon V. Default-mode network activity distinguishes Alzheimer's disease from healthy aging: evidence from functional MRI. Proc Natl Acad Sci U S A (2004) 101(13):4637-42. doi: 10.1073/pnas.0308627101 \\\",\\\"4. Buckner RL, Sepulcre J, Talukdar T, Krienen FM, Liu H, Hedden T, et al. Cortical Hubs Revealed by Intrinsic Functional Connectivity: Mapping, Assessment of Stability, and Relation to Alzheimer's Disease. J Neurosci (2009) 29(6):1860-73. doi: 10.1523/JNEUROSCI.5062-08.2009 \\\",\\\"5. Wang K, Liang M, Wang L, Tian L, Zhang X, Li K, et al. Altered functional connectivity in early Alzheimer's disease: A resting-state fMRI study. Hum Brain Mapp (2007) 28(10):967-78. doi: 10.1002/hbm.20324 \\\",\\\"6. Agosta F, Pievani M, Geroldi C, Copetti M, Frisoni GB, Filippi M. Resting state fMRI in Alzheimer's disease: Beyond the default mode network. Neurobiol Aging (2012) 33(8):1564-78. doi: 10.1016/j.neurobiolaging.2011.06.007 \\\",\\\"7. Hafkemeijer A, Moller C, Dopper EGP, Jiskoot LC, Schouten TM, van Swieten JC, et al. Resting state functional connectivity differences between behavioral variant frontotemporal dementia and Alzheimer's disease. Front Hum Neurosci (2015) 9(September):1-12. doi: 10.3389/fnhum.2015.00474 \\\",\\\"8. Brier M, Thomas JB, Snyder AZ, Benzinger TL, Zhang D. Loss of Intra- and Inter-Network Resting State Functional Connections with Alzheimer's Disease Progression. J Neurosci (2012) 32(26):8890-9. doi: 10.1523/JNEUROSCI.5698-11.2012.Loss \\\",\\\"9. Klaassens BL, van Gerven JMA, van der Grond J, de Vos F, Moller C, Rombouts SARB. Diminished posterior precuneus connectivity with the default mode network differentiates normal aging from Alzheimer's Disease. Front Aging Neurosci (2017) 9(APR):1-13. doi: 10.3389/fnagi.2017.00097 \\\",\\\"10. Celone KA, Calhoun VD, Dickerson BC, Atri A, Chua EF, Miller SL, et al. Alterations in Memory Networks in Mild Cognitive Impairment and Alzheimer's Disease: An Independent Component Analysis. J Neurosci (2006) 26(40):10222-31. doi: 10.1523/JNEUROSCI.2250-06.2006 \\\",\\\"11. Wang T, Shi F, Jin Y, Yap PT, Wee CY, Zhang J, et al. Multilevel Deficiency of White Matter Connectivity Networks in Alzheimer's Disease: A Diffusion MRI Study with DTI and HARDI Models. *Neural Plast* (2016). doi: 10.1155/2016/2947136 \\\",\\\"12. Jin Y, Huang C, Daianu M, Zhan L, Dennis EL, Reid RI, et al. 3D Tract-Specific Local and Global Analysis of White Matter Integrity in Alzheimer's Disease. Hum Brain Mapp (2017) 38(3):1191-207. doi: 10.1002/hbm.23448 \\\",\\\"13. Seeley WW, Crawford RK, Zhou J, Miller BL, Greicius MD. Neurodegenerative Diseases Target Large-Scale Human Brain Networks. Neuron (2009) 62(1):42-52. doi: 10.1016/j.neuron.2009.03.024 \\\",\\\"14. Hafkemeijer A, Altmann-Schneider I, de Craen AJM, Slagboom PE, van der Grond J, Rombouts SARB. Associations between age and gray matter volume \\\"]\",\n \"page_idx\": \"7\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[512,292,924,913]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"in anatomical brain networks in middle-aged to older adults. Aging Cell (2014) 13(6):1068-74. doi: 10.1111/accel.12271 \\\",\\\"15. Montembeault M, Joubert S, Doyon J, Carrier J, Gagnon JF, Monchi O, et al. The impact of aging on gray matter structural covariance networks. Neuroimage (2012) 63(2):754-9. doi: 10.1016/j.neuroimage.2012.06.052 \\\",\\\"16. Koini M, Duering M, Gesierich BG, Rombouts SARB, Ropele S, Wagner F, et al. Grey-matter network disintegration as predictor of cognitive and motor function with aging. Brain Struct Funct (2018) 0(0):0. doi: 10.1007/s00429-018-1642-0 \\\",\\\"17. Foster-Dingley JC, Hafkemeijer A, van den Berg-Huysmans AA, Moonen JEF, de Ruijter W, de Craen AJM, et al. Structural Covariance Networks and Their Association with Age, Features of Cerebral Small-Vessel Disease, and Cognitive Functioning in Older Persons. Brain Connect (2016) 6(9):681-90. doi: 10.1089/brain.2016.0434 \\\",\\\"18. Hafkemeijer A, Möller C, Dopper EGP, Jiskoot LC, van den Berg-Huysmans AA, van Swieten JC, et al. Differences in structural covariance brain networks between behavioral variant frontotemporal dementia and Alzheimer's disease. Hum Brain Mapp (2016) 37(3):978-88. doi: 10.1002/hbm.23081 \\\",\\\"19. Li K, Luo X, Zeng Q, Huang P, Shen Z, Xu X, et al. Gray matter structural covariance networks changes along the Alzheimer's disease continuum. NeuroImage Clin (2019) 23(October 2018):101828. doi: 10.1016/j.nicl.2019.101828 \\\",\\\"20. Seiler S, Schmidt H, Lechner A, Benke T, Sanin G, Ransmayr G, et al. Driving Cessation and Dementia: Results of the Prospective Registry on Dementia in Austria (PRODEM). *PloS One* (2012) 7(12):e52710. doi: 10.1371/journal.pone.0052710 \\\",\\\"21. Schmidt R, Ropele S, Enzinger C, Petrovic K, Smith S, Schmidt H, et al. White matter lesion progression, brain atrophy, and cognitive decline: The Austrian stroke prevention study. Ann Neurol (2005) 58(4):610-6. doi: 10.1002/ana.20630 \\\",\\\"22. McKhann G, Drachman D, Folstein M, Katzman R, Price D, Stadlan EM. Clinical diagnosis of Alzheimer's disease: Report of the NINCDS-ADRDA Work Group* under the auspices of Department of Health and Human Services Task Force on Alzheimer's Disease. *Neurology* (1984) 34(7):939. doi: 10.1212/WNL.34.7.939 \\\",\\\"23. Jenkinson M, Beckmann CF, Behrens TEJ, Woolrich MW, Smith SM. Fsl. Neuroimage (2012) 62(2):782-90. doi: 10.1016/j.neuroimage.2011.09.015 \\\",\\\"24. Douaud G, MacKay C, Andersson J, James S, Quested D, Ray MK, et al. Schizophrenia delays and alters maturation of the brain in adolescence. Brain (2009) 132(9):2437-48. doi: 10.1093/brain/awp126 \\\",\\\"25. Smith SM. Fast robust automated brain extraction. Hum Brain Mapp (2002) 17(3):143-55. doi: 10.1002/hbm.10062 \\\",\\\"26. Ashburner J, Friston KJ. Voxel-based morphometry - The methods. Neuroimage (2000) 11(6 I):805-21. doi: 10.1006/nimg.2000.0582 \\\",\\\"27. Andersson JLR, Jenkinson M, Smith S. Non-linear registration, aka spatial normalisation. In: . FMRIB Technical Report TR07JA2. UK: Oxford Cent Funct Magn Reson Imaging Brain, Dep Clin Neurol Oxford Univ Oxford (2007). p. 22. \\\",\\\"28. Good CD, Johnsrude IS, Ashburner J, Henson RNA, Friston KJ. Frackowiak RSJ. A voxel-based morphometric study of ageing in 465 normal adult human brains. Neuroimage (2001) 14(1 Pt 1):21-36. doi: 10.1006/nimg.2001.0786 \\\"]\",\n \"page_idx\": \"7\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[73,35,142,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Wagner et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[731,35,924,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Structural Covariance Networks in AD \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[73,949,297,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Psychiatry | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[495,951,502,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"8 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[734,949,922,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"May 2020 | Volume 11 | Article 360 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[73,77,485,636]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"29. Filippini N, MacIntosh BJ, Hough MG, Goodwin GM, Frisoni GB, Smith SM, et al. Distinct patterns of brain activity in young carriers of the APOE-epsilon4 allele. PNAS (2009) 106(17):7209-14. doi: 10.1073/pnas.0811879106 \\\",\\\"30. Scheltens P, Leys D, Barkhof F, Hugo D, Weinstein HC, Vermersch P, et al. Atrophy of medial temporal lobes on MRI in \\\\\\\"probable\\\\\\\" Alzheimer's disease and normal ageing: diagnostic value and neuropsychological correlates. J Neurol Neurosurg Psychiatry (1992) 55:967-72. doi: 10.1136/jnnp.55.10.967 \\\",\\\"31. Fischl B. Freesurfer. Neuroimage (2012) 62(2):774-81. doi: 10.1016/j.neuroimage.2012.01.021 \\\",\\\"32. Morris JC, Heyman A, Mohs RC, Hughes JP, van Belle G, Fillenbaum G, et al. The Consortium to Establish a Registry for Alzheimer's Disease (CERAD). Neurology (1989) 39(9):1159-65. doi: 10.1212/WNL.39.9.1159 \\\",\\\"33. Chandler MJ, Lacritz LH, Hynan LS, Barnard HD, Allen G, Deschner M, et al. A total score for the CERAD neuropsychological battery. *Neurology* (2005) 65 (1):102-6. doi: 10.1212/01.wnl.0000167607.63000.38 \\\",\\\"34. R-Core-Team. A language and environment for statistical computing. R Found Stat Comput (2016). \\\",\\\"35. Breiman L. Random forests. Mach Learn (2001) 45(1):5-32. doi: 10.1023/A:1010933404324 \\\",\\\"36. Hothorn T, Buhlmann P, Dudoit S, Molinaro A, Van Der Laan MJ. Survival ensembles. Biostatistics (2006) 7(3):355-73. doi: 10.1093/biostatistics/kxj011 \\\",\\\"37. Strobl C, Malley J, Tutz G. An Introduction to Recursive Partitioning: An Introduction to Recursive Partitioning: Rationale. Psychol Methods (2009) 14 (4):323-38. doi: 10.1037/a0016973. Technical Report Number 55. \\\",\\\"38. Strobl C, Boulesteix AL, Zeileis A, Hothorn T. Bias in random forest variable importance measures: Illustrations, sources and a solution. BMC Bioinf (2007) 8:25. doi: 10.1186/1471-2105-8-25 \\\",\\\"39. Strobl C, Boulesteix AL, Kneib T, Augustin T, Zeileis A. Conditional variable importance for random forests. BMC Bioinf (2008) 9:1-11. doi: 10.1186/1471-2105-9-307 \\\",\\\"40. Kuhn M, Wing J, Weston S, Williams A, Keefer C, Engelhardt A, et al. caret: Classification and Regression Training. (2018). \\\",\\\"41. Zhou L, Wang Y, Li Y, Yap PT, Shen D. Alzheimer's Disease Neuroimaging Initiative (ADNI). Hierarchical anatomical brain networks for MCI prediction: Revisiting volumetric measures. PloS One (2011) 6(7):1-14. doi: 10.1371/journal.pone.0021935 \\\",\\\"42. Reid AT, Evans AC. Structural networks in Alzheimer's disease. Eur Neuropsychopharmacol. (2013) 23(1):63-77. doi: 10.1016/j.euroneuro.2012.11.010 \\\",\\\"43. Scheff SW, Price DA, Schmitt FA, Scheff MA, Mufson EJ. Synaptic Loss in the Inferior Temporal Gyrus in Mild Cognitive Impairment and Alzheimer Disease. J Alzheimer's Dis (2011) 24(3):547-57. doi: 10.3233/JAD-2011-101782 \\\",\\\"44. Jones BF, Barnes J, Uylings HBM, Fox NC, Frost C, Witter MP, et al. Differential regional atrophy of the cingulate gyrus in Alzheimer disease: A volumetric MRI study. Cereb Cortex (2006) 16(12):1701-8. doi: 10.1093/cercor/bhj105 \\\"]\",\n \"page_idx\": \"8\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[510,75,924,490]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"45. Frisoni GB, Testa C, Zorzan A, Sabattoli F, Beltramello A, Soininen H, et al. Detection of grey matter loss in mild Alzheimer's disease with voxel based morphometry. J Neurol Neurosurg Psychiatry (2002) 73(6):657-64. doi: 10.1136/jnnp.73.6.657 \\\",\\\"46. Steven AE, Hyman BT, Van Hoesen GW. Neuropathologic Changes of the Temporal Pole in Alzheimer's Disease and Pick's Disease. Arch Neurol (1994) 51(2):145-50. doi: 10.1001/archneur.1994.00540140051014 \\\",\\\"47. Seeley WW, Crawford RK, Zhou J, Miller BL, Greicius MD. Neurodegenerative Diseases Target Large-Scale Human Brain Networks. Neuron (2009) 62(1):42-52. doi: 10.1016/j.neuron.2009.03.024 \\\",\\\"48. Montembeault M, Rouleau I, Provost JS, Brambati SM. Altered Gray Matter Structural Covariance Networks in Early Stages of Alzheimer's Disease. Cereb Cortex (2016) 26(6):2650-62. doi: 10.1093/cercor/bhv105 \\\",\\\"49. Spreng RN, Turner GR. Structural Covariance of the Default Network in Healthy and Pathological Aging. J Neurosci (2013) 33(38):15226-34. doi: 10.1523/JNEUROSCI.2261-13.2013 \\\",\\\"50. Fillmore PT, Phillips-Meek MC, Richards JE. Age-specific MRI brain and head templates for healthy adults from 20 through 89 years of age. Front Aging Neurosci (2015) 7(MAR):1-14. doi: 10.3389/fnagi.2015.00044 \\\",\\\"51. Huang CM, Lee SH, Hsiao IT, Kuan WC, Wai YY, Ko HJ, et al. Study-specific EPI template improves group analysis in functional MRI of young and older adults. J Neurosci Methods (2010) 189(2):257-66. doi: 10.1016/j.jneumeth.2010.03.021 \\\",\\\"52. Koini M, Duering M, Gesierich BG, Rombouts SARB, Ropele S, Wagner F, et al. Grey-matter network disintegration as predictor of cognitive and motor function with aging. Brain Struct Funct (2018) 0(0):0. doi: 10.1007/s00429-018-1642-0 \\\",\\\"53. Bijsterbosch J, Smith SM, Beckmann CF. Introduction to resting state FMRI functional connectivity. Oxford: Oxford University Press (2017). \\\",\\\"54. Jovicich J, Czanner S, Han X, Salat D, van der Kouwe A, Quinn B, et al. MRI-derived measurements of human subcortical, ventricular and intracranial brain volumes: Reliability effects of scan sessions, acquisition sequences, data analyses, scanner upgrade, scanner vendors and field strengths. NeuroImage [Internet] (2009) 46(1):177-92. doi: 10.1016/j.neuroimage.2009.02.010 \\\"]\",\n \"page_idx\": \"8\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[510,503,922,540]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conflict of Interest: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,551,924,637]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Copyright © 2020 Wagner, Duering, Gesierich, Enzinger, Ropele, Dal-Bianco, Mayer, Schmidt and Koini. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,35,142,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Wagner et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[729,35,924,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Structural Covariance Networks in AD \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[73,949,297,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Psychiatry | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[495,951,502,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"9 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[734,949,922,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"May 2020 | Volume 11 | Article 360 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.3389/fpsyt.2020.00360"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Life Sciences"},"sha256":{"kind":"string","value":"9528a59226dc78cf4a2fa32c4b7ecec5c464be4d873be1aa94938df29350b36b"},"title":{"kind":"string","value":"Gray Matter Covariance Networks as Classifiers and Predictors of Cognitive Function in Alzheimer’s Disease"}}},{"rowIdx":48,"cells":{"abstract":{"kind":"string","value":"Pre-eclampsia (PE) is usually defined as new-onset hypertension with albuminuria or other organ damage. Herein, the role and mechanism of long noncoding RNA (lncRNA) gastric carcinoma high expressed transcript 1 (GHET1) during PE are investigated. Expression of GHET1 in PE pregnancies was evaluated using quantitative real-time polymerase chain reaction (qRT-PCR). Proliferation and cell cycle of extravillous trophoblasts were assessed by Cell Counting Kit-8 (CCK-8), colony formation, 5-Ethynyl-2'-deoxyuridine (EdU) assays, and flow cytometry, respectively. Migration, invasion, and network formation of trophoblasts were measured by wound healing, transwell system, and tube formation assays. RNA immunoprecipitation (RIP), RNA pull-down, and chromatin immunoprecipitation (ChIP) assays were used to confirm the molecular interaction. GHET1 was markedly decreased in the placenta of PE patients. GHET1 promoted the proliferation and cell cycle of extravillous trophoblasts, as well as migration, invasion, and network formation in vitro. Metallothionein 2A (MT2A) functioned as a downstream effector of GHET1, which was negatively correlated with GHET1 in PE. GHET1 directly bound with zeste 2 polycomb repressive complex 2/lysine-specific demethylase 1 (EZH2/LSD1). Knockdown of GHET1 reduced the occupancies of H3K27me3 and H3K4me2 in the MT2A promoter region by recruiting EZH2 and LSD1. MT2A knockdown reversed GHET1 inhibition mediated biological functions. GHET1 regulates extravillous trophoblastic phenotype via EZH2/LSD1-mediated MT2A epigenetic suppression in PE."},"author":{"kind":"string","value":"Pengyun Wan|Jia Huang|Wenting Liu|Xiaoyan Su|Bei Zhao|Xianggang Wang|Lu Zhao"},"content_list":{"kind":"list like","value":[{"bbox":"[70,134,816,221]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"lncRNA GHET1 regulates extravillous trophoblastic phenotype via EZH2/LSD1-mediated MT2A epigenetic suppression in pre-eclampsia ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[68,248,840,292]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Pengyun Wan $^{1}$ | Jia Huang $^{2}$ | Wenting Liu $^{1}$ | Xiaoyan Su $^{3}$ | Bei Zhao $^{4}$ | Xianggang Wang $^{5}$ | Lu Zhao $^{1}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,320,317,571]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"$^{1}$ Department of Obstetrics and Gynecology, Second Affiliated Hospital of Nanchang University, Nanchang, Jiangxi Province, People's Republic of China \",\"$^{2}$ Reproductive Health Department, Jiangxi Provincial Maternal and Child Health Hospital, Nanchang, Jiangxi Province, People's Republic of China \",\"<sup>3</sup>Pathology Department, Second Affiliated Hospital of Nanchang University, Nanchang, Jiangxi Province, People's Republic of China \",\"$^{4}$ Traditional Chinese Medicine Department, Duchang County People's Hospital, Jiujiang, Jiangxi Province, People's Republic of China \",\"Medical Examination Department, Second Affiliated Hospital of Nanchang University, Nanchang, Jiangxi Province, People's Republic of China \"]","page_idx":"0","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[70,588,164,599]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Correspondence ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,602,314,664]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lu Zhao, Department of Obstetrics and Gynecology, Second Affiliated Hospital of Nanchang University, No.1, Minde Rd, Nanchang 330006, Jiangxi Province, People's Republic of China. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[72,666,240,677]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Email: ndefy11068@ncu.edu.cn ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[347,331,418,346]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[344,351,915,750]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Pre-eclampsia (PE) is usually defined as new-onset hypertension with albuminuria or other organ damage. Herein, the role and mechanism of long noncoding RNA (lncRNA) gastric carcinoma high expressed transcript 1 (GHET1) during PE are investigated. Expression of GHET1 in PE pregnancies was evaluated using quantitative real-time polymerase chain reaction (qRT-PCR). Proliferation and cell cycle of extravillous trophoblasts were assessed by Cell Counting Kit-8 (CCK-8), colony formation, 5-Ethynyl-2'-deoxyuridine (EdU) assays, and flow cytometry, respectively. Migration, invasion, and network formation of trophoblasts were measured by wound healing, transwell system, and tube formation assays. RNA immunoprecipitation (RIP), RNA pull-down, and chromatin immunoprecipitation (ChIP) assays were used to confirm the molecular interaction. GHET1 was markedly decreased in the placenta of PE patients. GHET1 promoted the proliferation and cell cycle of extravillous trophoblasts, as well as migration, invasion, and network formation in vitro. Metallothionein 2A (MT2A) functioned as a downstream effector of GHET1, which was negatively correlated with GHET1 in PE. GHET1 directly bound with zeste 2 polycomb repressive complex 2/lysine-specific demethylase 1 (EZH2/LSD1). Knockdown of GHET1 reduced the occupancies of H3K27me3 and H3K4me2 in the MT2A promoter region by recruiting EZH2 and LSD1. MT2A knockdown reversed GHET1 inhibition mediated biological functions. GHET1 regulates extravillous trophoblastic phenotype via EZH2/LSD1-mediated MT2A epigenetic suppression in PE. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[347,768,438,780]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"KEYWORDS ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[347,783,665,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"EZH2/LSD1, GHET1, IncRNA, MT2A, Pre-eclampsia ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,883,919,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abbreviations: BCA, Bicinchoninic acid; CCAT1, Colon cancer-associated transcript 1; CCK-8, cell counting kit-8; ChIP, chromatin immunoprecipitation; EdU, 5-Ethynyl-2'-deoxyuridine; EED, embryonic ectoderm development; EVTs, extra villous trophoblasts; EZH2, zeste 2 polycomb repressive complex 2; FITC, Fluorescein isotiocyanate; GAPDH, Glyceraldehyde-3-phosphate dehydrogenase; IncRNA, long noncoding RNA; LSD1, lysine specific demethylase1; MT2A, metallothionein 2A; MTs, metallothioneins; pcDNA, pcDNA 3.1; PE, Pre-eclampsia; PFA, paraformaldehyde; PRC, polycomb-group protein; PTEN, phosphatase and tensin homolog; qRT-PCR, quantitative real-time polymerase chain reaction; RIP, RNA immunoprecipitation; SUZ12, suppressor of zeste 12; TUG1, taurine up-regulated gene 1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[872,0,895,16]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"# ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[899,3,998,12]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[72,34,206,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Received: 9 March 2023 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[228,34,354,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Accepted: 6 June 2023 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[72,52,211,63]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DOI: 10.1002/mrd.23693 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[778,58,922,74]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Molecular Reproduction ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[813,74,922,92]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Development ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[70,84,263,99]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RESEARCH ARTICLE ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[73,955,97,965]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"758 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[134,953,300,966]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"© 2023 Wiley Periodicals LLC. ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[405,955,591,966]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"wileyonlinelibrary.com/journal/mrd ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[734,955,922,966]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Mol Reprod Dev. 2023;90:758-770. ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[72,61,273,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 | INTRODUCTION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,97,485,359]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Pre-eclampsia (PE), a multisystem progressive disorder unique to pregnancy, is characterized by elevated blood pressure after 20 weeks of gestation, accompanied by albuminuria or organ damage with high maternal and infant morbidity and mortality (Hao et al., 2019; Moufarrej et al., 2022), affecting $3\\% -5\\%$ of pregnancies in the world (Mol et al., 2016). There are several factors that resulted in PE, including a history of PE, chronic hypertension, pregestational diabetes mellitus, antiphospholipid syndrome, and obesity, among others (Bartsch et al., 2016). Recently, the dysfunction of placental trophoblast cells was considered as the key factor to trigger PE (Gyselaers et al., 2019). It has long been believed that inadequate transformation of the maternal uterine vasculature is a consequence of primary defective extravillous trophoblast cell invasion and leads to the development of PE (Ridder et al., 2019). The abnormal ability of migration and invasion of trophoblasts induce obstacles of spiral artery remodeling, thus driving PE progression (Xu et al., 2020). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,363,485,609]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Long noncoding RNA (lncRNA) refers to noncoding RNA with a length greater than 200 nts (Wu et al., 2020) with various regulatory functions (Long et al., 2019). It has been widely reported that lncRNA is closely related to human diseases, including PE (Gong et al., 2020; He et al., 2019; Jiang & Zhao, 2018; Li et al., 2018). As proof, Li et al. revealed that lncRNA colon cancer-associated transcript 1 was markedly upregulated in PE patients, and its knockdown markedly increased trophoblast proliferation (Li et al., 2018). In addition, lncRNA uc003fir was reported to be upregulated in placental tissues of PE patients, and it could negatively regulate trophoblast proliferation (Pengjie et al., 2018). However, the studies on the role of lncRNAs in PE are still limited. This indicates that the study of PE-related lncRNAs and their molecular mechanisms is of great significance for the understanding of the pathogenesis of PE and the improvement of treatment status. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,612,485,907]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Gastric carcinoma highly expressed transcript 1 (GHET1) is a lncRNA that is located on chromosome 12q14.1 with the length about 1567 nts (Hui, Ji, et al., 2019; Tao et al., 2020). It has been widely demonstrated that GHET1 functions in regulating various cellular processes, such as cell proliferation, migration, and invasion (Yang et al., 2018; Zhou et al., 2016). For instance, Yang et al. revealed that GHET1 knockdown inhibited osteosarcoma cell proliferation, invasion, and migration (Yang et al., 2018). Importantly, it was previously described that GHET1 overexpression could restore hypoxia-induced decrease in cell invasion and migration abilities in PE (Li et al., 2019), revealing the important roles of GHET1 in PE. Nevertheless, the role and underlying mechanism of GHET1 in PE needs to be further investigated. As illustrated, lncRNAs epigenetically regulate the expression regulation of downstream target genes by binding to the enhancer of zeste 2 polycomb repressive complex 2 (EZH2) and lysine-specific demethylase1 (LSD1) (Ding et al., 2017; Huang et al., 2017; Lian et al., 2016). Herein, our work aims to explore whether GHET1 affects PE progress by combining with EZH2 and LSD1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,911,485,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Herein, we draw the conclusion that GHET1 regulated the extravillous trophoblastic phenotype via EZH2/LSD1-mediated ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,63,924,143]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"metallothionein 2A (MT2A) epigenetic suppression in PE. We revealed the essential role and epigenetic regulatory network of GHET1 in the development of PE. These findings might help to deepen our understanding of PE pathogenesis and provide useful ideas for PE treatment. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,177,823,193]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 MATERIALS AND METHODS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,212,815,227]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sample collection and ethics statement ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,246,926,475]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Thirty-one pregnant women with intact membranes, no clinical evidence of intrauterine infection, singleton pregnancy, and who delivered by cesarean section (C-section) with no evidence of active labor were included in the study and divided into two study groups: normal group $(n = 11)$ constituted by women with uncomplicated pregnancies at term who delivered normal weight newborns and PE group $(n = 20)$ . Placental tissues were collected at Second Affiliated Hospital of Nanchang University. All tissues were stored at $-80^{\\circ}\\mathrm{C}$ . The study was authorized by Second Affiliated Hospital of Nanchang University Ethics Committee. All patients who met the inclusion criteria provided written informed consent. Disease diagnosis was performed according to the 2018 ACOG criteria. Clinical and anthropometric data from the study groups are represented in Table 1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,510,665,525]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.1 | Cell culture ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,546,924,643]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Human umbilical vein endothelial cells (HUVEC-C) and human placenta cells (HTR-8/SVneo, JEG3, BeWo, and Wish cells), obtained from the ATCC, were cultured in RPMI 1640 (REF: C11875500B, LOT: 8118016) containing $10\\%$ fetal bovine serum (FBS) (FBS, Invitrogen), $100\\mathrm{U / mL}$ penicillin and $100\\mathrm{U / mL}$ streptomycin (Invitrogen) at $37^{\\circ}\\mathrm{C}$ with $5\\%$ $\\mathrm{CO}_{2}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,677,702,691]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2 Cell transfection ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,712,924,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Small interfering negative control (si-NC), si-GHET1, si-EZH2, si-LSD1, pcDNA (pcDNA 3.1)-GHET1, as well as short-hairpin negative control (sh-NC) and sh-MT2A were purchased from GenePharma. When cell fusion reached $80\\%$ , cells were transfected with the above plasmids by using Lipofectamine 2000 (Invitrogen) following the protocols of manufacturer. After $48\\mathrm{h}$ , the transfection efficiency was identified using quantitative real-time polymerase chain reaction (qRT-PCR). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,875,828,892]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.3 | Cell counting kit-8 (CCK-8) assay ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,911,924,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"During the logarithmic growth phase, HTR-8/SVneo and JEG3 cells were planted to 96-well plates $(2\\times 10^{3}$ cells/well) and cultured for ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,26,137,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"WAN ET AL. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[715,21,862,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Molecular Reproduction ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[752,37,860,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Development ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[899,26,922,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"759 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[73,65,643,253]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0633/4394c4798ccd49fbec9a90eda063c4aa25e4d07ab7aa7a59cc560c76dd6d2d14.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":"<table><tr><td>Characteristics</td><td>Normal (n = 11)</td><td>PE (n = 20)</td><td>p value</td></tr><tr><td>Maternal age (years)</td><td>28.97 ± 5.15</td><td>30.85 ± 5.66</td><td>>0.05</td></tr><tr><td>Maternal weight (kg)</td><td>78.99 ± 5.98</td><td>75.56 ± 6.71</td><td>>0.05</td></tr><tr><td>Fetal birth weight (g)</td><td>3586.45 ± 506.88</td><td>2856.52 ± 501.12</td><td><0.001</td></tr><tr><td>Smoking</td><td>0</td><td>0</td><td>>0.05</td></tr><tr><td>Systolic blood pressure (mmHg)</td><td>112.89 ± 8.15</td><td>148.52 ± 10.15</td><td><0.001</td></tr><tr><td>Diastolic blood pressure (mmHg)</td><td>76.65 ± 8.52</td><td>108.56 ± 9.13</td><td><0.001</td></tr><tr><td>Proteinuria (g/day)</td><td><0.3</td><td>>0.3</td><td><0.01</td></tr><tr><td>Gestational age (week)</td><td>38.05 ± 1.33</td><td>34.21 ± 3.12</td><td><0.01</td></tr></table>","table_caption":"[\"TABLE 1 Clinical features of PE and normal pregnant women. \"]","table_footnote":"[\"Note: Data were presented as mean ± SD (Student's t test). \",\"Abbreviation: PE, pre-eclamncia. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[70,329,485,393]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"24 h. Cells were subsequently incubated with $10\\mu \\mathrm{L}$ of CCK-8 solution (Cat: 96992, Sigma-Aldrich) at $37^{\\circ}C$ for $2h$ . Absorbance was examined at $450nm$ with a microplate spectrophotometer (Bioteke). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,428,317,443]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.4 Colony formation assay ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,462,485,576]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cells were cultured in a six-well plate with 500 cells/well for 2 weeks. After that, cells were washed twice in phosphate-buffered saline (PBS), fixed in the $4\\%$ paraformaldehyde $(4\\%$ PFA) for $15\\mathrm{min}$ , and stained in the Giemsa solution for $10\\mathrm{min}$ . The colonies were dried and observed under the microscope following the removal of Giemsa solution. The colonies (a diameter $\\geq 100\\mu \\mathrm{m}$ ) were counted in triplicate assays. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,609,428,626]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.5 | 5-Ethynyl- $2^{\\prime}$ -deoxyuridine (EdU) assay ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,644,485,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cells were inoculated in a 24-well plate with $4 \\times 10^{4}$ cells/well. EdU solution (10 $\\mu$ M, RiboBio) was added to the cells and cultured for 2 h at $37^{\\circ} \\mathrm{C}$ . Subsequently, cells were dyed with EdU labeling/detection kit (RiboBio) following the instructions of the manufacturer. After fixation using $4 \\%$ formaldehyde for $20 \\mathrm{~min}$ , the EdU-stained cells were examined. Next, the cells were treated with Hoechst 33342 (Sangon) for $20 \\mathrm{~min}$ , and the results were visualized under a fluorescence microscope (Olympus). The percentage of EdU-positive cells was calculated as EdU-positive cell number/Hoechst-positive cell number. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,843,258,859]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.6 | Flow cytometry ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,877,485,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cells $(1 \\times 10^{6}$ cells/mL) were centrifuged at $500g$ for $5\\mathrm{min}$ after two washes in binding buffer. Next, the cells were incubated in the dark in $100\\mu L$ of $1\\times$ binding buffer supplemented with $5\\mu L$ of Annexin V-Fluorescein isotiocyanate and $5\\mu L$ of propidium iodide (PI) at room ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,329,922,377]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"temperature for $15 \\mathrm{~min}$ and subjected to flow cytometry (BD Biosciences). Early-stage (Annexin V +/PI-) and late-stage (Annexin V +/PI+) apoptosis rates were calculated. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,411,739,428]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.7 Wound healing assay ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,446,924,576]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Wound healing assays were performed utilizing Wound Healing Assay Kit (ab242285, Abcam) following the instructions of the manufacturer. In brief, when cells reached $80\\%$ confluence, cell suspension was added to the well with insert in place and then incubated for $48\\mathrm{h}$ . Cells were cultured in the well with insert in place until a monolayer form, then the insert was removed to generate a $0.9\\mathrm{mm}$ wound field. After fixing, the ability of cell migration was determined by microscopic imaging (Olympus). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,609,695,626]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.8 | Transwell assay ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,645,924,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Transwell chambers (24-well, $8\\mu \\mathrm{m}$ pore size, $6.5\\mathrm{mm}$ diameter; Corning) were used to detect the invasion ability of cells. The upper chamber was precoated with Matrigel (Corning) at a 1:8 ratio. The lower chamber of the experiment device was filled with $500\\mu \\mathrm{L}$ RPMI Medium $1640\\%$ and $10\\%$ FBS. Cells $(200\\mu \\mathrm{L}, 5000$ cells per well) were collected, resuspended in medium without serum, and then shifted to the hydrated matrigel chambers for $24\\mathrm{h}$ . Cells on the upper surface were scraped, whereas the invasive cells on the lower surface were fixed and colored with $0.1\\%$ crystal violet for half an hour. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,843,769,859]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.9 Network formation assay ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,878,922,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To investigate the network formation of cells, we performed an in vitro culture system. Transwell cell culture inserts (12 mm diameter, 0.4 mM pore size; Corning) were coated with $100\\mu \\mathrm{L}$ of the growth factor reduced Matrigel (BD Biosciences) and allowed to set at $37^{\\circ}C$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,26,97,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"760 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[136,21,280,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Molecular Reproduction ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[858,26,922,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"WAN ET AL. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[174,37,280,52]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Development ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[70,63,485,194]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for at least $1 \\mathrm{~h}$ . HTR-8/SVneo and JEG3 cells were cultured in the upper layer of the chamber with a complete medium ( $2 \\times 10^{3}$ cells/ well) for $24 \\mathrm{~h}$ . After that, the chamber was washed with PBS for at least three times to remove the cells that did not form a network. Subsequently, cells were fixed with methanol, and then stained with $1 \\%$ crystal violet staining solution (Sigma-aldrich). Cells were washed twice using sterile water and were imaging using an inverted microscope (Olympus). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,228,453,260]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.10 | Quantitative real-time polymerase chain reaction (qRT-PCR) ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,278,485,411]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Total RNA was extracted with TRIzol (Sigma-Aldrich), chloroform (Sigma-Aldrich), and isopropanol (Sigma-Aldrich) following the instructions of the manufacturer. Subsequently, $200\\mathrm{ng}$ extracted RNA was reverse-transcribed into complementary DNAs (cDNAs) by ReverTra Ace qPCR RT Kit (Toyobo) and subjected to qRT-PCR assay with SYBR (Invitrogen) in an ABI 7500 system. The data were analyzed with $2^{-\\Delta \\Delta G_{T}}$ method. GAPDH was used as the reference gene. The primers were listed as follows $(5^{\\prime} - 3^{\\prime})$ : ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[90,428,356,443]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"GHET1 (F): CAACAAAGCAGGTAAACATTGG ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,446,342,459]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"GHET1 (R): GCAAAGGCAGAGTGAAAGGT 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","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,612,322,625]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ESM1 (R): CTGGGAAACATGAAGAGCG ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,629,349,641]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"GAPDH (F): GGAGCGAGATCCCTCCAAAAT ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,645,359,658]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"GAPDH (R): GGCTGTTGTCATACTTCTCATGG 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Bicinchoninic acid (BCA) kit (Beyotime) was employed to quantify the protein concentration. Subsequently, $20\\mu \\mathrm{g}$ protein samples were separated by $10\\%$ sodium dodecyl sulfate-polyacrylamide gel electrophoresis (SDS-PAGE) gel electrophoresis, transferred to polyvinylidene fluoride (PVDF) membranes (Millipore). The membranes were sealed with $5\\%$ skim milk for $1\\mathrm{h}$ at room temperature and incubated overnight with antibodies against MT2A (Abcam, 1:1000, ab12228) and GAPDH (Abcam, 1:5000, ab8245) at $4^{\\circ}\\mathrm{C}$ , then hybridized with secondary antibodies (Abcam, 1:2000, ab6728) for $2\\mathrm{h}$ at $37^{\\circ}\\mathrm{C}$ . Blots were visualized on an Odyssey Infrared Imaging System (LI-COR Biosciences) and subsequently analyzed with ImageJ software. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,62,826,78]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.12 RNA immunoprecipitation (RIP) ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[507,97,924,310]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RIP was carried out by utilizing a Magna $\\mathsf{RIP}^{\\mathsf{TM}}$ RNA-Binding Protein Immunoprecipitation Kit (17-700, Sigma-Aldrich). Briefly, $1\\times 10^{7}$ cells were lysed with RIP lysis buffer, and the lysates were incubated with indicated antibodies including anti-enhancer of zeste homolog 2 (EZH2) (Abcam, 1:30, ab191250), anti-suppressor of zeste 12 (SUZ12) (Abcam, 1:30, ab175187), anti-histone lysine-specific demethylase 1 (LSD1) (Abcam, 1:30, ab129195), and immunoglobulin G (IgG) (Santa Cruz Biotechnology,1:30, #3900) and protein A/G magnetic beads at $4^{\\circ}\\mathrm{C}$ overnight. The recovered RNA-protein complex was washed six times with wash buffer and protein denaturation at $55^{\\circ}\\mathrm{C}$ . RNA was purified from the complex. cDNA was synthesized and used for qRT-PCR or gel electrophoresis assay. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,343,702,360]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.13 RNA pull-down ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[507,379,922,542]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RNA pull-down was performed by using Pierce™ Magnetic RNA-Protein Pull-Down Kit (20164, Thermofisher Scientific). Briefly, biotinlabeled full-length GHET1 or its antisense RNA was in vitro transcribed, treated with RNase-free DNase I on column during RNA purification with RNA Clean & Concentrator-25 (Zymo Research), and coincubated with cell lysis buffer to form RNA-protein complex. $3^{\\prime}$ -untranslated-region of androgen receptor (AR) RNA containing UC-rich regions for HuR was used as a positive control. Subsequently, proteins were eluted from the complex and were then detected by western blot. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,576,919,593]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.14 Chromatin immunoprecipitation (ChIP) assay ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[507,612,922,774]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ChIP assay was performed by using ChIP Assay Kit (17-295, Sigma-Aldrich). In brief, cells were cross-linked with formaldehyde and sonicated to an average size of 200-to-500 basepairs. Cell lysates were precleared with protein A/G beads before incubated with protein A/G beads coated with the anti-EZH2 (Millipore), anti-H3K27me3 (Millipore), anti-LSD1 (Millipore), and anti-H3K4me2 (Millipore) antibodies. Anti-rabbit IgG was used as a negative control. Cross-linked DNA released from the protein-DNA complex was purified by DNA Extraction Kit (Sango) and the eluted DNA was further subjected to qRT-PCR. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,809,719,824]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.15 | Statistics analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[507,845,922,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"All our data were obtained from three independent experiments. Statistical data were analyzed by Prism Graphpad (Version 8.0) and presented as mean $\\pm$ standard deviation (SD). Between-group differences and multigroup comparisons were determined using Student's t test and one-way analysis of variance (ANOVA), respectively. Pearson correlation coefficients were calculated using Pearson ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,26,134,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"WAN ET AL. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[715,21,860,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Molecular Reproduction ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[752,37,858,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Development ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[899,26,922,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"761 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[70,63,485,93]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"correlation analysis by Prism. $p$ value which lower than 0.05 was referred as significant difference. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,127,205,141]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 | RESULTS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,161,475,194]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.1 GHET1 was decreased in the placenta of PE patients ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,212,485,459]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We first explored the role of GHET1 in PE at the clinical level. To investigate whether GHET1 was dysregulated in the PE development, normal placental tissues $(n = 11)$ and placental tissues from patients diagnosed with PE $(n = 20)$ were collected. The data of qRT-PCR presented a lower abundance of GHET1 in the placenta of PE pregnancies relative to the normal placental tissues (Figure 1a), suggesting the potential role of GHET1 in PE progression. Meanwhile, it was observed that GHET1 expression was markedly increased in human placenta cell lines (HTR-8/ SVneo, JEG3, BeWo, and Wish cells) compared with that in HUVEC-C cells (Figure 1b). Notably, among five human placenta cell lines, the levels of GHET1 were relatively higher in HTR-8/ SVneo cells and relatively lower in JEG3 cells, therefore HTR-8/ SVneo and JEG3 cell lines were selected to further investigate the role of GHET1 in vitro. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,494,480,526]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.2 GHET1 modulated the proliferation and cycle of extravillous trophoblast cells ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,546,485,625]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To explore the latent biological function of GHET1 in PE progression, small interfering RNA (siRNAs) specifically target GHET1 or its overexpressing plasmids were applied in HTR-8/Svneo and JEG3 cells, respectively. As expected, transfection of si-GHET1 1# 2# efficiently caused the inhibition of GHET1 expression in ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,63,924,359]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"HTR-8/Svneo cells (Figure 2a). Similarly, GHET1 overexpressing plasmids also induced the significant upregulation of GHET1 in JEG3 cells (Figure 2a). After the successful transfection efficiency, biological effects of GHET1 on extravillous trophoblasts were evaluated. As revealed in Figure 2b, HTR-8/SVneo cell viability was markedly reduced by GHET1 knockdown, whereas the viability was enhanced within GHET1 overexpression in JEG3. Meanwhile, colony formation and EdU assays also revealed that GHET1 knockdown markedly inhibited the cell proliferation in HTR-8/SVneo cells, while GHET1 overexpression exerted the opposite trend in JEG3 cells (Figure 2c,d). Flow cytometry was employed to investigate whether GHET1 regulates cell proliferation via changing cell cycle, and the results displayed that silencing of GHET1 resulted in the significant accumulation of G0/G1 phase in HTR-8/SVneo cells. Conversely, GHET1 overexpression decreased the number of JEG3 cells in the accumulation of G0/G1 phase (Figure 2e). Collectively, GHET1 acted as a facilitator for the growth and proliferation of extravillous trophoblast cells. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,393,919,427]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.3 GHET1 regulated the migration, invasion, and network formation of extravillous trophoblast cells ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[507,446,922,625]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We next explore the effect of GHET1 on migration, invasion, and network formation of extravillous trophoblasts. The results from wound healing assay showed GHET1 knockdown markedly suppressed HTR-8/SVneo cell migration, whereas GHET1 upregulation enhanced JEG3 cell migration (Figure 3a). In addition, transwell assay results demonstrated that HTR-8/SVneo cell invasion was markedly inhibited by GHET1 silencing, whereas JEG3 cell invasion was markedly increased following GHET1 overexpression (Figure 3b). The results from network formation assay subsequently demonstrated that transfection of si-GHET1 #1 #2 reduced the number of capillary-like networks significantly in HTR-8/SVneo cells, whereas ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[191,656,490,831]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0687/8aeedb9b5ab1d0aa1bbb6d95e28873f1b7c4d49382ec5fdc56ae6b73d35e3f1b.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[512,656,803,856]","code_body":null,"code_caption":null,"image_caption":"[\"FIGURE 1 GHET1 was decreased in the placenta of PE patients. (a) Normal placental tissue $(n = 11)$ and placental tissue from patients diagnosed with PE were collected $(n = 20)$ . qRT-PCR assay was performed to examine GHET1 expression in tissues. (b) qRT-PCR analysis was performed to detect GHET1 expression in HTR-8/SVneo, JEG3, BeWo, and Wish cells compared with HUVEC-C cells. All experiments were performed in biological triplicates with three technical replicates. $^{*}p < 0.05$ , $^{**}p < 0.01$ , $^{***}p < 0.001$ . HUVEC-C, Human umbilical vein endothelial cells; GHET1, gastric carcinoma high expressed transcript 1; PE, pre-eclampsia; qRT-PCR, quantitative real-time polymerase chain reaction. \"]","image_footnote":"[]","img_path":"images/0524/e87597e787828987dcefe4b1cd693c0f48bcb45bae90852278e483bc66633572.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[73,26,97,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"762 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[136,21,280,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Molecular Reproduction ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[174,37,280,51]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Development ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[858,26,922,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"WAN ET AL. 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(a) Transfection of siRNA specifically targeting GHET1 (si-GHET1) and GHET1 overexpressing plasmid (pcDNA-GHET1) was conducted in HTR-8/SVneo and JEG3 cells, respectively. qRT-PCR assay was subjected to evaluate the transfection efficiency. (b) CCK-8 assay was employed to measure extravillous trophoblast cell viability. (c) Colony formation was performed to evaluate extravillous trophoblast cells proliferation. (d) EdU staining assay showed that percentage of EdU-positive cells. (e) Flow cytometry was performed to analyze the cell cycle. All experiments were performed in biological triplicates with three technical replicates. $^{*}p < 0.05$ , $^{**}p < 0.01$ , $^{***}p < 0.001$ . CCK-8, cell counting kit-8; EdU, 5-Ethynyl-2'-deoxyuridine; GHET1, gastric carcinoma high expressed transcript 1; qRT-PCR, quantitative real-time polymerase chain reaction; siRNA, small interfering RNA. \"]","image_footnote":"[]","img_path":"images/0145/f3fd833d560c4b7134648b002d19f9c89811bf914ed9b7ec940d88d3dd21c2b9.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[73,26,134,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"WAN ET AL. 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(a) The capacity of migration in extravillous trophoblast cells was determined using would healing assay. (b) Transwell system was performed to measure the invasion capacity in si-GHET1-treated or pcDNA-GHET1-treated extravillous trophoblast cells. (c) Performing network formation, cells transfected with si-GHET1-treated or pcDNA-GHET1 were detected. All experiments were performed in biological triplicates with three technical replicates. $^{**}p < 0.01$ , $^{***}p < 0.001$ . GHET1, gastric carcinoma high expressed transcript 1. \"]","image_footnote":"[]","img_path":"images/0145/29c2bd68b50658c4ff48deee4ed9bacc5be03e39a270ab46e9cb2450ecdcbaa9.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[70,778,485,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"transfection of pcDNA-GHET1 enhanced the number of capillary-like networks significantly in JEG3 cells (Figure 3c). All these results suggested that GHET1 increased extravillous trophoblast cell migration, invasion, and angiogenesis. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,875,480,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.4 MT2A was a downstream effector of GHET1 ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,911,485,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To further investigate whether there is transcriptional regulation of GHET1 on downstream target genes, we preliminarily detected the ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,778,924,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"expression of several target genes involved in the regulation of cell proliferation and migration. It was found that the change of MT2A was most obvious after interfering GHET1. MT2A expression was markedly increased after the transfection of si-GHET1 in HTR-8/ SVneo and JEG3 cells (Figure 4a). These results revealed that the dysregulated MT2A may be an important regulatory factor of GHET1. As shown in Figure 4b, GHET1 knockdown increased MT2A protein level in HTR-8/SVneo cells, whereas GHET1 overexpression reduced MT2A expression in JEG3 cells. Moreover, it was found that MT2A was increased in the placenta of PE patients ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,26,95,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"764 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[137,21,280,48]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Molecular Reproduction Development ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[858,26,922,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"WAN ET AL. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[147,79,428,225]","code_body":null,"code_caption":null,"image_caption":"[\"(a) \"]","image_footnote":"[]","img_path":"images/0036/8b47c650fb245d414b80288c0bf643a871b2307ef45ddd637e6cb694cf643474.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[452,63,724,225]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0838/7c06e8b67a1f8fac19d0ecdd699d6abbfea29a1ec18253400e19a30196fa3c52.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[137,241,423,384]","code_body":null,"code_caption":null,"image_caption":"[\"(b) \"]","image_footnote":"[]","img_path":"images/0971/33e35100358e7d418ccdad2efe871d8c65c7d82af70bc12f9aace846456acd87.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[405,230,613,411]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0095/d8306a8521a512a4bba8a04f2f2c82fda8a88eb342d682163ebe381f5d912709.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[621,236,860,384]","code_body":null,"code_caption":null,"image_caption":"[\"(c) \"]","image_footnote":"[]","img_path":"images/0942/824f0a36dc20f9ed98dc1d43cabbe9c54a43ed2156df34d852632f4c5b93eb22.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[137,418,505,589]","code_body":null,"code_caption":null,"image_caption":"[\"(d) \"]","image_footnote":"[]","img_path":"images/0836/3a4b86ce205a3d6c436747c7d0b145d179369df4c7800badeadd5172b067cb29.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[521,430,788,572]","code_body":null,"code_caption":null,"image_caption":"[\"(e) \",\"FIGURE 4 MT2A was a downstream effector of GHET1. (a) MT2A, RND3, KLF2, PLK3, and ESM1 expressions in si-GHET1-treated extravillous trophoblast cells were determined using qRT-PCR. (b) Western blot was performed in HTR-8/SVneo and JEG3 cells after the transfection of si-GHET1 and pcDNA-GHET1, respectively. (c) qRT-PCR assay was performed to quantify the relative MT2A expression in placental tissue of PE patients $(n = 20)$ and normal tissue $(n = 11)$ . (d) Western blot assay was performed to quantify the protein level of MT2A in placental tissue of PE patients and normal tissue. (e) Pearson correlation analysis was performed, and the result showed that relative MT2A expression was negatively correlated with relative GHET1 expression. All experiments were performed in biological triplicates with three technical replicates. $^*p < 0.05$ , $^{**}p < 0.01$ . GHET1, gastric carcinoma high expressed transcript 1; MT2A, Metallothionein 2A; qRT-PCR, quantitative real-time polymerase chain reaction. \",\"(Figure 4c,d), which was negatively correlated with GHET1 (Figure 4e). \"]","image_footnote":"[]","img_path":"images/0451/cef16170ab77db6b10312b484bfa3581a2cfe8d3faee1a30f982cfd3b161ddc9.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[70,809,465,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.5 GHET1 suppressed MT2A transcription via recruiting EZH2/LSD1 ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,861,485,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To further investigate the mechanism by which GHET1 regulates MT2A-mediated PE development, we analyzed the binding relationship between GHET1 and polycomb-group protein (PRC)2 complexes (EZH2, LSD1, and SUZ12). As demonstrated by RIP assay, EZH2 and LSD1 could directly bind to GHET1, while there was no binding relationship between ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,745,924,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"SUZ12 and GHET1 (Figure 5a). Consistently, EZH2 and LSD1 were pulled down by labeled GHET1, which further validated the binding relationship between GHET1 and EZH2/LSD1 (Figure 5b). In addition, silencing of EZH2 or LSD1 increased MT2A protein level in extravillous trophoblast cells (Figure 5c). The result of ChIP assay subsequently displayed that knockout of GHET1 reduced the levels of EZH2, LSD1, and H3K27me3 recruited in the MT2A promoter in HTR-8/SVneo cells, while increased the enrichment of H3K4me2 (Figure 5d). Consistently, overexpression of GHET1 increased the enrichment of EZH2, LSD1, H3K27me3 and decreased H3K4me2 enrichment in the MT2A promoter of JEG3 cells (Figure 5d). The above indicated that GHET1 mediated the MT2A promoter region H3K27me3 by recruiting EZH2 and LSD1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,26,134,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"WAN ET AL. 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(a) RIP assay was employed to analyze the interaction between GHET1 and EZH2/LSD1. (b) RNA pull-down was performed to confirm the binding of GHET1 to EZH2 and LSD1. (c) Western blot assay was performed to quantify the protein level of MT2A. (d) The enrichment of EZH2, H3K27me3, LSD1, and H3K4me2 in the MT2A promoter region was detected by ChIP assay. All experiments were performed in biological triplicates with three technical replicates. $^* p < 0.05$ , $^{**}p < 0.01$ , $^{***}p < 0.001$ . ChIP, chromatin immunoprecipitation; EZH2, zeste 2 polycomb repressive complex 2; GHET1, gastric carcinoma high expressed transcript 1; MT2A, Metallothionein 2A; LSD1, lysine-specific demethylase 1. \"]","image_footnote":"[]","img_path":"images/0924/11f9da7075cef2e31daa76e91c91ba4e2f752f08d6285042cb1282ac18bfb531.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[70,643,473,693]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.6 | EZH2 or LSD1 knockdown reversed GHET1 overexpression-mediated biological function in extravillous trophoblast cells ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,712,485,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As shown in Supporting Information: Figure S1A-B, EZH2 and LSD1 were lowly expressed in the placenta of PE pregnancies relative to the normal placental tissues. To investigate the role of EZH2 and LSD1 in GHET1-mediated biological function during PE progression, both GHET1 overexpression and EZH2/LSD1 knockdown were induced in JEG3 cells. It was observed that GHET1 overexpression reduced MT2A expression in JEG3 cells, while this change was reversed by EZH2 knockdown or LSD1 knockdown (Supporting Information: Figure S1C). CCK-8 result subsequently demonstrated that JEG3 cell viability was significantly increased by GHET1 upregulation, which was abolished by EZH2 knockdown or LSD1 knockdown (Supporting Information: Figure S1D). In addition, GHET1 overexpression promoted JEG3 cell proliferation, while this effect was abrogated by EZH2 knockdown or LSD1 knockdown (Supporting ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,644,924,759]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Information: Figure S1E). Moreover, JEG3 cell migration and invasion were markedly facilitated by GHET1 overexpression, whereas these changes were eliminated by EZH2 knockdown or LSD1 knockdown (Supporting Information: Figure S1F-G). All these results suggested that EZH2 knockdown or LSD1 knockdown reversed the promoting effect of GHET1 overexpression on extravillous trophoblast cell proliferation, migration, and invasion. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,792,847,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.7 | MT2A knockdown reversed GHET1 inhibition-mediated biological functions in extravillous trophoblast cells ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[507,861,922,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To further validate that MT2A participates in the GHET1-mediated functional regulation as a downstream target, si-GHET1 and shMT2A were cotransfected into extravillous trophoblast cells. It was observed that sh-MT2A transfection reduced MT2A expression in extravillous trophoblast cells (Figure 6a), suggesting that the ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,26,95,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"766 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[136,21,280,52]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Molecular Reproduction Development ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[858,26,922,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"WAN ET AL. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[137,76,314,191]","code_body":null,"code_caption":null,"image_caption":"[\"(a) \"]","image_footnote":"[]","img_path":"images/0515/795d3c04f8ad39f463efaf6e4979662a8621df875d13da1483cf4471524366bc.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[321,81,527,196]","code_body":null,"code_caption":null,"image_caption":"[\"(b) \"]","image_footnote":"[]","img_path":"images/0664/a3bfae2acc4f4a200c88f2cbbc0e1bbbde2a4f7c4f898806fed8c4c7ac69f3be.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[579,63,737,237]","code_body":null,"code_caption":null,"image_caption":"[\"(c) \"]","image_footnote":"[]","img_path":"images/0348/c2c0bed348ac68f2ab07bd1d7ca5c2bcf640483b7076ffc48223e8ba6db27f4f.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[137,219,505,352]","code_body":null,"code_caption":null,"image_caption":"[\"(d) \"]","image_footnote":"[]","img_path":"images/1005/38eb2dd9439892ca87a0b4aa937a53d7fdeca532b3a1c32c23b29db42ba42fbd.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[532,253,862,324]","code_body":null,"code_caption":null,"image_caption":"[\"(e) \"]","image_footnote":"[]","img_path":"images/0905/a9a4f7d2417b7372d3ba851a8a1f2d482dcee498f23967ff5ec318f014c8c2e2.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[141,383,329,489]","code_body":null,"code_caption":null,"image_caption":"[\"(f) \"]","image_footnote":"[]","img_path":"images/0484/44eebc3ea95a0263a27cf05ac93dd749fafa95ebc7b7626dae055cc07faffafd.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[352,359,517,512]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0390/d46596e6902c91c8f3414bc20934a989204a2bbb4441a43d5c33df58c6b63335.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[549,365,715,512]","code_body":null,"code_caption":null,"image_caption":"[\"FIGURE 6 MT2A knockdown reversed GHET1 inhibition-mediated biological functions in extravillous trophoblasts. (a) MT2A expression in HTR-8/SVneo cells after sh-NC or sh-MT2A transfection was determined by qRT-PCR. (b,c) CCK-8 and colony formation assays were employed to analyze cell viability and proliferation. (d) Wound healing assay was performed to detect cell migration. (e) Tube formation assays were performed to detect tube formation in HTR-8/SVneo cells. (f) Western blot analysis was subjected to assess the alteration of MT2A expression levels. All experiments were performed in biological triplicates with three technical replicates. $^{*}p < 0.05$ , $^{**}p < 0.01$ . CCK-8, Cell Counting Kit-8; GHET1, gastric carcinoma high expressed transcript 1; MT2A, Metallothionein 2A; qRT-PCR, quantitative real-time polymerase chain reaction. \"]","image_footnote":"[]","img_path":"images/0414/a9255b2c47ab649a82ed7844c3a3a55c84084550dea1875dc21ad79308795dba.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[70,645,485,824]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"transfection was successful. As shown in Figure 6b,c, GHET1 knockdown's inhibition effect on cell proliferation was abolished after MT2A knockdown. It was also turned out that GHET1 knockdown decreased cell migration, whereas knockdown of MT2A rescued this inhibiting effect (Figure 6d). In addition, the silencing of MT2A weakened the suppressive efficiency of tube formation induced by the silencing of GHET1 (Figure 6e). Knockdown of GHET1 enhanced the expression of MT2A, whereas this effect was blocked by silencing of MT2A (Figure 6f). Collectively, these results indicated that GHET1 achieved its biological roles in extravillous trophoblast cells by inhibiting MT2A. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,859,238,873]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 | DISCUSSION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,893,485,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As widely described, IncRNAs are closely related to PE progression. For instance, IncRNA uc003fir suppressed trophoblast proliferation and migration in PE by enhancing CCL5 expression ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,645,924,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(Pengjie et al., 2018). Overexpression of PUM1 impaired the invasion of trophoblast cells in PE by inhibiting IncRNA HOTAIR expression (Zhang et al., 2020). Upregulation of IncRNA TCL6 expression promoted the progression of PE via phosphatase and tensin homolog (Wu et al., 2019). Nevertheless, the role of IncRNAs in PE has not been fully clarified, more differential expression IncRNAs need to be further explored and investigated. GHET1, as an oncogene, functions in facilitating cancer cell proliferation, migration, and invasion (Han et al., 2019; Liu et al., 2021). Notably, it has been reported that GHET1 is involved in PE progression, specifically its overexpression-enhanced cell invasion and migration abilities in PE (Li et al., 2019). However, the role of GHET1 in PE has only been reported in one report. Herein, our work revealed downstream regulatory mechanisms by which GHET1 accelerated extravillous trophoblast proliferation, cell cycle, migration, invasion, and network formation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,893,924,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is well known that nearly $20\\%$ human IncRNAs can bind with PRC2 (which contains EZH2, SUZ12, and embryonic ectoderm development [EED]), thus erecting their epigenetic regulation of ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,26,134,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"WAN ET AL. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[715,21,860,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Molecular Reproduction ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[752,37,860,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Development ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[899,26,922,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"767 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[70,63,485,427]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"downstream targets through interacting with RNA-binding proteins, as well as LSD1 (Ding et al., 2017; Huang et al., 2017; Lian et al., 2016). Histone lysine methylation is closely related to gene expression, chromatin remodeling, and cell differentiation, histone lysine methyltransferase (KMTs) and histone lysine demethylase (KDMs) can dynamically regulate this process. LSD1 is the first identified histone lysine-specific demethylase, which is known to catalyze the removal of monomethylation and dimethylation at H3K4 and H3K9 sites, thus inhibiting or activating gene transcription. The main component of PRC2 complex, EZH2, is a methyltransferase of H3K27, which can inhibit gene transcription by methylating H3K27 (Ding et al., 2017; Huang et al., 2017; Lian et al., 2016). To further explore whether GHET1 could mediate the transcriptional regulation of downstream targets, we conducted a serious of experiments in this study. In this study, it was turned out that GHET1 directly bound to EZH2 and LSD1, and the knockdown of EZH2 and LSD1 induced the upregulation of MT2A. Furthermore, the ChIP assay demonstrated that GHET1 mediated H3K27me3 and H3K4me2 in the MT2A promoter region by recruiting EZH2 and LSD1. To sum up, we concluded that GHET1 inhibited the transcriptional level of MT2A by regulating EZH2/LSD1-mediated H3K27me3 levels in the MT2A promoter region. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,429,485,826]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Metallothioneins (MTs) are a family of metal-binding proteins that widely exist in organisms (Hui, Xu, et al., 2019). There are four subfamilies in human MT, namely MT1, MT2A, MT3, and MT4 respectively (Mangelinck et al., 2019). MT2A is one of the major MT subfamily, which are highly conserved and exists in almost all types of soft tissues (Ma et al., 2015). MTs are involved in different kinds of cellular functions, such as cell differentiation, proliferation, and apoptosis. MTs dysregulation have also been linked to tumor progression (Hui, Xu, et al., 2019; Ma et al., 2015; Mangelinck et al., 2019). For instance, upregulation of IncRNA taurine up-regulated gene 1 regulated pancreatic cancer cell malignant behaviors via RND3 and MT2A (Hui, Xu, et al., 2019). In addition, it was previously illustrated that HMBOX 1 could regulate vascular endothelial cell autophagy and apoptosis by interacting with MT2A (Ma et al., 2015). MT2A is considered as a biomarker for stress-related diseases/disorders including hypertension (Sekovanic et al., 2020). Notably, it is revealed by Wei et al., the genetic variation of rs10636 G locus of MT2A was associated with the susceptibility of hypertensive disorders of pregnancy, especially PE (Wei et al., 2022). Nevertheless, the function of MT2A in regulating extravillous trophoblastic phenotype during PE progression remains unclear. Herein, we demonstrated that GHET1 silencing inhibited proliferation, migration, invasion, and angiogenesis of cells, whereas silencing of MT2A rescued this inhibition. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,828,485,925]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Taken together, GHET1 epigenetically suppressed MT2A expression by recruiting EZH2/LSD1, thus promoting the proliferation, migration, invasion, and angiogenesis of extravillous trophoblasts in PE (Figure 7). Our findings may provide innovative research findings and therapeutic targets for PE treatment. However, there are some limitations in our research, such as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[515,63,922,300]","code_body":null,"code_caption":null,"image_caption":"[\"FIGURE 7 Graphical abstract of GHET1 regulated extravillous trophoblastic phenotype via EZH2/LSD1-mediated MT2A epigenetic suppression in pre-eclampsia. EZH2, zeste 2 polycomb repressive complex 2; GHET1, gastric carcinoma high expressed transcript 1; LSD1, lysine-specific demethylase 1; MT2A, metallothionein 2A. \"]","image_footnote":"[]","img_path":"images/0281/0395cc4c35289deb3d5c60abce110d6f308cb924a4acec90e8f9855ce4ab4a4a.jpg","list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[507,411,924,492]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"the lack of verification experiments in mouse models. Besides, whether there are other functional downstream targets remains unclear. In addition, more PE-related IncRNAs and the various molecular mechanisms of PE need to be further investigated for the prevention and treatment of PE. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,512,714,524]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"AUTHOR CONTRIBUTIONS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[507,529,924,643]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Pengyun Wan: Conceptualization; methodology; validation; investigation; visualization; writing—original draft; formal analysis. Jia Huang: Software. Wenting Liu: Data curation. Xiaoyan Su: Resources. Bei Zhao: Formal analysis. Xianggang Wang: Methodology. Lu Zhao: Conceptualization; funding acquisition; writing—original draft; writing—review & editing; project administration; supervision. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,662,680,673]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ACKNOWLEDGMENTS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,679,922,709]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The authors would like to give our sincere gratitude to the reviewers for their constructive comments. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,728,794,741]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CONFLICT OF INTEREST STATEMENT ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,745,773,758]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The authors declare no conflict of interest. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,778,766,790]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DATA AVAILABILITY STATEMENT ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[507,795,922,824]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"All data generated or analyzed during this study are included in this published article. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,845,668,856]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ETHICS STATEMENT ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[507,861,924,925]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The study was authorized by Second Affiliated Hospital of Nanchang University Ethics Committee. All patients who met the inclusion criteria provided written informed consent. The informed consent obtained from study participants. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,26,97,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"768 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[137,21,280,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Molecular Reproduction ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[173,37,280,51]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Development ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[858,26,922,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"WAN ET AL. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[73,63,171,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"REFERENCES ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[73,81,485,925]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"Bartsch, E., Medcalf, K. E., Park, A. L., & Ray, J. G. (2016). Clinical risk factors for pre-eclampsia determined in early pregnancy: Systematic review and meta-analysis of large cohort studies. BMJ, 353, i1753. \",\"Ding, J., Xie, M., Lian, Y., Zhu, Y., Peng, P., Wang, J., Wang, L., & Wang, K. (2017). Long noncoding RNA HOXA-AS2 represses P21 and KLF2 expression transcription by binding with EZH2, LSD1 in colorectal cancer. Oncogenesis, 6(1), e288. \",\"Gong, F., Cheng, H., Shi, Y., Cui, L., & Jia, G. (2020). LncRNA TDRG1/miR214-5p axis affects preeclampsia by modulating trophoblast cells. Cell Biochemistry and Function, 38(4), 352-361. \",\"Gyselaers, W., Lees, C., Valensise, H., & Thilaganathan, B. (2020). Preeclampsia: the role of persistent endothelial cells in uteroplacental arteries. Brosens I, Brosens JJ, Muter J, Puttemans P, Benagiano G. Am J Obstet Gynecol 2019;221:219-26. Am J Obstet Gynecol, 6, 633. \",\"Han, M., Wang, Y., Gu, Y., Ge, X., Seng, J., Guo, G., Zhang, X., Zhao, Y., & Dou, D. (2019). IncRNA GHET1 knockdown suppresses breast cancer activity in vitro and in vivo. American Journal of Translational Research, 11(1), 31-44. \",\"Hao, J., Hassen, D., Hao, Q., Graham, J., Paglia, M. J., Brown, J., Cooper, M., Schlieder, V., & Snyder, S. R. (2019). Maternal and infant health care costs related to preeclampsia. Obstetrics & Gynecology, 134(6), 1227-1233. \",\"He, T., Qiao, Y., Lv, Y., Wang, J., Hu, R., & Cao, Y. (2019). IncRNA FAM99A is downregulated in preeclampsia and exerts a regulatory effect on trophoblast cell invasion, migration and apoptosis. Molecular Medicine Reports, 20(2), 1451-1458. \",\"Huang, M., Hou, J., Wang, Y., Xie, M., Wei, C., Nie, F., Wang, Z., & Sun, M. (2017). Long noncoding RNA LINC00673 is activated by SP1 and exerts oncogenic properties by interacting with LSD1 and EZH2 in gastric cancer. Molecular Therapy, 25(4), 1014-1026. \",\"Hui, B., Ji, H., Xu, Y., Wang, J., Ma, Z., Zhang, C., Wang, K., & Zhou, Y. (2019). RREB1-induced upregulation of the IncRNA AGAP2-AS1 regulates the proliferation and migration of pancreatic cancer partly through suppressing ANKRD1 and ANGPTL4. Cell Death & Disease, 10(3), 207. \",\"Hui, B., Xu, Y., Zhao, B., Ji, H., Xu, S., Ma, Z., He, Z. Y., Wang, K., & Lu, J. (2019). Overexpressed long noncoding RNA TUG1 affects the cell cycle, proliferation, and apoptosis of pancreatic cancer partly through suppressing RND3 and MT2A. OncoTargets and Therapy, 12, 1043-1057. \",\"Jiang, J., & Zhao, Z. M. (2018). LncRNA HOXD-AS1 promotes preeclampsia progression via MAPK pathway. European Review for Medical and Pharmacological Sciences, 22(24), 8561-8568. \",\"Li, J. L., Li, R., Gao, Y., Guo, W. C., Shi, P. X., & Li, M. (2018). LncRNA CCAT1 promotes the progression of preeclampsia by regulating CDK4. European Review for Medical and Pharmacological Sciences, 22(5), 1216-1223. \",\"Li, S., Liu, J., Kong, F., Wang, Y., Li, N., & Zou, Y. (2019). IncRNA GHET1 has effects in development of pre-eclampsia. Journal of Cellular Biochemistry, 120(8), 12647-12652. \",\"Lian, Y., Wang, J., Feng, J., Ding, J., Ma, Z., Li, J., Peng, P., De, W., & Wang, K. (2016). Long non-coding RNA IRAIN suppresses apoptosis and promotes proliferation by binding to LSD1 and EZH2 in pancreatic cancer. *Tumor Biology*, 37(11), 14929-14937. \",\"Liu, Y., Shi, P., Fang, H., Zhao, Z., Yang, F., Zhang, J., Jing, S., & Geng, C. (2021). Long non-coding RNA GHET1 promotes thyroid cancer cell proliferation and invasion. Translational Cancer Research, 10(9), 4148-4157. \",\"Long, J., Xiong, J., Bai, Y., Mao, J., Lin, J., Xu, W., Zhang, H., Chen, S., & Zhao, H. (2019). Construction and investigation of a \"]","page_idx":"11","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[512,65,924,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"lncRNA-associated ceRNA regulatory network in cholangiocarcinoma. Frontiers in Oncology, 9, 649. \",\"Ma, H., Su, L., Yue, H., Yin, X., Zhao, J., Zhang, S., Kung, H., Xu, Z., & Miao, J. (2015). HMBOX 1 interacts with MT2A to regulate autophagy and apoptosis in vascular endothelial cells. Scientific Reports, 5, 15121. \",\"Mangelinck, A., da Costa, M. E. M., Stefanovska, B., Bawa, O., Polrot, M., Gaspar, N., & Fromigue, O. (2019). MT2A is an early predictive biomarker of response to chemotherapy and a potential therapeutic target in osteosarcoma. Scientific Reports, 9(1), 12301. \",\"Mol, B. W. J., Roberts, C. T., Thangaratinam, S., Magee, L. A., de Groot, C. J. M., & Hofmeyr, G. J. (2016). Pre-eclampsia. The Lancet, 387(10022), 999-1011. \",\"Moufarrej, M. N., Vorperian, S. K., Wong, R. J., Campos, A. A., Quaintance, C. C., Sit, R. V., Tan, M., Detweiler, A. M., Mekonen, H., Neff, N. F., Baruch-Gravett, C., Litch, J. A., Druzin, M. L., Winn, V. D., Shaw, G. M., Stevenson, D. K., & Quake, S. R. (2022). Early prediction of preeclampsia in pregnancy with cell-free RNA. Nature, 602(7898), 689-694. \",\"Pengjie, Z., Xionghui, C., Yueming, Z., Ting, X., Na, L., Jianying, T., & Zhice, X. (2018). LncRNA uc003fir promotes CCL5 expression and negatively affects proliferation and migration of trophoblast cells in preeclampsia. *Pregnancy Hypertension*, 14, 90-96. \",\"Ridder, A., Giorgione, V., Khalil, A., & Thilaganathan, B. (2019). Preeclampsia: the relationship between uterine artery blood flow and trophoblast function. International Journal of Molecular Sciences, 20(13), 3263. \",\"Sekovanic, A., Jurasovic, J., & Piasek, M. (2020). Metallothionein 2A gene polymorphisms in relation to diseases and trace element levels in humans. Archives of Industrial Hygiene and Toxicology, 71(1), 27-47. \",\"Tao, Y., Tang, Y., Yang, Z., Wu, F., Wang, L., Yang, L., Lei, L., Jing, Y., Jiang, X., Jin, H., Bai, Y., & Zhang, L. (2020). Exploration of serum exosomal LncRNA TBILA and AGAP2-AS1 as promising biomarkers for diagnosis of Non-Small cell lung cancer. International Journal of Biological Sciences, 16(3), 471-482. \",\"Wei, S., Yu, X., Wen, X., Zhang, M., Lang, Q., Zhong, P., & Huang, B. (2022). Genetic variations in metallothionein genes and susceptibility to hypertensive disorders of pregnancy: A case-control study. Frontiers in Genetics, 13, 830446. \",\"Wu, J., Huang, H., Huang, W., Wang, L., Xia, X., & Fang, X. (2020). Analysis of exosomal lncRNA, miRNA and mRNA expression profiles and ceRNA network construction in endometriosis. Epigenomics, 12, 1193-1213. \",\"Wu, J. L., Wang, Y. G., Gao, G. M., Feng, L., Guo, N., & Zhang, C. X. (2019). Overexpression of IncRNA TCL6 promotes preeclampsia progression by regulating PTEN. European Review for Medical and Pharmacological Sciences, 23(10), 4066-4072. \",\"Xu, Y., Xia, X., Jiang, Y., Wu, D., Wang, S., Fu, S., Yang, N., Zhang, Y., & Sun, L. (2020). Down-regulated IncRNA AGAP2-AS1 contributes to pre-eclampsia as a competing endogenous RNA for JDP2 by impairing trophoblastic phenotype. Journal of Cellular and Molecular Medicine, 24(8), 4557-4568. \",\"Yang, W., Shan, Z., Zhou, X., Peng, L., Zhi, C., Chai, J., Liu, H., Yang, J., & Zhang, Z. (2018). Knockdown of IncRNA GHET1 inhibits osteosarcoma cells proliferation, invasion, migration and EMT in vitro and in vivo. Cancer Biomarkers: Section A of Disease markers, 23(4), 589-601. \",\"Zhang, Y., He, X. Y., Qin, S., Mo, H. Q., Li, X., Wu, F., Zhang, J., Li, X., Mao, L., Peng, Y. Q., Guo, Y. N., Lin, Y., & Tian, F. J. (2020). Upregulation of PUM1 expression in preeclampsia impairs trophoblast invasion by negatively regulating the expression of the IncRNA HOTAIR. Molecular Therapy, 28(2), 631-641. \"]","page_idx":"11","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[73,26,134,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"WAN ET AL. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[715,21,860,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Molecular Reproduction ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[752,37,858,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Development ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[899,26,922,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"769 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[72,63,487,118]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Zhou, J., Li, X., Wu, M., Lin, C., Guo, Y., & Tian, B. (2016). Knockdown of long noncoding RNA GHET1 inhibits cell proliferation and invasion of colorectal cancer. Oncology Research Featuring Preclinical and Clinical Cancer Therapeutics, 23(6), 303-309. ","text_format":null,"text_level":null,"type":"ref_text"},{"bbox":"[72,147,292,159]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"SUPPORTING INFORMATION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[72,164,485,194]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Additional supporting information can be found online in the Supporting Information section at the end of this article. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[532,79,892,176]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"How to cite this article: Wan, P., Huang, J., Liu, W., Su, X., Zhao, B., Wang, X., & Zhao, L. (2023). IncRNA GHET1 regulates extravillous trophoblastic phenotype via EZH2/LSD1-mediated MT2A epigenetic suppression in pre-eclampsia. Molecular Reproduction and Development, 90, 758-770. https://doi.org/10.1002/mrd.23693 ","text_format":null,"text_level":null,"type":"ref_text"},{"bbox":"[73,26,97,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"770 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[137,21,280,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Molecular Reproduction ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[174,37,280,52]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Development ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[858,26,922,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"WAN ET AL. ","text_format":null,"text_level":null,"type":"header"}],"string":"[\n {\n \"bbox\": \"[70,134,816,221]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"lncRNA GHET1 regulates extravillous trophoblastic phenotype via EZH2/LSD1-mediated MT2A epigenetic suppression in pre-eclampsia \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[68,248,840,292]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Pengyun Wan $^{1}$ | Jia Huang $^{2}$ | Wenting Liu $^{1}$ | Xiaoyan Su $^{3}$ | Bei Zhao $^{4}$ | Xianggang Wang $^{5}$ | Lu Zhao $^{1}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,320,317,571]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"$^{1}$ Department of Obstetrics and Gynecology, Second Affiliated Hospital of Nanchang University, Nanchang, Jiangxi Province, People's Republic of China \\\",\\\"$^{2}$ Reproductive Health Department, Jiangxi Provincial Maternal and Child Health Hospital, Nanchang, Jiangxi Province, People's Republic of China \\\",\\\"<sup>3</sup>Pathology Department, Second Affiliated Hospital of Nanchang University, Nanchang, Jiangxi Province, People's Republic of China \\\",\\\"$^{4}$ Traditional Chinese Medicine Department, Duchang County People's Hospital, Jiujiang, Jiangxi Province, People's Republic of China \\\",\\\"Medical Examination Department, Second Affiliated Hospital of Nanchang University, Nanchang, Jiangxi Province, People's Republic of China \\\"]\",\n \"page_idx\": \"0\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[70,588,164,599]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Correspondence \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,602,314,664]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lu Zhao, Department of Obstetrics and Gynecology, Second Affiliated Hospital of Nanchang University, No.1, Minde Rd, Nanchang 330006, Jiangxi Province, People's Republic of China. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[72,666,240,677]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Email: ndefy11068@ncu.edu.cn \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[347,331,418,346]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[344,351,915,750]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Pre-eclampsia (PE) is usually defined as new-onset hypertension with albuminuria or other organ damage. Herein, the role and mechanism of long noncoding RNA (lncRNA) gastric carcinoma high expressed transcript 1 (GHET1) during PE are investigated. Expression of GHET1 in PE pregnancies was evaluated using quantitative real-time polymerase chain reaction (qRT-PCR). Proliferation and cell cycle of extravillous trophoblasts were assessed by Cell Counting Kit-8 (CCK-8), colony formation, 5-Ethynyl-2'-deoxyuridine (EdU) assays, and flow cytometry, respectively. Migration, invasion, and network formation of trophoblasts were measured by wound healing, transwell system, and tube formation assays. RNA immunoprecipitation (RIP), RNA pull-down, and chromatin immunoprecipitation (ChIP) assays were used to confirm the molecular interaction. GHET1 was markedly decreased in the placenta of PE patients. GHET1 promoted the proliferation and cell cycle of extravillous trophoblasts, as well as migration, invasion, and network formation in vitro. Metallothionein 2A (MT2A) functioned as a downstream effector of GHET1, which was negatively correlated with GHET1 in PE. GHET1 directly bound with zeste 2 polycomb repressive complex 2/lysine-specific demethylase 1 (EZH2/LSD1). Knockdown of GHET1 reduced the occupancies of H3K27me3 and H3K4me2 in the MT2A promoter region by recruiting EZH2 and LSD1. MT2A knockdown reversed GHET1 inhibition mediated biological functions. GHET1 regulates extravillous trophoblastic phenotype via EZH2/LSD1-mediated MT2A epigenetic suppression in PE. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[347,768,438,780]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"KEYWORDS \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[347,783,665,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"EZH2/LSD1, GHET1, IncRNA, MT2A, Pre-eclampsia \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,883,919,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abbreviations: BCA, Bicinchoninic acid; CCAT1, Colon cancer-associated transcript 1; CCK-8, cell counting kit-8; ChIP, chromatin immunoprecipitation; EdU, 5-Ethynyl-2'-deoxyuridine; EED, embryonic ectoderm development; EVTs, extra villous trophoblasts; EZH2, zeste 2 polycomb repressive complex 2; FITC, Fluorescein isotiocyanate; GAPDH, Glyceraldehyde-3-phosphate dehydrogenase; IncRNA, long noncoding RNA; LSD1, lysine specific demethylase1; MT2A, metallothionein 2A; MTs, metallothioneins; pcDNA, pcDNA 3.1; PE, Pre-eclampsia; PFA, paraformaldehyde; PRC, polycomb-group protein; PTEN, phosphatase and tensin homolog; qRT-PCR, quantitative real-time polymerase chain reaction; RIP, RNA immunoprecipitation; SUZ12, suppressor of zeste 12; TUG1, taurine up-regulated gene 1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[872,0,895,16]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"# \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[899,3,998,12]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[72,34,206,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Received: 9 March 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[228,34,354,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Accepted: 6 June 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[72,52,211,63]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DOI: 10.1002/mrd.23693 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[778,58,922,74]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Molecular Reproduction \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[813,74,922,92]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Development \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[70,84,263,99]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RESEARCH ARTICLE \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[73,955,97,965]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"758 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[134,953,300,966]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"© 2023 Wiley Periodicals LLC. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[405,955,591,966]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"wileyonlinelibrary.com/journal/mrd \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[734,955,922,966]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Mol Reprod Dev. 2023;90:758-770. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[72,61,273,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 | INTRODUCTION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,97,485,359]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Pre-eclampsia (PE), a multisystem progressive disorder unique to pregnancy, is characterized by elevated blood pressure after 20 weeks of gestation, accompanied by albuminuria or organ damage with high maternal and infant morbidity and mortality (Hao et al., 2019; Moufarrej et al., 2022), affecting $3\\\\% -5\\\\%$ of pregnancies in the world (Mol et al., 2016). There are several factors that resulted in PE, including a history of PE, chronic hypertension, pregestational diabetes mellitus, antiphospholipid syndrome, and obesity, among others (Bartsch et al., 2016). Recently, the dysfunction of placental trophoblast cells was considered as the key factor to trigger PE (Gyselaers et al., 2019). It has long been believed that inadequate transformation of the maternal uterine vasculature is a consequence of primary defective extravillous trophoblast cell invasion and leads to the development of PE (Ridder et al., 2019). The abnormal ability of migration and invasion of trophoblasts induce obstacles of spiral artery remodeling, thus driving PE progression (Xu et al., 2020). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,363,485,609]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Long noncoding RNA (lncRNA) refers to noncoding RNA with a length greater than 200 nts (Wu et al., 2020) with various regulatory functions (Long et al., 2019). It has been widely reported that lncRNA is closely related to human diseases, including PE (Gong et al., 2020; He et al., 2019; Jiang & Zhao, 2018; Li et al., 2018). As proof, Li et al. revealed that lncRNA colon cancer-associated transcript 1 was markedly upregulated in PE patients, and its knockdown markedly increased trophoblast proliferation (Li et al., 2018). In addition, lncRNA uc003fir was reported to be upregulated in placental tissues of PE patients, and it could negatively regulate trophoblast proliferation (Pengjie et al., 2018). However, the studies on the role of lncRNAs in PE are still limited. This indicates that the study of PE-related lncRNAs and their molecular mechanisms is of great significance for the understanding of the pathogenesis of PE and the improvement of treatment status. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,612,485,907]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Gastric carcinoma highly expressed transcript 1 (GHET1) is a lncRNA that is located on chromosome 12q14.1 with the length about 1567 nts (Hui, Ji, et al., 2019; Tao et al., 2020). It has been widely demonstrated that GHET1 functions in regulating various cellular processes, such as cell proliferation, migration, and invasion (Yang et al., 2018; Zhou et al., 2016). For instance, Yang et al. revealed that GHET1 knockdown inhibited osteosarcoma cell proliferation, invasion, and migration (Yang et al., 2018). Importantly, it was previously described that GHET1 overexpression could restore hypoxia-induced decrease in cell invasion and migration abilities in PE (Li et al., 2019), revealing the important roles of GHET1 in PE. Nevertheless, the role and underlying mechanism of GHET1 in PE needs to be further investigated. As illustrated, lncRNAs epigenetically regulate the expression regulation of downstream target genes by binding to the enhancer of zeste 2 polycomb repressive complex 2 (EZH2) and lysine-specific demethylase1 (LSD1) (Ding et al., 2017; Huang et al., 2017; Lian et al., 2016). Herein, our work aims to explore whether GHET1 affects PE progress by combining with EZH2 and LSD1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,911,485,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Herein, we draw the conclusion that GHET1 regulated the extravillous trophoblastic phenotype via EZH2/LSD1-mediated \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,63,924,143]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"metallothionein 2A (MT2A) epigenetic suppression in PE. We revealed the essential role and epigenetic regulatory network of GHET1 in the development of PE. These findings might help to deepen our understanding of PE pathogenesis and provide useful ideas for PE treatment. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,177,823,193]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 MATERIALS AND METHODS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,212,815,227]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sample collection and ethics statement \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,246,926,475]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Thirty-one pregnant women with intact membranes, no clinical evidence of intrauterine infection, singleton pregnancy, and who delivered by cesarean section (C-section) with no evidence of active labor were included in the study and divided into two study groups: normal group $(n = 11)$ constituted by women with uncomplicated pregnancies at term who delivered normal weight newborns and PE group $(n = 20)$ . Placental tissues were collected at Second Affiliated Hospital of Nanchang University. All tissues were stored at $-80^{\\\\circ}\\\\mathrm{C}$ . The study was authorized by Second Affiliated Hospital of Nanchang University Ethics Committee. All patients who met the inclusion criteria provided written informed consent. Disease diagnosis was performed according to the 2018 ACOG criteria. Clinical and anthropometric data from the study groups are represented in Table 1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,510,665,525]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.1 | Cell culture \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,546,924,643]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Human umbilical vein endothelial cells (HUVEC-C) and human placenta cells (HTR-8/SVneo, JEG3, BeWo, and Wish cells), obtained from the ATCC, were cultured in RPMI 1640 (REF: C11875500B, LOT: 8118016) containing $10\\\\%$ fetal bovine serum (FBS) (FBS, Invitrogen), $100\\\\mathrm{U / mL}$ penicillin and $100\\\\mathrm{U / mL}$ streptomycin (Invitrogen) at $37^{\\\\circ}\\\\mathrm{C}$ with $5\\\\%$ $\\\\mathrm{CO}_{2}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,677,702,691]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2 Cell transfection \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,712,924,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Small interfering negative control (si-NC), si-GHET1, si-EZH2, si-LSD1, pcDNA (pcDNA 3.1)-GHET1, as well as short-hairpin negative control (sh-NC) and sh-MT2A were purchased from GenePharma. When cell fusion reached $80\\\\%$ , cells were transfected with the above plasmids by using Lipofectamine 2000 (Invitrogen) following the protocols of manufacturer. After $48\\\\mathrm{h}$ , the transfection efficiency was identified using quantitative real-time polymerase chain reaction (qRT-PCR). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,875,828,892]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.3 | Cell counting kit-8 (CCK-8) assay \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,911,924,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"During the logarithmic growth phase, HTR-8/SVneo and JEG3 cells were planted to 96-well plates $(2\\\\times 10^{3}$ cells/well) and cultured for \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,26,137,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"WAN ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[715,21,862,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Molecular Reproduction \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[752,37,860,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Development \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[899,26,922,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"759 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[73,65,643,253]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0633/4394c4798ccd49fbec9a90eda063c4aa25e4d07ab7aa7a59cc560c76dd6d2d14.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Characteristics</td><td>Normal (n = 11)</td><td>PE (n = 20)</td><td>p value</td></tr><tr><td>Maternal age (years)</td><td>28.97 ± 5.15</td><td>30.85 ± 5.66</td><td>>0.05</td></tr><tr><td>Maternal weight (kg)</td><td>78.99 ± 5.98</td><td>75.56 ± 6.71</td><td>>0.05</td></tr><tr><td>Fetal birth weight (g)</td><td>3586.45 ± 506.88</td><td>2856.52 ± 501.12</td><td><0.001</td></tr><tr><td>Smoking</td><td>0</td><td>0</td><td>>0.05</td></tr><tr><td>Systolic blood pressure (mmHg)</td><td>112.89 ± 8.15</td><td>148.52 ± 10.15</td><td><0.001</td></tr><tr><td>Diastolic blood pressure (mmHg)</td><td>76.65 ± 8.52</td><td>108.56 ± 9.13</td><td><0.001</td></tr><tr><td>Proteinuria (g/day)</td><td><0.3</td><td>>0.3</td><td><0.01</td></tr><tr><td>Gestational age (week)</td><td>38.05 ± 1.33</td><td>34.21 ± 3.12</td><td><0.01</td></tr></table>\",\n \"table_caption\": \"[\\\"TABLE 1 Clinical features of PE and normal pregnant women. \\\"]\",\n \"table_footnote\": \"[\\\"Note: Data were presented as mean ± SD (Student's t test). \\\",\\\"Abbreviation: PE, pre-eclamncia. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[70,329,485,393]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"24 h. Cells were subsequently incubated with $10\\\\mu \\\\mathrm{L}$ of CCK-8 solution (Cat: 96992, Sigma-Aldrich) at $37^{\\\\circ}C$ for $2h$ . Absorbance was examined at $450nm$ with a microplate spectrophotometer (Bioteke). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,428,317,443]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.4 Colony formation assay \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,462,485,576]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cells were cultured in a six-well plate with 500 cells/well for 2 weeks. After that, cells were washed twice in phosphate-buffered saline (PBS), fixed in the $4\\\\%$ paraformaldehyde $(4\\\\%$ PFA) for $15\\\\mathrm{min}$ , and stained in the Giemsa solution for $10\\\\mathrm{min}$ . The colonies were dried and observed under the microscope following the removal of Giemsa solution. The colonies (a diameter $\\\\geq 100\\\\mu \\\\mathrm{m}$ ) were counted in triplicate assays. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,609,428,626]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.5 | 5-Ethynyl- $2^{\\\\prime}$ -deoxyuridine (EdU) assay \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,644,485,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cells were inoculated in a 24-well plate with $4 \\\\times 10^{4}$ cells/well. EdU solution (10 $\\\\mu$ M, RiboBio) was added to the cells and cultured for 2 h at $37^{\\\\circ} \\\\mathrm{C}$ . Subsequently, cells were dyed with EdU labeling/detection kit (RiboBio) following the instructions of the manufacturer. After fixation using $4 \\\\%$ formaldehyde for $20 \\\\mathrm{~min}$ , the EdU-stained cells were examined. Next, the cells were treated with Hoechst 33342 (Sangon) for $20 \\\\mathrm{~min}$ , and the results were visualized under a fluorescence microscope (Olympus). The percentage of EdU-positive cells was calculated as EdU-positive cell number/Hoechst-positive cell number. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,843,258,859]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.6 | Flow cytometry \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,877,485,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cells $(1 \\\\times 10^{6}$ cells/mL) were centrifuged at $500g$ for $5\\\\mathrm{min}$ after two washes in binding buffer. Next, the cells were incubated in the dark in $100\\\\mu L$ of $1\\\\times$ binding buffer supplemented with $5\\\\mu L$ of Annexin V-Fluorescein isotiocyanate and $5\\\\mu L$ of propidium iodide (PI) at room \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,329,922,377]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"temperature for $15 \\\\mathrm{~min}$ and subjected to flow cytometry (BD Biosciences). Early-stage (Annexin V +/PI-) and late-stage (Annexin V +/PI+) apoptosis rates were calculated. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,411,739,428]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.7 Wound healing assay \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,446,924,576]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Wound healing assays were performed utilizing Wound Healing Assay Kit (ab242285, Abcam) following the instructions of the manufacturer. In brief, when cells reached $80\\\\%$ confluence, cell suspension was added to the well with insert in place and then incubated for $48\\\\mathrm{h}$ . Cells were cultured in the well with insert in place until a monolayer form, then the insert was removed to generate a $0.9\\\\mathrm{mm}$ wound field. After fixing, the ability of cell migration was determined by microscopic imaging (Olympus). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,609,695,626]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.8 | Transwell assay \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,645,924,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Transwell chambers (24-well, $8\\\\mu \\\\mathrm{m}$ pore size, $6.5\\\\mathrm{mm}$ diameter; Corning) were used to detect the invasion ability of cells. The upper chamber was precoated with Matrigel (Corning) at a 1:8 ratio. The lower chamber of the experiment device was filled with $500\\\\mu \\\\mathrm{L}$ RPMI Medium $1640\\\\%$ and $10\\\\%$ FBS. Cells $(200\\\\mu \\\\mathrm{L}, 5000$ cells per well) were collected, resuspended in medium without serum, and then shifted to the hydrated matrigel chambers for $24\\\\mathrm{h}$ . Cells on the upper surface were scraped, whereas the invasive cells on the lower surface were fixed and colored with $0.1\\\\%$ crystal violet for half an hour. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,843,769,859]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.9 Network formation assay \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,878,922,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To investigate the network formation of cells, we performed an in vitro culture system. Transwell cell culture inserts (12 mm diameter, 0.4 mM pore size; Corning) were coated with $100\\\\mu \\\\mathrm{L}$ of the growth factor reduced Matrigel (BD Biosciences) and allowed to set at $37^{\\\\circ}C$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,26,97,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"760 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[136,21,280,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Molecular Reproduction \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[858,26,922,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"WAN ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[174,37,280,52]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Development \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[70,63,485,194]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for at least $1 \\\\mathrm{~h}$ . HTR-8/SVneo and JEG3 cells were cultured in the upper layer of the chamber with a complete medium ( $2 \\\\times 10^{3}$ cells/ well) for $24 \\\\mathrm{~h}$ . After that, the chamber was washed with PBS for at least three times to remove the cells that did not form a network. Subsequently, cells were fixed with methanol, and then stained with $1 \\\\%$ crystal violet staining solution (Sigma-aldrich). Cells were washed twice using sterile water and were imaging using an inverted microscope (Olympus). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,228,453,260]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.10 | Quantitative real-time polymerase chain reaction (qRT-PCR) \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,278,485,411]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Total RNA was extracted with TRIzol (Sigma-Aldrich), chloroform (Sigma-Aldrich), and isopropanol (Sigma-Aldrich) following the instructions of the manufacturer. Subsequently, $200\\\\mathrm{ng}$ extracted RNA was reverse-transcribed into complementary DNAs (cDNAs) by ReverTra Ace qPCR RT Kit (Toyobo) and subjected to qRT-PCR assay with SYBR (Invitrogen) in an ABI 7500 system. The data were analyzed with $2^{-\\\\Delta \\\\Delta G_{T}}$ method. GAPDH was used as the reference gene. The primers were listed as follows $(5^{\\\\prime} - 3^{\\\\prime})$ : \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[90,428,356,443]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"GHET1 (F): CAACAAAGCAGGTAAACATTGG \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,446,342,459]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"GHET1 (R): GCAAAGGCAGAGTGAAAGGT \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,462,327,475]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"MT2A (F): ATGGATCCCAACTGCTCCTG \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,480,327,492]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"MT2A (R): AGCAGCAGCTTTTCTTGCAG \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,496,319,508]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RND3 (F): GCTCCATGTCTTCGCCAAG \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,512,344,525]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RND3 (R):AAAACTGGCCGTGTAATTCTCA \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,529,312,540]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"KLF2 (F): TTCGGTCTCTTCGACGACG \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,546,329,558]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"KLF2 (R): TGCGAACTCTTGGTGTAGGTC \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,563,315,575]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLK3 (F): AGCGCCTACGCTGTCAAAG \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,579,334,592]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLK3 (R): CTCAAAGTGGTGCGAAAAACG \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,595,321,608]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ESM1 (F): GCCTGAGACTGTGCGGTAG \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,612,322,625]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ESM1 (R): CTGGGAAACATGAAGAGCG \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,629,349,641]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"GAPDH (F): GGAGCGAGATCCCTCCAAAAT \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,645,359,658]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"GAPDH (R): GGCTGTTGTCATACTTCTCATGG \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,693,248,708]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.11 Western blot \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,728,485,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Total protein was first extracted using RIPA Lysis and Extraction Buffer (Thermofisher Scientific). Bicinchoninic acid (BCA) kit (Beyotime) was employed to quantify the protein concentration. Subsequently, $20\\\\mu \\\\mathrm{g}$ protein samples were separated by $10\\\\%$ sodium dodecyl sulfate-polyacrylamide gel electrophoresis (SDS-PAGE) gel electrophoresis, transferred to polyvinylidene fluoride (PVDF) membranes (Millipore). The membranes were sealed with $5\\\\%$ skim milk for $1\\\\mathrm{h}$ at room temperature and incubated overnight with antibodies against MT2A (Abcam, 1:1000, ab12228) and GAPDH (Abcam, 1:5000, ab8245) at $4^{\\\\circ}\\\\mathrm{C}$ , then hybridized with secondary antibodies (Abcam, 1:2000, ab6728) for $2\\\\mathrm{h}$ at $37^{\\\\circ}\\\\mathrm{C}$ . Blots were visualized on an Odyssey Infrared Imaging System (LI-COR Biosciences) and subsequently analyzed with ImageJ software. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,62,826,78]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.12 RNA immunoprecipitation (RIP) \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,97,924,310]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RIP was carried out by utilizing a Magna $\\\\mathsf{RIP}^{\\\\mathsf{TM}}$ RNA-Binding Protein Immunoprecipitation Kit (17-700, Sigma-Aldrich). Briefly, $1\\\\times 10^{7}$ cells were lysed with RIP lysis buffer, and the lysates were incubated with indicated antibodies including anti-enhancer of zeste homolog 2 (EZH2) (Abcam, 1:30, ab191250), anti-suppressor of zeste 12 (SUZ12) (Abcam, 1:30, ab175187), anti-histone lysine-specific demethylase 1 (LSD1) (Abcam, 1:30, ab129195), and immunoglobulin G (IgG) (Santa Cruz Biotechnology,1:30, #3900) and protein A/G magnetic beads at $4^{\\\\circ}\\\\mathrm{C}$ overnight. The recovered RNA-protein complex was washed six times with wash buffer and protein denaturation at $55^{\\\\circ}\\\\mathrm{C}$ . RNA was purified from the complex. cDNA was synthesized and used for qRT-PCR or gel electrophoresis assay. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,343,702,360]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.13 RNA pull-down \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,379,922,542]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RNA pull-down was performed by using Pierce™ Magnetic RNA-Protein Pull-Down Kit (20164, Thermofisher Scientific). Briefly, biotinlabeled full-length GHET1 or its antisense RNA was in vitro transcribed, treated with RNase-free DNase I on column during RNA purification with RNA Clean & Concentrator-25 (Zymo Research), and coincubated with cell lysis buffer to form RNA-protein complex. $3^{\\\\prime}$ -untranslated-region of androgen receptor (AR) RNA containing UC-rich regions for HuR was used as a positive control. Subsequently, proteins were eluted from the complex and were then detected by western blot. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,576,919,593]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.14 Chromatin immunoprecipitation (ChIP) assay \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,612,922,774]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ChIP assay was performed by using ChIP Assay Kit (17-295, Sigma-Aldrich). In brief, cells were cross-linked with formaldehyde and sonicated to an average size of 200-to-500 basepairs. Cell lysates were precleared with protein A/G beads before incubated with protein A/G beads coated with the anti-EZH2 (Millipore), anti-H3K27me3 (Millipore), anti-LSD1 (Millipore), and anti-H3K4me2 (Millipore) antibodies. Anti-rabbit IgG was used as a negative control. Cross-linked DNA released from the protein-DNA complex was purified by DNA Extraction Kit (Sango) and the eluted DNA was further subjected to qRT-PCR. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,809,719,824]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.15 | Statistics analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,845,922,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"All our data were obtained from three independent experiments. Statistical data were analyzed by Prism Graphpad (Version 8.0) and presented as mean $\\\\pm$ standard deviation (SD). Between-group differences and multigroup comparisons were determined using Student's t test and one-way analysis of variance (ANOVA), respectively. Pearson correlation coefficients were calculated using Pearson \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,26,134,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"WAN ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[715,21,860,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Molecular Reproduction \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[752,37,858,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Development \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[899,26,922,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"761 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[70,63,485,93]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"correlation analysis by Prism. $p$ value which lower than 0.05 was referred as significant difference. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,127,205,141]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 | RESULTS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,161,475,194]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.1 GHET1 was decreased in the placenta of PE patients \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,212,485,459]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We first explored the role of GHET1 in PE at the clinical level. To investigate whether GHET1 was dysregulated in the PE development, normal placental tissues $(n = 11)$ and placental tissues from patients diagnosed with PE $(n = 20)$ were collected. The data of qRT-PCR presented a lower abundance of GHET1 in the placenta of PE pregnancies relative to the normal placental tissues (Figure 1a), suggesting the potential role of GHET1 in PE progression. Meanwhile, it was observed that GHET1 expression was markedly increased in human placenta cell lines (HTR-8/ SVneo, JEG3, BeWo, and Wish cells) compared with that in HUVEC-C cells (Figure 1b). Notably, among five human placenta cell lines, the levels of GHET1 were relatively higher in HTR-8/ SVneo cells and relatively lower in JEG3 cells, therefore HTR-8/ SVneo and JEG3 cell lines were selected to further investigate the role of GHET1 in vitro. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,494,480,526]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.2 GHET1 modulated the proliferation and cycle of extravillous trophoblast cells \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,546,485,625]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To explore the latent biological function of GHET1 in PE progression, small interfering RNA (siRNAs) specifically target GHET1 or its overexpressing plasmids were applied in HTR-8/Svneo and JEG3 cells, respectively. As expected, transfection of si-GHET1 1# 2# efficiently caused the inhibition of GHET1 expression in \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,63,924,359]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"HTR-8/Svneo cells (Figure 2a). Similarly, GHET1 overexpressing plasmids also induced the significant upregulation of GHET1 in JEG3 cells (Figure 2a). After the successful transfection efficiency, biological effects of GHET1 on extravillous trophoblasts were evaluated. As revealed in Figure 2b, HTR-8/SVneo cell viability was markedly reduced by GHET1 knockdown, whereas the viability was enhanced within GHET1 overexpression in JEG3. Meanwhile, colony formation and EdU assays also revealed that GHET1 knockdown markedly inhibited the cell proliferation in HTR-8/SVneo cells, while GHET1 overexpression exerted the opposite trend in JEG3 cells (Figure 2c,d). Flow cytometry was employed to investigate whether GHET1 regulates cell proliferation via changing cell cycle, and the results displayed that silencing of GHET1 resulted in the significant accumulation of G0/G1 phase in HTR-8/SVneo cells. Conversely, GHET1 overexpression decreased the number of JEG3 cells in the accumulation of G0/G1 phase (Figure 2e). Collectively, GHET1 acted as a facilitator for the growth and proliferation of extravillous trophoblast cells. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,393,919,427]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.3 GHET1 regulated the migration, invasion, and network formation of extravillous trophoblast cells \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,446,922,625]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We next explore the effect of GHET1 on migration, invasion, and network formation of extravillous trophoblasts. The results from wound healing assay showed GHET1 knockdown markedly suppressed HTR-8/SVneo cell migration, whereas GHET1 upregulation enhanced JEG3 cell migration (Figure 3a). In addition, transwell assay results demonstrated that HTR-8/SVneo cell invasion was markedly inhibited by GHET1 silencing, whereas JEG3 cell invasion was markedly increased following GHET1 overexpression (Figure 3b). The results from network formation assay subsequently demonstrated that transfection of si-GHET1 #1 #2 reduced the number of capillary-like networks significantly in HTR-8/SVneo cells, whereas \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[191,656,490,831]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0687/8aeedb9b5ab1d0aa1bbb6d95e28873f1b7c4d49382ec5fdc56ae6b73d35e3f1b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[512,656,803,856]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIGURE 1 GHET1 was decreased in the placenta of PE patients. (a) Normal placental tissue $(n = 11)$ and placental tissue from patients diagnosed with PE were collected $(n = 20)$ . qRT-PCR assay was performed to examine GHET1 expression in tissues. (b) qRT-PCR analysis was performed to detect GHET1 expression in HTR-8/SVneo, JEG3, BeWo, and Wish cells compared with HUVEC-C cells. All experiments were performed in biological triplicates with three technical replicates. $^{*}p < 0.05$ , $^{**}p < 0.01$ , $^{***}p < 0.001$ . HUVEC-C, Human umbilical vein endothelial cells; GHET1, gastric carcinoma high expressed transcript 1; PE, pre-eclampsia; qRT-PCR, quantitative real-time polymerase chain reaction. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0524/e87597e787828987dcefe4b1cd693c0f48bcb45bae90852278e483bc66633572.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[73,26,97,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"762 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": 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\"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"WAN ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[134,63,386,221]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0756/27f13adf861e77d842abd3eac71ec50a0427442bc34adca2217694ea2a925ae7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[400,63,573,225]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0260/b75d01182fdbdd95c3f0a9efe9fa36afe68ff028ce43eba7f4b8f1993a81da25.jpg\",\n 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\"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[453,611,569,746]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0297/1121948338d252ac3b95db9fb5ef5631900ecabf2339f7cfa66e85015f4e5ef9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[628,604,858,717]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIGURE 2 GHET1 modulated the proliferation and cycle of extravillous trophoblast cells. (a) Transfection of siRNA specifically targeting GHET1 (si-GHET1) and GHET1 overexpressing plasmid (pcDNA-GHET1) was conducted in HTR-8/SVneo and JEG3 cells, respectively. qRT-PCR assay was subjected to evaluate the transfection efficiency. (b) CCK-8 assay was employed to measure extravillous trophoblast cell viability. (c) Colony formation was performed to evaluate extravillous trophoblast cells proliferation. (d) EdU staining assay showed that percentage of EdU-positive cells. (e) Flow cytometry was performed to analyze the cell cycle. All experiments were performed in biological triplicates with three technical replicates. $^{*}p < 0.05$ , $^{**}p < 0.01$ , $^{***}p < 0.001$ . CCK-8, cell counting kit-8; EdU, 5-Ethynyl-2'-deoxyuridine; GHET1, gastric carcinoma high expressed transcript 1; qRT-PCR, quantitative real-time polymerase chain reaction; siRNA, small interfering RNA. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0145/f3fd833d560c4b7134648b002d19f9c89811bf914ed9b7ec940d88d3dd21c2b9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[73,26,134,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"WAN ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n 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null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Development \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[899,26,922,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"763 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[68,62,492,245]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0808/e7ceb9eca1f36b08accf71c3998a7edee0a00be329532911f040df217b982bf2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n 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null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[500,249,922,466]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0739/c95689c1a17dbd9742dd51c19b61a644090a2706135c467f209a3e8d55a70b0c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[95,476,295,663]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0390/ffbe1b978cbdb77041de78837d623fd35204a1a19e542aaa8a3202a2482d3c3c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[304,476,490,659]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0516/290666f5b13ad8a71b403bfae7a1fe46a631bb3c3707b424070e9c8b2bf7e0f0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[519,476,710,663]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0763/e6dd043cc2c5370b6e20b7ca26810e74f32ef76d7e986770b4672fc3d50610b0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[726,476,915,659]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIGURE 3 GHET1 regulated the migration, invasion, and network formation of extravillous trophoblast cells. (a) The capacity of migration in extravillous trophoblast cells was determined using would healing assay. (b) Transwell system was performed to measure the invasion capacity in si-GHET1-treated or pcDNA-GHET1-treated extravillous trophoblast cells. (c) Performing network formation, cells transfected with si-GHET1-treated or pcDNA-GHET1 were detected. All experiments were performed in biological triplicates with three technical replicates. $^{**}p < 0.01$ , $^{***}p < 0.001$ . GHET1, gastric carcinoma high expressed transcript 1. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0145/29c2bd68b50658c4ff48deee4ed9bacc5be03e39a270ab46e9cb2450ecdcbaa9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[70,778,485,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"transfection of pcDNA-GHET1 enhanced the number of capillary-like networks significantly in JEG3 cells (Figure 3c). All these results suggested that GHET1 increased extravillous trophoblast cell migration, invasion, and angiogenesis. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,875,480,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.4 MT2A was a downstream effector of GHET1 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,911,485,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To further investigate whether there is transcriptional regulation of GHET1 on downstream target genes, we preliminarily detected the \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,778,924,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"expression of several target genes involved in the regulation of cell proliferation and migration. It was found that the change of MT2A was most obvious after interfering GHET1. MT2A expression was markedly increased after the transfection of si-GHET1 in HTR-8/ SVneo and JEG3 cells (Figure 4a). These results revealed that the dysregulated MT2A may be an important regulatory factor of GHET1. As shown in Figure 4b, GHET1 knockdown increased MT2A protein level in HTR-8/SVneo cells, whereas GHET1 overexpression reduced MT2A expression in JEG3 cells. Moreover, it was found that MT2A was increased in the placenta of PE patients \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,26,95,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"764 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[137,21,280,48]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Molecular Reproduction Development \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[858,26,922,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"WAN ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[147,79,428,225]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(a) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0036/8b47c650fb245d414b80288c0bf643a871b2307ef45ddd637e6cb694cf643474.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[452,63,724,225]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0838/7c06e8b67a1f8fac19d0ecdd699d6abbfea29a1ec18253400e19a30196fa3c52.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[137,241,423,384]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(b) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0971/33e35100358e7d418ccdad2efe871d8c65c7d82af70bc12f9aace846456acd87.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[405,230,613,411]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0095/d8306a8521a512a4bba8a04f2f2c82fda8a88eb342d682163ebe381f5d912709.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[621,236,860,384]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(c) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0942/824f0a36dc20f9ed98dc1d43cabbe9c54a43ed2156df34d852632f4c5b93eb22.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[137,418,505,589]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(d) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0836/3a4b86ce205a3d6c436747c7d0b145d179369df4c7800badeadd5172b067cb29.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[521,430,788,572]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(e) \\\",\\\"FIGURE 4 MT2A was a downstream effector of GHET1. (a) MT2A, RND3, KLF2, PLK3, and ESM1 expressions in si-GHET1-treated extravillous trophoblast cells were determined using qRT-PCR. (b) Western blot was performed in HTR-8/SVneo and JEG3 cells after the transfection of si-GHET1 and pcDNA-GHET1, respectively. (c) qRT-PCR assay was performed to quantify the relative MT2A expression in placental tissue of PE patients $(n = 20)$ and normal tissue $(n = 11)$ . (d) Western blot assay was performed to quantify the protein level of MT2A in placental tissue of PE patients and normal tissue. (e) Pearson correlation analysis was performed, and the result showed that relative MT2A expression was negatively correlated with relative GHET1 expression. All experiments were performed in biological triplicates with three technical replicates. $^*p < 0.05$ , $^{**}p < 0.01$ . GHET1, gastric carcinoma high expressed transcript 1; MT2A, Metallothionein 2A; qRT-PCR, quantitative real-time polymerase chain reaction. \\\",\\\"(Figure 4c,d), which was negatively correlated with GHET1 (Figure 4e). \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0451/cef16170ab77db6b10312b484bfa3581a2cfe8d3faee1a30f982cfd3b161ddc9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[70,809,465,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.5 GHET1 suppressed MT2A transcription via recruiting EZH2/LSD1 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,861,485,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To further investigate the mechanism by which GHET1 regulates MT2A-mediated PE development, we analyzed the binding relationship between GHET1 and polycomb-group protein (PRC)2 complexes (EZH2, LSD1, and SUZ12). As demonstrated by RIP assay, EZH2 and LSD1 could directly bind to GHET1, while there was no binding relationship between \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,745,924,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"SUZ12 and GHET1 (Figure 5a). Consistently, EZH2 and LSD1 were pulled down by labeled GHET1, which further validated the binding relationship between GHET1 and EZH2/LSD1 (Figure 5b). In addition, silencing of EZH2 or LSD1 increased MT2A protein level in extravillous trophoblast cells (Figure 5c). The result of ChIP assay subsequently displayed that knockout of GHET1 reduced the levels of EZH2, LSD1, and H3K27me3 recruited in the MT2A promoter in HTR-8/SVneo cells, while increased the enrichment of H3K4me2 (Figure 5d). Consistently, overexpression of GHET1 increased the enrichment of EZH2, LSD1, H3K27me3 and decreased H3K4me2 enrichment in the MT2A promoter of JEG3 cells (Figure 5d). The above indicated that GHET1 mediated the MT2A promoter region H3K27me3 by recruiting EZH2 and LSD1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,26,134,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"WAN ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[715,21,860,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Molecular Reproduction \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[752,37,779,52]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[781,37,858,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Development \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[899,26,922,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"765 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[134,62,473,208]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0958/e0814087ca9504361c4bc7cefa1b2eef20d5b9982158d681e5d5e7238b3e4cf8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[477,62,863,210]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0882/91ffed67818cecd15755112b99069aa415a57d541a1da1ffc0ddfd0f0b76ced1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[132,222,465,336]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0319/0e6b63bd6f355910fb37f6011a8435554ddfa23c7d2d1529f549bca74ab4fa0c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[478,213,816,352]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0835/c05bf9004491bbac813f9c612c6df0730abc7c16f49ce6edb80dd2a0e3409e30.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[132,352,401,510]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0041/720c5785313a1d335f78c9f706cffaac85f9fa1071fd7c8f54619ac7658bb490.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[443,356,729,510]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIGURE 5 GHET1 suppressed MT2A transcription via recruiting EZH2/LSD1. (a) RIP assay was employed to analyze the interaction between GHET1 and EZH2/LSD1. (b) RNA pull-down was performed to confirm the binding of GHET1 to EZH2 and LSD1. (c) Western blot assay was performed to quantify the protein level of MT2A. (d) The enrichment of EZH2, H3K27me3, LSD1, and H3K4me2 in the MT2A promoter region was detected by ChIP assay. All experiments were performed in biological triplicates with three technical replicates. $^* p < 0.05$ , $^{**}p < 0.01$ , $^{***}p < 0.001$ . ChIP, chromatin immunoprecipitation; EZH2, zeste 2 polycomb repressive complex 2; GHET1, gastric carcinoma high expressed transcript 1; MT2A, Metallothionein 2A; LSD1, lysine-specific demethylase 1. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0924/11f9da7075cef2e31daa76e91c91ba4e2f752f08d6285042cb1282ac18bfb531.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[70,643,473,693]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.6 | EZH2 or LSD1 knockdown reversed GHET1 overexpression-mediated biological function in extravillous trophoblast cells \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,712,485,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As shown in Supporting Information: Figure S1A-B, EZH2 and LSD1 were lowly expressed in the placenta of PE pregnancies relative to the normal placental tissues. To investigate the role of EZH2 and LSD1 in GHET1-mediated biological function during PE progression, both GHET1 overexpression and EZH2/LSD1 knockdown were induced in JEG3 cells. It was observed that GHET1 overexpression reduced MT2A expression in JEG3 cells, while this change was reversed by EZH2 knockdown or LSD1 knockdown (Supporting Information: Figure S1C). CCK-8 result subsequently demonstrated that JEG3 cell viability was significantly increased by GHET1 upregulation, which was abolished by EZH2 knockdown or LSD1 knockdown (Supporting Information: Figure S1D). In addition, GHET1 overexpression promoted JEG3 cell proliferation, while this effect was abrogated by EZH2 knockdown or LSD1 knockdown (Supporting \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,644,924,759]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Information: Figure S1E). Moreover, JEG3 cell migration and invasion were markedly facilitated by GHET1 overexpression, whereas these changes were eliminated by EZH2 knockdown or LSD1 knockdown (Supporting Information: Figure S1F-G). All these results suggested that EZH2 knockdown or LSD1 knockdown reversed the promoting effect of GHET1 overexpression on extravillous trophoblast cell proliferation, migration, and invasion. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,792,847,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.7 | MT2A knockdown reversed GHET1 inhibition-mediated biological functions in extravillous trophoblast cells \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,861,922,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To further validate that MT2A participates in the GHET1-mediated functional regulation as a downstream target, si-GHET1 and shMT2A were cotransfected into extravillous trophoblast cells. It was observed that sh-MT2A transfection reduced MT2A expression in extravillous trophoblast cells (Figure 6a), suggesting that the \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,26,95,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"766 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[136,21,280,52]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Molecular Reproduction Development \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[858,26,922,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"WAN ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[137,76,314,191]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(a) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0515/795d3c04f8ad39f463efaf6e4979662a8621df875d13da1483cf4471524366bc.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[321,81,527,196]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(b) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0664/a3bfae2acc4f4a200c88f2cbbc0e1bbbde2a4f7c4f898806fed8c4c7ac69f3be.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[579,63,737,237]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(c) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0348/c2c0bed348ac68f2ab07bd1d7ca5c2bcf640483b7076ffc48223e8ba6db27f4f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[137,219,505,352]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(d) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/1005/38eb2dd9439892ca87a0b4aa937a53d7fdeca532b3a1c32c23b29db42ba42fbd.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[532,253,862,324]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(e) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0905/a9a4f7d2417b7372d3ba851a8a1f2d482dcee498f23967ff5ec318f014c8c2e2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[141,383,329,489]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(f) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0484/44eebc3ea95a0263a27cf05ac93dd749fafa95ebc7b7626dae055cc07faffafd.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[352,359,517,512]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0390/d46596e6902c91c8f3414bc20934a989204a2bbb4441a43d5c33df58c6b63335.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[549,365,715,512]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIGURE 6 MT2A knockdown reversed GHET1 inhibition-mediated biological functions in extravillous trophoblasts. (a) MT2A expression in HTR-8/SVneo cells after sh-NC or sh-MT2A transfection was determined by qRT-PCR. (b,c) CCK-8 and colony formation assays were employed to analyze cell viability and proliferation. (d) Wound healing assay was performed to detect cell migration. (e) Tube formation assays were performed to detect tube formation in HTR-8/SVneo cells. (f) Western blot analysis was subjected to assess the alteration of MT2A expression levels. All experiments were performed in biological triplicates with three technical replicates. $^{*}p < 0.05$ , $^{**}p < 0.01$ . CCK-8, Cell Counting Kit-8; GHET1, gastric carcinoma high expressed transcript 1; MT2A, Metallothionein 2A; qRT-PCR, quantitative real-time polymerase chain reaction. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0414/a9255b2c47ab649a82ed7844c3a3a55c84084550dea1875dc21ad79308795dba.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[70,645,485,824]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"transfection was successful. As shown in Figure 6b,c, GHET1 knockdown's inhibition effect on cell proliferation was abolished after MT2A knockdown. It was also turned out that GHET1 knockdown decreased cell migration, whereas knockdown of MT2A rescued this inhibiting effect (Figure 6d). In addition, the silencing of MT2A weakened the suppressive efficiency of tube formation induced by the silencing of GHET1 (Figure 6e). Knockdown of GHET1 enhanced the expression of MT2A, whereas this effect was blocked by silencing of MT2A (Figure 6f). Collectively, these results indicated that GHET1 achieved its biological roles in extravillous trophoblast cells by inhibiting MT2A. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,859,238,873]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 | DISCUSSION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,893,485,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As widely described, IncRNAs are closely related to PE progression. For instance, IncRNA uc003fir suppressed trophoblast proliferation and migration in PE by enhancing CCL5 expression \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,645,924,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(Pengjie et al., 2018). Overexpression of PUM1 impaired the invasion of trophoblast cells in PE by inhibiting IncRNA HOTAIR expression (Zhang et al., 2020). Upregulation of IncRNA TCL6 expression promoted the progression of PE via phosphatase and tensin homolog (Wu et al., 2019). Nevertheless, the role of IncRNAs in PE has not been fully clarified, more differential expression IncRNAs need to be further explored and investigated. GHET1, as an oncogene, functions in facilitating cancer cell proliferation, migration, and invasion (Han et al., 2019; Liu et al., 2021). Notably, it has been reported that GHET1 is involved in PE progression, specifically its overexpression-enhanced cell invasion and migration abilities in PE (Li et al., 2019). However, the role of GHET1 in PE has only been reported in one report. Herein, our work revealed downstream regulatory mechanisms by which GHET1 accelerated extravillous trophoblast proliferation, cell cycle, migration, invasion, and network formation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,893,924,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is well known that nearly $20\\\\%$ human IncRNAs can bind with PRC2 (which contains EZH2, SUZ12, and embryonic ectoderm development [EED]), thus erecting their epigenetic regulation of \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,26,134,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"WAN ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[715,21,860,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Molecular Reproduction \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[752,37,860,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Development \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[899,26,922,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"767 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[70,63,485,427]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"downstream targets through interacting with RNA-binding proteins, as well as LSD1 (Ding et al., 2017; Huang et al., 2017; Lian et al., 2016). Histone lysine methylation is closely related to gene expression, chromatin remodeling, and cell differentiation, histone lysine methyltransferase (KMTs) and histone lysine demethylase (KDMs) can dynamically regulate this process. LSD1 is the first identified histone lysine-specific demethylase, which is known to catalyze the removal of monomethylation and dimethylation at H3K4 and H3K9 sites, thus inhibiting or activating gene transcription. The main component of PRC2 complex, EZH2, is a methyltransferase of H3K27, which can inhibit gene transcription by methylating H3K27 (Ding et al., 2017; Huang et al., 2017; Lian et al., 2016). To further explore whether GHET1 could mediate the transcriptional regulation of downstream targets, we conducted a serious of experiments in this study. In this study, it was turned out that GHET1 directly bound to EZH2 and LSD1, and the knockdown of EZH2 and LSD1 induced the upregulation of MT2A. Furthermore, the ChIP assay demonstrated that GHET1 mediated H3K27me3 and H3K4me2 in the MT2A promoter region by recruiting EZH2 and LSD1. To sum up, we concluded that GHET1 inhibited the transcriptional level of MT2A by regulating EZH2/LSD1-mediated H3K27me3 levels in the MT2A promoter region. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,429,485,826]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Metallothioneins (MTs) are a family of metal-binding proteins that widely exist in organisms (Hui, Xu, et al., 2019). There are four subfamilies in human MT, namely MT1, MT2A, MT3, and MT4 respectively (Mangelinck et al., 2019). MT2A is one of the major MT subfamily, which are highly conserved and exists in almost all types of soft tissues (Ma et al., 2015). MTs are involved in different kinds of cellular functions, such as cell differentiation, proliferation, and apoptosis. MTs dysregulation have also been linked to tumor progression (Hui, Xu, et al., 2019; Ma et al., 2015; Mangelinck et al., 2019). For instance, upregulation of IncRNA taurine up-regulated gene 1 regulated pancreatic cancer cell malignant behaviors via RND3 and MT2A (Hui, Xu, et al., 2019). In addition, it was previously illustrated that HMBOX 1 could regulate vascular endothelial cell autophagy and apoptosis by interacting with MT2A (Ma et al., 2015). MT2A is considered as a biomarker for stress-related diseases/disorders including hypertension (Sekovanic et al., 2020). Notably, it is revealed by Wei et al., the genetic variation of rs10636 G locus of MT2A was associated with the susceptibility of hypertensive disorders of pregnancy, especially PE (Wei et al., 2022). Nevertheless, the function of MT2A in regulating extravillous trophoblastic phenotype during PE progression remains unclear. Herein, we demonstrated that GHET1 silencing inhibited proliferation, migration, invasion, and angiogenesis of cells, whereas silencing of MT2A rescued this inhibition. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,828,485,925]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Taken together, GHET1 epigenetically suppressed MT2A expression by recruiting EZH2/LSD1, thus promoting the proliferation, migration, invasion, and angiogenesis of extravillous trophoblasts in PE (Figure 7). Our findings may provide innovative research findings and therapeutic targets for PE treatment. However, there are some limitations in our research, such as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[515,63,922,300]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIGURE 7 Graphical abstract of GHET1 regulated extravillous trophoblastic phenotype via EZH2/LSD1-mediated MT2A epigenetic suppression in pre-eclampsia. EZH2, zeste 2 polycomb repressive complex 2; GHET1, gastric carcinoma high expressed transcript 1; LSD1, lysine-specific demethylase 1; MT2A, metallothionein 2A. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0281/0395cc4c35289deb3d5c60abce110d6f308cb924a4acec90e8f9855ce4ab4a4a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[507,411,924,492]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"the lack of verification experiments in mouse models. Besides, whether there are other functional downstream targets remains unclear. In addition, more PE-related IncRNAs and the various molecular mechanisms of PE need to be further investigated for the prevention and treatment of PE. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,512,714,524]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"AUTHOR CONTRIBUTIONS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,529,924,643]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Pengyun Wan: Conceptualization; methodology; validation; investigation; visualization; writing—original draft; formal analysis. Jia Huang: Software. Wenting Liu: Data curation. Xiaoyan Su: Resources. Bei Zhao: Formal analysis. Xianggang Wang: Methodology. Lu Zhao: Conceptualization; funding acquisition; writing—original draft; writing—review & editing; project administration; supervision. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,662,680,673]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ACKNOWLEDGMENTS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,679,922,709]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The authors would like to give our sincere gratitude to the reviewers for their constructive comments. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,728,794,741]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CONFLICT OF INTEREST STATEMENT \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,745,773,758]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The authors declare no conflict of interest. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,778,766,790]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DATA AVAILABILITY STATEMENT \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,795,922,824]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"All data generated or analyzed during this study are included in this published article. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,845,668,856]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ETHICS STATEMENT \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,861,924,925]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The study was authorized by Second Affiliated Hospital of Nanchang University Ethics Committee. All patients who met the inclusion criteria provided written informed consent. The informed consent obtained from study participants. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,26,97,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"768 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[137,21,280,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Molecular Reproduction \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[173,37,280,51]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Development \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[858,26,922,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"WAN ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[73,63,171,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"REFERENCES \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,81,485,925]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"Bartsch, E., Medcalf, K. E., Park, A. L., & Ray, J. G. (2016). Clinical risk factors for pre-eclampsia determined in early pregnancy: Systematic review and meta-analysis of large cohort studies. BMJ, 353, i1753. \\\",\\\"Ding, J., Xie, M., Lian, Y., Zhu, Y., Peng, P., Wang, J., Wang, L., & Wang, K. (2017). Long noncoding RNA HOXA-AS2 represses P21 and KLF2 expression transcription by binding with EZH2, LSD1 in colorectal cancer. Oncogenesis, 6(1), e288. \\\",\\\"Gong, F., Cheng, H., Shi, Y., Cui, L., & Jia, G. (2020). LncRNA TDRG1/miR214-5p axis affects preeclampsia by modulating trophoblast cells. Cell Biochemistry and Function, 38(4), 352-361. \\\",\\\"Gyselaers, W., Lees, C., Valensise, H., & Thilaganathan, B. (2020). Preeclampsia: the role of persistent endothelial cells in uteroplacental arteries. Brosens I, Brosens JJ, Muter J, Puttemans P, Benagiano G. Am J Obstet Gynecol 2019;221:219-26. Am J Obstet Gynecol, 6, 633. \\\",\\\"Han, M., Wang, Y., Gu, Y., Ge, X., Seng, J., Guo, G., Zhang, X., Zhao, Y., & Dou, D. (2019). IncRNA GHET1 knockdown suppresses breast cancer activity in vitro and in vivo. American Journal of Translational Research, 11(1), 31-44. \\\",\\\"Hao, J., Hassen, D., Hao, Q., Graham, J., Paglia, M. J., Brown, J., Cooper, M., Schlieder, V., & Snyder, S. R. (2019). Maternal and infant health care costs related to preeclampsia. Obstetrics & Gynecology, 134(6), 1227-1233. \\\",\\\"He, T., Qiao, Y., Lv, Y., Wang, J., Hu, R., & Cao, Y. (2019). IncRNA FAM99A is downregulated in preeclampsia and exerts a regulatory effect on trophoblast cell invasion, migration and apoptosis. Molecular Medicine Reports, 20(2), 1451-1458. \\\",\\\"Huang, M., Hou, J., Wang, Y., Xie, M., Wei, C., Nie, F., Wang, Z., & Sun, M. (2017). Long noncoding RNA LINC00673 is activated by SP1 and exerts oncogenic properties by interacting with LSD1 and EZH2 in gastric cancer. Molecular Therapy, 25(4), 1014-1026. \\\",\\\"Hui, B., Ji, H., Xu, Y., Wang, J., Ma, Z., Zhang, C., Wang, K., & Zhou, Y. (2019). RREB1-induced upregulation of the IncRNA AGAP2-AS1 regulates the proliferation and migration of pancreatic cancer partly through suppressing ANKRD1 and ANGPTL4. Cell Death & Disease, 10(3), 207. \\\",\\\"Hui, B., Xu, Y., Zhao, B., Ji, H., Xu, S., Ma, Z., He, Z. Y., Wang, K., & Lu, J. (2019). Overexpressed long noncoding RNA TUG1 affects the cell cycle, proliferation, and apoptosis of pancreatic cancer partly through suppressing RND3 and MT2A. OncoTargets and Therapy, 12, 1043-1057. \\\",\\\"Jiang, J., & Zhao, Z. M. (2018). LncRNA HOXD-AS1 promotes preeclampsia progression via MAPK pathway. European Review for Medical and Pharmacological Sciences, 22(24), 8561-8568. \\\",\\\"Li, J. L., Li, R., Gao, Y., Guo, W. C., Shi, P. X., & Li, M. (2018). LncRNA CCAT1 promotes the progression of preeclampsia by regulating CDK4. European Review for Medical and Pharmacological Sciences, 22(5), 1216-1223. \\\",\\\"Li, S., Liu, J., Kong, F., Wang, Y., Li, N., & Zou, Y. (2019). IncRNA GHET1 has effects in development of pre-eclampsia. Journal of Cellular Biochemistry, 120(8), 12647-12652. \\\",\\\"Lian, Y., Wang, J., Feng, J., Ding, J., Ma, Z., Li, J., Peng, P., De, W., & Wang, K. (2016). Long non-coding RNA IRAIN suppresses apoptosis and promotes proliferation by binding to LSD1 and EZH2 in pancreatic cancer. *Tumor Biology*, 37(11), 14929-14937. \\\",\\\"Liu, Y., Shi, P., Fang, H., Zhao, Z., Yang, F., Zhang, J., Jing, S., & Geng, C. (2021). Long non-coding RNA GHET1 promotes thyroid cancer cell proliferation and invasion. Translational Cancer Research, 10(9), 4148-4157. \\\",\\\"Long, J., Xiong, J., Bai, Y., Mao, J., Lin, J., Xu, W., Zhang, H., Chen, S., & Zhao, H. (2019). Construction and investigation of a \\\"]\",\n \"page_idx\": \"11\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[512,65,924,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"lncRNA-associated ceRNA regulatory network in cholangiocarcinoma. Frontiers in Oncology, 9, 649. \\\",\\\"Ma, H., Su, L., Yue, H., Yin, X., Zhao, J., Zhang, S., Kung, H., Xu, Z., & Miao, J. (2015). HMBOX 1 interacts with MT2A to regulate autophagy and apoptosis in vascular endothelial cells. Scientific Reports, 5, 15121. \\\",\\\"Mangelinck, A., da Costa, M. E. M., Stefanovska, B., Bawa, O., Polrot, M., Gaspar, N., & Fromigue, O. (2019). MT2A is an early predictive biomarker of response to chemotherapy and a potential therapeutic target in osteosarcoma. Scientific Reports, 9(1), 12301. \\\",\\\"Mol, B. W. J., Roberts, C. T., Thangaratinam, S., Magee, L. A., de Groot, C. J. M., & Hofmeyr, G. J. (2016). Pre-eclampsia. The Lancet, 387(10022), 999-1011. \\\",\\\"Moufarrej, M. N., Vorperian, S. K., Wong, R. J., Campos, A. A., Quaintance, C. C., Sit, R. V., Tan, M., Detweiler, A. M., Mekonen, H., Neff, N. F., Baruch-Gravett, C., Litch, J. A., Druzin, M. L., Winn, V. D., Shaw, G. M., Stevenson, D. K., & Quake, S. R. (2022). Early prediction of preeclampsia in pregnancy with cell-free RNA. Nature, 602(7898), 689-694. \\\",\\\"Pengjie, Z., Xionghui, C., Yueming, Z., Ting, X., Na, L., Jianying, T., & Zhice, X. (2018). LncRNA uc003fir promotes CCL5 expression and negatively affects proliferation and migration of trophoblast cells in preeclampsia. *Pregnancy Hypertension*, 14, 90-96. \\\",\\\"Ridder, A., Giorgione, V., Khalil, A., & Thilaganathan, B. (2019). Preeclampsia: the relationship between uterine artery blood flow and trophoblast function. International Journal of Molecular Sciences, 20(13), 3263. \\\",\\\"Sekovanic, A., Jurasovic, J., & Piasek, M. (2020). Metallothionein 2A gene polymorphisms in relation to diseases and trace element levels in humans. Archives of Industrial Hygiene and Toxicology, 71(1), 27-47. \\\",\\\"Tao, Y., Tang, Y., Yang, Z., Wu, F., Wang, L., Yang, L., Lei, L., Jing, Y., Jiang, X., Jin, H., Bai, Y., & Zhang, L. (2020). Exploration of serum exosomal LncRNA TBILA and AGAP2-AS1 as promising biomarkers for diagnosis of Non-Small cell lung cancer. International Journal of Biological Sciences, 16(3), 471-482. \\\",\\\"Wei, S., Yu, X., Wen, X., Zhang, M., Lang, Q., Zhong, P., & Huang, B. (2022). Genetic variations in metallothionein genes and susceptibility to hypertensive disorders of pregnancy: A case-control study. Frontiers in Genetics, 13, 830446. \\\",\\\"Wu, J., Huang, H., Huang, W., Wang, L., Xia, X., & Fang, X. (2020). Analysis of exosomal lncRNA, miRNA and mRNA expression profiles and ceRNA network construction in endometriosis. Epigenomics, 12, 1193-1213. \\\",\\\"Wu, J. L., Wang, Y. G., Gao, G. M., Feng, L., Guo, N., & Zhang, C. X. (2019). Overexpression of IncRNA TCL6 promotes preeclampsia progression by regulating PTEN. European Review for Medical and Pharmacological Sciences, 23(10), 4066-4072. \\\",\\\"Xu, Y., Xia, X., Jiang, Y., Wu, D., Wang, S., Fu, S., Yang, N., Zhang, Y., & Sun, L. (2020). Down-regulated IncRNA AGAP2-AS1 contributes to pre-eclampsia as a competing endogenous RNA for JDP2 by impairing trophoblastic phenotype. Journal of Cellular and Molecular Medicine, 24(8), 4557-4568. \\\",\\\"Yang, W., Shan, Z., Zhou, X., Peng, L., Zhi, C., Chai, J., Liu, H., Yang, J., & Zhang, Z. (2018). Knockdown of IncRNA GHET1 inhibits osteosarcoma cells proliferation, invasion, migration and EMT in vitro and in vivo. Cancer Biomarkers: Section A of Disease markers, 23(4), 589-601. \\\",\\\"Zhang, Y., He, X. Y., Qin, S., Mo, H. Q., Li, X., Wu, F., Zhang, J., Li, X., Mao, L., Peng, Y. Q., Guo, Y. N., Lin, Y., & Tian, F. J. (2020). Upregulation of PUM1 expression in preeclampsia impairs trophoblast invasion by negatively regulating the expression of the IncRNA HOTAIR. Molecular Therapy, 28(2), 631-641. \\\"]\",\n \"page_idx\": \"11\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[73,26,134,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"WAN ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[715,21,860,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Molecular Reproduction \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[752,37,858,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Development \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[899,26,922,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"769 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[72,63,487,118]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Zhou, J., Li, X., Wu, M., Lin, C., Guo, Y., & Tian, B. (2016). Knockdown of long noncoding RNA GHET1 inhibits cell proliferation and invasion of colorectal cancer. Oncology Research Featuring Preclinical and Clinical Cancer Therapeutics, 23(6), 303-309. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"ref_text\"\n },\n {\n \"bbox\": \"[72,147,292,159]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"SUPPORTING INFORMATION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[72,164,485,194]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Additional supporting information can be found online in the Supporting Information section at the end of this article. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[532,79,892,176]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"How to cite this article: Wan, P., Huang, J., Liu, W., Su, X., Zhao, B., Wang, X., & Zhao, L. (2023). IncRNA GHET1 regulates extravillous trophoblastic phenotype via EZH2/LSD1-mediated MT2A epigenetic suppression in pre-eclampsia. Molecular Reproduction and Development, 90, 758-770. https://doi.org/10.1002/mrd.23693 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"ref_text\"\n },\n {\n \"bbox\": \"[73,26,97,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"770 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[137,21,280,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Molecular Reproduction \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[174,37,280,52]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Development \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[858,26,922,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"WAN ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.1002/mrd.23693"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Medicine and Health Sciences"},"sha256":{"kind":"string","value":"59d6ce0dfc65b7672bdd997a8841f09dc82508fc88d2c55692aca5fcdc5e61cf"},"title":{"kind":"string","value":"lncRNA GHET1 regulates extravillous trophoblastic phenotype via EZH2/LSD1‐mediated MT2A epigenetic suppression in pre‐eclampsia"}}},{"rowIdx":49,"cells":{"abstract":{"kind":"string","value":"\n \nWe experimentally demonstrate the power scaling of optical vortices by the coherent beam combining, encompassing topological charges ranging from l=1 to l=5 realized on the basis of a Yb-doped fiber short-pulsed laser system. The combining efficiency varies from 83.2 to 96.9% depending on the topological charge and beam pattern quality generated by the spatial light modulators. These results open a pathway to high-intensity optical vortices with enormous potential applications in science and industry by utilizing advances in light-matter interactions.\n "},"author":{"kind":"string","value":"Mikko Narhi|Regina Gumenyuk|Hossein Fathi"},"content_list":{"kind":"list like","value":[{"bbox":"[78,155,767,184]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Coherent beam combining of optical vortices ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[78,200,633,220]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"HOSSEIN FATHI, $^{1*}$ MIKKO NÄRHI $^{1}$ , REGINA GUMENYUK $^{1,2}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,229,807,243]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$^{1}$ Laboratory of Photonics, Physics Unit, Faculty of Engineering and Natural Sciences, Tampere University, 33720 Tampere, Finland. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,243,633,256]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$^{2}$ Tampere Institute for Advanced Study, Tampere University, Kalevantie 4, Tampere 33100, Finland ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,268,334,282]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"*Corresponding author: hossein.fathi@tuni.fi ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,290,895,303]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,327,480,479]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We experimentally demonstrate the power scaling of optical vortices by the coherent beam combining, encompassing topological charges ranging from $\\ell = 1$ to $\\ell = 5$ realized on the basis of a Yb-doped fiber short-pulsed laser system. The combining efficiency varies from 83.2 to $96.9\\%$ depending on the topological charge and beam pattern quality generated by the spatial light modulators. These results open a pathway to high-intensity optical vortices with enormous potential applications in science and industry by utilizing advances in light-matter interactions. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,510,480,771]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Optical vortices (OVs), spiral-shaped light beams with orbital angular momentum (OAM), have undergone substantial advancements and gained extensive interest since their introduction in 1989 [1-3]. By precisely tailoring the spatial structure of the beam, researchers can manipulate its properties and achieve functionalities beyond those of traditional Gaussian beams. OVs characterized by Hilbert factor $\\exp (i\\ell \\varphi)$ , $\\ell$ is an integer number and $\\varphi$ is the azimuthal angle, possess a spiral phase front and carry OAM equivalent to $\\ell \\hbar$ per photon, which is $\\ell$ times larger than spin angular momentum (SAM), equivalent to $\\pm \\hbar$ per photon. These unique properties make them highly promising in a variety of applications such as optical tweezers [4-6], high-capacity optical communications [7-9], super-resolution microscopy [10, 11], and laser-plasma interaction [12, 13]. Moreover, improving the tunability of OVs, including spectral, temporal, topological charge (TC), chirality and singularity, spurred progress in various fields of advanced research [14-19]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,772,480,859]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There are two main ways to generate OVs [20], the first is by direct generation inside the laser cavity [21, 22], and the second by indirect mode conversion based on phase front modulators such as spatial light modulators (SLMs) as computer-generated holograms [23, 24], spiral phase plates [25], q-plates [26], and cylindrical lens [27]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,859,480,917]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There has been a sustained interest in generating high-power spatially structured beams to fully exploit their potential. For instance, in addressing the attenuation challenges encountered in free-space optical communication, enabling powerful ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[516,327,916,718]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"optical trapping and manipulation, and facilitating high-power laser material processing. [4-13, 28, 29]. However, generating high-power OVs encounters limitations with all the previously mentioned conventional methods due to thermal damage of components. To address this issue, we employ coherent beam combining (CBC) as a versatile technique for power scaling OVs while preserving their desired characteristics for the first time, to the best of our knowledge. CBC is a method for increasing the power output of lasers by combining multiple laser amplifiers initially seeded by a common source [30]. The main idea involves splitting a seed laser into multiple replicas (N channels), amplifying each replica to its maximum power/energy through separate amplifier sections, and subsequently merging them into a single high-intensity beam while maintaining the beam's quality. Along with preserving the spatial properties, CBC also ensures that the spectral properties of the lasers are retained. This technique relies on establishing a phase relationship between the laser amplifiers, allowing them to operate effectively as a single laser amplifier. CBC technique has been fully established for both continuous wave (CW) and pulsed lasers, and to date, more than $100\\mathrm{kW}$ average power, a few tens of mJ in pulse energy and a few tens of GW peak power have been experimentally demonstrated [31-33]. Furthermore, CBC reveals significant potential for scaling the number of channels, CBC of 107 beams has been experimentally demonstrated [34]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[516,719,916,907]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There are several reports on the generation of OVs using the CBC technique after the first demonstration in 2009 [35-40]. These studies consistently applied the CBC technique within a tiled-aperture configuration. This approach involves adjusting the intensity weights and the piston phase distributions of fundamental Gaussian array beams to create a helical phase structure, ultimately leading to generating the desired optical vortices in the far field. However, in this paper, we present a first-ever experimental demonstration of an active coherent beam combination of OVs via filled-aperture configuration. Our experiment involves the CBC of linearly polarized OVs with topological charges spanning from $\\ell = 1$ to $\\ell = 5$ , all carried by short pulses. The beam combining ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[81,73,919,381]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 1. Scheme of the experimental setup for the generation and coherent combination of optical vortices. The schematic process of the generation of the LG (3, 0) out of LG (0, 0) is shown on the right side of the picture. The Amp: Amplifier, PM: Phase Modulator, VDL: Variable Delay Line, QWP: Quarter Wave Plate, HWP: Half Wave Plate, SLM: Spatial Light Modulator, HRM: High Reflective Mirror, MM-PA: Motorized Mirror equipped with Piezo Actuators, IBS: Intensity Beam splitter, BS: Beam Sampler. \"]","image_footnote":"[]","img_path":"images/0518/47147c67fd0878ba7cfc1d884eb3c48a2451b1c162430b569e4207dbbd20901d.jpg","list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[75,445,483,503]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"efficiency depends on the quality and spatial pattern complexity of the generated vortices and can be as high as $\\sim 96\\%$ , roughly analogous to the combining efficiency for Gaussian beams. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,511,486,910]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Fig. 1 illustrates the experimental setup employed for the generation and coherent combination of OVs. The scheme highlights the key components and techniques utilized in generating and combining OVs. The seed source consists of an ultrafast gain-switched laser that emits pulses at a repetition rate of $20\\mathrm{MHz}$ with a pulse duration of $55~\\mathrm{ps}$ $(\\pm 5$ ps). Following amplification in the initial stage, the seed laser is split into two channels using a 50:50 fiber coupler. Subsequently, each channel undergoes a second amplification stage, delivering an average power of up to $100\\mathrm{mW}$ . The second channel is equipped with a $\\mathrm{LiNbO_3}$ electrooptic phase modulator (PM) to precisely control and adjust the phase of the OVs, ensuring a stable and coherent combination. To achieve an efficient coherent combination, a variable delay line (VDL) is incorporated to equalize the optical path lengths of the OVs. Using two collimators at the end of each channel, the output beams were collimated to propagate in free space. Two Spatial light modulators (SLMs) are utilized to introduce the desired phase patterns, thereby generating Laguerre-Gaussian (LG) optical vortex beams with different TCs $(\\ell = 1-5$ and $p = 0)$ . For illustrative purposes, the schematic process for generating LG $(3,0)$ out of LG $(0,0)$ is depicted on the right side of Fig. 1. Since the SLMs are polarization-dependent, we utilize a pair of quarter-wave plate (QWP) and half-wave plate (HWP) to achieve the optimal polarization state. Each generated OV, with an output power of $50\\mathrm{mW}$ , is then directed towards a set of high reflection mirror (HRM) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,445,921,722]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"mirrors that steer the beams into the combining elements. To ensure precise spatial overlapping and maximize the combining efficiency, two high-reflection motorized mirrors, equipped with piezo actuators were utilized in each channel. These actuators enable fine adjustments of the mirror positions, ensuring accurate alignment of the OVs. The filled-aperture configuration, utilizing near-field beam combining, has been employed in this setup. The combining element is an intensity beam splitter (IBS), which allows the two OVs to spatially overlap and combine coherently. The coherently combined beam emerges from a predetermined port of the intensity beam splitter. Using a beam sampler (BS), a portion of the combined beam is extracted for measuring the beam quality factor. The uncombined light that emerges from the idler port of the IBS is utilized as feedback for the active control system. The beam phase control system employed here is a customized commercial feedback loop system, based on a top-of-fringe stabilization technique (Laselock, TEM Messetechnik). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,729,921,910]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Through an extensive experimental investigation, five distinct linearly polarized Laguerre-Gaussian optical vortices, denoted as LG ( $\\ell = 1 - 5$ , $p = 0$ ) and Gaussian beam ( $\\ell = 0$ , $p = 0$ ) were coherently combined. The LG (0, 0) beams were coherently combined with an efficiency of $98.3\\%$ for the reference of the technique. The first five Laguerre-Gaussian OVs, each possessing distinct topological charges of $\\ell = 1$ , 2, 3, 4, and 5, were combined with corresponding efficiencies of $95.5\\%$ , $86.9\\%$ , $83.2\\%$ , $78\\%$ , and $68.1\\%$ . Combining efficiency is calculated by dividing the power of the combined beam by the sum of the output power of each laser beam. The decrease in combining efficiency for high-order OVs can ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,75,482,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"predominantly be attributed to the imperfection of the generated OVs by SLMs, as the field experiences some undesirable distortions during SLM-induced modulation. Furthermore, differences in collimation quality resulting in different divergences for the two beams, as well as an imbalance in the free-space optical paths, could also lead to a decrease in efficiency. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,185,486,657]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Fig. 2 illustrates the beam profiles of five distinct linearly polarized Laguerre-Gaussian optical vortices and a Gaussian beam for two different channels and their coherent combination output, along with the corresponding phase patterns applied on SLM to generate them from the LG (0,0). They show the time-varying beam profiles of the combined beams when the phase control system is on (coherently combination) and off (random combination). Additionally, the videos include separate beam profiles for each individual channel beam of the replica. Other LG OVs ( $\\ell = 2 - 4$ , $p = 0$ ) demonstrated similar behaviour during combination. The output beam qualities of all combined LG OVs were analyzed. We would like to note that the combined beam demonstrated higher uniformity of intensity distribution. This could be interpreted as an improvement of the beam quality by CBC. Fig. 3 depicts the ISO 11146-compliant $\\mathbf{M}^2$ measurement of the combined beams utilizing the $4\\sigma$ -method, along with the corresponding combining efficiencies. Fig. 3(b), 3(c), and 3(d) illustrate the $\\mathbf{M}^2$ measurement of the coherently combined beams for LG (0,0), LG (1,0), and LG (5,0), respectively, accompanied by the corresponding near-field beam profiles as insets. We present three of six beams as examples, and other OVs demonstrated similar performance. The typical optical spectrum of both channels and the combined one for LG (1,0) are depicted in Fig. 4(a). For further analysis, we characterized the optical pulse durations of both individual channels and the combined output in the time domain using a 25-GHz photodetector, as illustrated in Fig. 4(b). The full width at half maximum (FWHM) pulse duration was $55\\pm 5$ ps. We have not noticed any spectral or temporal profile changes for other LG modes. Polarization and output power stability assessment was conducted on the ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[66,666,483,840]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 2. Profiles of five distinct linearly polarized Laguerre-Gaussian optical vortices, LG OVs $(\\\\ell = 1 - 5, p = 0)$ and a Gaussian beam $(\\\\ell = 0, p = 0)$ , for two different channels and their coherent combination, along with the corresponding phase patterns applied on SLM. \"]","image_footnote":"[]","img_path":"images/0723/053250feb8ceae6a158a457f6048b1ecbac2b621e8340dd745bad415fcda7b3a.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[522,82,727,204]","code_body":null,"code_caption":null,"image_caption":"[\"(a) \"]","image_footnote":"[]","img_path":"images/0478/3b35506ea7a790b43ed3c118a056e5365e09bbcef18d7081637fa8834f40febf.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[743,82,926,205]","code_body":null,"code_caption":null,"image_caption":"[\"(b) \"]","image_footnote":"[]","img_path":"images/0597/17862fed2c80cc549043e2a4f52dba290deacc28ef7e2b95d99be4b9b4d859e0.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[522,220,710,342]","code_body":null,"code_caption":null,"image_caption":"[\"(c) \"]","image_footnote":"[]","img_path":"images/0043/a238785665ea4efd4536aadfcd5c8043158e7c6d6cf845debc3d686ca344c7b2.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[746,223,926,342]","code_body":null,"code_caption":null,"image_caption":"[\"(d) \"]","image_footnote":"[]","img_path":"images/0600/6900e2fae309f9ac6fd7e173be68ebb0e19b53c212cfd57586f3a7556f21fbfa.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[516,450,723,579]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 3. Analysis of beam quality and combining efficiency of the combined beams. (a): ISO 11146-compliant M2 -measurement of the combined beams with the $4\\\\sigma$ -method with corresponding combining efficiencies, (b), (c), and(d): M2 -measurement of the combined beams of LG (0.0), LG (1.0), and LG (5.0), respectively (Insets: near field beam profiles of the combined beams). \",\"(a) \"]","image_footnote":"[]","img_path":"images/0831/c537c09dc8abb2f18cc642b3712c8a4b0a37f82187b8e7020b99cf147108a43f.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[725,450,926,579]","code_body":null,"code_caption":null,"image_caption":"[\"(b) \",\"Fig. 4. Optical spectrum and pulse duration measurements of the output of channel 1, channel 2 and the combined output of both channels for LG (1,0). (a) Normalized optical spectra, (b) time-domain envelope measured using a 25-GHz photodetector. \"]","image_footnote":"[]","img_path":"images/0583/7aaf831a38235d13aa43fbf571b5bce5a2a675f6216a3754b5ea522ce6d2986a.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[511,666,921,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"combined OV beam of LG (1,0) over 20 minutes using a commercial polarimeter (PAX1000IR2/M), presenting a common performance for all OV beams. The results corresponding to this assessment are shown in Fig. 5, providing a clear illustration of the excellent polarization and power stability of the CBC system. The power fluctuation was less than $3.7\\%$ , while DOP variations did not exceed $0.2\\%$ . The ellipticity of the combined beam demonstrated negligible changes. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,804,921,907]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In conclusion, we presented the first experimental demonstration, to the best of our knowledge, of coherent beam combining for Laguerre-Gaussian optical vortices, achieving high combining efficiencies of up to $\\sim 96\\%$ in a filled-aperture configuration. Remarkably, the combined beam demonstrated improved uniformity of the spatial intensity distribution for all OVs, which could be interpreted as improved beam ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[71,36,480,251]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 5. Polarization and output power measurements stability of the output of coherent beam combination of two LG (1, 0) beams. (a) Output power stability, (b) degree of polarization (DOP), and (c) the Ellipticity. \"]","image_footnote":"[]","img_path":"images/0453/ad15bb4f59eef9a270c831509d8812da2e19d6e240ee08e0f8a062099882d996.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[75,308,483,555]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"quality. These results mark a pioneering achievement in the potential for power/energy scaling of optical vortices in short-pulsed laser systems, thereby paving the way for exploring novel applications in the field of high-intensity light-matter interactions. The CBC technique is highly scalable, revealing promising benefits for fields that require high-power OAM beams, as a case in point, overcoming signal attenuation in long-distance free-space optical communications. To maximize combining efficiency, it is essential to generate high-quality OVs, especially for higher-order OVs, where efficiency decreases. Therefore, attention should be given to the quality of SLMs or phase plates used, as well as free-space beam propagation principles. Ongoing research and development endeavours in this domain will further refine the implementation of CBC for extremely high-power OVs, unlocking new possibilities for various scientific and technological applications. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[76,561,482,588]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Funding. Horizon Europe EIC Pathfinder-OPEN (101096317), Research Council of Finland (320165). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[76,594,480,621]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acknowledgements. The authors thank Rafael Barros, Hassan Asgharzadeh, and Ebrahim Aghayari for fruitful discussions. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[76,627,418,640]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Disclosures. The authors declare no conflicts of interest. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[76,647,480,686]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Data availability. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,702,163,715]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[89,720,482,916]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. P. Couillet, L. Gil, and F. Rocca, Opt. Commun. 73, 403-408 (1989). \",\"2. L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185-8189 (1992). \",\"3. Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, Light Sci Appl. 8, 90 (2019). \",\"4. D. G. Grier, Nature 424, 810-816 (2003). \",\"5. X. W. Zhuang, Science 305, 188-190 (2004). \",\"6. M. Padgett, and R. Bowman, Nat. Photonics 5, 343-348 (2011). \",\"7. J. T. Barreiro, T. C. Wei, and P. G. Kwiat, Nat. Phys. 4, 282-286 (2008). \",\"8. J. Wang et al., Nat, Photonics 6, 488-496 (2012). \",\"9. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, Science 340, 1545-1548 (2013). \",\"10. F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, Phys. Rev. Lett. 97, 163903 (2006). \",\"11. Y. Kozawa, D. Matsunaga, and S. Sato, Optica 5, 86-92 (2018). \"]","page_idx":"3","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[527,75,919,796]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"12. C. Brabetz, S. Busold, T. Cowan, O. Deppert, D. Jahn, O. Kester, M. Roth, D. Schumacher, and V. Bagnoud, Phys. Plasmas 22, 013105 (2015). \",\"13. A. Denoecd, L. Chopineau, A. Leblanc, and F. Quéré, Phys. Rev. Lett. 118, 033902 (2017). \",\"14. R. C. Devlin, A. Ambrosio, N. A. Rubin, J. P. B. Mueller, and F. Capasso, Science 358, 896-901 (2017). \",\"15. F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, P. B. Corkum, Nat. Commun. 8, 14970 (2017). \",\"16. D. Gauthier et al., Nat. Commun. 8, 14971 (2017). \",\"17. Z. Xie, T. Lei, F. Li, H. Qiu, Z. Zhang, H. Wang, C. Min, L. Du, Z. Li, and X. Yuan, Light Sci. Appl. 7, 18001 (2018). \",\"18. N. C. Zambon et al., Nat. Photonics 13, 283-288 (2019). \",\"19. L. Rego et al., Science 364, eaaw9486 (2019). \",\"20. X. Wang, Z. Nie, Y. Liang, J. Wang, T. Li, and B. Jia, Nanophotonics 7 (4), 1533-1556 (2018). \",\"21. D. J. Kim, J. W. Kim, and W. A. Clarkson, Appl. Phys. B 117, 459-464 (2014). \",\"22. A. Mock, D. Sounas, and A. Alù, Phys. Rev. Lett. 121, 173004 (2018). \",\"23. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, Opt. Lett. 17, 221-223 (1992). \",\"24. J. E. Curtis and D. G. Grier, Opt. Lett. 28, 872-874 (2003). \",\"25. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun., 112(5), 321-327, (1994). \",\"26. L. Yan, P. Gregg, E. Karimi, A. Rubano, L. Marrucci, R. Boyd, and S. Ramachandran, Optica 2, 900-903 (2015). \",\"27. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96(1-3), 123-132 (1993). \",\"28. G. Xie, L. Li, Y. Ren, H. Huang, Y. Yan, N. Ahmed, Z. Zhao, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, Optica 2(4), 357-364 (2015). \",\"29. A. Goffin, I. Larkin, A. Tartaro, A. Schweinsberg, A. Valenzuela, E. W. Rosenthal, and H. M. Milchberg, Phys. Rev. X 13, 011006 (2023). \",\"30. H. Fathi, M. Narhi, R. Gumenyuk, Photonics. 8, 566 (2021). \",\"31. S. J. McNaught et al., Frontiers in Optics 2009/Laser Science XXV/Fall 2009 OSA Optics & Photonics Technical Digest, OSA Technical Digest (CD) (Optica Publishing Group, 2009), paper FThD2 (2009). \",\"32. M. Müller, C. Aleshire, A. Klenke, E. Haddad, F. Légare, A. Tünnermann, and J. Limpert, Opt. Lett. 45, 3083-3086 (2020). \",\"33. H. Stark, J. Buldt, M. Müller, A. Klenke, A. Tünnermann, and J. Limpert, Opt. Lett. 44, 5529-5532 (2019). \",\"34. H. Chang, Q. Chang, J. Xi, T. Hou, R. Su, P. Ma, J. Wu, C. Li, M. Jiang, Y. Ma, and P. Zhou, Photon. Res. 8, 1943-1948 (2020). \",\"35. L. Wang, L. Wang, and S. Zhu, Opt. Commun. 282(6), 1088-1094 (2009). \",\"36. T. Hou, D. Zhi, R. Tao, Y. Ma, P. Zhou, and Z. Liu, Opt. Express 26, 14945-14958 (2018). \",\"37. D. Zhi, T. Hou, P. Ma, Y. Ma, P. Zhou, R. Tao, X. Wang, and L. Si, Laser Science and Engineering. 7, E33(2019). \",\"38. M. Veinhard, S. Bellanger, L. Daniault, I. Fsaifes, J. Bourderionnet, C. Larat, E. Lallier, A. Brignon, and J. C. Chanteloup, Opt. Lett. 46, 25-28 (2021). \",\"39. X. Guo, X. Zhang, D. Xu, W. Chen, Y. Guo, K. Lan, and B. Shen, Sci Rep. 13, 1104 (2023). \",\"40. J. Long, K. Jin, Q. Chen, H. Chang, Q. Chang, Y. Ma, J. Wu, R. Su, P. Ma, and P. Zhou, Opt. Lett. 48, 5021-5024 (2023). \"]","page_idx":"3","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[514,839,601,852]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[526,857,919,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. P. Coulet, L. Gil, and F. Rocca, \"Optical vortices,\" Opt. Commun. 73 (5), 403-408 (1989). https://doi.org/10.1016/0030-4018(89)90180-6 ","text_format":null,"text_level":null,"type":"ref_text"},{"bbox":"[89,75,482,902]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"2. L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, \\\"Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,\\\" Phys. Rev. A 45, 8185-8189 (1992). https://link.aps.org/doi/10.1103/PhysRevA.45.8185 \",\"3. Shen, Y., Wang, X., Xie, Z. et al. Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities. Light Sci App/8, 90 (2019). https://doi.org/10.1038/s41377-019-0194-2 \",\"4. D. G. Grier, \\\"revolution in optical manipulation,\\\" Nature 424, 810-816 (2003). https://doi.org/10.1038/nature01935 \",\"5. X. W. Zhuang, \\\"Unraveling DNA Condensation with Optical Tweezers,\\\" Science 305, 188-190 (2004). https://science.org/doi/10.1126/science.1100603 \",\"6. M. Padgett, and R. Bowman, \\\"Tweezers with a twist,\\\" Nat. Photonics 5, 343-348 (2011). https://doi.org/10.1038/nphoton.2011.81 \",\"7. J. T. Barreiro, T. C. Wei, and P. G. Kwiat, \\\"Beating the channel capacity limit for linear photonic superdense coding,\\\" Nat. Phys. 4, 282-286 (2008). https://doi.org/10.1038/nphys919 \",\"8. J. Wang et al., \\\"Terabit free-space data transmission employing orbital angular momentum multiplexing,\\\" Nat, Photonics 6, 488-496 (2012). https://doi.org/10.1038/nphoton.2012.138 \",\"9. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, \\\"Terabit-scale orbital angular momentum mode division multiplexing in fibers,\\\" Science 340, 1545-1548 (2013). https://www.science.org/doi/10.1126/science.1237861 \",\"10. F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, \\\"Overcoming the Rayleigh Criterion Limit with Optical Vortices,\\\" Phys. Rev. Lett. 97, 163903 (2006). https://link.aps.org/doi/10.1103/PhysRevLett.97.163903 \",\"11. Y. Kozawa, D. Matsunaga, and S. Sato, \\\"Superresolution imaging via superoscillation focusing of a radially polarized beam,\\\" Optica 5, 86-92 (2018). https://doi.org/10.1364/OPTICA.5.000086 \",\"12. C. Brabetz, S. Busold, T. Cowan, O. Deppert, D. Jahn, O. Kester, M. Roth, D. Schumacher, and V. Bagnoud, \\\"Laser-driven ion acceleration with hollow laser beams,\\\" Phys. Plasmas 22, 013105 (2015). https://doi.org/10.1063/1.4905638 \",\"13. A. Denoecd, L. Chopineau, A. Leblanc, and F. Quere, \\\"Interaction of Ultraintense Laser Vortices with Plasma Mirrors,\\\" Phys. Rev. Lett. 118, 033902 (2017). https://link.aps.org/doi/10.1103/PhysRevLett.118.033902 \",\"14. R. C. Devlin, A. Ambrosio, N. A. Rubin, J. P. B. Mueller, and F. Capasso, \\\"Arbitrary spin-to-orbital angular momentum conversion of light,\\\" Science 358, 896-901 (2017). https://www.science.org/doi/10.1126/science.aao5392 \",\"15. F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, P. B. Corkum, \\\"Controlling the orbital angular momentum of high harmonic vortices,\\\" Nat. Commun. 8, 14970 (2017). https://www.nature.com/articles/ncomms14970 \",\"16. D. Gauthier et al., \\\"Tunable orbital angular momentum in high-harmonic generation,\\\" Nat. Commun. 8, 14971 (2017). https://doi.org/10.1038/ncomms14971 \",\"17. Z. Xie, T. Lei, F. Li, H. Qiu, Z. Zhang, H. Wang, C. Min, L. Du, Z. Li, and X. Yuan, \\\"Ultra-broadband on-chip twisted light emitter for optical communications,\\\" Light Sci. Appl. 7, 18001 (2018). https://doi.org/10.1038/lsa.2018.1 \",\"18. N. C. Zambon et al., \\\"Optically controlling the emission chirality of microlasers,\\\" Nat. Photonics 13, 283-288 (2019). https://doi.org/10.1038/s41566-019-0380-z \"]","page_idx":"4","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[527,75,918,914]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"19. L. Rego et al., \\\"Generation of extreme-ultraviolet beams with time-varying orbital angular momentum,\\\" Science 364, eaaw9486 (2019). https://doi.org/10.1126/science.aaw9486 \",\"20. X. Wang, Z. Nie, Y. Liang, J. Wang, T. Li, and B. Jia, \\\"Recent advances on optical vortex generation,\\\" Nanophotonics 7 (4), 1533-1556 (2018). https://doi.org/10.1515/nanoph-2018-0072 \",\"21. D. J. Kim, J. W. Kim, and W. A. Clarkson, \\\"High-power master-oscillator power-amplifier with optical vortex output,\\\" Appl. Phys. B 117, 459-464 (2014). https://doi.org/10.1007/s00340-014-5855-5 \",\"22. A. Mock, D. Sounas, and A. Alù, \\\"Tunable Orbital Angular Momentum Radiation from Angular-Momentum-Biased Microcavities,\\\" Phys. Rev. Lett. 121, 103901 (2018). https://link.aps.org/doi/10.1103/PhysRevLett.121.103901 \",\"23. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, \\\"Generation of optical phase singularities by computer-generated holograms,\\\" Opt. Lett. 17, 221-223 (1992). https://doi.org/10.1364/OL.17.000221 \",\"24. J. E. Curtis and D. G. Grier, \\\"Modulated optical vortices,\\\" Opt. Lett. 28, 872-874 (2003). https://doi.org/10.1364/OL.28.000872 \",\"25. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, \\\"Helical-wavefront laser beams produced with a spiral phaseplates,\\\" Opt. Commun., 112(5), 321-327, (1994). https://doi.org/10.1016/0030-4018(94)90638-6 \",\"26. L. Yan, P. Gregg, E. Karimi, A. Rubano, L. Marrucci, R. Boyd, and S. Ramachandran, \\\"Q-plate enabled spectrally diverse orbital-angular-momentum conversion for stimulated emission depletion microscopy,\\\" Optica 2, 900-903 (2015). https://doi.org/10.1364/OPTICA.2.000900 \",\"27. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, \\\"Astigmatic laser mode converters and transfer of orbital angular momentum,\\\" Opt. Commun. 96(1-3), 123-132 (1993). https://doi.org/10.1016/0030-4018(93)90535-D \",\"28. G. Xie, L. Li, Y. Ren, H. Huang, Y. Yan, N. Ahmed, Z. Zhao, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, \\\"Performance metrics and design considerations for a free-space optical orbital-angular-momentum-multiplexed communication link,\\\" Optica 2(4), 357-364 (2015). https://doi.org/10.1364/OPTICA.2.000357 \",\"29. A. Goffin, I. Larkin, A. Tartaro, A. Schweinsberg, A. Valenzuela, E. W. Rosenthal, and H. M. Milchberg, \\\"Optical Guiding in 50-Meter-Scale Air Waveguides,\\\" Phys. Rev. X 13, 011006 (2023). https://link.aps.org/doi/10.1103/PhysRevX.13.011006 \",\"30. H. Fathi, M. Narhi, R. Gumenyuk, \\\"Towards Ultimate High-Power Scaling: Coherent Beam Combining of Fiber Lasers,\\\" Photonics 8, 566 (2021). https://doi.org/10.3390/photonics8120566 \",\"31. S. J. McNaught et al., \\\"100-kW coherently combined Nd: YAG MOPA laser array,\\\" Frontiers in Optics 2009/Laser Science XXV/Fall 2009 OSA Optics & Photonics Technical Digest, OSA Technical Digest (CD) (Optica Publishing Group, 2009), paper FThD2 (2009). https://doi.org/10.1364/FIO.2009.FThD2 \",\"32. M. Müller, C. Aleshire, A. Klenke, E. Haddad, F. Légare, A. Tünnermann, and J. Limpert, \\\"10.4 kW coherently combined ultrafast fiber laser,\\\" Opt. Lett. 45, 3083-3086 (2020). https://doi.org/10.1364/OL.392843 \",\"33. H. Stark, J. Buldt, M. Müller, A. Klenke, A. Tünnermann, and J. Limpert, \\\"23 mJ high-power fiber CPA system using electro-optically controlled divided-pulse amplification,\\\" Opt. Lett. 44, 5529-5532 (2019). https://doi.org/10.1364/OL.44.005529 \",\"34. H. Chang, Q. Chang, J. Xi, T. Hou, R. Su, P. Ma, J. Wu, C. Li, M. Jiang, Y. Ma, and P. Zhou, \\\"First experimental demonstration of coherent beam combining of more than 100 beams,\\\" Photon. Res. 8, 1943-1948 (2020). \"]","page_idx":"4","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[89,75,483,455]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"https://doi.org/10.1364/PRJ.409788 \",\"35. L. Wang, L. Wang, and S. Zhu, \\\"Formation of optical vortices using coherent laser beam arrays,\\\" Opt. Commun. 282(6), 1088-1094 (2009). https://doi.org/10.1016/j.optcom.2008.12.004 \",\"36. T. Hou, D. Zhi, R. Tao, Y. Ma, P. Zhou, and Z. Liu, \\\"Spatially-distributed orbital angular momentum beam array generation based on greedy algorithms and coherent combining technology,\\\" Opt. Express 26, 14945-14958 (2018). https://doi.org/10.1364/OE.26.014945 \",\"37. D. Zhi, T. Hou, P. Ma, Y. Ma, P. Zhou, R. Tao, X. Wang, and L. Si, \\\"Comprehensive investigation on producing high-power orbital angular momentum beams by coherent combining technology. High Power Laser Science and Engineering,\\\" Laser Science and Engineering. 7, E33(2019). https://doi.org/10.1017/hpl.2019.17 \",\"38. M. Veinhard, S. Bellanger, L. Daniault, I. Fsaifes, J. Bourderionnet, C. Larat, E. Lallier, A. Brignon, and J. C. Chanteloup, \\\"Orbital angular momentum beams generation from 61 channels coherent beam combining femtosecond digital laser,\\\" Opt. Lett. 46, 25-28 (2021). https://doi.org/10.1364/OL.405975 \",\"39. X. Guo, X. Zhang, D. Xu, W. Chen, Y. Guo, K. Lan, and B. Shen, \\\"Intense vortical-field generation using coherent superposition of multiple vortex beams,\\\" Sci Rep. 13, 1104 (2023). https://doi.org/10.1038/s41598-023-28216-9 \",\"40. J. Long, K. Jin, Q. Chen, H. Chang, Q. Chang, Y. Ma, J. Wu, R. Su, P. Ma, and P. Zhou, \\\"Generating the 1.5 kW mode-tunable fractional vortex beam by a coherent beam combining system,\\\" Opt. Lett. 48, 5021-5024 (2023). https://doi.org/10.1364/OL.502321 \"]","page_idx":"5","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"}],"string":"[\n {\n \"bbox\": \"[78,155,767,184]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Coherent beam combining of optical vortices \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,200,633,220]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"HOSSEIN FATHI, $^{1*}$ MIKKO NÄRHI $^{1}$ , REGINA GUMENYUK $^{1,2}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,229,807,243]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$^{1}$ Laboratory of Photonics, Physics Unit, Faculty of Engineering and Natural Sciences, Tampere University, 33720 Tampere, Finland. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,243,633,256]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$^{2}$ Tampere Institute for Advanced Study, Tampere University, Kalevantie 4, Tampere 33100, Finland \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,268,334,282]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"*Corresponding author: hossein.fathi@tuni.fi \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,290,895,303]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,327,480,479]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We experimentally demonstrate the power scaling of optical vortices by the coherent beam combining, encompassing topological charges ranging from $\\\\ell = 1$ to $\\\\ell = 5$ realized on the basis of a Yb-doped fiber short-pulsed laser system. The combining efficiency varies from 83.2 to $96.9\\\\%$ depending on the topological charge and beam pattern quality generated by the spatial light modulators. These results open a pathway to high-intensity optical vortices with enormous potential applications in science and industry by utilizing advances in light-matter interactions. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,510,480,771]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Optical vortices (OVs), spiral-shaped light beams with orbital angular momentum (OAM), have undergone substantial advancements and gained extensive interest since their introduction in 1989 [1-3]. By precisely tailoring the spatial structure of the beam, researchers can manipulate its properties and achieve functionalities beyond those of traditional Gaussian beams. OVs characterized by Hilbert factor $\\\\exp (i\\\\ell \\\\varphi)$ , $\\\\ell$ is an integer number and $\\\\varphi$ is the azimuthal angle, possess a spiral phase front and carry OAM equivalent to $\\\\ell \\\\hbar$ per photon, which is $\\\\ell$ times larger than spin angular momentum (SAM), equivalent to $\\\\pm \\\\hbar$ per photon. These unique properties make them highly promising in a variety of applications such as optical tweezers [4-6], high-capacity optical communications [7-9], super-resolution microscopy [10, 11], and laser-plasma interaction [12, 13]. Moreover, improving the tunability of OVs, including spectral, temporal, topological charge (TC), chirality and singularity, spurred progress in various fields of advanced research [14-19]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,772,480,859]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There are two main ways to generate OVs [20], the first is by direct generation inside the laser cavity [21, 22], and the second by indirect mode conversion based on phase front modulators such as spatial light modulators (SLMs) as computer-generated holograms [23, 24], spiral phase plates [25], q-plates [26], and cylindrical lens [27]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,859,480,917]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There has been a sustained interest in generating high-power spatially structured beams to fully exploit their potential. For instance, in addressing the attenuation challenges encountered in free-space optical communication, enabling powerful \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[516,327,916,718]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"optical trapping and manipulation, and facilitating high-power laser material processing. [4-13, 28, 29]. However, generating high-power OVs encounters limitations with all the previously mentioned conventional methods due to thermal damage of components. To address this issue, we employ coherent beam combining (CBC) as a versatile technique for power scaling OVs while preserving their desired characteristics for the first time, to the best of our knowledge. CBC is a method for increasing the power output of lasers by combining multiple laser amplifiers initially seeded by a common source [30]. The main idea involves splitting a seed laser into multiple replicas (N channels), amplifying each replica to its maximum power/energy through separate amplifier sections, and subsequently merging them into a single high-intensity beam while maintaining the beam's quality. Along with preserving the spatial properties, CBC also ensures that the spectral properties of the lasers are retained. This technique relies on establishing a phase relationship between the laser amplifiers, allowing them to operate effectively as a single laser amplifier. CBC technique has been fully established for both continuous wave (CW) and pulsed lasers, and to date, more than $100\\\\mathrm{kW}$ average power, a few tens of mJ in pulse energy and a few tens of GW peak power have been experimentally demonstrated [31-33]. Furthermore, CBC reveals significant potential for scaling the number of channels, CBC of 107 beams has been experimentally demonstrated [34]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[516,719,916,907]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There are several reports on the generation of OVs using the CBC technique after the first demonstration in 2009 [35-40]. These studies consistently applied the CBC technique within a tiled-aperture configuration. This approach involves adjusting the intensity weights and the piston phase distributions of fundamental Gaussian array beams to create a helical phase structure, ultimately leading to generating the desired optical vortices in the far field. However, in this paper, we present a first-ever experimental demonstration of an active coherent beam combination of OVs via filled-aperture configuration. Our experiment involves the CBC of linearly polarized OVs with topological charges spanning from $\\\\ell = 1$ to $\\\\ell = 5$ , all carried by short pulses. The beam combining \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[81,73,919,381]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 1. Scheme of the experimental setup for the generation and coherent combination of optical vortices. The schematic process of the generation of the LG (3, 0) out of LG (0, 0) is shown on the right side of the picture. The Amp: Amplifier, PM: Phase Modulator, VDL: Variable Delay Line, QWP: Quarter Wave Plate, HWP: Half Wave Plate, SLM: Spatial Light Modulator, HRM: High Reflective Mirror, MM-PA: Motorized Mirror equipped with Piezo Actuators, IBS: Intensity Beam splitter, BS: Beam Sampler. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0518/47147c67fd0878ba7cfc1d884eb3c48a2451b1c162430b569e4207dbbd20901d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[75,445,483,503]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"efficiency depends on the quality and spatial pattern complexity of the generated vortices and can be as high as $\\\\sim 96\\\\%$ , roughly analogous to the combining efficiency for Gaussian beams. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,511,486,910]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Fig. 1 illustrates the experimental setup employed for the generation and coherent combination of OVs. The scheme highlights the key components and techniques utilized in generating and combining OVs. The seed source consists of an ultrafast gain-switched laser that emits pulses at a repetition rate of $20\\\\mathrm{MHz}$ with a pulse duration of $55~\\\\mathrm{ps}$ $(\\\\pm 5$ ps). Following amplification in the initial stage, the seed laser is split into two channels using a 50:50 fiber coupler. Subsequently, each channel undergoes a second amplification stage, delivering an average power of up to $100\\\\mathrm{mW}$ . The second channel is equipped with a $\\\\mathrm{LiNbO_3}$ electrooptic phase modulator (PM) to precisely control and adjust the phase of the OVs, ensuring a stable and coherent combination. To achieve an efficient coherent combination, a variable delay line (VDL) is incorporated to equalize the optical path lengths of the OVs. Using two collimators at the end of each channel, the output beams were collimated to propagate in free space. Two Spatial light modulators (SLMs) are utilized to introduce the desired phase patterns, thereby generating Laguerre-Gaussian (LG) optical vortex beams with different TCs $(\\\\ell = 1-5$ and $p = 0)$ . For illustrative purposes, the schematic process for generating LG $(3,0)$ out of LG $(0,0)$ is depicted on the right side of Fig. 1. Since the SLMs are polarization-dependent, we utilize a pair of quarter-wave plate (QWP) and half-wave plate (HWP) to achieve the optimal polarization state. Each generated OV, with an output power of $50\\\\mathrm{mW}$ , is then directed towards a set of high reflection mirror (HRM) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,445,921,722]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"mirrors that steer the beams into the combining elements. To ensure precise spatial overlapping and maximize the combining efficiency, two high-reflection motorized mirrors, equipped with piezo actuators were utilized in each channel. These actuators enable fine adjustments of the mirror positions, ensuring accurate alignment of the OVs. The filled-aperture configuration, utilizing near-field beam combining, has been employed in this setup. The combining element is an intensity beam splitter (IBS), which allows the two OVs to spatially overlap and combine coherently. The coherently combined beam emerges from a predetermined port of the intensity beam splitter. Using a beam sampler (BS), a portion of the combined beam is extracted for measuring the beam quality factor. The uncombined light that emerges from the idler port of the IBS is utilized as feedback for the active control system. The beam phase control system employed here is a customized commercial feedback loop system, based on a top-of-fringe stabilization technique (Laselock, TEM Messetechnik). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,729,921,910]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Through an extensive experimental investigation, five distinct linearly polarized Laguerre-Gaussian optical vortices, denoted as LG ( $\\\\ell = 1 - 5$ , $p = 0$ ) and Gaussian beam ( $\\\\ell = 0$ , $p = 0$ ) were coherently combined. The LG (0, 0) beams were coherently combined with an efficiency of $98.3\\\\%$ for the reference of the technique. The first five Laguerre-Gaussian OVs, each possessing distinct topological charges of $\\\\ell = 1$ , 2, 3, 4, and 5, were combined with corresponding efficiencies of $95.5\\\\%$ , $86.9\\\\%$ , $83.2\\\\%$ , $78\\\\%$ , and $68.1\\\\%$ . Combining efficiency is calculated by dividing the power of the combined beam by the sum of the output power of each laser beam. The decrease in combining efficiency for high-order OVs can \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,75,482,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"predominantly be attributed to the imperfection of the generated OVs by SLMs, as the field experiences some undesirable distortions during SLM-induced modulation. Furthermore, differences in collimation quality resulting in different divergences for the two beams, as well as an imbalance in the free-space optical paths, could also lead to a decrease in efficiency. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,185,486,657]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Fig. 2 illustrates the beam profiles of five distinct linearly polarized Laguerre-Gaussian optical vortices and a Gaussian beam for two different channels and their coherent combination output, along with the corresponding phase patterns applied on SLM to generate them from the LG (0,0). They show the time-varying beam profiles of the combined beams when the phase control system is on (coherently combination) and off (random combination). Additionally, the videos include separate beam profiles for each individual channel beam of the replica. Other LG OVs ( $\\\\ell = 2 - 4$ , $p = 0$ ) demonstrated similar behaviour during combination. The output beam qualities of all combined LG OVs were analyzed. We would like to note that the combined beam demonstrated higher uniformity of intensity distribution. This could be interpreted as an improvement of the beam quality by CBC. Fig. 3 depicts the ISO 11146-compliant $\\\\mathbf{M}^2$ measurement of the combined beams utilizing the $4\\\\sigma$ -method, along with the corresponding combining efficiencies. Fig. 3(b), 3(c), and 3(d) illustrate the $\\\\mathbf{M}^2$ measurement of the coherently combined beams for LG (0,0), LG (1,0), and LG (5,0), respectively, accompanied by the corresponding near-field beam profiles as insets. We present three of six beams as examples, and other OVs demonstrated similar performance. The typical optical spectrum of both channels and the combined one for LG (1,0) are depicted in Fig. 4(a). For further analysis, we characterized the optical pulse durations of both individual channels and the combined output in the time domain using a 25-GHz photodetector, as illustrated in Fig. 4(b). The full width at half maximum (FWHM) pulse duration was $55\\\\pm 5$ ps. We have not noticed any spectral or temporal profile changes for other LG modes. Polarization and output power stability assessment was conducted on the \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[66,666,483,840]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 2. Profiles of five distinct linearly polarized Laguerre-Gaussian optical vortices, LG OVs $(\\\\\\\\ell = 1 - 5, p = 0)$ and a Gaussian beam $(\\\\\\\\ell = 0, p = 0)$ , for two different channels and their coherent combination, along with the corresponding phase patterns applied on SLM. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0723/053250feb8ceae6a158a457f6048b1ecbac2b621e8340dd745bad415fcda7b3a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[522,82,727,204]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(a) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0478/3b35506ea7a790b43ed3c118a056e5365e09bbcef18d7081637fa8834f40febf.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[743,82,926,205]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(b) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0597/17862fed2c80cc549043e2a4f52dba290deacc28ef7e2b95d99be4b9b4d859e0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[522,220,710,342]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(c) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0043/a238785665ea4efd4536aadfcd5c8043158e7c6d6cf845debc3d686ca344c7b2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[746,223,926,342]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(d) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0600/6900e2fae309f9ac6fd7e173be68ebb0e19b53c212cfd57586f3a7556f21fbfa.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[516,450,723,579]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 3. Analysis of beam quality and combining efficiency of the combined beams. (a): ISO 11146-compliant M2 -measurement of the combined beams with the $4\\\\\\\\sigma$ -method with corresponding combining efficiencies, (b), (c), and(d): M2 -measurement of the combined beams of LG (0.0), LG (1.0), and LG (5.0), respectively (Insets: near field beam profiles of the combined beams). \\\",\\\"(a) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0831/c537c09dc8abb2f18cc642b3712c8a4b0a37f82187b8e7020b99cf147108a43f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[725,450,926,579]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(b) \\\",\\\"Fig. 4. Optical spectrum and pulse duration measurements of the output of channel 1, channel 2 and the combined output of both channels for LG (1,0). (a) Normalized optical spectra, (b) time-domain envelope measured using a 25-GHz photodetector. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0583/7aaf831a38235d13aa43fbf571b5bce5a2a675f6216a3754b5ea522ce6d2986a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[511,666,921,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"combined OV beam of LG (1,0) over 20 minutes using a commercial polarimeter (PAX1000IR2/M), presenting a common performance for all OV beams. The results corresponding to this assessment are shown in Fig. 5, providing a clear illustration of the excellent polarization and power stability of the CBC system. The power fluctuation was less than $3.7\\\\%$ , while DOP variations did not exceed $0.2\\\\%$ . The ellipticity of the combined beam demonstrated negligible changes. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,804,921,907]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In conclusion, we presented the first experimental demonstration, to the best of our knowledge, of coherent beam combining for Laguerre-Gaussian optical vortices, achieving high combining efficiencies of up to $\\\\sim 96\\\\%$ in a filled-aperture configuration. Remarkably, the combined beam demonstrated improved uniformity of the spatial intensity distribution for all OVs, which could be interpreted as improved beam \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[71,36,480,251]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 5. Polarization and output power measurements stability of the output of coherent beam combination of two LG (1, 0) beams. (a) Output power stability, (b) degree of polarization (DOP), and (c) the Ellipticity. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0453/ad15bb4f59eef9a270c831509d8812da2e19d6e240ee08e0f8a062099882d996.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[75,308,483,555]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"quality. These results mark a pioneering achievement in the potential for power/energy scaling of optical vortices in short-pulsed laser systems, thereby paving the way for exploring novel applications in the field of high-intensity light-matter interactions. The CBC technique is highly scalable, revealing promising benefits for fields that require high-power OAM beams, as a case in point, overcoming signal attenuation in long-distance free-space optical communications. To maximize combining efficiency, it is essential to generate high-quality OVs, especially for higher-order OVs, where efficiency decreases. Therefore, attention should be given to the quality of SLMs or phase plates used, as well as free-space beam propagation principles. Ongoing research and development endeavours in this domain will further refine the implementation of CBC for extremely high-power OVs, unlocking new possibilities for various scientific and technological applications. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[76,561,482,588]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Funding. Horizon Europe EIC Pathfinder-OPEN (101096317), Research Council of Finland (320165). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[76,594,480,621]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acknowledgements. The authors thank Rafael Barros, Hassan Asgharzadeh, and Ebrahim Aghayari for fruitful discussions. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[76,627,418,640]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Disclosures. The authors declare no conflicts of interest. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[76,647,480,686]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data availability. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,702,163,715]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[89,720,482,916]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. P. Couillet, L. Gil, and F. Rocca, Opt. Commun. 73, 403-408 (1989). \\\",\\\"2. L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185-8189 (1992). \\\",\\\"3. Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, Light Sci Appl. 8, 90 (2019). \\\",\\\"4. D. G. Grier, Nature 424, 810-816 (2003). \\\",\\\"5. X. W. Zhuang, Science 305, 188-190 (2004). \\\",\\\"6. M. Padgett, and R. Bowman, Nat. Photonics 5, 343-348 (2011). \\\",\\\"7. J. T. Barreiro, T. C. Wei, and P. G. Kwiat, Nat. Phys. 4, 282-286 (2008). \\\",\\\"8. J. Wang et al., Nat, Photonics 6, 488-496 (2012). \\\",\\\"9. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, Science 340, 1545-1548 (2013). \\\",\\\"10. F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, Phys. Rev. Lett. 97, 163903 (2006). \\\",\\\"11. Y. Kozawa, D. Matsunaga, and S. Sato, Optica 5, 86-92 (2018). \\\"]\",\n \"page_idx\": \"3\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[527,75,919,796]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"12. C. Brabetz, S. Busold, T. Cowan, O. Deppert, D. Jahn, O. Kester, M. Roth, D. Schumacher, and V. Bagnoud, Phys. Plasmas 22, 013105 (2015). \\\",\\\"13. A. Denoecd, L. Chopineau, A. Leblanc, and F. Quéré, Phys. Rev. Lett. 118, 033902 (2017). \\\",\\\"14. R. C. Devlin, A. Ambrosio, N. A. Rubin, J. P. B. Mueller, and F. Capasso, Science 358, 896-901 (2017). \\\",\\\"15. F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, P. B. Corkum, Nat. Commun. 8, 14970 (2017). \\\",\\\"16. D. Gauthier et al., Nat. Commun. 8, 14971 (2017). \\\",\\\"17. Z. Xie, T. Lei, F. Li, H. Qiu, Z. Zhang, H. Wang, C. Min, L. Du, Z. Li, and X. Yuan, Light Sci. Appl. 7, 18001 (2018). \\\",\\\"18. N. C. Zambon et al., Nat. Photonics 13, 283-288 (2019). \\\",\\\"19. L. Rego et al., Science 364, eaaw9486 (2019). \\\",\\\"20. X. Wang, Z. Nie, Y. Liang, J. Wang, T. Li, and B. Jia, Nanophotonics 7 (4), 1533-1556 (2018). \\\",\\\"21. D. J. Kim, J. W. Kim, and W. A. Clarkson, Appl. Phys. B 117, 459-464 (2014). \\\",\\\"22. A. Mock, D. Sounas, and A. Alù, Phys. Rev. Lett. 121, 173004 (2018). \\\",\\\"23. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, Opt. Lett. 17, 221-223 (1992). \\\",\\\"24. J. E. Curtis and D. G. Grier, Opt. Lett. 28, 872-874 (2003). \\\",\\\"25. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun., 112(5), 321-327, (1994). \\\",\\\"26. L. Yan, P. Gregg, E. Karimi, A. Rubano, L. Marrucci, R. Boyd, and S. Ramachandran, Optica 2, 900-903 (2015). \\\",\\\"27. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96(1-3), 123-132 (1993). \\\",\\\"28. G. Xie, L. Li, Y. Ren, H. Huang, Y. Yan, N. Ahmed, Z. Zhao, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, Optica 2(4), 357-364 (2015). \\\",\\\"29. A. Goffin, I. Larkin, A. Tartaro, A. Schweinsberg, A. Valenzuela, E. W. Rosenthal, and H. M. Milchberg, Phys. Rev. X 13, 011006 (2023). \\\",\\\"30. H. Fathi, M. Narhi, R. Gumenyuk, Photonics. 8, 566 (2021). \\\",\\\"31. S. J. McNaught et al., Frontiers in Optics 2009/Laser Science XXV/Fall 2009 OSA Optics & Photonics Technical Digest, OSA Technical Digest (CD) (Optica Publishing Group, 2009), paper FThD2 (2009). \\\",\\\"32. M. Müller, C. Aleshire, A. Klenke, E. Haddad, F. Légare, A. Tünnermann, and J. Limpert, Opt. Lett. 45, 3083-3086 (2020). \\\",\\\"33. H. Stark, J. Buldt, M. Müller, A. Klenke, A. Tünnermann, and J. Limpert, Opt. Lett. 44, 5529-5532 (2019). \\\",\\\"34. H. Chang, Q. Chang, J. Xi, T. Hou, R. Su, P. Ma, J. Wu, C. Li, M. Jiang, Y. Ma, and P. Zhou, Photon. Res. 8, 1943-1948 (2020). \\\",\\\"35. L. Wang, L. Wang, and S. Zhu, Opt. Commun. 282(6), 1088-1094 (2009). \\\",\\\"36. T. Hou, D. Zhi, R. Tao, Y. Ma, P. Zhou, and Z. Liu, Opt. Express 26, 14945-14958 (2018). \\\",\\\"37. D. Zhi, T. Hou, P. Ma, Y. Ma, P. Zhou, R. Tao, X. Wang, and L. Si, Laser Science and Engineering. 7, E33(2019). \\\",\\\"38. M. Veinhard, S. Bellanger, L. Daniault, I. Fsaifes, J. Bourderionnet, C. Larat, E. Lallier, A. Brignon, and J. C. Chanteloup, Opt. Lett. 46, 25-28 (2021). \\\",\\\"39. X. Guo, X. Zhang, D. Xu, W. Chen, Y. Guo, K. Lan, and B. Shen, Sci Rep. 13, 1104 (2023). \\\",\\\"40. J. Long, K. Jin, Q. Chen, H. Chang, Q. Chang, Y. Ma, J. Wu, R. Su, P. Ma, and P. Zhou, Opt. Lett. 48, 5021-5024 (2023). \\\"]\",\n \"page_idx\": \"3\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[514,839,601,852]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[526,857,919,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. P. Coulet, L. Gil, and F. Rocca, \\\"Optical vortices,\\\" Opt. Commun. 73 (5), 403-408 (1989). https://doi.org/10.1016/0030-4018(89)90180-6 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"ref_text\"\n },\n {\n \"bbox\": \"[89,75,482,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"2. L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, \\\\\\\"Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,\\\\\\\" Phys. Rev. A 45, 8185-8189 (1992). https://link.aps.org/doi/10.1103/PhysRevA.45.8185 \\\",\\\"3. Shen, Y., Wang, X., Xie, Z. et al. Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities. Light Sci App/8, 90 (2019). https://doi.org/10.1038/s41377-019-0194-2 \\\",\\\"4. D. G. Grier, \\\\\\\"revolution in optical manipulation,\\\\\\\" Nature 424, 810-816 (2003). https://doi.org/10.1038/nature01935 \\\",\\\"5. X. W. Zhuang, \\\\\\\"Unraveling DNA Condensation with Optical Tweezers,\\\\\\\" Science 305, 188-190 (2004). https://science.org/doi/10.1126/science.1100603 \\\",\\\"6. M. Padgett, and R. Bowman, \\\\\\\"Tweezers with a twist,\\\\\\\" Nat. Photonics 5, 343-348 (2011). https://doi.org/10.1038/nphoton.2011.81 \\\",\\\"7. J. T. Barreiro, T. C. Wei, and P. G. Kwiat, \\\\\\\"Beating the channel capacity limit for linear photonic superdense coding,\\\\\\\" Nat. Phys. 4, 282-286 (2008). https://doi.org/10.1038/nphys919 \\\",\\\"8. J. Wang et al., \\\\\\\"Terabit free-space data transmission employing orbital angular momentum multiplexing,\\\\\\\" Nat, Photonics 6, 488-496 (2012). https://doi.org/10.1038/nphoton.2012.138 \\\",\\\"9. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, \\\\\\\"Terabit-scale orbital angular momentum mode division multiplexing in fibers,\\\\\\\" Science 340, 1545-1548 (2013). https://www.science.org/doi/10.1126/science.1237861 \\\",\\\"10. F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, \\\\\\\"Overcoming the Rayleigh Criterion Limit with Optical Vortices,\\\\\\\" Phys. Rev. Lett. 97, 163903 (2006). https://link.aps.org/doi/10.1103/PhysRevLett.97.163903 \\\",\\\"11. Y. Kozawa, D. Matsunaga, and S. Sato, \\\\\\\"Superresolution imaging via superoscillation focusing of a radially polarized beam,\\\\\\\" Optica 5, 86-92 (2018). https://doi.org/10.1364/OPTICA.5.000086 \\\",\\\"12. C. Brabetz, S. Busold, T. Cowan, O. Deppert, D. Jahn, O. Kester, M. Roth, D. Schumacher, and V. Bagnoud, \\\\\\\"Laser-driven ion acceleration with hollow laser beams,\\\\\\\" Phys. Plasmas 22, 013105 (2015). https://doi.org/10.1063/1.4905638 \\\",\\\"13. A. Denoecd, L. Chopineau, A. Leblanc, and F. Quere, \\\\\\\"Interaction of Ultraintense Laser Vortices with Plasma Mirrors,\\\\\\\" Phys. Rev. Lett. 118, 033902 (2017). https://link.aps.org/doi/10.1103/PhysRevLett.118.033902 \\\",\\\"14. R. C. Devlin, A. Ambrosio, N. A. Rubin, J. P. B. Mueller, and F. Capasso, \\\\\\\"Arbitrary spin-to-orbital angular momentum conversion of light,\\\\\\\" Science 358, 896-901 (2017). https://www.science.org/doi/10.1126/science.aao5392 \\\",\\\"15. F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, P. B. Corkum, \\\\\\\"Controlling the orbital angular momentum of high harmonic vortices,\\\\\\\" Nat. Commun. 8, 14970 (2017). https://www.nature.com/articles/ncomms14970 \\\",\\\"16. D. Gauthier et al., \\\\\\\"Tunable orbital angular momentum in high-harmonic generation,\\\\\\\" Nat. Commun. 8, 14971 (2017). https://doi.org/10.1038/ncomms14971 \\\",\\\"17. Z. Xie, T. Lei, F. Li, H. Qiu, Z. Zhang, H. Wang, C. Min, L. Du, Z. Li, and X. Yuan, \\\\\\\"Ultra-broadband on-chip twisted light emitter for optical communications,\\\\\\\" Light Sci. Appl. 7, 18001 (2018). https://doi.org/10.1038/lsa.2018.1 \\\",\\\"18. N. C. Zambon et al., \\\\\\\"Optically controlling the emission chirality of microlasers,\\\\\\\" Nat. Photonics 13, 283-288 (2019). https://doi.org/10.1038/s41566-019-0380-z \\\"]\",\n \"page_idx\": \"4\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[527,75,918,914]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"19. L. Rego et al., \\\\\\\"Generation of extreme-ultraviolet beams with time-varying orbital angular momentum,\\\\\\\" Science 364, eaaw9486 (2019). https://doi.org/10.1126/science.aaw9486 \\\",\\\"20. X. Wang, Z. Nie, Y. Liang, J. Wang, T. Li, and B. Jia, \\\\\\\"Recent advances on optical vortex generation,\\\\\\\" Nanophotonics 7 (4), 1533-1556 (2018). https://doi.org/10.1515/nanoph-2018-0072 \\\",\\\"21. D. J. Kim, J. W. Kim, and W. A. Clarkson, \\\\\\\"High-power master-oscillator power-amplifier with optical vortex output,\\\\\\\" Appl. Phys. B 117, 459-464 (2014). https://doi.org/10.1007/s00340-014-5855-5 \\\",\\\"22. A. Mock, D. Sounas, and A. Alù, \\\\\\\"Tunable Orbital Angular Momentum Radiation from Angular-Momentum-Biased Microcavities,\\\\\\\" Phys. Rev. Lett. 121, 103901 (2018). https://link.aps.org/doi/10.1103/PhysRevLett.121.103901 \\\",\\\"23. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, \\\\\\\"Generation of optical phase singularities by computer-generated holograms,\\\\\\\" Opt. Lett. 17, 221-223 (1992). https://doi.org/10.1364/OL.17.000221 \\\",\\\"24. J. E. Curtis and D. G. Grier, \\\\\\\"Modulated optical vortices,\\\\\\\" Opt. Lett. 28, 872-874 (2003). https://doi.org/10.1364/OL.28.000872 \\\",\\\"25. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, \\\\\\\"Helical-wavefront laser beams produced with a spiral phaseplates,\\\\\\\" Opt. Commun., 112(5), 321-327, (1994). https://doi.org/10.1016/0030-4018(94)90638-6 \\\",\\\"26. L. Yan, P. Gregg, E. Karimi, A. Rubano, L. Marrucci, R. Boyd, and S. Ramachandran, \\\\\\\"Q-plate enabled spectrally diverse orbital-angular-momentum conversion for stimulated emission depletion microscopy,\\\\\\\" Optica 2, 900-903 (2015). https://doi.org/10.1364/OPTICA.2.000900 \\\",\\\"27. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, \\\\\\\"Astigmatic laser mode converters and transfer of orbital angular momentum,\\\\\\\" Opt. Commun. 96(1-3), 123-132 (1993). https://doi.org/10.1016/0030-4018(93)90535-D \\\",\\\"28. G. Xie, L. Li, Y. Ren, H. Huang, Y. Yan, N. Ahmed, Z. Zhao, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, \\\\\\\"Performance metrics and design considerations for a free-space optical orbital-angular-momentum-multiplexed communication link,\\\\\\\" Optica 2(4), 357-364 (2015). https://doi.org/10.1364/OPTICA.2.000357 \\\",\\\"29. A. Goffin, I. Larkin, A. Tartaro, A. Schweinsberg, A. Valenzuela, E. W. Rosenthal, and H. M. Milchberg, \\\\\\\"Optical Guiding in 50-Meter-Scale Air Waveguides,\\\\\\\" Phys. Rev. X 13, 011006 (2023). https://link.aps.org/doi/10.1103/PhysRevX.13.011006 \\\",\\\"30. H. Fathi, M. Narhi, R. Gumenyuk, \\\\\\\"Towards Ultimate High-Power Scaling: Coherent Beam Combining of Fiber Lasers,\\\\\\\" Photonics 8, 566 (2021). https://doi.org/10.3390/photonics8120566 \\\",\\\"31. S. J. McNaught et al., \\\\\\\"100-kW coherently combined Nd: YAG MOPA laser array,\\\\\\\" Frontiers in Optics 2009/Laser Science XXV/Fall 2009 OSA Optics & Photonics Technical Digest, OSA Technical Digest (CD) (Optica Publishing Group, 2009), paper FThD2 (2009). https://doi.org/10.1364/FIO.2009.FThD2 \\\",\\\"32. M. Müller, C. Aleshire, A. Klenke, E. Haddad, F. Légare, A. Tünnermann, and J. Limpert, \\\\\\\"10.4 kW coherently combined ultrafast fiber laser,\\\\\\\" Opt. Lett. 45, 3083-3086 (2020). https://doi.org/10.1364/OL.392843 \\\",\\\"33. H. Stark, J. Buldt, M. Müller, A. Klenke, A. Tünnermann, and J. Limpert, \\\\\\\"23 mJ high-power fiber CPA system using electro-optically controlled divided-pulse amplification,\\\\\\\" Opt. Lett. 44, 5529-5532 (2019). https://doi.org/10.1364/OL.44.005529 \\\",\\\"34. H. Chang, Q. Chang, J. Xi, T. Hou, R. Su, P. Ma, J. Wu, C. Li, M. Jiang, Y. Ma, and P. Zhou, \\\\\\\"First experimental demonstration of coherent beam combining of more than 100 beams,\\\\\\\" Photon. Res. 8, 1943-1948 (2020). \\\"]\",\n \"page_idx\": \"4\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[89,75,483,455]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"https://doi.org/10.1364/PRJ.409788 \\\",\\\"35. L. Wang, L. Wang, and S. Zhu, \\\\\\\"Formation of optical vortices using coherent laser beam arrays,\\\\\\\" Opt. Commun. 282(6), 1088-1094 (2009). https://doi.org/10.1016/j.optcom.2008.12.004 \\\",\\\"36. T. Hou, D. Zhi, R. Tao, Y. Ma, P. Zhou, and Z. Liu, \\\\\\\"Spatially-distributed orbital angular momentum beam array generation based on greedy algorithms and coherent combining technology,\\\\\\\" Opt. Express 26, 14945-14958 (2018). https://doi.org/10.1364/OE.26.014945 \\\",\\\"37. D. Zhi, T. Hou, P. Ma, Y. Ma, P. Zhou, R. Tao, X. Wang, and L. Si, \\\\\\\"Comprehensive investigation on producing high-power orbital angular momentum beams by coherent combining technology. High Power Laser Science and Engineering,\\\\\\\" Laser Science and Engineering. 7, E33(2019). https://doi.org/10.1017/hpl.2019.17 \\\",\\\"38. M. Veinhard, S. Bellanger, L. Daniault, I. Fsaifes, J. Bourderionnet, C. Larat, E. Lallier, A. Brignon, and J. C. Chanteloup, \\\\\\\"Orbital angular momentum beams generation from 61 channels coherent beam combining femtosecond digital laser,\\\\\\\" Opt. Lett. 46, 25-28 (2021). https://doi.org/10.1364/OL.405975 \\\",\\\"39. X. Guo, X. Zhang, D. Xu, W. Chen, Y. Guo, K. Lan, and B. Shen, \\\\\\\"Intense vortical-field generation using coherent superposition of multiple vortex beams,\\\\\\\" Sci Rep. 13, 1104 (2023). https://doi.org/10.1038/s41598-023-28216-9 \\\",\\\"40. J. Long, K. Jin, Q. Chen, H. Chang, Q. Chang, Y. Ma, J. Wu, R. Su, P. Ma, and P. Zhou, \\\\\\\"Generating the 1.5 kW mode-tunable fractional vortex beam by a coherent beam combining system,\\\\\\\" Opt. Lett. 48, 5021-5024 (2023). https://doi.org/10.1364/OL.502321 \\\"]\",\n \"page_idx\": \"5\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.48550/arxiv.2312.09824"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Physics"},"sha256":{"kind":"string","value":"ac8b87d442924b3b7ed0d17e856e4d92a26768d1148c74906e737eda3257b428"},"title":{"kind":"string","value":"Coherent beam combining of optical vortices"}}},{"rowIdx":50,"cells":{"abstract":{"kind":"string","value":"\n \n This article develops new tools and new statistical theory for a statistical\nproblem we call Scale Reliant Inference (SRI). Many scientific fields collect\nmultivariate data that lack scale: where the size, sum, or total of each\nmeasurement is arbitrary and is not representative of the scale of the\nunderlying system being measured. For example, in the analysis of\nhigh-throughput sequencing data, it is well known that the number of sequencing\nreads (the sequencing depth) varies substantially due non-biological\n(technical) factors. This article develops a formal problem statement for SRI\nwhich unifies problems seen in multiple scientific fields. Informally, we\ndefine SRI as an estimation problem in which an estimand of interest cannot be\nuniquely identified due to the lack of scale information in the observed data.\nThis problem statement represents a reformulation of the related field of\nCompositional Data Analysis and allows us to prove fundamental limits on SRI.\nFor example, we prove that inferential criteria such as consistency,\ncalibration, and bias are unattainable for common SRI tasks. Moreover, we show\nthat common methods often applied to SRI implicitly assume infinite knowledge\nof the system scale and can lead to a troubling phenomena termed unacknowledged\nbias. Counter-intuitively, we show that this problem worsens with more data and\ncan lead to substantially elevated Type-I and Type-II error rates. Still, we\nshow that rigorous statistical inference is possible so long as models\nacknowledge the fundamental uncertainty in the system scale. We introduce a\nclass of models we call Scale Simulation Random Variables (SSRVs) as flexible,\nrigorous, and computationally efficient approach to SRI.\n\n "},"author":{"kind":"string","value":"Lawrence A. David|Nicole A. Lazar|Justin D. Silverman|Jeffrey Letourneau|Sayan Mukherjee|Michelle Pistner Nixon"},"content_list":{"kind":"list like","value":[{"bbox":"[305,148,691,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Scale Reliant Inference ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[166,198,831,243]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Michelle Pistner Nixon1, Jeffrey Letourneau’ Lawrence A. David $_ 3$ 4 \nNicole A. Lazar; Sayan Mukherjee $^ { 4 6 } ,$ and Justin D. Silverman12478 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[459,281,537,297]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[166,304,828,659]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This article develops new tools and new statistical theory for a statistical problem we call Scale Reliant Inference (SRI). Many scientific fields collect multivariate data that lack scale: where the size, sum, or total of each measurement is arbitrary and is not representative of the scale of the underlying system being measured. For example, in the analysis of high-throughput sequencing data, it is well known that the number of sequencing reads (the sequencing depth） varies substantially due non-biological (technical) factors. This article develops a formal problem statement for SRI which unifies problems seen in multiple scientific fields. Informally,we define SRI as an estimation problem in which an estimand of interest cannot be uniquely identified due to the lack of scale information in the observed data. This problem statement represents a reformulation of the related field of Compositional Data Analysis and allows us to prove fundamental limits on SRI. For example, we prove that inferential criteria such as consistency, calibration,and bias are unattainable for common SRI tasks. Moreover，we show that common methods often applied to SRI implicitly assume infinite knowledge of the system scale and can lead to a troubling phenomena termed unacknowledged bias. Counter-intuitively, we show that this problem worsens with more data and can lead to substantially elevated Type-I and Type-II error rates. Still, we show that rigorous statistical inference is possible so long as models acknowledge the fundamental uncertainty in the system scale. We introduce a class of models we call Scale Simulation Random Variables (SSRVs) as flexible, rigorous, and computationally efficient approach to SRI. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[122,671,877,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1Equal contributions 2Collge of Information Sciences and Technology, Pennsylvania State University 3Department of Molecular Genetics and Microbiology, Duke University 4Center for Genomic and Computational Biology, Duke University 5Department of Statistics, Pennsylvania State University $_ 6$ Departments of Statistical Science,Mathematics, Computer Science, Biostatistics & Bioinformatics, Duke University 7Department of Medicine, Pennsylvania State University $^ { \\mathrm { ~ 8 ~ } }$ Corresponding author: JustinSilverman@psu.edu ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[22,256,62,722]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[491,829,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[491,829,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,85,890,122]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords: Multivariate Analysis, Identifiability, Uncertainty Quantification, Sequence Count Data ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[122,80,343,104]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,119,877,375]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This article develops new theory and tools for a statistical problem we call Scale Reliant Inference (SRI). Consider an unobserved physical system that can be represented as a positive-valued $D \\times M$ matrix $W$ with elements $W _ { d m }$ representing the amount of entity $d$ present in configuration $m$ .For example, in macro-ecology, $W _ { d m }$ might represent the abundance of species $d \\in \\{ 1 , \\dots , D \\}$ in forest $m \\in \\{ 1 , \\ldots , M \\}$ .We call $W$ a scaled system as it can be decomposed uniquely into scale and compositional components: $W _ { d m } =$ $W _ { d m } ^ { \\parallel } W _ { m } ^ { \\perp }$ where $W _ { d m } ^ { \\parallel }$ denotesthesi $\\begin{array} { r } { W _ { d m } ^ { \\parallel } = W _ { d m } / \\sum _ { d = 1 } ^ { D } W _ { d m } ) } \\end{array}$ of entities in configuration $m$ that are of type $d$ and $W _ { m } ^ { \\perp }$ denotes the scale (i.e.， sum total, $\\begin{array} { r } { W ^ { \\perp } = \\sum _ { d = 1 } ^ { D } W _ { d m } ) } \\end{array}$ of configuration $m$ .A scaled system is not directly observable but instead must be measured (e.g., sampled or surveyed). Yet due to limitations of the measurement process, these measurements often do not provide information about the system scale $W ^ { \\perp } = ( W _ { 1 } ^ { \\perp } , \\ldots , W _ { M } ^ { \\perp } )$ . Informally, SRI is an estimation problem in which an estimand of interest cannot be uniquely identified due to the lack of scale information in the observed data. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,376,877,702]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A major contribution of this work is the formulation of SRI which provides a unified representation of inferential problems seen in numerous scientific fields. In macro-ecology researchers often want to infer population dynamics of one or more animal species yet only have access to survey data that lacks scale information due to an inability to exhaustively survey all animals in a forest (e.g.， Murphy et al. (2016)). In demography, researchers may want to estimate the size of difficult to measure populations yet be limited to survey methods that do not exuastively survey the population leading to confounded or biased estimates (Rossi et al.， 1987). In the study of human gut microbiota, researchers may want to estimate pairwise covariances between each of $D$ bacterial species yet only be able to measure these communities using 16S rRNA sequencing where the total number of sequences measured in each sample is unrepresentative of total biological load in the sampled microbial communities (e.g., Paulson et al. (20l3)). In all three of these cases, the inferential problem can be framed as an example of SRI: as an estimation problem where the estimand of interest cannot be uniquely determined from the observed data due to the lack of scale information.By providing a formal definition of this problem framed in terms of probabilistic identifiability, we are able to provide new theoretical results on the limits of SRI. We show that for many common SRI tasks Frequentist criteria such as consistency, unbiasedness,and even calibration are impossible. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,704,875,795]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A second contribution of this work is to provide a unified language,and key theoretical constructs,to facilitate discussion of the many modeling assumptions commonly used to circumvent the challenges of SRI. These modeling assumptions are often implicit and poorly understood. As an example, consider the use of data normalizations common in the analysis of microbiome 16S rRNA sequencing data. Recognizing that the total number of 16S rRNA sequences in each sequenced sample (the sequencing depth) varies due to non-biological (i.e., technical) factors, many researchers choose to apply various data transformations intended to “normalize” the data and make “make the microbial abundances comparable across samples” Srinivasan et al. (2021). Often, the transformed data is then analyzed as if the measurements reflect direct measurements of microbial abundances: as if it contained both scale and compositional information. Our formulation of SRI allows us to provide new insights into the implications of these approaches, the assumptions behind them,and to study the sensitivity of study conclusions to errors in these assumptions. For example, we find that the common approach of data normalization in sequence count data analysis fails to acknowledge uncertainty in the system scale and that this lack of uncertainty leads to a troubling phenomena called unacknowledged bias which in turn can lead to substantially elevated Type-I error rates in practice Gustafson (2015). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[491,829,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,877,304]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,306,877,523]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Beyond studying modeling assumptions and proving fundamental limits on SRI, our problem formulation allows us to develop new statistical tools that respect those limits and produce statistically rigorous statistical inference that has both theoretical and practical advantages compared to existing methods. By framing SRI as a problem of identifiability, we are able to leverage existing literature on Bayesian Partially Identified Models (Bayesian PIMs) to develop models that acknowledge their own biases, reduce Type-I error rates. To address computational problems with posterior inference of Bayesian PIMs,we introduce a related class of models we term Scale Simulation Random Variables (SSRVs) which can often by sampled using direct Monte-Carlo simulation without the need to establish a Markov-Chain. In addition, SSRVs allow researchers to incorporate additional information about scale into analyses in ways that are possible with many existing tools. We show examples of how this additional information can even reduce Type-II error rates. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,525,877,705]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The structure of this work is as follows. In Section 2，we provide the definitions and basic results needed to formally state the problem of scale reliant inference and contrast this problem with problems studied in the related field of Compositional Data Analysis. In Section 3, we use the preceding problem formulation to prove fundamental limits on SRI in terms of consistency, calibration,and bias. In Section 4, we define Scale Simulation Random Variables (SSRVs). In Sections 4.1, 4.2, and 4.3, we provide key properties of SSRVs which illustrate their utility in the setting of SRI. In Section 5,we turn our attention to empirical and theoretical studies that contrast SSRVs with existing methods used for SRI in the context of both simulated and real data. Finally, we close with a discussion in Section 6 including a brief discussion of potential connections between our work and Causal Inference. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[491,829,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,80,465,104]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2Problem Formulation ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,118,877,301]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We define Scale Reliant Inference (SRI) as an estimation problem where our ability to uniquely determine the estimand of interest is limited due to a lack of scale information in the observed data. To build this definition，we first need to define two relationships between three constructs (Section 2.1): 1. the relationship between the system we are studying (first construct） and the quantity we wish to estimate (second construct） and 2. the relationship between the system we are studying and our measurements of the system (third construct). We call these three constructs the scaled system, target estimand, and compositional survey respectively. Together these two relationships then allow us to link our measurements to our estimand of interest and define scale reliant inference (Section 2.3). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,325,370,345]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.1 Core Constructs ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,356,875,410]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A scaled system is a representation of the physical system of scientifc interest. It is central to our problem formulation as it provides the context for the questions: “What is being measured?” and “What is being estimated?\". ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,422,877,478]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 1 (Scaled System). A scaled system $W \\in \\mathbf { W }$ isa $D$ -dimensional positive-valued quantity defined over a set of configurations $m \\in \\mathbf { M }$ such that $W _ { d m }$ denotes a scientifically meaningful amount of entity d in configuration $m$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,489,877,671]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To reflect the diverse goals of different studies in different fields of science, we purposefully leave the definitions of configurations, entities, and amounts flexible. While we use notation $W _ { d m }$ for elements of a scaled system,we do not even require that $W$ is a finite dimensional matrix. For concreteness,consider a researcher interested in human-microbe interactions who is studying how the abundance of different bacterial taxa at the epithelial surface of the colon change over time. Supposing this researcher will ultimately want to em-ploy a continuous-time model of these abundances, they might choose to represent the set of configurations $M$ as a continuous interval of time over which the study was performed, entities as a set of $D$ -different bacterial genera, and amounts as the average concentration per square centimeter of bacterial cells at the colonic epithelial surface. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,672,875,746]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Regardless of the chosen definitions of configuration，entity, and amount, $W$ can be uniquely decomposed into in terms of its scale (summed amounts) and composition (pro-portional amounts). The scale of the system ( $W ^ { \\perp } = ( W _ { 1 } ^ { \\perp } , \\ldots W _ { M } ^ { \\perp } ) )$ is an $M$ -vector repre-senting the summed amount of each entity in each configuration $m$ ： ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[429,758,565,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0141/9733d75de895ede0ac510a4c7dfa69b231ce56dcf0e9b7d43c6b0cb279029d43.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nW _ { m } ^ { \\perp } = \\sum _ { i = 1 } ^ { D } W _ { d m } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[491,829,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,875,122]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The composition of the system $( W ^ { \\parallel } )$ is a $D \\times M$ -dimensional quantity, defined by a $\\boldsymbol { D }$ -vector at each $m \\in \\mathbf { M }$ , representing the proportional amounts of each entity in each configuratin: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[382,128,614,169]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0392/c64bf2f0ad4a0b3b263f243bf2655c695dc85f0fa699483a1861161505819755.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nW _ { \\cdot m } ^ { \\parallel } = \\left( \\frac { W _ { 1 m } } { W _ { m } ^ { \\perp } } , \\ldots , \\frac { W _ { D m } } { W _ { m } ^ { \\perp } } \\right) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[122,172,875,208]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Together these definitions define the set of $D \\times M$ dimensional scaled systems as a Cartesian product space: $\\mathbf { W } = \\mathbf { W } ^ { \\parallel } \\times \\mathbf { W } ^ { \\perp }$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,210,875,300]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The scale of a system plays a special role in scientific inference. Note that the compo-sition of a system is a unit-less quantity, all the information defining the units of $W$ (the definition of amount) is contained in the system scale. The scale of the system therefore is intricately tied to the scientific goals of a study in a way that the composition is not. This feature also helps motivate our study of scale reliant inference. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,301,875,356]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The target estimand is the second construct of SRI and is the answer to the question “What is being estimated?\". It is defined through its relationship to the scaled system, a relationship called the target estimator. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,361,877,454]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 2 (Target Estimand and Estimator). Let $\\theta \\in \\Theta$ denote a target estimand: a quantity of scientific interest to a researcher studying a scaled system $W$ . The target estimand is an expression of the scaled system along with potential covariates $X$ such that it can be written as $\\theta = f ( W , X )$ where $f$ is a function $f : \\mathbf { W } \\times \\mathbf { X } \\to \\Theta$ called the target estimator. For notational simplicity, we will often exclude $X$ and simply write $\\theta = f ( W )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,460,872,497]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As an example, consider the following two target estimands which reflect common estimation problems in the analysis of sequence count data: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[313,505,681,556]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0420/aa14ebd5e47082bac781c1340fc5abdd335bd6648d025b15cc6c5b73420d29ab.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { \\begin{array} { l } { \\theta _ { i } = \\underset { m : x _ { m } = 1 } { \\mathrm { m e a n } } \\left( \\log W _ { d m } \\right) - \\underset { m : x _ { m } = 0 } { \\mathrm { m e a n } } ( \\log W _ { d m } ) } \\\\ { \\Sigma _ { d _ { 1 } , d _ { 2 } } = c o v ( \\log W _ { d _ { 1 } \\cdot } , \\log W _ { d _ { 2 } \\cdot } ) . } \\end{array} } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,560,877,671]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The first (Equation 3) can be equated to the Log-Fold-Change (LFC) of an entity (i.e., gene or microbe） between two biological conditions $x \\in \\{ 0 , 1 \\}$ . Estimation of LFCs is a central problem in gene expression and microbiome studies and is often called: differential abundance or differential expression analysis Fernandes et al. (2014). The second is the log-scale covariation between two entities which is a common statistic of interest in microecology Friedman and Alm (2012). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,671,872,707]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finally, a compositional survey is the measurement process used to study a scaled system. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,713,879,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3 (Compositional Survey). Let $\\mathcal { V } \\in \\mathbf { Y }$ denote a random variable describing the measurement of a scaled system $W$ such that we can denote by ${ \\mathcal { N } } _ { m }$ as a stochastic measurement of the system W at the configuration $m \\in \\mathbf { M }$ and $P _ { W } ( \\mathcal { V } )$ as the measurement model. V is a compositional survey if the re-parameterized model $P _ { W ^ { \\perp } , W \\parallel } ( \\mathcal { V } )$ identifies the composition of the system ( $W ^ { \\parallel }$ ） but not the scale $( W ^ { \\perp } )$ · ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[490,829,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,877,250]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In discussing compositional surveys, we will often refer to them as being scale limited, this term refer to the fact that they do not identify the system scale. We also distinguish between two types of compositional surveys: partial and full compositional surveys. For a full compositional surveys, the set $W ^ { \\perp }$ is fully unidentified by the measurement model: any value of $W ^ { \\perp } \\in \\mathbf { W } ^ { \\perp }$ could result in the same measurement model. For a partial compositional survey, while $W ^ { \\parallel }$ is not uniquely identified, there exists at least one value of $W ^ { \\perp } \\in \\mathbf { W } ^ { \\perp }$ that could not result in the same measurement model. In this article we primarily discuss fully compositional surveys and leave partial compositional surveys as a topic for future research. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,272,743,295]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2Invertibility as a Generalization of Identifiability ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,304,877,540]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our goal is to define Scale Reliant Inference (SRI) as an estimation problem where our ability to uniquely determine the estimand of interest is limited due to a lack of scale information in the observed data. The prior subsection already provided the relationship between a compositional survey and a scaled system (the measurement model, $P _ { W ^ { \\perp } , W \\parallel } ( \\mathcal { V } ) )$ and the relationship between a scaled system and a target estimand (the target estimator, $\\theta \\ = \\ f ( W ^ { \\perp } , W ^ { \\parallel } ) )$ .Moreover, the prior section has already expressed scale limitations of $\\mathcal { V }$ in terms of model identifiability. It may therefore seem natural to again use the concept identifiability to define the phrase “our ability to uniquely determine the estimand is limited\". For example, some readers may consider defining SRI as problem of inferring $\\theta$ when $\\theta$ is unidentifiable in a reparameterized model $P _ { f ^ { - 1 } ( \\theta ) } ( \\mathcal { V } )$ .Unfortunately, this approach is problematic. Many target estimands (e.g., Equations 3 and 4) are functions of $W ^ { \\parallel }$ and $W ^ { \\perp }$ without a unique inverse. Instead, we need to use to a more general notion of identifiability which we call invertibility. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,541,877,758]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A family of probability models $P _ { W } ( \\mathcal { V } )$ is a function $g$ linking a parameter space $\\mathbf { W }$ to a set of measures $\\mathbf { P }$ defined over the set $\\mathbf { Y }$ ： $P _ { W } ( \\mathcal { V } ) \\equiv g : \\mathbf { W } \\to \\mathbf { P }$ . From this perspective, the identifiability (or unidentifability) of a particular model $P \\in \\mathbf { P }$ is a statement on the size of the pre-image of the function $g$ . We denote the pre-image of $g$ as $g ^ { - 1 }$ If $g ^ { - 1 } ( P )$ is a singleton set，we say that the model $P$ (or colloquially the model parameter $W$ ）is identifiable. Otherwise, it is unidentifiable. The distinction between fully and partially unidentifiable depends on whether $g ^ { - 1 } ( P ) = \\mathbf { W }$ (fully unidentifiable） or $g ^ { - 1 } ( P ) \\subsetneq \\mathbf { W }$ (partially unidentifable). In fact, this pre-image is precisely the identification region of the model (Basse and Bojinov, 202O). Yet,by expressing identifability in this form,it is now natural to generalize this concept to arbitrary maps $h : \\mathbf { A } \\mathbf { B }$ . In the following definition, for a map $h : \\mathbf { A } \\mathbf { B }$ we use the notation $h ^ { - 1 } ( \\mathbf { b } )$ to represent the pre-image of the map at the set $\\mathbf { b } \\subset \\mathbf { B }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[125,770,872,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 4 (Invertibility). Consider a map $h : \\mathbf { A } \\mathbf { B }$ and a subset $\\mathbf { b } \\subseteq \\mathbf { B }$ ， we say that: 1. h is invertible at b if $h ^ { - 1 } ( \\mathbf { b } )$ is a singleton set. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[491,829,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[142,85,836,122]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. h is partially uninvertible at b if $h ^ { - 1 } ( \\mathbf { b } )$ is not a singleton set and $h ^ { - 1 } ( \\mathbf { b } ) \\subset \\mathbf { A }$ \n3. h is fully uninvertible at $\\mathbf { b }$ if $h ^ { - 1 } ( \\mathbf { b } )$ is not a singleton set and $h ^ { - 1 } ( \\mathbf { b } ) = \\mathbf { A }$ · ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,136,875,190]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Identifiability is therefore a special case of invertibility where the domain is a set of parameters, the co-domain is a set of probability measures, $h$ is a function, and our set $\\mathbf { b }$ is a singleton element. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,212,658,234]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.3Scale Reliant and Scale Reliant Inference ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[122,243,872,280]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We are finally in a position to define SRI as the problem of uniquely determining the target estimand given the scale limitations of a compositional survey. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,281,877,371]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From our three constructs we have two maps: the target estimand ( $f : \\mathbf { W } ^ { \\parallel } \\times \\mathbf { W } ^ { \\perp } \\to \\Theta$ ） and the map representation of the measurement model ( $g : \\mathbf { W } ^ { \\parallel } \\times \\mathbf { W } ^ { \\perp } \\to \\mathbf { P }$ where $\\mathbf { P }$ denotes a set of probability measures over the set $\\mathbf { Y }$ ). We define scale reliance in terms of the invertibility of the composite map $h = g \\circ f ^ { - 1 } : \\Theta \\to \\mathbf { P }$ where $f ^ { - 1 }$ denotes the pre-image map of $f$ · ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,383,877,438]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 5 (Scale Reliant Inference). Let $f$ and $g$ represent the maps defined by the target estimator and measurement model respectively and $h = g \\circ f ^ { - 1 }$ the composite map. Given a compositional survey $\\mathcal { V }$ with distribution $P \\in \\mathbf { P }$ ， we define inference of $\\theta \\in \\Theta$ as: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[143,440,812,493]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Scale Invariant Inference if $h$ is invertible at $P$ \n2. Partially Scale Reliant Inference if $h$ is partially uninvertible at $P$ \n3. Fully Scale Reliant Inference if $h$ is fully uninvertible at $P$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,506,877,616]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An important and unique feature of our definition of SRI is its reliance on the scientific question being asked. Quite intuitively，whether or not an inferential problem is scale reliant or scale invariant depends not only on the available data (through the term $g$ ）but also on the scientific question (through the term $f ^ { - 1 }$ ). As we discuss in Section 2.5, this feature is absent from the related but distinct feld of Compositional Data Analysis which can lead to problems when evaluating inference. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,638,486,660]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.4Tools for Establishing SRI ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,667,877,779]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The prior section defined SRI as the lack of invertibility of the composite map $h = g \\circ f ^ { - 1 }$ . However,it can be challenging to establish if a given estimation problem is scale reliant inference using that definition directly. Instead, this section introduces a simpler characterization of target estimands (whether the estimand is scale reliant or scale invariant) and then shows how this characterization can be used to easily establish when an estimation problem constitutes SRI. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[490,829,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"8 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,877,140]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Quite simply, we say a target estimand (or the target estimator) is scale reliant if fixing the value of $W ^ { \\parallel }$ is insufficient to uniquely determine $\\theta$ . The following definitions formalizes this notion. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,152,879,224]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 6 (Scale Reliance of Target Estimands and Estimators). For a target estimand and estimator $\\theta \\ : = \\ : f ( W ^ { \\parallel } , W ^ { \\perp } )$ ， $f _ { W ^ { \\parallel } }$ denote a restriction of the target estimator to the domain $W ^ { \\parallel } \\times \\mathbf { W } ^ { \\perp }$ for a particular value of $W ^ { \\| } \\in \\mathbf { W } ^ { \\| }$ . The following three statements are equivalent: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,226,879,373]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. $f$ is a scale invariant target estimator and $\\theta$ is a scale invariant target estimand. 2. $f _ { W ^ { \\parallel } }$ is invertible at $W ^ { \\parallel }$ 3. $A t \\ W ^ { \\parallel }$ there exists a single unique value of $\\theta \\in \\Theta$ that satisfies $\\theta \\ : = \\ : f ( W ^ { \\parallel } , W ^ { \\perp } )$ regardless of the value of $W ^ { \\perp } \\in { \\bf W } ^ { \\perp }$ \nIf $f$ (and $\\theta$ ) are not scale invariant, then they are scale reliant. There are two types of \nscale reliant target estimators (and estimands). ·If $f _ { W ^ { \\parallel } }$ is partially uninvertible then $f$ (and $\\theta$ ） are partially scale reliant. · If $f _ { W ^ { \\parallel } }$ is fully uninvertible then $f$ (and $\\theta$ ) are fully scale reliant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,385,877,530]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Using this definition, Table 1 summaries a number of propositions and proofs provided in Supplementary Section B.2 which categorize inference as scale invariant, partially scale reliant, or fully scale reliant, depending on whether the data comes from a partial or full compositional survey and whether the target estimand is scale invariant， partially scale reliant,or fully scale reliant. The combination of partially compositional surveys with a partially or fully scale reliant target estimand is substantially more complicated and can induce a composite map $h$ that is invertible,partially uninvertible,or fully uninvertable depending on the specific target estimand and the specific measurement model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,531,877,622]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As examples, note that the target estimands (LFC and Covariance estimands) provided in Equations 3 and 4 are fully and partially scale reliant respectively (proofs provided in Supplementary Section B.2). For example, for LFC estimation we can place no restrictions on the set of possible values of $\\theta \\in ( - \\infty , \\infty )$ using knowledge of the system compositional alone.In contrast consider that a target estimator of the form ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[401,633,591,674]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0340/92c56002ae1660a421a54e5f3f715f46bd1c0a025404196a5c25c2f8da0b8c9f.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\zeta _ { i } = \\operatorname* { m e a n } _ { m } \\left( \\log \\frac { W _ { d m } } { W _ { j m } } \\right)\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,688,446,722]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is scale invariant as $\\begin{array} { r } { \\log { \\frac { W _ { d m } } { W _ { j m } } } = \\log { \\frac { W _ { d m } ^ { \\parallel } } { W _ { j m } ^ { \\parallel } } } } \\end{array}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,739,704,762]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.5 Connections to Compositional Data Analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,770,875,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"SRI can be understood as a reformulation and refocusing of Compositional Data Analysis (CoDA). Both SRI and CoDA focus on the challenge of analyzing of multivariate data in the absence of scale information yet they define this focus in very different ways. The prior section shows how SRI defnes this challenge through two relationships, both centered around the system being studied: the target estimator that defines the scientific goal, and the measurement model that describes the observed data. In contrast, the core of CoDA does not have any notion of the system being studied. Rather than centering on the system, CoDA is centered on the observed data which it defines mathematically as simplicial vectors (vectors $x \\in \\mathbb { R } ^ { D + }$ that sum to a constant value). In fact，while early work on CoDA discussed the notion of a base-scale, a construct that is conceptually similar to SRI's scaled systems, the base-scale is introduced largely as an special-topic and is neither a central part of CoDA theory nor is part of the central problem formulation of CoDA (Aitchison,1986, Chapter 9). Without the emphasis on the scaled system, it is also not possible for CoDA to defne a construct analogous to SRI's target estimands and estimators: the scientific goal is not an integral part of CoDA theory. Instead, the definition of the estimation task is typically discussed in the context of an applied model $P _ { \\theta } ( Y )$ or an applied estimator ${ \\hat { \\theta } } = r ( Y )$ : constructs that directly relate and estimand with the data. As a result, applied estimators (or models) typically serve two duties in CoDA: they both define the estimation task and define an approach to estimation. In contrast, in SRI we distinguish between the estimation task (the target estimand and estimator) and the approach to estimating the target estimand (the applied estimator). The separation of these two concepts is necessary in order to establish the limits on inference and to properly evaluate applied estimators. In fact, challenges arise when trying to evaluate applied estimators from a CoDA perspective. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[490,829,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"9 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,83,892,234]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0400/38ded8f9154a52ef9093a119ddf2a27639cd75ec05db378edb40010653a59898.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":"<table><tr><td></td><td>PropositionCompositional Survey (V)</td><td>Target Estimand (0)</td><td>Inference of 0 from observations of Y</td></tr><tr><td>D.3</td><td>Full</td><td>Scale Invariant</td><td>Scale Invariant</td></tr><tr><td>D.6</td><td>Full</td><td></td><td>Partially Scale ReliantPartially Scale Reliant</td></tr><tr><td>D.5</td><td>Full</td><td>Fully Scale Reliant</td><td>Fully Scale Reliant</td></tr><tr><td>D.3</td><td>Partial</td><td>Scale Invariant</td><td>Scale Invariant</td></tr><tr><td></td><td>Partial</td><td>Partially Scale Reliant</td><td></td></tr><tr><td></td><td>Partial</td><td>Fully Scale Reliant</td><td></td></tr></table>","table_caption":"[]","table_footnote":"[\"Table 1: Inference of $\\\\theta$ depending on the type of the data (partial or full compositional survey) and target estimand (scale invariant or scale reliant). \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[119,306,877,689]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,690,875,799]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In SRI, the target estimand gives context to the question “what is being estimated?\" and thereby forms a foundation on which an applied estimand can be evaluated. Without an analogous construct, CoDA instead proposes three axioms (or principles) that it argues any applied estimator of compositional data should satisfy (Pawlowsky-Glahn et al., 2015). The most important of these three is the principle of scale inuariance which is nicely summarized by Lovell et al. (2020): ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[488,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[166,85,828,159]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Scale Invariance is a fundamental principle of CoDA. Put simply, scale invariance means that the functions we use to analyze compositions should return identical results when we scale compositions by multiplying all components by a constant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[130,171,834,189]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"'he following example helps to illustrate the problems with this axiomatic approach. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,190,877,356]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Consider an ecologist studying the effect of an invasive beetle on three different species of plants in 10 different forests (half with, $x _ { m } = 1$ , and half without, $x _ { m } = 0$ , the beetle). Suppose that，due to experimental limitations, the ecologist can only measure the pro-portional abundance of each of the plant species in each forest, leading to measurements $Y _ { \\cdot 1 } , \\dots , Y _ { \\cdot D } \\in \\mathbb { S } ^ { D }$ constrained to the $D$ -dimensional simplex. For rhetorical simplicity, let us pretend there is no measurement noise such that $Y _ { \\cdot m } = W _ { \\cdot m } ^ { \\parallel }$ .Familiar with the field of CoDA,this ecologist decides to first apply the Centered Log-Ratio (CLR) transform to these measurements Wdm $W _ { d m } ^ { \\parallel * } = W _ { d m } ^ { \\parallel } / G ( W . m )$ (where $G ( \\cdot )$ denotes the geometric mean) and then for each species quantify the effect of the invasive species as a difference in means: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[328,367,666,424]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0054/86defdf1bb4ff4ee047940fafeca9045e625f5831632190efa0dc80def77533f.jpg","list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\hat { \\theta } _ { d } = \\underset { m : x _ { m } = 1 } { \\mathrm { m e a n } } ( \\log W _ { d m } ^ { \\parallel * } ) - \\underset { m : x _ { m } = 0 } { \\mathrm { m e a n } } ( \\log W _ { d m } ^ { \\parallel * } ) } \\\\ & { \\quad = r ( W ^ { \\parallel } ) . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,434,877,563]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This applied estimator is scale invariant per the CoDA definition: $r ( Y ) = r ( \\alpha Y )$ for any constant $\\alpha \\in \\mathsf { \\Gamma } ( 0 , \\infty )$ . In fact, this estimator even abides by CoDAs two other defining principles (permutation invariance and sub-compositional coherence） (Pawlowsky-Glahn et al., 2015). Yet,this is not sufficient to argue that this is a principled or rigorous analysis of these data. Evaluating this applied estimator (i.e., determining if it is principled, rigorous, or useful) must depend on the precise goals of the researcher: what precisely does the researcher want to estimate? The language of SRI allows us to discuss this issue clearly. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,564,877,763]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Suppose the researcher's target estimand is given by Equation 6: they want to under-stand how the invasive beetle changes the mean CLR normalized abundance of each species. Then researchers applied estimator is clearly a principled analysis of these data as the ap-plied methods are guaranteed to exactly output the target estimand. Suppose instead that the researcher's target estimand is given by Equation 3: they care about how the actual abundance of each species changes in forests with or without the beetle and not the change in CLR normalized abundance. In this case, the target estimand is (fully) scale reliant yet somehow the applied estimator is scale invariant.1 In fact, the assumption is that what is being estimated by the applied estimator is answering the scientifc question implied by the target estimand, that $\\theta _ { d } = \\hat { \\theta } _ { d }$ . With minimal algebra this assumption can be reduced to an assumption that the diference in mean log geometric means of the compositions is exactly equal to the difference in mean log scales: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[122,775,872,805]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1other than the qualifier /fully/ which is unique to SRI, this sentence holds from the perspective of the SRI definition of scale reliance/invariance of an estimator or the CoDA definition. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[488,829,506,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"11 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,537,104]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[160,117,816,145]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0042/e6cede6b370a5931f3e5bebca423794d38ddf39125f3cf71919da62289e908d1.jpg","list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { m e a n } _ { m : x _ { m } = 1 } ( \\log W _ { d m } ^ { \\perp } ) - \\operatorname* { m e a n } _ { m : x _ { m } = 0 } ( \\log W _ { d m } ^ { \\perp } ) = \\operatorname* { m e a n } _ { m : x _ { m } = 1 } ( g ( \\log W _ { \\cdot m } ^ { \\parallel } ) ) - \\operatorname* { m e a n } _ { m : x _ { m } = 0 } ( g ( \\log W _ { \\cdot m } ^ { \\parallel } ) ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[122,159,877,194]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We call this the CLR assumption and will return to it later in Section 5.2, when we show this same assumption is at play in popular tools used to analyze Sequence Count Data. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,195,877,323]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This example illustrates how SRI allows for a degree of subtlety, a consideration of the scientific question, not present in CoDAs axiomatic approach. The next section emphasizes this subtlety and shows how more well established concepts such as consistency, calibration, and bias can be used within SRI to establish the limits of inference and evaluate estimators instead of CoDAs axioms. Moreover, in later sections we show how SRI can even provide a framework for showing the functional consequences of violating modeling assumptions and using applied estimators that may be inconsistent with the scientific goal. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,348,620,373]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3Limit on Scale Reliant Inference ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,388,875,424]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"With our problem formulation, we can now state and prove key results on the limits on scale reliant inference in terms of consistency, calibration, and bias. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,425,877,570]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $\\theta = f ( W )$ denote a target estimand and estimator pairing, and let $P _ { W } ( \\mathcal { V } )$ denote a measurement model for a compositional survey $\\mathcal { V }$ . The following results are stated in terms of the composite map $h = g \\circ f ^ { - 1 }$ where $g$ denotes the mapping $g \\equiv P _ { W } ( \\mathcal { V } ) : \\mathbf { W } \\to \\mathbf { P }$ .In Supplementary Section B.3, we prove the following statements: (1） that invertibility is a necessary condition for consistency, (2) that invertibility is a necessary condition for unbi-asedness,and (3) that full uninvertability is incompatable with the existence of calibrated confidence sets. Furthermore, we show that those three results imply the following three limits on scale reliant inference. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[122,582,874,619]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 1 (Consistency). For partially or fully scale reliant inference， there does not exist a sequence of estimators for θ that is consistent. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[122,631,870,667]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2 (Calibration). For fully scale reliant inference, there does not exist a calibratec confidence set for $\\theta$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[122,680,867,699]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3 (Bias). For partially or fully scale reliant inference, any estimator is biased. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,712,875,766]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"These results highlight the difficulty of scale reliant inference. In fact, they show that Frequentist ideals such as consistency， calibration， and unbiasedness are fundamentally impossible for fully scale reliant inference. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[488,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"12 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,82,787,107]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4Scale Simulation Random Variables (SSRVs) ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,121,877,212]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The previous section established limits on Frequentist ideals such as unbiasedness, consistency, and calibration in scale reliant inference. In this section, we develop Scale Simulation Random Variables (SSRVs) which operate under a Bayesian inference paradigm. We show that these random variables satisfy weaker Bayesian notions of calibration and are closely related to a class of models called Bayesian Partially Identified Models (Bayesian PIMs). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,213,877,449]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In moving towards a Bayesian viewpoint, we now treat $W$ as a random quantity (which we emphasize with script notation $\\mathcal { W }$ ). Similarly, we wil sometimes discuss a compositional survey as a fixed (i.e.， observed, denoted $Y$ ）rather than random (denoted $\\mathcal { V }$ ）quantity. Following this shift in viewpoint，we will switch from using notation such as $P _ { W } ( \\mathcal { V } )$ to instead using conditional probability or sampling notation $P ( \\mathcal { P } | W )$ or $\\mathcal { V } \\sim P ( W )$ ，either of which are more common in the Bayesian literature. We will start to denote probability densities or probability mass functions with a lowercase $p$ as opposed to the uppercase $P$ which we reserve for measures. Finally, we will denote with an $( a )$ (e.g, $p ^ { ( a ) }$ ）probability models specified by an analyst and contrast that to the true distribution of a random quantity which we denote with a dagger (e.g., $p ^ { \\dagger }$ ). In particular,we call $p ^ { \\dagger } ( \\mathcal { V } | W )$ the target measurement model which represents the true relationship between the distribution of $\\mathcal { V }$ and the value of $W$ . When we wish to explicitly refer to the true distribution of $\\mathcal { V }$ ， which is conditioned on the true value of $W$ ， we will denote this as $p ^ { \\dagger } ( \\mathcal { V } | W ^ { \\dagger } )$ · ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,460,879,570]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 7 (Scale Simulation Random Variables). Let $\\theta = f ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } )$ denote a tar-get estimand and estimator pairing. Let $p ^ { ( a ) } ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\parallel } )$ denote an applied scale model and $p ^ { ( a ) } ( \\mathcal { W } ^ { \\parallel } | Y )$ denote an applied measurement model for an observed compositional survey $Y$ . We define a scale simulation random variable $\\theta$ as the transformation of the random variable $( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } )$ via the target estimator. This can be written in hierarchical notation $a s$ ： ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[419,582,576,656]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0582/23d58efb6c9303f7af3cd12da0680e0f860b325aadcc1372c836bc4414627672.jpg","list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { c } { { \\theta = f ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } ) } } \\\\ { { \\mathcal { W } ^ { \\perp } \\sim p ^ { ( a ) } ( \\mathcal { W } ^ { \\parallel } ) } } \\\\ { { \\mathcal { W } ^ { \\parallel } \\sim p ^ { ( a ) } ( Y ) } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,667,663,686]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"or alternatively as an integral transformation for any set $\\mathbf { A } \\subset \\Theta$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[212,698,782,738]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0130/c14e97008cef98ccdb47afb5e5b7bbc82d348312c48d1abee14dfa3346394e38.jpg","list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nP ^ { ( a ) } ( \\theta \\in \\mathbf { A } | Y ) = \\int _ { ( \\mathcal { W } ^ { \\bot } , \\mathcal { W } ^ { \\| } ) \\in f ^ { - 1 } ( \\mathbf { A } ) } p ^ { ( a ) } ( \\mathcal { W } ^ { \\bot } | \\mathcal { W } ^ { \\| } ) p ^ { ( a ) } ( \\mathcal { W } ^ { \\| } | Y ) d \\mathcal { W } ^ { \\| } d \\mathcal { W } ^ { \\bot } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,770,874,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Less formally, Scale Simulation Random Variables (SSRVs) are random variables defined over the same space as a target estimand that capture our uncertainty in both the system scale and the system composition. The name scale simulation random variables comes from the fact that they can be readily simulated by Monte-Carlo Simulation without the need to establish a Markov Chain (See Supplementary Algorithm 1). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[488,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"13 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,875,140]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,142,875,213]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A special quantity that we will refer to throughout this work is the Scale-Conditional Random Variable (SCRV） which can be thought of as an SSRV conditional on $\\mathcal { W } ^ { \\perp }$ ,i.e., without incorporating uncertainty from the applied measurement model. Using the above notation, the SCRV can be defined as: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[235,227,754,267]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0160/6e781f0c5486a2e574588e6ac14dc88a295170d282864deae5877fc7d4d65362.jpg","list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nP ^ { ( a ) } ( \\theta \\in \\mathbf { A } | Y , W ^ { \\perp } ) = \\int _ { ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } ) \\in f ^ { - 1 } ( \\mathbf { A } ) : \\mathcal { W } ^ { \\perp } = W ^ { \\perp } } p ^ { ( a ) } ( \\mathcal { W } ^ { \\parallel } | Y ) d \\mathcal { W } ^ { \\parallel } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,279,874,314]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Later in this work we will show how existing applied estimators such as ALDEx2 can be related to the framework of SSRVs by equating them to SCRVs. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[122,315,870,348]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The mean of SSRVs, which we will denote as $\\bar { \\theta }$ have an appealingly simple and intuitive form: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[331,349,665,372]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0248/ee41a7ce9a260f13a9f5b620429a37a8c7f9e1d1aaef89c20cd0c871e1ef27fa.jpg","list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { \\overline { { \\theta } } = \\mathbb { E } _ { p ( \\mathcal { W } ^ { \\perp } \\mid \\mathcal { W } ^ { \\parallel } ) } \\left[ \\mathbb { E } _ { p ( \\mathcal { W } ^ { \\parallel } \\mid Y ) } \\left[ f ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } ) \\right] \\right] . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,377,875,433]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This is simply the expectation of the target estimand with respect to the measurement and scale models. Moreover, $\\tilde { \\theta }$ can be readily estimated as the mean of Monte-Carlo samples obtained from Supplementary Algorithm 1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[122,455,787,477]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4.1Relationship to Bayesian Partially Identified Models ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,484,877,651]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bayesian Partially Identified Models (Bayesian PIMs) are a special class of Bayesian hierarchical models where the likelihood identifies only a subset of the parameters that the prior is defined over (Gustafson, 2O15). In the present context, the likelihood is defined with respect to parameters $\\begin{array} { r } { \\mathcal { W } ^ { \\| } \\left( p ( \\boldsymbol { Y } | \\mathcal { W } ^ { \\| } ) \\right) } \\end{array}$ while the prior is defined over the possible values of both $\\mathcal { W } ^ { \\parallel }$ and $\\mathcal { W } ^ { \\parallel }$ ， $( p ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } ) )$ .These models are remarkable in that they maintain their calibration properties even in the setting of partially identified inference,a feature that many other models lack (Gustafson, 2015). The following proposition establishes that SSRVs can be seen as the transformed posterior distribution of a Bayesian PIMs (proof provided in Supplementary Section C.1). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,661,879,791]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 1. Let $p ( \\mathcal { W } ^ { \\perp } , \\mathcal { W } ^ { \\parallel } | Y )$ denote the posterior of a Bayesian Partially Identified model with likelihood $p ( Y | \\mathcal { W } ^ { \\| } )$ and prior $p ( \\mathcal { W } ^ { \\perp } , \\mathcal { W } ^ { \\parallel } )$ Let $\\theta = f ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } )$ denote a tar-get estimand and estimator pairing. The transformation of the posterior distribution for $( W ^ { \\parallel } , W ^ { \\perp } )$ via the target estimator is a scale simulation random variable with the measure-ment model corresponding to the marginal posterior distribution $p ( W ^ { \\parallel } | Y )$ ， and scale model corresponding to the conditional prior $p ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\parallel } )$ which is equal to the conditional posterior $p ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\parallel } , Y )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[488,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"14 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,877,323]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Rather than introducing SSRVs as transformations of Bayesian PIMs, we instead introduce them as in Definition 7 for two reasons. First, as we show in Supplementary Algorithm 1, the modular specifcation of SSRVs tends to produce models that are readily amenable to Monte-Carlo simulation-based inference without the need to establish a Markov Chain. This is in sharp contrast to Bayesian PIMs which do not generally permit direct Monte-Carlo simulation. This computational challenge is often exacerbated by highly anisotropic posteriors that arise due to the only partial updating of the prior (Gustafson, 2O15). Second,we find it simpler to specify these estimators modularly. Since the data $Y$ provides little to no information about the scale of the system, we believe that specifying uncertainty over $\\mathcal { W } ^ { \\perp }$ should occupy a dedicated part of the model design process. Such modularity also enhances reproducibility as all assumptions about $\\mathcal { W } ^ { \\perp }$ are clearly stated in the scale model $p ( \\mathcal { W } ^ { \\perp } | W ^ { \\parallel } )$ . In contrast, specifying a prior over $p ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } )$ may involve less obvious assumptions about the conditional relation between $\\mathcal { W } ^ { \\parallel }$ and $\\mathcal { W } ^ { \\perp }$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,345,387,367]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4.2SSRV Calibration ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,377,877,686]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"While we have already shown that Frequentist calibration is unattainable for fully scale reliant inference, the relationship between SSRVs and Bayesian PIMs implies that SSRVs enjoy the same Bayesian calibration properties as Bayesian PIMs. Beyond the notion that a Bayesian $1 - \\alpha$ credible interval is calibrated in a subjective probability sense, Bayesian credible intervals have a type of calibration that connects to Frequentist concepts of coverage. For example, Gustafson (2015) (pg. 41) shows that for any Bayesian $( 1 - \\alpha )$ posterior credible interval, the expected probability that the true parameter value exists in the interval is exactly equal to $( 1 - \\alpha )$ . While this Bayesian calibration is comparable to the notion of Frequentist coverage, it is weaker: this coverage is only guaranteed on average where the averaging is weighted according the prior distribution. Connecting this back to SSRVs, we can say that any $( 1 - \\alpha )$ probability interval of an SSRV enjoys the typical notion of Bayesian calibration in terms of subjective probability as well as this weaker form of Frequentist calibration. Still， while these calibration properties are appealing, they are not our primary motivation for developing SSRVs (or alternatively introducing Bayesian PIMs) for scale reliant inference: that motivation is best highlighted through the asymptotic properties of SSRVs which show how they can help mitigate inferential errors despite the inevitability of a large-sample bias. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,710,560,731]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4.3Asymptotic Properties of SSRVs ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,741,875,795]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Classically, the variance of an estimator (including the variance of a Bayesian posterior) decreases monotonically with increasing data, approaching zero in the asymptotic limit. Yet on an intuitive level, such behavior is troubling in the setting of scale reliant inference where limitations of the data place fundamental limits on our knowledge. Especially considering that any estimator for a target estimand $\\theta$ in the setting of scale reliant inference must be biased (Theorem 3),an asymptotic variance that tends to zero despite a non-zero bias clearly points to a substantial risk for Type I errors. This combination of non-zero bias in the setting of zero asymptotic variance has been called an “unacknowledged bias” in the Bayesian PIM literature (Gustafson, 2015, pg. 35). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[488,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"15 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,877,195]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,196,877,321]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section, we show that,unlike classical models, SSRVs do not suffer from such an unacknowledged bias: so long as the scale model is not degenerate, the asymptotic variance of a SSRV is non-zero reflecting the limits on inference in the setting of uninvertability. In fact, we show that the scale model of an SSRV can be thought of as providing a model for the bias of the SSRV mean and lead to an asymptotic variance that is directly tied to the modeled bias. This leads us to say that SSRVs demonstrate an acknowledged rather than unacknowledged bias. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,325,877,578]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The following results should come as no surprise to readers familiar with the study of Bayesian PIMs where nearly identical properties are wel established. In fact, our treatment of the asymptotic properties of SSRVs is modeled after the discussion of unacknowledged versus acknowledged bias in Gustafson (2015). Still, we include this presentation to highlight these properties and to provide conditions under which identical conditions hold for SSRVs. In particular, to ensure the correspondence between Bayesian PIMs and SSRVs we must placed mild regularity conditions on the target estimator. In brief,we assume that target estimator is bounded and that its expectation with respect to the joint distribution $P ( Y , W ^ { \\parallel } , W ^ { \\perp } )$ exists (See Supplementary Section C.2 for formal statement). With these conditions,we can state the asymptotic properties in a concise form analogous to the results presented in Gustafson (2015) (pg. 52). In addition, to avoid muddling our presentation with non-scale-related sources of bias and unidentifiability, we assume that the measurement model is faithful such that in the asymptotic limit it converges to a point mass about the true value of the system composition. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,580,875,689]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We use the following short-hand notations to state our results concisely. We denote the true value of the target estimand as $\\theta ^ { \\dagger } \\equiv f ( W ^ { \\parallel \\dagger } , W ^ { \\perp \\dagger } )$ , the true measurement model $p ^ { \\dagger } \\equiv p ^ { \\dagger } ( \\mathcal { V } | W ^ { \\parallel \\dagger } , W ^ { \\perp \\dagger } )$ ， and the joint of the applied measurement and scale models $p ^ { ( a ) } \\equiv$ $p ^ { ( a ) } ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } | \\mathcal { V } ) = p ^ { ( a ) } ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\parallel } ) p ^ { ( a ) } ( \\mathcal { W } ^ { \\parallel } | \\mathcal { V } )$ . Finally, we will use the following notation for bias and variance which emphasizes the model with which expectations are taken: for an estimator $\\hat { \\phi }$ and a model $P$ we define $\\mathrm { b i a s } _ { P } ( \\hat { \\phi } ) = \\mathbb { E } _ { P } [ \\hat { \\phi } ] - \\phi ^ { \\dagger }$ and $\\mathrm { v a r } _ { P } ( \\hat { \\phi } ) = \\mathbb { E } _ { P } [ ( \\hat { \\phi } - \\mathbb { E } _ { p } ( \\hat { \\phi } ) ) ^ { 2 } ]$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,689,875,743]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We first show that the asymptotic bias with respect to the true measurement model $( p ^ { \\dagger } )$ (the Frequentist bias)is equal to the bias calculated with respect to the posterior $p ^ { ( a ) }$ (the Bayesian bias). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,752,875,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 4. Let $\\theta = f ( \\mathcal { W } )$ denote a target estimand and estimator pairing. Let $\\theta$ denote a faithful scale simulation estimator that satisfies the regularity conditions provided in Definition E.2. Then the asymptotic bias of $\\overline { { \\theta } }$ with respect to the target measurement model $p ^ { \\dagger }$ or with respect to the applied measurement and scale models $p ^ { ( a ) }$ are equal ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[488,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"16 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[117,85,745,104]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[284,117,714,145]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0029/d86c0a5c7db5a1fbd19532ef64ad558220d4033aa4f01440d25b56b4cbd93e61.jpg","list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { l i m } _ { n \\to \\infty } \\operatorname { b i a s } ( { \\overline { { \\theta } } } | W ^ { \\perp \\dagger } ) = \\operatorname* { l i m } _ { n \\to \\infty } \\operatorname { b i a s } _ { p ^ { \\dagger } } ( { \\overline { { \\theta } } } ) = \\operatorname* { l i m } _ { n \\to \\infty } \\operatorname { b i a s } _ { p ^ { ( a ) } } ( { \\overline { { \\theta } } } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,157,230,175]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and given by ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[236,188,761,217]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0732/fc06c512bddbbbf3c6263ad3ef9ef2561b9283dfca6d43ab4d301374657cca8b.jpg","list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { l i m } _ { n \\to \\infty } b i a s ( \\overline { { { \\theta } } } | W ^ { \\perp \\dagger } ) = \\mathbb { E } _ { p ^ { ( a ) } ( \\mathcal { W } ^ { \\parallel } | W ^ { \\parallel \\dagger } ) } [ f ( W ^ { \\parallel \\dagger } , \\mathcal { W } ^ { \\perp } ) ] - f ( W ^ { \\parallel \\dagger } , W ^ { \\perp \\dagger } ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,229,875,304]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our notation $\\scriptstyle \\operatorname* { l i m } _ { n \\to \\infty } \\operatorname { b i a s } ( \\theta | W ^ { \\perp \\dagger } )$ introduced the condition on $W ^ { \\perp \\dagger }$ in order to emphasize which parts of this bias required knowledge of the true value of the system scale to calculate. Next, we show that the scale model provides a model for this bias and dictates the asymptotic variance. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,315,874,353]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 5. The asymptotic variance of a faithful scale simulation random variable under regularity conditions provided in Definition E.2 is: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[267,366,730,393]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0279/917bb660860d4a4dda69eca903297ec7a960566a0963ae3fe6ea73169f3c45c1.jpg","list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { l i m } _ { n \\to \\infty } \\operatorname { v a r } _ { p ^ { ( a ) } ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } | Y _ { n } ) } ( \\theta ) = \\mathbb { E } _ { p ^ { ( a ) } ( \\mathcal { W } ^ { \\perp } | W ^ { \\parallel \\dagger } ) } \\left[ \\operatorname { b i a s } ( \\bar { \\theta } | \\mathcal { W } ^ { \\perp } ) ^ { 2 } \\right] .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,405,877,534]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This result establishes what we mean when we say that SSRVs acknowledge their own bias: while the mean of an SSRV is a biased point estimator, the variance of an SSRV depends on the bias. As the true value of $W ^ { \\perp }$ is not known, the SSRV models its own bias based on beliefs about the system scale encoded in the scale model. So long as the scale model has non-zero variance (is not degenerate), the asymptotic variance of an SSRV will be non-zero - acknowledging its bias. Next, we show that common models used to analyze sequence count data do not acknowledge their bias. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,559,875,613]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5Empirical Studies of SSRVs applied to Sequence Counting ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,626,877,790]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section, we show that common models used in the analysis of sequence count data can demonstrate elevated rates of Type-1 errors due to unacknowledged bias. We then focus our attention on a single model, ALDEx2, in order to provide intuition behind these results and explain why they should be expected from many currently used methods. To strengthen our interpretation of these results, we show how the ALDEx2 model can be viewed as a SSRV and modified to acknowledge its bias and reduce false positives. In fact, through applications to both real and simulated data, we show that the flexibility provided by SSRVs even allows us to incorporate additional knowledge about the system scale that can reduce Type-II errors as well. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[488,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"17 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,875,140]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"17","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For concreteness and brevity, in this section we present al results and discusson relative to a population-level (as opposed to sample-level） version of the LFC estimand given in Equation 3: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[330,141,666,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0382/a033ede5dfad437ebe464724bf4358e36cd101bb62113537c238c078320d0fb4.jpg","list_items":null,"page_idx":"17","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\theta _ { d } = \\mathbb { E } [ \\log W _ { d m : x _ { m } = 1 } ] - \\mathbb { E } [ \\log W _ { d m : x _ { m } = 0 } ] .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,167,874,204]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"17","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We switch to a population based estimator to be consistent with the ALDEx2 model definition (presented below). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,226,723,247]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"17","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5.1Unacknowledged Bias in ALDEx2 and DESeq2 ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,256,877,366]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"17","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As described by Gustafson (2015) (pg. 35)，a hallmark of unacknowledged bias is an increasing rate of Type-1 errors with increasing data. This phenomena is troubling as it defies standard logic: typically， we think of inference improving with more data, not worsening. Figure 1 illustrates the intuition behind this phenomena. As we show next, two popular methods for differential abundance and differential expression analysis of sequence count data display this hallmark of unacknowledged bias. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[316,378,679,569]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 1: The relationship between unacknowledged bias and sample size can be understood as a symp-tom of model misspecification. Typical model-based estimators demonstrate decreasing uncertainty (e.g., decreasing variance)with increasing data,approaching zero in the asymptotic limit. Now consider a model that is misspecified such that a corresponding estimator displays a large-sample bias.When the sample size is small,the uncertainty can be large and encapsulate the bias: large uncertainty arising from even a misspecified model may bufer against the inferential effects of the bias.Yet with increasing sample size the model becomes increasingly certain in a biased estimate; in turn the impact of the bias becomes magnified. In the present context, model misspecification refers to models failing to incorporate uncertainty from the lack of scale information in the observed data. \"]","image_footnote":"[]","img_path":"images/0840/3a46d23df54c705c062e4b629d756865a8142d8a4ef92f4ea8751149a8471f09.jpg","list_items":null,"page_idx":"17","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[119,734,875,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"17","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To demonstrate this phenomena we simulated a scaled system with $D = 2 0$ entities defined over $M$ configurations equally split between two conditions. For concreteness, let us imagine this is a microbiome system consisting of $D$ taxa in the gut of $M$ individuals half of which were randomly assigned to a narrow-spectrum antibiotic treatment and the other half to placebo. For simplicity,we simulate $W _ { d m } \\sim \\mathrm { P o i s s o n } ( \\lambda _ { i x _ { m } } )$ . Of the 20 taxa, we set $\\lambda _ { d 0 } = \\lambda _ { d 1 }$ in all but four taxa,reflecting no association between their abundance and treatment. In the remaining four, we set $\\lambda _ { d 0 } < \\lambda _ { d 1 }$ simulating decreased abundance associated with antibiotic treatment. To simulate the loss of scale information，we then simulated observations $Y _ { \\cdot m }$ as a multivariate hyper-geometric draw from the corresponding $W _ { \\cdot m }$ . We then applied both the ALDEx2 (Fernandes et al., 2014) and DESeq2 (Love et al., 2014) models to the resampled data. We take as our target estimand $\\theta _ { d }$ defined as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[488,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"17","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"18 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,877,214]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[253,227,740,246]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0287/d2c8d3ead102baf24e10f2a061313351aae8c69dc6ed26f82357402e7336ab21.jpg","list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\theta _ { d } = \\log \\lambda _ { d 1 } - \\log \\lambda _ { d 0 } = \\log \\operatorname { \\mathbb { E } } [ W _ { d m : x _ { m } = 1 } ] - \\log \\operatorname { \\mathbb { E } } [ W _ { d m : x _ { m } = 0 } ]\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,257,877,330]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and therefore define a false positive as estimating a taxon $d$ is differentially abundant when $\\lambda _ { i 0 } = \\lambda _ { i 1 }$ . We repeated this process simulating different numbers of samples, $M \\in ( 0 , 2 0 0 ]$ ， to study how the False Discovery Rate (FDR) changes as a function of sample size. Full simulation details can be found in Supplementary Section F.1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,330,877,421]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The results are shown in Figure 2 and demonstrate the hallmark of unacknowledged bias: the FDR increases with increasing sample size. With sample sizes as small as 20, both methods demonstrate false discovery rates greater than 6O%. Moreover, in the asymptotic limit both methods display rates of 80% which corresponds to calling every taxon significantly differentially abundant². ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,443,524,465]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5.2Examining ALDEx2 in Detail ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,474,875,546]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We examine the form of the ALDEx2 model to identify the origin of this unacknowledged bias. The ALDEx2 model is defined procedurally in 3 parts. In the first part,each mul-tivariate observation $Y _ { \\cdot m }$ is modeled using an independent Bayesian Multinomial-Dirichlet model: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[390,558,606,606]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0665/eaea922b682aa983aadb2386f4fa04d2244a61ddc2cae3b013e6703aef7b424a.jpg","list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { Y _ { \\cdot m } \\sim \\mathrm { M u l t i n o m i a l } ( W _ { \\cdot m } ^ { \\parallel } ) } \\\\ & { W _ { \\cdot m } ^ { \\parallel } \\sim \\mathrm { D i r i c h l e t } ( \\alpha ) . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,614,877,669]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Posterior samples of $W _ { \\cdot m }$ are then normalized using the Centered Log-Ratio (CLR) transform For each transformed posterior sample and for each $i \\in \\{ 1 , \\ldots , D \\}$ ，ALDEx2 then uses a t-test to test the null hypothesis that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[369,683,622,702]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0135/48777617c5bad441f4c191cefe008673eb4bd49ed83a3972423986cfe2214954.jpg","list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n0 = \\mathbb { E } [ \\eta _ { d m : x _ { m = 1 } } ] - \\mathbb { E } [ \\eta _ { d m : x _ { m = 0 } } ] . ^ { 3 }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,713,875,751]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In short, ALDEx2 is using the CLR transform as a parameter normalization and rather than testing the null hypothesis that $\\theta _ { d } = 0$ (with $\\theta _ { d }$ defined in Eq. 18) is instead testing ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,760,875,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2The maximum is not $1 0 0 \\%$ due to 4 true positives. 3Ultimately the p-value that is reported by ALDEx2 is the mean of the p-values computed on each posterior sample although this step is not central to our discussion. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[488,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"19 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[145,88,838,353]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 2: False discovery rate versus sample size for ALDEx2 and DESeq2. For each method, false discovery rate increases even in the presence of infinite data. The maximum false discovery rate was 0.80, as four of 2O taxa were truly differentially expressed. Sequencing depth was set to $5 , 0 0 0$ . Simulation details are in Supplementary Section D.1. \"]","image_footnote":"[]","img_path":"images/0371/ced333e56f0ada8767c46a2dd65bedb52cfb498f1fca2ae128f6866ba29f981d.jpg","list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[119,460,323,477]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"the null hypothesis that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[254,489,736,568]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0614/29e9e77cde5fd6376dab84bb0f0efe00647bbc9085538ba0ba76bbadceed1651.jpg","list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { 0 = \\hat { { \\boldsymbol \\theta } } _ { d } = \\hat { { \\boldsymbol \\theta } } _ { i } = \\mathbb { E } [ \\eta _ { d m : x _ { m } = 1 } ] - \\mathbb { E } [ \\eta _ { d m : x _ { m } = 0 } ] } \\\\ & { \\qquad = \\mathbb { E } \\left[ \\log \\frac { W _ { d m : x _ { m } = 1 } ^ { \\parallel } } { g ( W _ { \\cdot m : x _ { m } = 1 } ^ { \\parallel } ) } \\right] - \\mathbb { E } \\left[ \\log \\frac { w _ { d m : x _ { m } = 0 } ^ { \\parallel } } { g ( w _ { \\cdot m : x _ { m } = 0 } ^ { \\parallel } ) } \\right] } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,579,733,598]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To develop intuition about the assumption that $\\theta _ { d } = \\hat { \\theta } _ { d }$ , we approximate ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[393,613,604,632]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0885/e76bbfb06b54f85ac7b637f0162a446a08edf3ee6261d84c780f1b01995e9ad1.jpg","list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\log \\mathbb { E } [ W _ { d m } ] \\approx \\mathbb { E } [ \\log W _ { d m } ]\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,646,875,702]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"based on the Taylor approximation given by Teh et al. (2006) which is highly accurate for a Poisson random variable with rate parameter much greater than 1. With this approximation, the assumption $\\theta _ { d } = \\hat { \\theta } _ { d }$ reduces to: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[166,714,795,737]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0248/3a6fba98a0e2ff13ef4bd17e680be7c0a9cd42ebfde6f9dc642c7c5377b63bba.jpg","list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { \\mathbb { E } [ \\log W _ { m : x _ { m } = 1 } ^ { \\perp } ] - \\mathbb { E } [ \\log W _ { m : x _ { m } = 0 } ^ { \\perp } ] = \\mathbb { E } [ \\log g ( W _ { \\cdot m : x _ { m = 0 } } ^ { \\parallel } ) ] - \\mathbb { E } [ \\log g ( W _ { \\cdot m : x _ { m = 1 } } ^ { \\parallel } ) ] . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,750,877,804]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In other words, ALDEx2 assumes that expected difference in log scales between conditions can be imputed exactly (without error） from the system composition. This is simply the population-level analogue of the CLR assumption presented in Equation 8. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"20 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,85,400,104]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"20","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5.2.1 Generalizing ALDEx2 ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,114,875,167]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"20","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"With the ALDEx2 scale assumption exposed， we are now in a position to generalize ALDEx2 and show that introducing even small amounts of uncertainty into this assumption is sufficient to alleviate the unacknowledged bias and mitigate Type-I errors. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,169,875,321]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"20","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The ALDEx2 model can be easily generalized to a SSRV. In ALDEx2, instead of defining $\\eta$ in terms of the CLR transform of the system composition: let us instead define $\\eta _ { d m } = \\log W _ { d m } ^ { \\parallel } + \\log W _ { m } ^ { \\perp }$ .This modified ALDEx2 represents a scale conditional random variable, conditional because the value of $\\log W _ { m } ^ { \\perp }$ is left unspecified. This SCRV can then be combined with a scale model to form an SSRV which can be easily sampling using Algorithm C.2. For example, in Supplemental Figure 1 and 2 we show that we can recover the original ALDEx2 model by using the following point-mass as a scale model: $p ( \\log W _ { m } ^ { \\perp } ) \\sim \\delta _ { - \\log G \\in \\cal W _ { \\cdot m } ^ { \\parallel } }$ where $\\delta _ { x }$ denotes a point-masscntered on $x$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,323,875,412]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"20","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Viewing ALDEx2 as an SSRV highlights the source of its unacknowledged bias. There is no uncertainty in the system scale: the scale model is degenerate. The system scale is not precisely equal to the negative geometric mean of the composition yet the model is certain that it is. As we showed in Section 4.3,a degenerate scale model essentially guarantees an unacknowledged bias. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,412,875,522]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"20","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Some readers may question whether it is in fact the lack of scale uncertainty in ALDEx2 that is responsible for the unacknowledged bias and not some other factors such as the misspecification of Multinomial-Dirichlet model to our simulation procedure. To address this concern and to further reinforce that scale uncertainty is the root of the elevated Type-1 error rates, we use our ALDEx2 based SSRV with a non-degenerate scale model. We generalize ALDEx2's point-mass scale model as the following log-normal model: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[356,526,640,549]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0854/23b6d5db1d4c8bdf85d47cc1c1753a32c59ef99b179b833abc4cd09a73a849d5.jpg","list_items":null,"page_idx":"20","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { \\log ( W _ { m } ^ { \\perp } ) \\sim N ( - \\log g \\left( W _ { \\cdot m } ^ { \\parallel } \\right) , \\alpha ^ { 2 } ) . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,554,877,717]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"20","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As $\\alpha 0$ this model reduces to the ALDEx2 point mass model. Yet when $\\alpha ^ { 2 } \\ > \\ 0$ ， this model adds to the ALDEx2 point mass model uncertainty in the system scale. In Supplementary Section D.1, we show that adding even small amounts of uncertainty ( $\\alpha >$ 0.2) is sufficient to remove allof the false positives while retaining 3 of the 4 true positives results. Moreover, we show that the optimal window for the choice of $\\alpha ^ { 2 }$ is large in that this SSRV continues to identify true positives so long as $\\alpha < 1 . 8$ . To provide context for this value, consider that for $\\alpha > 1 . 8$ we would have to believe that, for any individual in the study, there is a greater than 5% probability that the CLR Estimate of Scale is in error by more than a factor of $e ^ { 1 . 9 6 \\times 1 . 8 ^ { \\circ } } \\approx 5 7 3$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[122,739,681,762]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"20","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5.3ALDEx2 and CoDA as Opposing Extremes ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[122,770,872,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"20","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Before progressing to our study of real data, we briefly point out that the above ALDEx2 SSRV can also allow us to contextualize CoDA principle of scale invariance as an extreme opposite of the assumptions of ALDEx2. Using the log-normal scale model above，we recovered ALDEx2's point mass scale model in the limit as $\\alpha 0$ .This limit implies infinite knowledge (zero uncertainty) in the system scale given the system composition. In the other extreme,as $\\alpha \\to \\infty$ we assume zero knowledge (infinite uncertainty) in the system scale. Unsurprisingly, in this extreme,any non-zero credible interval calculated from the corresponding SSRV must cover the entire parameter space - enforcing restricting us to only ask scale invariant questions. While this is technically not the same as CoDA's scale invariance axiom (which enforces scale invariance of an applied estimator not the target estimator), it does echo the common view in CoDA that only questions that can be framed in terms of log-ratio can be answered from data that lack scale Aitchison (1994). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,506,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"20","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"21 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,877,267]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,270,875,342]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Between these two extremes (the point mass model of ALDEx2 and CoDA's infinite uncertainty), SSRVs allow us to use express non-zero yet finite uncertainty in the system scale. We believe this view emphasizes the utility of SSRVs, as a middle ground between these two opposing extremes. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[122,364,710,386]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5.4Application to a novel 16S rRNA-Seq dataset ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,395,877,631]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Artificial gut systems (bioreactors) are seeded with human feces and often used as a model for host-associated microbiota. Yet， many bioreactors undergo a two week equilibration period in which the inoculated communities diverge from those seen in human hosts and adapt to their new ex vivo environment. To evaluate the limits of these model systems we ran and collected data for 8 bioreactors over the course of the first two weeks after inoculation. Each bioreactor was seeded with 100mL of fecal slurry and maintained at a constant 400mL of growth media over the course of the experiment. Our primary interest was in determining which of the microbial taxa present in the innoculum were able to grow in the bioreactors. We quantify this interest using the differential abundance target estimator given in Equation 3 with $x _ { m } = 0$ denoting day 1 of the experiment and $x _ { m } = 1$ denoting day 14. In this case,we were primarily interested in the de facto total scale, i.e., we wanted to identify taxa that increased or decreased in total cellular number over the two week period. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,633,877,796]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Two features of this experimental system allowed us to obtain high-quality measurements of the total scale in these bioreactors. First,because these vessels were kept at a constant 400mL measuring total cellular concentration allowed us to estimate total cellular number. Second, per protocol (Silverman et al., 2018)， these bioreactors are constantly stirred to produce homogeneous growth conditions throughout the bioreactor; this ensured that estimation of concentration from a sample of material reflected the concentration of the overall bioreactor. For each of the 8 bioreactors,at each of the two time-points,we obtained triplicate flow-cytometry measurements of bacterial cellular concentration. Trip-licate measurements were obtained so that we could estimate uncertainty in the scale due ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"22 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,87,307,103]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"to measurement noise. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,104,877,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For analysis of the resultant data, we compared 5 models: 3 SSRVs, ALDEx2 and DESeq2. To introduce our 3 SSRVs, we frst consider the following multinomial log-normal Bayesian PIM which is an extension of Bayesian multinomial logistic-normal models that have become popular for analysis of microbiota survey data (Silverman et al., 2022): ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[388,184,607,253]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0212/3260d029706c283a7679b396a371f24ed57dc5abae810cf2c010588cd70278f8.jpg","list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { Y _ { \\cdot m } \\sim \\mathrm { M u l t i n o m i a l } ( \\mathcal { W } _ { \\cdot m } ^ { \\parallel } ) } \\\\ & { \\mathcal { W } _ { \\cdot m } = e ^ { \\psi _ { \\cdot m } } } \\\\ & { \\psi _ { \\cdot m } \\sim N ( B X _ { \\cdot m } , \\Omega ) } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,258,877,405]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"with priors $B \\sim \\mathit { N } ( M , \\Omega , V )$ and $\\Omega \\sim I W ( u , \\Xi )$ (See Supplementary Section D.2 for prior specifications). This is a Bayesian Partially Identified Model since the multinomial likelihood is parameterized in terms of only the parameter $\\mathcal { W } ^ { \\parallel }$ whereas the prior is defined over $\\mathbf { W } = \\mathbf { W } ^ { \\parallel } \\times \\mathbf { W } ^ { \\perp }$ . Due to computational limitations, it is infeasible to sample this model directly using standard approaches (e.g., MCMC). In contrast, we show in Supplementary Section D.3, that this model can be decomposed into a measurement model and a scale model and sampled efficiently as an SSRV using Algorithm C.1. This SSRV representation of the above Bayesian PIM is the first of the three SSRVs we consider in this section. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,406,877,515]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our next two SSRVs are based upon the above Bayesian Multinomial log-normal model. When written as an SSRV the above Bayesian Multinomial log-normal model has a Multinomial Logistic Normal measurement model of the form studied in Silverman et al. (2022). We created two more SSRVs using this measurement model with diferent scale models. Letting $m \\in \\mathbf { M }$ denote a given vessel at a given time-point,we build two scale models around the following family of log-normal models: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[393,522,606,542]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0867/ff69182da905f41f3fc931cafc944f14b63a7f28b3a8a76a1679c0dbc0e0ab91.jpg","list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\log W _ { m } ^ { \\perp } \\sim N ( c + \\mu _ { m } , \\alpha _ { m } ^ { 2 } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,550,877,770]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $c$ is some constant ad $\\mu$ is subjectgthe consuraint $\\sum _ { m = 1 } ^ { M } \\mu _ { m } = 0$ The constraint is not a critical requirement mathematically or statistically, but we include it to simplify the task of specifying the scale model. Consider that for diffrential analysis (e.g., target estimand in Equation 3),we do not actually need to know the abundance of each entity in each configuration but only the relative scale between configurations (i.e., the scale up to a constant factor $e ^ { c }$ ). This simplifies our task as biologically we often have prior knowledge about the relative scales between conditions yet have substantially less information about the absolute scale of any one condition. To illustrate this claim, consider that for a healthy individual, we believe that the total number of microbes in that individuals mouth is likely the same today as it is tomorrow, yet we have little prior knowledge about the total number of microbes on either day in isolation. In short,for differential analysis we need only know a composition over the scale and can simply set $c = 1$ with no loss of generality. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,771,872,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Of the two scale models we pair with the measurement model $p ( \\mathcal { W } ^ { \\parallel } | Y )$ ， the first is considered our gold standard (our benchmark against which we measure false positives), defined by letting $\\mu _ { m }$ and $\\alpha _ { m } ^ { 2 }$ equal the mean and variance of our flow cytometry measurements. We call the resultant SSRV the flow model. Our second scale model is built based on our knowledge of the experimental design of our study. Since, we know that each vessel contained 400mL of growth media and the innoculum consisted of $1 0 0 ~ \\mathrm { m L }$ of fecal slurry, we expect that the total microbial load would, on average, increase from the day 1 to day 14 by a factor of approximately 4. We encapsulated these beliefs in the scale model ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"23 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,877,195]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[367,210,629,252]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0190/b9517480ba9eb95f9bf3daffdff5ca4b7b4e5794b0328f9626083f1c3ad69c52.jpg","list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\frac { f ( W _ { j , \\mathrm { t = 1 } } ^ { \\perp } ) } { f ( W _ { j , \\mathrm { t = 1 4 } } ^ { \\perp } ) } \\sim \\mathrm { L o g N o r m a l } ( \\bar { d } , \\tau ^ { 2 } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[122,265,826,284]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $d = \\log ( 1 0 0 / 4 0 0 )$ and $\\tau = 0 . 5$ .4 We call the resultant SSRV the design model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,285,877,593]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To summarise, we created 3 SSRVs all of which used the same Multinomial Logistic-Normal measurement model but three different scaled models: (1） a scale model chosen to ensure that the resultant SSRV was identical to the above Multinomial Log-Normal model,(2) A scale model built using the flow cytometry measurements,and (3) a scale model built using only our knowledge of the experimental design. We applied the resultant 3 SSRVs as well as ALDEx2 and DESeq2, the results are summarized in Figure 3. As in Section 5.1, we again find that DESeq2 follows the patterns of errors displayed by ALDEx2. Yet in contrast to Section 5.1, here we find that both ALDEx2 and DESeq2 demonstrate reasonable specificities (66.7% and $1 0 0 \\%$ respectively） yet poor sensitivities (14.2% and 17.8% respectively). Notably what differs here from prior sections is that here many taxa are truly changing in abundance, growing to occupy the larger 400mL bioreactor. In addition, here we have small sample sizes ( $M = 1 6$ ） compared to the range of sample sizes considered in Section 5.1 ( $M \\in ( 1 0 , 1 0 0 ) )$ . Still, by bootstrap resampling we can simulate how results would change had we used more bioreactors. With 25 bioreactors ( $M = 5 0$ ）， we again see the hallmark increasing Type-I error rate of unacknowledged bias: DESeq2 and ALDEx2 give FDRs on the order of $4 0 \\%$ and $4 5 \\%$ respectively (Supplementary Figure 4). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,594,875,685]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In contrast, across all tested scenarios (i.e., with 8 vessels or 50), the design-based SSRV remains calibrated and displays better sensitivity, FDR,and specifcity than ALDEx2 or DESeq2: FDR of O%, sensitivity of $1 0 0 \\%$ ， and specificity of $1 0 0 \\%$ . Similar results were found for the Bayesian PIM based scale model which demonstrated a sensitivity of 92.9%, a specificity of $0 \\%$ , and a false discovery rate of $1 0 . 3 \\%$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,707,382,724]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5.4.1 Sensitivity Analyses ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,734,875,771]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Whereas measurement models are built around real data and therefore have some degree of robustness to miss-specification, scale models (such as our design based scale model above) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[135,784,705,799]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 $\\tau ^ { 2 }$ can be compared to $\\alpha ^ { 2 }$ in the flow estimator by the relationship $\\tau ^ { 2 } = 2 \\alpha ^ { 2 }$ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"24 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[140,0,839,284]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 3: Comparison of results from ALDEx2, DESeq2,and 3 SSRVs using real data from an ex vivo artificial gut study.“GS\\\" denotes the gold-standard model against which true/false positives/negatives are defined.Our goal standard (the SSRV using the Flow scale model) is the only model that could use the replicate flow-cytometry measurements collcted in this study. For brevity,taxa that were not significant with respect to any of the 5 models are not shown. \"]","image_footnote":"[]","img_path":"images/0052/5ec970e554097d57e9c51d8f581854ce9024f9c050e7031fe2cf4e332da33032.jpg","list_items":null,"page_idx":"24","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[119,406,875,515]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"24","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"can be purely expressions of subjective belief. It is therefore natural to question whether inferences drawn from an SSRV are sensitive to the chosen scale model: how robust are our claims about sensitivity and specificity of the design model above if we had picked a different value of $\\tau ^ { 2 }$ ? When working with SSRVs specified as a target estimand, a measurement model and a scale model, the structure of SSRVs often allows for computationally efficient global sensitivity analyses. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,517,875,626]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"24","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We expect that the most computationally limiting part of Algorithm D.1 will be sampling from the measurement model which is often high-dimensional and conditioned on observed data. Yet to study how the distribution of an SSRV changes with different scale models, the samples from the measurement model need only be obtained once and can be reused for each measurement model. This type of memoization algorithm allows for scalable global sensitivity analyses. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,626,875,752]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"24","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In Figure 4, we show that our results from the design-based SSRV are largely insensitive to our choice of the parameter $\\tau$ . So long as $\\tau$ is greater than O.35, the specificity of the design-based SSRV will be equal to $1 0 0 \\%$ . Moreover， so long as we choose $\\tau$ less than 6, then the sensitivity of the design-based SSRV will be greater than that of either DESeq2 or ALDEx2. Consider, that a value of $\\tau \\geq 6$ would require the unreasonable belief that there is a greater than $5 \\%$ chance that the number of microbial cells at day 14 was greater than or equal to $4 e ^ { 1 . 9 6 * 7 ^ { 2 } } \\approx 2 \\times 1 0 ^ { 4 2 }$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,753,870,789]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"24","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For completeness, we also demonstrate a sensitivity analysis about the parameter $\\alpha$ used in SSRV analysis of the simulated dataset from Section 5.2. Those results are provided and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"24","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"25 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[261,85,730,286]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 4: Sensitivity analysis of design-based total model as a function of $\\\\tau$ . The estimated effect size as a function of $\\\\tau$ is shown for each taxa in the artificial gut study. The effect size is colored black if the corresponding credible set excluded zero（\\\"significant\\\"） and grey otherwise (\\\"not significant\\\"). As an illstration, we highlight the 5 taxa that are the most clearly diffrentially abundant in the sense that their credible sets exclude zero over the largest range of $\\\\tau$ . For comparison,we depict the value of $\\\\tau$ used in Figure 3 ( $\\\\tau = 0 . 5$ ）as a black dashed line. \"]","image_footnote":"[]","img_path":"images/0587/4e20f84b3c7e7f6834de611ba7080a04dee53a1b96d03cffc5b15740d61ebe4d.jpg","list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[120,425,459,443]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"discussed in Supplementary Section D.1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,465,527,488]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5.5Generalizing beyond ALDEx2 ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,496,877,787]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"So far,we have focused on providing intuition explaining the origin of ALDEx2's unacknowledged bias. Yet from both Sections 5.1 and 5.4 it is clear that other models such as DESeq2 sufer from unacknowledged bias as well. Unfortunately, whereas the form of the ALDEx2 model permitted in depth dissection, the form of DESeq2 is more complex. For example,whereas the CLR normalization used by ALDEx2 can be expressed as a function of $\\mathcal { W } ^ { \\perp }$ ， DESeq's median-of-ratios method represents a normalization based on the observed count data（ $Y$ ). It is unclear how or even if this could be represented as a closed form scale model. Still our theoretical results from Section 3 can be applied. DESeq2 is a standard parametric (specifically a Negative Binomial empirical Bayesian model) and has a posterior variance that will collapse to zero in the asymptotic limit. In other words, DE-Seq2 is clearly lacking any uncertainty in the system scale that act against this asymptotic certainty: whatever the scale model used by DESeq2, it is likely degenerate. Moreover, given that any point estimator is biased (Theorem 3), it follows that any model/estimator that aims to perform scale reliant inference yet has a variance that collapses to zero in the asymptotic limit will display an unacknowledged bias. We expect that most models currently used for scale reliant inference fallinto this category. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[137,790,874,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Even beyond sequence counting, we expect unacknowledged bias to be prevalent. For example, in mark-recapture studies it is often assumed (without error) that allindividuals have the same probability of being captured in the second sample as the first (Krebs,1999). We expect that this lack of uncertainty in a critical assumption that can be related to the system scale equates to a degenerate scale model when viewed through the lens of SSRVs. We provided an extended discusson and an example of SRI in the context of ecology in Supplementary Section E. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"26 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,877,195]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,222,313,247]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6Discussion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,261,877,406]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This article has introduced a formal statement of the problem of scale reliant inference. This problem statement was then used to highlight the critical role of modeling assumptions in scale reliant inference. Most notably, for the sub-class of fully scale reliant inference, we show that Frequentist criteria such as consistency， calibration，and unbiasedness are unattainable and even Bayesian notions of calibration require specialized models in order to avoid unacknowledged bias. Still, we showed that there scale reliant inference is possible and that attractive approaches (such as the use of SSRVs) may help avoid the pitfalls of current methods. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,409,875,497]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There are numerous avenues for future expansion and refinement of this work, we re-strict our discussion to four potential future directions: (1) exploring connections to other statistical fields, (2) studying partially scale reliant or biased compositional surveys, (3) developing new tools to quantify the impact of unacknowledged bias, and (4) reevaluating standard attitudes towards experimental design and sample size calculations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,500,877,790]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this article, we have shown how our problem statement represents a reformulation of compositional data analysis and represents an application of Bayesian Partially Identified Models. Still, there are other important connections between scale reliant inference and other statistical fields that should be explored. Perhaps the most notable is the connection between our work and the Potential Outcomes Formulation (POF） of Causal Inference as discussed by Imbens and Rubin (2015). We notice the following connections. Both the POF and scale reliant inference center around clear statements of the scientific question being asked in the form of an estimand (called a causal estimand in the POF and a target estimand here). In both problem formulations,the central challenge is a lack of requisite information (inability to observe both potential outcomes in the POF, lack of knowledge of the system scale here). In both problem formulations,key methods attempt to directly model and quantify uncertainty in assumptions used to address the lack of knowledge (Bayesian model based methods in the POF and Bayesian PIMs or SSRVs here). Beyond the potential for sharing ideas and solutions between the POF and scale reliant inference, we expect that some studies will ask questions that fall at the boundary between these two fields. For example, a researcher studying differential abundance may want to use an estimand that is both a causal estimand and a target estimand: “How would the abundance of taxa $d$ change if that patient had been given the drug versus if they had not been given the drug?” Further work is necessary to answer such questions. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"27 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,87,875,140]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,142,877,431]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A core motivation in this work has been to provide discussion on the limits of scale reliant inference and to relate this back to applied problems in bioinformatics, ecology, and social sciences. While we have discussed limits on scale reliant inference, the relation back to the analysis of real data is non-trivial. In reality, compositional surveys are simply a model and cannot hope to explain the many intricacies and complexities of real data. For example, in practice, we expect that any survey data provides at least minuscule amount of information about the system scale making partially compositional surveys the norm and fully compositional surveys an extreme. Yet with partially compositional surveys conclu-sions about whether inference is fully or partially scale reliant cannot be solely based on the question being asked but requires a more careful consideration of the precise question and the precise information about scale present in the observed data. Beyond this, our definition of compositional surveys assumed that the measurement process identified the system composition. In practice, we expect that measurement bias, such as PCR bias in 16S rRNA sequencing (Silverman et al., 2021), implies that there are other sources of non-invertibility or non-identifiability at play beyond the lack of scale information. Exploring how these other factors impact scale reliant inference would surely be a worthwhile endeavor. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,434,877,650]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A critical limitation of the present work is our inability to adequately answer the ques-tion: “When will unacknowledged bias due to scale limitations lead to problems in an analysis?” Especially in the analysis of sequence count data, this remains and outstanding problem. While there has been some work showing that commonly used methods display elevated Type-1 error rates in a wide variety of real and simulated datasets (Roche and Mukherjee, 2022)， these analyses have all been dependent on the particular simulation model or on assumed fidelity of qPCR or flow cytometry measurements without consideration of potential error in those measurements themselves. Still, the sensitivity analyses we discuss in Section 5.4.1 may find use here. Given a real dataset, these sensitivity analyses allow an analyst to quantify the degree to which their conclusions would change when there is error in their assumptions. Overall, we expect this is a fruitful avenue for future work and development. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,652,877,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Scale reliant inference is a remarkable problem that can defy standard statistical logic. Suppose that a research decides to use ALDEx2 or DESeq2 despite the unacknowledged bias. Standard statistical attitudes towards the value of data might argue “that's probably fine so long as that researcher has sufficient data.” Yet this is the exact opposite of what we have shown: the effect of the bias problems will get worse with more data. Key questions remain: are there instances where having a small number of samples will allow researchers to ignore scale uncertainty and simply using existing tools like DESeq2 and ALDEx2? Again, we expect sensitivity analyses such as those described in prior sections to be helpful in identifying these instances. More broadly, the impact of sample size on experimental design is well described in the Bayesian PIM literature (e.g., see Chapter 3 and Section 7.2 or Gustafson (2015)). Interestingly, in that literature, authors discuss diminishing returns that occur with increasing data: since the model uncertainty asymptotically remains non-zero at a certain point, collcting more data cannot appreciably decrease uncertainty beyond the limiting value. Overall, we expect that further consideration of how scale uncertainty and unacknowledged bias interact with sample size will provide actionable insights. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"28 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,877,214]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[122,239,669,265]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acknowledgments and Funding Sources ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,279,877,424]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We thank Rachel Silverman， Juan Jose Egozcue, Vera Pawlosky-Glahn, Gregory Gloor, Michael Love,and James Morton for their manuscript comments. LAD acknowledges support from the the Global Probiotics Council, a Searle Scholars Award, the Hartwell Foundation,an Alfred P. Sloan Research Fellowship,the Translational Research Institute through Cooperative Agreement NNX16AO69A, the Damon Runyon Cancer Research Foundation, and NIH 1R01DK116187-01. SM would like to acknowledge the support of grants NSF IIS-1546331, NSF DMS-1418261， NSF IIS-1320357, NSF DMS-1045153,and NSF DMS-1613261. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[122,452,604,477]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Data Access and Code Availability ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,491,870,526]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"All data and R scripts necessary to replicate all analyses in the manuscript and supplement are located at https://github.com/michellepistner/ScaleSimulation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,554,269,578]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[116,592,874,628]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Aitchison, J. (1986). The Statistical Analysis of Compositional Data. Chapman and Hall, New York. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,641,733,660]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Aitchison, J. (1994). Principles of compositional data analysis. 24:73-81. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,671,872,708]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Basse,G. and Bojinov, I. (202O). A general theory of identification. arXiv preprint arXiv:2002.06041. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,720,877,795]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Fernandes, A. D., Reid, J. N., Macklaim, J. M., McMurrough, T. A., Edgel, D.R., and Gloor，G. B. (2014). Unifying the analysis of high-throughput sequencing datasets: characterizing RNA-seq, 16S rRNA gene sequencing and selective growth experiments by compositional data analysis. Microbiome, 2(1):1-13. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"29 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[112,55,880,804]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"29","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Friedman, J. and Alm, E. J. (2012). Inferring correlation networks from genomic survey data. PLoS Computational Biology, 8(9):e1002687. \nGustafson， P. (2O15). Bayesian inference for partially identified models: Exploring the limits of limited data, volume 140. CRC Press. \nImbens, G. W. and Rubin, D.B. (2015). Causal inference in statistics, social, and biomedical sciences. Cambridge University Press. \nKrebs, C. J. (1999). Ecological methodology. Benjamin/Cummings, Menlo Park, Calif, 2nd ed edition. \nLove,M. I., Huber, W.,and Anders, S. (2014). Moderated estimation of fold change and dispersion for RNA-seq data with DESeq2. Genome Biology, 15(12):1-21. \nLovell,D. R., Chua, X.-Y.， and McGrath, A. (2020). Counts: an outstanding challenge for log-ratio analysis of compositional data in the molecular biosciences. NAR Genomics and Bioinformatics, 2(2):lqaa040. \nMurphy， S. M.， Cox, J. J.，Augustine, B. C.， Hast, J. T.，Guthrie, J. M.， Wright, J., McDermott, J.， Maehr, S. C.，and Plaxico, J. H. (2016). Characterizing recolonization by a reintroduced bear population using genetic spatial capture-recapture. The Journal of Wildlife Management, 80(8):1390-1407. \nPaulson, J. N.， Stine, O. C.， Bravo, H. C.，and Pop,M. (2013). Diferential abundance analysis for microbial marker-gene surveys. Nature methods,10(12):1200-1202. \nPawlowsky-Glahn， V.， Egozcue, J. J.，and Tolosana-Delgado， R. (2015). Modeling and analysis of compositional data. John Wiley & Sons. \nRoche,K. E. and Mukherjee, S. (2022). The accuracy of absolute differential abundance analysis from relative count data. PLOs Computational Biology, 18(7):el010284. \nRossi,P. H.， Wright, J. D., Fisher,G. A.，and Wilis, G. (1987). The urban homeless: estimating composition and size. Science, 235(4794):1336-1341. \nSilverman, J. D., Bloom, R. J., Jiang, S., Durand, H. K., Dallow, E., Mukherjee, S., and David,L. A. (2021). Measuring and mitigating pcr bias in microbiota datasets. PLoS Computational Biology, 17(7):e1009113. \nSilverman, J. D., Durand, H. K., Bloom, R. J., Mukherjee, S.，and David, L. A. (2018). Dynamic linear models guide design and analysis of microbiota studies within artificial human guts. Microbiome, 6(1):1-20. \nSilverman， J. D.， Roche,K.， Holmes, Z. C.， David,L.A.，and Mukherjee, S. (2022). Bayesian multinomial logistic normal models through marginally latent matrix-t processes. J. Mach. Learn. Res., 23:7-1. \nSrinivasan，A.， Xue, L.，and Zhan， X. (2021). Compositional knockoff filter for highdimensional regression analysis of microbiome data. Biometrics, 77(3):984-995. \nTeh, Y., Newman, D.,and Welling, M. (2006). A Collapsed Variational Bayesian Inference Algorithm for Latent Dirichlet Allocation. In Advances in Neural Information Processing Systems, volume 19. MIT Press. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"29","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"30 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[116,85,879,257]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"30","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,506,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"30","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"31 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[161,154,833,183]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Supplement to “Scale Reliant Inference\" ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[173,215,823,260]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Michelle Pistner Nixon, Jefrey Letourneau, Lawrence A. David, Nicole A. Lazar, Sayan Mukherjee, and Justin D. Silverman ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,321,877,376]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this supplement,we present clarifying remarks and the proofs to our main results and additional claims made throughout the paper. In addition, we show additional details and results of our simulations and data analyses. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[124,404,857,428]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"AClarifying Comments on the Importance of Scale ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,441,875,513]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Here, we provide an extended discussion of the following two questions that we have received frequently in the development of this work: (1)\"Isn't this problem solved by [field-specific technology]?” and (2) “Why should I care about the totals anyways [in this field]? The totals may not be any more relevant than the proportional abundances.\". ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,516,877,733]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Before addressing either question, let us provide an example related to human gut microbiota ilustrating that there can be multiple scientifically meaningful definitions of scale. It can be context-dependent: one researcher may want $W _ { d m }$ to represent the concentration of bacterial taxon $d$ in the colon of human host $m$ , yet another may want $W _ { d m }$ to represent the number of bacterial cells of taxon $d$ in the colon of host $m$ . Either choice implies different scales, i.e., $W _ { m } ^ { \\perp }$ represents either the total number or total concentration of bacterial cels in the colon. Yet a third researcher may find both of these scales irrelevant noting the following two trivialities: larger people likely have more bacteria in their gut and the amount of insoluble fiber someone ate the day before (e.g.，wheat bran) can add to dry mass in the colon and therefore impact bacterial concentration. This third researcher may want to use some more esoteric notion of scale such as the total number of bacteria relative to the colonic surface area. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,734,875,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We believe this example helps to address both Questions 1 and 2. With respect to Question 1,we believe its fairly clear that whatever technology is used for measuring to-tal bacterial or total 16S rRNA concentration (e.g.， flow cytometry or qPCR） is unlikely to provide information about all possibly scientifically relevant scales (e.g., total bacterial number or total bacterial number relative to colonic surface area). Moreover, even when the question is stated in terms of a scale that can be measured (e.g., total bacterial concen-tration), the measurements may be noisy or biased (Props et al., 2017) again introducing uncertainty in the system scale. The tools we develop and our discussion can apply equally well to cases where scale measurements are relevant but sufficiently noisy to introduce uncertainty in the estimand of interest ( $\\theta$ ） ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[22,257,62,724]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[491,829,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,877,195]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,196,875,287]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"With respect to the second question, note that our work does not advocate for any one scale. In particular， we encourage the reader to not conclude that we are arguing that default total scale (the total number of entities in a configuration) is the scale of interest. Instead, what we develop here is theory and tools for working with any scale of scientific interest - whatever that scale might be. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,314,266,338]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"BProofs ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,352,718,371]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section, we present all of the proofs referenced in the main text. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,371,874,406]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $h = g \\circ f ^ { - 1 }$ denote the composite map formed by the measurement model and estimand as discussed in the main text. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,429,745,450]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B.1Proofs of Scale Reliance for Selected Estimators ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[117,460,856,478]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section, let $W$ denote a $\\boldsymbol { D }$ -dimensional scaled system over a set of M conditions. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,500,390,518]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B.1.1 Differential Analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[122,527,872,564]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We begin by defining differential analysis. Then, we prove that differential analysis is fully scale reliant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,575,877,632]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition B.1 (Differential Analysis). Let $x _ { m } \\in \\{ 0 , 1 \\}$ denote a binary covariate associated with each condition $m \\in \\mathbf { M }$ . For a given $i \\in \\{ 1 , \\ldots , D \\}$ ， differential analysis is defined as a target estimand of the form ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[315,643,679,689]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0233/966604da5d825774f5d464fa032ea05d014de3e4356ff0d825592a63dfe32dec.jpg","list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\theta _ { i } = \\frac { 1 } { n _ { 1 } } \\sum _ { m : x _ { m } = 1 } \\log W _ { d m } - \\frac { 1 } { n _ { 0 } } \\sum _ { m : x _ { m } = 0 } \\log W _ { d m }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,699,879,736]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $n _ { 0 }$ and $n _ { 1 }$ denote the number of conditions $m \\in \\mathbf { M }$ such that $x _ { m } = 1$ or $x _ { m } = 0$ respectively. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,747,633,767]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition B.1. Differential Analysis is fully scale reliant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[491,829,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,84,879,122]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Prof.Bythelearityftemeanadusigthation $\\log W _ { d m } = \\log W _ { d m } ^ { \\parallel } + \\log W _ { m } ^ { \\perp }$ it follows that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[202,135,790,190]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0646/8bd0f9e433159191114237623cd386e43cdae0ba00f130e371c4295486814c5e.jpg","list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\theta _ { d } = \\underset { x _ { m } = 1 } { \\mathrm { m e a n } } ( \\mathrm { l o g } W _ { d } ^ { \\parallel } ) - \\underset { x _ { m } = 0 } { \\mathrm { m e a n } } ( \\mathrm { l o g } W _ { d } ^ { \\parallel } ) + \\underset { x _ { m } = 1 } { \\mathrm { m e a n } } ( \\mathrm { l o g } W ^ { \\perp } ) - \\underset { x _ { m } = 0 } { \\mathrm { m e a n } } ( \\mathrm { l o g } W ^ { \\perp } ) } \\\\ & { \\quad = \\theta _ { d } ^ { \\parallel } + \\theta ^ { \\perp } . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,204,877,265]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note that $\\theta _ { d } ^ { \\parallel }$ is solely a function of $\\boldsymbol { W } _ { d } ^ { \\parallel }$ and $\\theta ^ { \\perp }$ is solely a function of $W ^ { \\perp }$ . In this form it is clear that for any fixed value of $\\theta _ { d } ^ { \\parallel }$ , we can always select a value of $\\theta ^ { \\perp } \\in \\mathbb { R }$ such that $\\theta _ { i }$ can take on any value in $\\mathbb { R }$ (i.e., in $\\Theta$ ). Thus differential analysis is fully scale reliant. $\\sqsubset$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,285,393,304]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B.1.2 Covariation Analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,313,879,348]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition B.2 (Covariation Analysis). Define the Covariation analysis target estimand$a s$ ：","text_format":null,"text_level":null,"type":"text"},{"bbox":"[370,353,624,369]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0330/448a899e720ccab87d6afeef3b97f4717211f11aac34595efeac088020a3370b.jpg","list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\Sigma _ { d _ { 1 } , d _ { 2 } } = c o v ( \\log W _ { d _ { 1 } \\cdot } , \\log W _ { d _ { 2 } \\cdot } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,380,741,401]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition B.2. Covariation Analysis $\\left( \\Sigma _ { d _ { 1 } , d _ { 2 } } \\right)$ is partially scale reliant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,410,877,450]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof. For notational convenience, let $\\overline { W }$ be a shorthand for $\\log W$ . We reparameterize $\\Sigma _ { d _ { 1 } d _ { 2 } }$ in terms of $\\overline { W } ^ { \\| }$ and $\\overline { W } ^ { \\perp }$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[189,463,808,521]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0291/b19cf83e11247fa4915e6bd956a420714579f5079846dbc055d3047bfe75787f.jpg","list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\Sigma _ { d _ { 1 } d _ { 2 } } = \\mathrm { c o v } ( \\overline { W } _ { d _ { 1 } } ^ { \\parallel } + \\overline { W } ^ { \\perp } , \\overline { W } _ { d _ { 2 } } ^ { \\parallel } + \\overline { W } ^ { \\perp } ) } \\\\ & { \\qquad = \\mathrm { c o v } ( \\overline { W } _ { d _ { 1 } } ^ { \\parallel } , \\overline { W } _ { d _ { 2 } } ^ { \\parallel } ) + \\mathrm { c o v } ( \\overline { W } _ { d _ { 1 } } ^ { \\parallel } , \\overline { W } ^ { \\perp } ) + \\mathrm { c o v } ( \\overline { W } ^ { \\perp } , \\overline { W } _ { d _ { 2 } } ^ { \\parallel } ) + \\mathrm { c o v } ( \\overline { W } ^ { \\perp } , \\overline { W } ^ { \\perp } ) } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,531,877,570]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We denote the standard deviation of a vector $x$ as $\\sigma _ { x } = { \\sqrt { \\operatorname { v a r } ( x ) } }$ and the covariation of two vectors $x$ and $y$ as $\\rho _ { x y } = c o r r ( x , y )$ . We can then write: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[160,583,816,643]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0733/a2dfefaa1ba24578af91d0ef04c826bf85f9c9ae46f5a492d31b7a39ae79209f.jpg","list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { { _ { 1 } d _ { 2 } } = \\sigma _ { \\overline { { W } } _ { d _ { 1 } } ^ { \\parallel } } \\sigma _ { \\overline { { W } } _ { d _ { 2 } } ^ { \\parallel } } \\rho ( \\overline { { W } } _ { d _ { 1 } } ^ { \\parallel } , \\overline { { W } } _ { d _ { 2 } } ^ { \\parallel } ) + \\sigma _ { \\overline { { W } } _ { d _ { 1 } } ^ { \\parallel } } \\sigma _ { \\overline { { W } } ^ { \\perp } } \\rho ( \\overline { { W } } _ { d _ { 1 } } ^ { \\parallel } , \\overline { { W } } ^ { \\perp } ) + \\sigma _ { \\overline { { W } } _ { d _ { 2 } } ^ { \\parallel } } \\sigma _ { \\overline { { W } } ^ { \\perp } } \\rho ( \\overline { { W } } _ { d _ { 2 } } ^ { \\parallel } , \\overline { { W } } ^ { \\perp } ) + } \\\\ & { \\quad = \\alpha + \\sigma _ { \\overline { { W } } ^ { \\perp } } \\beta + \\sigma _ { \\overline { { W } } ^ { \\perp } } \\gamma + \\sigma _ { \\overline { { W } } ^ { \\perp } } ^ { 2 } . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,654,877,715]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For a fixed value of $\\alpha$ , the minimum value that $\\Sigma _ { d _ { 1 } d _ { 2 } }$ can attain is $\\alpha - ( \\beta + \\gamma ) ^ { 2 } / 4$ which occurs when o2 $\\sigma _ { \\overline { { W } } ^ { \\perp } } ^ { 2 } = ( \\beta + \\gamma ) / 2$ .If $c < 0$ then the minimum value is $\\alpha - c$ which occurs at $\\sigma _ { W ^ { \\bot } } ^ { 2 } = 1$ . Covariation analysis is therefore partially scale reliant. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,734,467,756]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B.2Proofs of Scale Reliance ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,765,875,803]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section， we provide proofs related to the scale reliance of variaous compositional survey and target estimand pairings. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[490,828,504,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,131,890,284]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0443/0cddb78fa98e8fa9a476296cdd7b522b8ff7f7e052b9803fa2ed0db1ef7e90ac.jpg","list_items":null,"page_idx":"34","sub_type":null,"table_body":"<table><tr><td></td><td>Proposition Compositional Survey (V)</td><td>Target Estimand (0)</td><td>Inference of θ from observations of Y</td></tr><tr><td>D.3</td><td>Full</td><td>Scale Invariant</td><td>Scale Invariant</td></tr><tr><td>D.6</td><td>Full</td><td>Partially Scale ReliantPartially Scale Reliant</td><td></td></tr><tr><td>D.5</td><td>Full</td><td>Fully Scale Reliant</td><td>Fully Scale Reliant</td></tr><tr><td>D.3</td><td>Partial</td><td>Scale Invariant</td><td>Scale Invariant</td></tr><tr><td></td><td>Partial</td><td>Partially Scale Reliant</td><td></td></tr><tr><td></td><td>Partial</td><td>Fully Scale Reliant</td><td></td></tr></table>","table_caption":"[\"Table 1: Inference of $\\\\theta$ depending on the type of the data (partial or full compositional survey) and target estimand (scale invariant or scale reliant). \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[119,303,879,358]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"34","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition B.3. If $\\mathcal { V }$ is a full or partial compositional survey of a scaled system $W$ and $\\theta = f ( W )$ a scale invariant target estimand, then $h$ is invertible at $P$ (i.e., scale invariant inference). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,369,877,444]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"34","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof. By definition, if $\\mathcal { V }$ is either a partial or full compositional survey then there is a single value of $W ^ { \\parallel }$ in the pre-image $g ^ { - 1 } ( P )$ . Since the target estimand is scale invariant, then it is uniquely defined by $W ^ { \\parallel }$ . It follows that there exists a single element $\\theta \\in \\Theta$ that is in the pre-image $h ^ { - 1 } ( P )$ . Hence $h$ is invertible at $P$ □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,455,877,511]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"34","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition B.4. If $\\mathcal { V }$ is a full compositional survey of a scaled system $W$ and $\\theta = f ( W )$ is a scale reliant target estimand at $W ^ { \\parallel }$ ，then $h$ is uninvertible at $P$ (i.e.， scale reliant inference). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,523,877,559]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"34","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof. Since $\\theta$ is scale reliant,it follows that there must be at least two $W _ { 1 } ^ { \\perp } , W _ { 2 } ^ { \\perp } \\in { \\bf W } ^ { \\perp }$ such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[428,571,568,643]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0120/46914c8ead8fdd8c240070156809aadf62b7ccc4098f51da14a59365e21a0c71.jpg","list_items":null,"page_idx":"34","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { f ( W ^ { \\parallel } , W _ { 1 } ^ { \\perp } ) = \\theta _ { 1 } } \\\\ & { f ( W ^ { \\parallel } , W _ { 2 } ^ { \\perp } ) = \\theta _ { 2 } } \\\\ & { \\theta _ { 1 } \\neq \\theta _ { 2 } . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,654,877,709]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"34","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since $\\mathcal { V }$ is a full compositional survey then both $W _ { 1 } ^ { \\perp }$ and $W _ { 2 } ^ { \\perp }$ must be in the pre-image $g ^ { - 1 } ( P )$ . It follows that both values $\\theta _ { 1 }$ and $\\theta _ { 2 }$ must be in the pre-image $h ^ { - 1 } ( P )$ . Thus, $h$ is uninvertible. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,722,877,777]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"34","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition B.5. If Y is a full compositional survey of a scaled system $W$ and $\\theta = f ( W )$ is a fully scale reliant target estimand at $W ^ { \\parallel }$ ，then $h$ is fully uninvertible at $P$ (i.e.,， fully scale reliant inference). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[491,829,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"34","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,875,140]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"35","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof. Building on PropositionB.4, we know that $h$ is uninvertible. It suffices to show that with the additional constraint that $\\theta$ is fully scale reliant that all possible values of $\\theta$ must be in the pre-image $h ^ { - 1 } ( P )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,142,875,231]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"35","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Building on PropositionB.4,and using the fact that $\\theta$ is fully scale reliant at $W ^ { \\parallel }$ it follows that for any value of $\\theta \\in \\Theta$ ，there must exist some $W ^ { \\perp } \\in \\mathbf { W } ^ { \\perp }$ such that $f ( W ^ { \\parallel } , W ^ { \\perp } ) = \\theta$ . Again, since $\\mathcal { V }$ is a full compositional survey it follows that every such pair of $W ^ { \\parallel }$ and $W ^ { \\perp }$ must be in the pre-image $g ^ { - 1 } ( P )$ . Thus $h ^ { - 1 } ( P ( \\mathcal { V } ) ) = \\Theta$ which is the definition of fully uninvertible. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,243,877,299]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"35","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition B.6. If Y is a full compositional survey of a scaled system $W$ and $\\theta = f ( W )$ is a partially scale reliant target estimand, then $h$ is partially uninvertible at $P$ (i.e.,， partially scale reliant inference). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,310,875,366]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"35","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof. Building on Proposition B.4, we know that $h$ is uninvertible. It suffices to show that with the additional constraint that $\\theta$ is partially scale reliant that there must exist some $\\theta \\in \\Theta$ that is not in the pre-image $h ^ { - 1 } ( P ( \\mathcal { V } ) )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,367,875,455]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"35","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Building on Proposition B.4, and using the fact that $\\theta$ is partially scale reliant it follows that at $W ^ { \\parallel }$ ， there must be some value $\\theta \\in \\Theta$ such that there does not exist a value of $W ^ { \\perp } \\in { \\bf W } ^ { \\perp }$ that satisfies $\\theta = f ( W ^ { \\parallel } , W ^ { \\perp } )$ . It therefore follows that this value of $\\theta$ cannot be in the pre-image $h ^ { - 1 } ( P ( \\mathcal { V } ) )$ and therefore that $h ^ { - 1 } ( P ( \\mathcal { V } ) ) \\subsetneq \\Theta$ . Therefore $h$ is partially uninvertible. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,468,877,559]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"35","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finally, we do not provide any formal statements about the situation of a partially compositional survey and a partially or fully scale reliant target estimator. We simply note that these situations are substantially more complicated and that the composite map $h$ can be invertible, partially uninvertible, or fully uninvertible depending on the specific target estimand $\\theta = f ( W )$ and the specific measurement model $P$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,582,823,603]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"35","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B.3Proofs Related to the Limits of Scale Reliant Inference ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,612,875,685]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"35","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We will use the following declarations throughout this section. Let $x$ denote a random variable with distribution $P \\in \\textbf { P }$ Let $\\phi \\in \\Phi$ denote a quantity related to a random variable through a map $h : \\Phi { \\bf P }$ .Let $\\phi ^ { \\dagger }$ denote the true value of $\\phi$ .Finally, let $\\{ \\hat { \\phi } _ { n } \\}$ denote a sequence of estimators. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,686,874,739]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"35","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We use the following axiom which serves to exclude trivial estimators defined tautologically with respect to the true value $\\phi ^ { \\dagger }$ , e.g., to exclude the potential for an omniscient analyst. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,751,875,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"35","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition B.3 (Axiom of Learning). Our ability to learn the value of $\\phi \\in \\Phi$ is limited to the pre-image of h such that, were we to know $P$ and know $h$ ， then we must not be able to decide which of the values of $\\phi \\in h ^ { - 1 } ( P )$ is the true value. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[491,829,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"35","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,877,195]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We believe that this axiom reflects a common assumption that is often unstated in statistical theory. Letting $h$ denote a family of probability models such that $\\phi$ represents a parameter indexing the family, this axiom implies that our ability to learn about $\\phi$ is limited to the identification region of the model family at $P$ . Practically, this also implies that we cannot chose an estimator $\\hat { \\phi }$ that magically converges to the correct value $\\phi ^ { \\dagger }$ when $h$ is uninvertible. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,217,308,234]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B.3.1 Consistency ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[117,243,748,262]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our definition of consistency is standard, still we state it for completeness. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,270,877,309]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition B.4 (Consistent Estimator). $\\{ \\hat { \\phi } _ { n } \\}$ is a consistent sequence of estimators for $\\phi$ if and only if, for all $\\epsilon > 0$ ， ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[385,320,607,347]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0154/021b765316dcb078aa9e544c840ab71f1b17a595777c546df890ca6b015b2aea.jpg","list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { l i m } _ { n \\infty } P ( | | \\hat { \\phi } _ { n } - \\phi | | < \\epsilon ) = 1 .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,354,877,429]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Next,we prove that no estimator can be consistent if $h$ is not invertible. This result and proof represent an extension of the proof by Gabrielsen (1978) that identifiability is a necessary condition for consistency to the more general notion of invertibility rather than identifiability. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,436,877,474]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem B.1. If $h$ is uninvertible at $P$ then $\\left\\{ { \\hat { \\phi } } _ { n } \\right\\}$ cannot be a consistent sequence of estimators. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,484,877,559]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof. By contradiction， suppose that $h$ is uninvertible at $P$ then $h ^ { - 1 } ( P )$ must contain at least two distinct elements $\\phi _ { 1 } , \\phi _ { 2 } \\in h ^ { - 1 } ( P )$ . By the axiom of learning, the sequence of estimators cannot converge to $\\phi ^ { \\dagger }$ alone.In order to be consistent, it must therefore converge in probability to both $\\phi _ { 1 }$ and $\\phi _ { 2 }$ which is a contradiction. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,568,875,607]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary B.1. Let $\\theta$ be a Target estimand in scale reliant inference. There does not exist a consistent sequence of estimators $\\{ \\hat { \\phi } _ { n } \\}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,614,875,651]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof. The proof follows immediately from the definition of scale reliant inference in terms of uninvertability and Proposition B.4. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,671,303,690]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B.3.2Calibration ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,699,880,736]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition B.5 (Calibration). A set estimator $S \\in \\Phi$ is a confidence set calibrated at level $\\beta$ i ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[397,738,598,765]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0667/e5243bc85b1fddea1ea8b5176ce3626e5b2dc58374e078c7f4671ada51b2a2d3.jpg","list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { i n f } _ { \\phi \\in \\Phi , P \\in h ( \\phi ) } P ( \\phi \\in S ) \\ge \\beta\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,770,875,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and if the following two additional conditions hold: (1) $\\beta \\geq 0$ and (2) for any $\\phi \\in \\Phi$ there exist for each $P \\in h ( \\phi )$ ， some other $\\phi ^ { * } \\in \\Phi$ such $P ( \\phi ^ { * } \\notin S ) = 1$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[490,829,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,877,232]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note that our definition is nearly identical to the standard definition but includes the additional term $P \\in h ( \\phi )$ in the infimum to address situations in $h ( \\phi )$ is not a singleton set. Beyond this modification,our definition emphasizes two additional conditions that are often assumed but not stated. The first conditions, $\\beta > 0$ , is included to avoid trivially calling a set estimator calibrated if $\\beta = 0$ . The second conditions is added to avoid trivial set estimators that may be calibrated at level $\\beta > 0$ by returning the entire set $\\Phi$ . Essentially, this second conditions simply states that the set estimator must exclude at least one value of $\\phi ^ { * } \\in \\Phi$ from consideration in order to be called a calibrated confidence set. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,242,870,279]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem B.2.If $h$ is fully uninvertible for some $P \\in \\textbf { P }$ ， then there does not exist d calibrated confidence set for $\\phi$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,290,877,435]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof. By contradiction, suppose that a set estimator $S$ is a calibrated confidence set for $\\phi$ and that $h$ is fully uninvertible for some $P \\in \\mathbf { P }$ . Since $S$ is calibrated, there must then be some $\\phi \\in h ^ { - 1 } ( P )$ and some other $\\phi ^ { * } \\in \\Phi$ such that, $P ( \\phi ^ { * } \\notin S ) = 1$ .However，by definition, since $h$ is fully uninvertible at $P$ every value of $\\phi \\in \\Phi$ , including $\\phi ^ { * }$ , is in the set $h ^ { - 1 } ( P )$ .Therefore, if $\\phi ^ { * }$ is not in $S$ then $P ( \\phi ^ { * } \\in S ) = 0$ and the infimum becomes zero requiring $\\beta = 0$ . However, since we require that $\\beta > 0$ for $S$ to be calibrated, and by the axiom of learning, our supposition that $S$ is a calibrated confidence set implies that $\\phi ^ { * }$ must be in the set $S$ which is a contradiction. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[122,446,875,483]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary B.2. Let $\\theta$ denote a target estimator in fully scale reliant inference. then there does not exist a calibrated confidence set for $\\theta$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,493,872,531]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof. The proof follws immediately from the definition of fully scale reliant inference in terms of full uninvertability and Proposition B.5. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,551,369,569]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B.3.3 Bias and Variance ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[122,579,779,598]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition B.6 (Bias and Variance). We define the bias of an estimator $\\hat { \\phi }$ as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[400,609,593,630]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0238/76645de9b92939bdc82e40b72b767574ef7e50809070167afcd9c0e0d998f61c.jpg","list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathrm { b i a s } _ { P } ( \\hat { \\phi } ) = \\mathbb { E } _ { P } [ \\hat { \\phi } ] - \\phi ^ { \\dagger } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[122,642,477,660]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We define the variance of an estimator as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[379,672,614,693]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0409/570bf949d07db65d4f27635ae77bbd2074d816ac9c0ddbdff21fce5192864b04.jpg","list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname { v a r } _ { P } ( \\hat { \\phi } ) = \\mathbb { E } _ { P } [ ( \\hat { \\phi } - \\mathbb { E } _ { P } [ \\hat { \\phi } ] ) ^ { 2 } ] .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,704,758,723]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem B.3. If $h$ is uninvertible at $P$ ， then any estimator of $\\phi$ is biased. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,732,877,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof. For any probability model $P$ ， the expectation $\\mathbb { E } _ { P } [ \\hat { \\phi } ]$ must be a singleton set. If $h$ is uninvertible then $h ^ { - 1 } ( P )$ must contain at least two distinct elements. By the axiom of Learning, the estimator $\\hat { \\phi }$ cannot select the correct element of the set $h ^ { - 1 } ( P )$ and must therefore be biased. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[491,828,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,877,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary B.3. Let $\\theta$ denote a target estimand in scale reliant inference. Let $\\hat { \\theta }$ denote an estimator for $\\theta$ . Then $b i a s _ { P _ { W } } ( \\hat { \\theta } ) > 0$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,135,874,171]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof. The proof follws immediately from Proposition B.4 and the definition of scale reliance in terms of uninvertability. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,196,671,223]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CScale Simulation Random Variables ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,237,877,455]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section, the notation introduced in Section 4 of the manuscript applies. Briefly, this involves switching from using notation such as $P _ { W } ( \\mathcal { V } )$ to conditional probability notation $P ( \\mathcal { P } | W )$ which is more common in the Bayesian literature. This also involves specifying models with respect to probability densities or probability mass functions which we will denote with a lowercase $p$ as opposed to an upper case $P$ (which we reserve for probability measures). Finally, we will denote with a $( a )$ (e.g., $p ^ { ( a ) }$ or $P ^ { ( a ) }$ ）probability models that are specified by an analyst (applied models) and contrast this to the true distribution of a random quantity which we denote with a dagger (e.g., $p ^ { \\dagger }$ ，or $P ^ { \\dagger }$ ). In particular we call $P ^ { \\dagger } ( \\mathcal { V } | W )$ the target measurement model which represents the true relationship between the distribution of $\\mathcal { V }$ and the value of $W$ . When we wish to explicitly refer to the true distribution of $\\mathcal { V }$ , i.e.,the target measurement model at the true value of $W ^ { \\dagger }$ ，we will explicitly write $P ^ { \\dagger } ( \\mathcal { V } | W ^ { \\dagger } )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,457,875,511]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As suggested in the manuscript, the name scale simulation random variables highlights that this form is readily amenable to evaluation by Monte-Carlo Simulation (See Algorithm 1). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,513,872,683]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/1017/c795c9db184d8e134f881a56fcb76b9ffecacbc305671116d8f1b62d3c5467b7.jpg","list_items":null,"page_idx":"38","sub_type":null,"table_body":"<table><tr><td>Algorithm 1: Monte-Carlo Algorithm for Simulating Scale Simulation Random Variables</td></tr><tr><td>Data: Y</td></tr><tr><td>Result: S samples of 0</td></tr><tr><td>for s in {1,..., S} do</td></tr><tr><td>Sample W(s) ~ p(@)(Y)</td></tr><tr><td>Sample W⊥(s) ~ p(@)(W(s))</td></tr><tr><td>Calculate 0(s) = f(W(s),W⊥(s))</td></tr><tr><td>end</td></tr></table>","table_caption":"[]","table_footnote":"[\"This algorithm can also be adapted for use with Scale Conditional Random Variables (SCRVs） with density $p ^ { ( a ) } ( \\\\theta ~ \\\\vert ~ Y , W ^ { \\\\perp } )$ in combination with a scale model of the form \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[490,829,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"8 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,108,867,257]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0501/1c01eb5389206a87e206b52be4300818106f065328cbb5ed247d582758499622.jpg","list_items":null,"page_idx":"39","sub_type":null,"table_body":"<table><tr><td>Algorithm 2: Monte-Carlo Algorithm for Simulating Scale Simulation Random Variables from an SCRV</td></tr><tr><td>Data: Y</td></tr><tr><td>Result: S samples of 0</td></tr><tr><td>for s in {1,..., S} do</td></tr><tr><td>Sample W⊥(s) ~ p(@)(·)</td></tr><tr><td>Sample 0(s) ~ p(Y,W⊥(s))</td></tr><tr><td>end</td></tr></table>","table_caption":"[]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[117,260,877,352]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We will often assume the following regularity conditions on SSRVs. This regularity conditions are introduced so that we will be able to apply the dominated convergence theorem in proofs of the bias and variance of SSRVs. In essence, these regularity conditions serve to formally exclude from consideration pathological pairings of target estimands and applied and target models that we expect would be unlikely to be encountered in practice. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,361,879,455]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition C.1 (Regularity Conditions on Scale Simulation Estimators). Let $\\mathcal { V }$ denote a compositional survey of a scaled system $\\mathcal { W }$ with target measurement model $P ^ { \\dagger } ( \\mathcal { V } | W ^ { \\dagger } )$ . Let $\\mathcal { V } _ { n }$ denote a sequence of random variables that converges in probability to $\\mathcal { V }$ Let $\\theta ( \\mathcal { V } _ { n } )$ denote a scale simulation random variable. Let $P ( \\mathcal { V } , \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } )$ denote the joint probability distribution defined by: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[245,469,748,488]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0325/5d30f054b6c715e7b6d9607e6ca1c6663c4acb22d428fd4a070471ea893c2162.jpg","list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nP ( \\mathcal { V } , \\mathcal { W } ^ { \\| } , \\mathcal { W } ^ { \\bot } ) = P ^ { ( a ) } ( \\mathcal { W } ^ { \\bot } | \\mathcal { W } ^ { \\| } ) P ^ { ( a ) } ( \\mathcal { W } ^ { \\| } | \\mathcal { V } ) P ^ { \\dagger } ( \\mathcal { V } | \\mathcal { W } ^ { \\| \\dagger } , \\mathcal { W } ^ { \\bot \\dagger } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,501,877,556]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $P ^ { ( a ) } ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\parallel } ) P ^ { ( a ) } ( \\mathcal { W } ^ { \\parallel } | \\mathcal { V } )$ denote the applied measurement and scale model used in the scale simulation estimand. We assume the following three regularity conditions on scale simulation estimators. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[137,558,874,595]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. That each element of the sequence $\\{ \\theta ( \\mathcal { V } _ { n } ) \\}$ is integrable with respect to the joint distribution $P ( \\mathcal { V } _ { n } , \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[140,611,803,632]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. Under this joint, that the sequence $\\{ \\theta ( \\mathcal { V } _ { n } ) \\}$ converges in probability to $\\theta ( \\mathcal { V } )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[142,647,879,686]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3. That there exists a random variable $Z$ that is integrable with respect to the this joint which satisfies: $| \\theta ( \\mathcal { V } _ { n } ) | \\leq Z$ for all $n$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,696,875,751]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note that it follows from the dominated convergence theorem (See Theorem 1.1 in Shao (2003)） that under these regularity conditions we may interchange limits and integration, e.g., ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[232,750,761,787]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0247/4416c0886a01be91cbae97a8058cebc3f8898e43e29c2f6b9399b251835be5eb.jpg","list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { l i m } _ { n \\to \\infty } \\mathbb { E } _ { P ( \\mathcal { V } , \\mathcal { W } ^ { \\| } , \\mathcal { W } ^ { \\bot } ) } [ \\theta ( \\mathcal { V } _ { n } ) ] = \\int \\operatorname* { l i m } _ { n \\to \\infty } f ( \\mathcal { W } ^ { \\| } , \\mathcal { W } ^ { \\bot } ) d P ( \\mathcal { V } _ { n } , \\mathcal { W } ^ { \\| } , \\mathcal { W } ^ { \\bot } ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[490,829,503,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"9 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[117,87,555,104]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$p ^ { ( a ) } ( W ^ { \\perp } , \\cdot )$ where · denotes known hyperparameters ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,82,754,106]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C.1 Relationship between SSRVs and Bayesian PIMs ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,114,877,151]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Scale Simulation Random Variables, can alternatively be thought of as a transformation of the posterior distribution of a Bayesian Partially Identified Model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,159,877,217]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition C.1. Let $p ( \\mathcal { W } ^ { \\perp } , \\mathcal { W } ^ { \\parallel } | Y )$ denote the posterior of a Bayesian Partially Identified model with likelihood $p ( Y | \\mathcal { W } ^ { \\| } )$ and prior $p ( \\mathcal { W } ^ { \\perp } , \\mathcal { W } ^ { \\parallel } )$ Let $\\theta = f ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } )$ denote a target estimand and estimator pairing. The posterior of $\\theta$ is a scale simulation random variable. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,226,444,244]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof. Let us first rewrite the prior as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[361,257,632,276]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0855/13163306161807a6ec13729328e249e4a4659e02516e57b2b8307b10b27388a3.jpg","list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\np ( \\mathcal { W } ^ { \\perp } , \\mathcal { W } ^ { \\| } ) = p ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\| } ) p ( \\mathcal { W } ^ { \\| } ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,287,852,308]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $\\mathbf { A } \\subset \\Theta$ . The posterior under the transform $\\theta = f ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } )$ can then be written as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[274,319,720,359]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0424/ed96c962d1d49a9cf8534269e5f636ffe4a3e66e351876475f09f461e00969bf.jpg","list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nP ( \\theta \\in \\mathbf { A } | Y ) = \\int _ { \\mathcal { W } ^ { \\perp } , \\mathcal { W } \\parallel \\in f ^ { - 1 } ( \\mathbf { A } ) } p ( \\mathcal { W } ^ { \\perp } , \\mathcal { W } ^ { \\parallel } | Y ) d \\mathcal { W } ^ { \\parallel } d \\mathcal { W } ^ { \\perp }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,369,872,409]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since in the likelihood, $Y$ is conditionally independent of $\\mathcal { W } ^ { \\perp }$ the posterior $p ( \\mathcal { W } ^ { \\perp } , \\mathcal { W } ^ { \\parallel } | Y )$ can be factored as $p ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\parallel } ) p ( \\mathcal { W } ^ { \\parallel } | Y )$ leading to ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[246,417,748,458]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0132/71c2dcf778183cad1e2ea08c4ffefe3d0aeb7d4018d617c3c5e97289b021abcc.jpg","list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nP ( \\theta \\in \\mathbf { A } | Y ) = \\int _ { \\mathcal { W } ^ { \\perp } , \\mathcal { W } \\parallel \\in f ^ { - 1 } ( \\mathbf { A } ) } p ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\parallel } ) p ( \\mathcal { W } ^ { \\parallel } | Y ) d \\mathcal { W } ^ { \\parallel } d \\mathcal { W } ^ { \\perp } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,507,872,551]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C.2Asymptotic Properties of Bayesian Scale Simulation Random Variables ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,560,877,671]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Like any model, there are numerous ways in which scale simulation estimators can be misspecified. Still, our goal here is to focus on issues related to the lack of information about the system scale provided by compositional surveys. The definition of Faithful SSRVs serves to focus our attention on that goal and to deemphasizes issues with out ability to learn $W ^ { \\parallel }$ ， a topic that is both studied elsewhere and largely domain specific (Silverman et al.,2021, 2022; Grantham et al., 2020). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,680,879,773]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition C.2 (Faithful Scale Simulation Random Variables). Let $\\theta$ be a scale simulation random variables related to a target estimand $\\theta = f ( \\mathcal { W } )$ . We say that $\\theta$ is a faithful scale simulation random variable if the measurement model converges to the true value of the composition of the scaled system. That is, if for a sequence of random variables $\\{ \\mathcal { { D } } _ { n } \\}$ that converge in probability to $\\mathcal { V }$ that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[392,785,599,811]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0176/86ee9b6e6597916430e9dda489b659f2fc8bf3965f5debfd1165e4138ffb24b8.jpg","list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { l i m } _ { n \\to \\infty } p ^ { ( a ) } ( \\mathcal { W } ^ { \\parallel } | \\mathcal { V } _ { n } ) \\to \\delta _ { W ^ { \\parallel } } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[488,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,468,877,484]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,84,877,141]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\delta _ { W ^ { \\parallel \\dagger } }$ denotes a point mass centered on the true value of $\\mathcal { W } ^ { \\parallel }$ . Alternatively for a sequence of observations $\\left\\{ Y _ { n } \\right\\}$ of a random variable $\\mathcal { V }$ ， a faithful scale simulation random variable is one that satisfies: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[362,155,629,183]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0657/9c36650e6c0a71d433a176f3ea559f84ada50a59d9c3dc8e5b5e5e6a8a84cbfb.jpg","list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { l i m } _ { n \\to \\infty } p ^ { ( a ) } ( \\mathcal { W } ^ { \\parallel } | Y _ { 1 } , \\ldots , Y _ { n } ) \\to \\delta _ { W ^ { \\parallel } }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,193,877,289]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In what follows, we provide results about the asymptotic expectation and variance of scale simulation random variables. For notational convenience, we will denote the expecta-tion of a scale simulation random variable as $\\overline { { { \\theta } } } = \\mathbb { E } _ { P ^ { ( a ) } ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } | Y ) } [ f ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } ) ]$ . In addition, let us use the following short-hand notations: $p ^ { \\dagger } \\equiv p ^ { \\dagger } ( \\mathcal { V } | W ^ { \\parallel \\dagger } , W ^ { \\perp \\dagger } )$ ， $\\theta ^ { \\dagger } \\equiv f ( W ^ { \\parallel \\dagger } , W ^ { \\perp \\dagger } )$ ， and $p ^ { ( a ) } \\equiv p ^ { ( a ) } ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } | \\mathcal { V } ) = p ^ { ( a ) } ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\parallel } ) p ^ { ( a ) } ( \\mathcal { W } ^ { \\parallel } | \\mathcal { V } )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,297,877,354]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem C.1. Let $\\theta \\ : = \\ : f ( \\mathcal { W } )$ denote a target estimand and estimator pairing. Let $\\theta$ denote a faithful scale simulation estimator that satisfies the regularity conditions provided in Definition C.1. Then the asymptotic bias of $\\bar { \\theta }$ is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[300,367,699,395]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0739/952a435c8e82d1e5c782d4736c3c2c1d8ad250a02724c8b1473e3ddfc9a3ab73.jpg","list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { l i m } _ { n \\to \\infty } \\mathrm { b i a s } _ { p ^ { \\dag } } ( \\overline { { \\theta } } ) = \\mathbb { E } _ { p ^ { ( a ) } ( \\mathcal { W } ^ { \\perp } | W ^ { | \\mathfrak { t } } ) } [ f ( W ^ { | \\mathfrak { t } | } , \\mathcal { W } ^ { \\perp } ) ] - \\theta ^ { \\dag }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,406,490,425]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof. By the definition of asymptotic bias: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[359,439,633,465]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0235/ed4b7f8c8b1e51e23c0aa7584719dc463378535d3f3725f88f7f882ac5ef6d15.jpg","list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { l i m } _ { n \\to \\infty } \\operatorname { b i a s } _ { p ^ { \\dag } } ( \\overline { { \\theta } } ) = \\operatorname* { l i m } _ { n \\to \\infty } \\mathbb { E } _ { p ^ { \\dag } } ( \\overline { { \\theta } } ) - \\theta ^ { \\dag } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,477,874,513]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By the dominated convergence theorem in combination with our regularity conditions, we have that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[181,523,839,561]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0942/7c5e4a77919e5d840d443fd5256ae9fb4ef777034b8c7833d444a35752046b7d.jpg","list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n_ { p ^ { \\prime } } ( \\overline { { { \\theta } } } ) = \\int \\operatorname* { l i m } _ { n \\infty } f ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } ) p ^ { ( a ) } ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\parallel } ) p ^ { ( a ) } ( \\mathcal { W } ^ { \\parallel } | \\mathcal { V } _ { n } ) p ^ { \\dagger } ( \\mathcal { V } _ { n } | \\mathcal { W } ^ { \\parallel \\dagger } , \\mathcal { W } ^ { \\perp \\dagger } ) d \\mathcal { W } ^ { \\parallel } d \\mathcal { W } ^ { \\perp } d \\mathcal { W } ^ { \\perp } d \\mathcal { W } ^ { \\perp } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,571,633,590]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since $\\theta$ is a faithful scale simulation estimator it follows that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[184,599,812,703]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0229/c736cb61fe551a7cf3a6abade25ee17d0f01f5e300528ab08501e75e5631e192.jpg","list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { l } { \\displaystyle \\operatorname* { l i m } _ { n \\to \\infty } E _ { p ^ { \\dagger } } ( \\overline { { \\theta } } ) = \\int f ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } ) p ^ { ( a ) } ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\parallel } ) \\delta _ { W ^ { \\parallel \\dagger } } p ^ { \\dagger } ( \\mathcal { V } | \\mathcal { W } ^ { \\parallel \\dagger } , \\mathcal { W } ^ { \\perp \\dagger } ) d \\mathcal { W } ^ { \\parallel } d \\mathcal { W } ^ { \\perp } d \\mathcal { Y } } \\\\ { \\displaystyle \\quad \\quad = \\int f ( \\mathcal { W } ^ { \\parallel \\dagger } , \\mathcal { W } ^ { \\perp } ) p ^ { ( a ) } ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\parallel \\dagger } ) d \\mathcal { W } ^ { \\perp } \\int p ^ { ( a ) } ( \\mathcal { V } | \\mathcal { W } ^ { \\parallel \\dagger } , \\mathcal { W } ^ { \\perp \\dagger } ) d \\mathcal { V } } \\\\ { \\displaystyle \\quad = \\mathbb { E } _ { p ^ { ( a ) } ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\parallel \\dagger } ) } \\left[ f ( \\mathcal { W } ^ { \\parallel \\dagger } , \\mathcal { W } ^ { \\perp } ) \\right] . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,713,241,731]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It follows that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[292,729,700,757]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0122/00b052c7c4dbe300730c528e77e14d62b91e888ebb69ac3cf0d9c2391ca1134f.jpg","list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { l i m } _ { n \\to \\infty } \\operatorname { b i a s } _ { p ^ { \\dag } } ( \\overline { { \\theta } } ) = \\mathbb { E } _ { p ^ { ( a ) } ( \\mathcal { W } ^ { \\bot } | W ^ { | \\dag } ) } \\left[ f ( W ^ { | \\dag } , \\mathcal { W } ^ { \\bot } ) \\right] - \\theta ^ { \\dag }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"11 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,762,877,777]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"□ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,877,214]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The above result should come as no surprise given Theorem B.3, in the setting of scale reliant inference (i.e., when $h$ is not invertible), the mean of SSRVs are a biased estimator. Still, there are at least two remarkable facts that tie into this result.First，we will show that the bias is identical when we shift to a Bayesian perspective and calculate bias with respect to the posterior $\\operatorname* { l i m } _ { n \\to \\infty }$ ${ \\mathrm { b 1 a s } } _ { p ^ { ( a ) } ( { \\mathcal { W } } \\parallel , { \\mathcal { W } } ^ { \\perp } | Y _ { n } ) }$ . Second, we will show that the asymptotic variance of the SSRV actually includes a model for this bias and remains non-zero even in the asymptotic limit. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,226,877,281]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem C.2. Let $\\theta \\ : = \\ : f ( \\mathcal { W } )$ denote a target estimand and estimator pairing. Let $\\theta$ denote a faithful scale simulation estimator that satisfies the regularity conditions provided in Definition C.1. Then the asymptotic bias of $\\bar { \\theta }$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[294,295,700,320]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0997/ac3f4ea5a59193dfc9aedfad8e53ee979c308c2498b4a9e49b4f5c33f7b8cec1.jpg","list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { l i m } _ { n \\to \\infty } \\mathrm { b i a s } _ { p ^ { ( a ) } } ( \\overline { { \\theta } } ) = \\mathbb { E } _ { p ^ { ( a ) } ( \\mathcal { W } ^ { \\bot } | W ^ { | | \\dagger } ) } [ f ( W ^ { | | \\dagger } , \\mathcal { W } ^ { \\bot } ) ] - \\theta ^ { \\dagger }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,332,392,351]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof. By the definition of bias, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[184,361,810,433]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0536/634eef011cf17fa6f659983416df9a3d3e9155cf2e8b9cdd32c3753d5ba77589.jpg","list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\underset { n \\infty } { \\operatorname* { l i m } } \\mathrm { b i a s } _ { p ^ { ( a ) } } ( \\overline { { \\theta } } ) = \\underset { n \\infty } { \\operatorname* { l i m } } \\mathbb { E } _ { p ^ { ( a ) } } [ f ( \\mathcal { W } ^ { \\| } , \\mathcal { W } ^ { \\perp } ) ] - \\theta ^ { \\dagger } } \\\\ & { \\mathbb { E } _ { p ^ { ( a ) } } [ f ( \\mathcal { W } ^ { \\| } , \\mathcal { W } ^ { \\perp } ) ] = \\underset { n \\infty } { \\operatorname* { l i m } } \\displaystyle \\int f ( \\mathcal { W } ^ { \\| } , \\mathcal { W } ^ { \\perp } ) p ^ { ( a ) } ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\| } ) p ^ { ( a ) } ( \\mathcal { W } ^ { \\| } | Y _ { n } ) d \\mathcal { W } ^ { \\| } d \\mathcal { W } ^ { \\perp } . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,440,874,479]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By the dominated convergence theorem (applicable through regularity conditions） and using the fact that $\\theta$ is a faithful SSRV, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[186,489,808,593]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0426/a025acae85654ffa4131356d7d2d0508c7356929539e39702d57fe5521b539e1.jpg","list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\mathbb { E } _ { p ^ { ( a ) } } [ f ( \\mathcal { W } ^ { \\| } , \\mathcal { W } ^ { \\bot } ) ] = \\displaystyle \\int \\operatorname* { l i m } _ { n \\to \\infty } f ( \\mathcal { W } ^ { \\| } , \\mathcal { W } ^ { \\bot } ) p ^ { ( a ) } ( \\mathcal { W } ^ { \\bot } | \\mathcal { W } ^ { \\| } ) p ^ { ( a ) } ( \\mathcal { W } ^ { \\| } | Y _ { n } ) d \\mathcal { W } ^ { \\| } d \\mathcal { W } ^ { \\bot } } \\\\ & { \\quad \\quad \\quad \\quad = \\displaystyle \\int f ( W ^ { \\| \\dag } , \\mathcal { W } ^ { \\bot } ) p ^ { ( a ) } ( \\mathcal { W } ^ { \\bot } | W ^ { \\| \\dag } ) d \\mathcal { W } ^ { \\bot } } \\\\ & { \\quad \\quad \\quad = \\mathbb { E } _ { p ^ { ( a ) } ( \\mathcal { W } ^ { \\bot } | W ^ { \\| \\dag } ) } [ f ( W ^ { \\| \\dag } , \\mathcal { W } ^ { \\bot } ) ] . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,603,207,621]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Therefore, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[290,619,702,647]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0485/dc436aeeee7a6a6b2bf0a57541208c9ab14c1639ad14c9a6e23f33df4db72e6b.jpg","list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { l i m } _ { n \\to \\infty } \\mathrm { b i a s } _ { p ^ { ( a ) } } ( \\overline { { \\theta } } ) = \\mathbb { E } _ p ^ { ( a ) } ( \\mathcal { W } ^ { \\bot } | W ^ { | \\} ) } [ f ( W ^ { | | \\dagger } , \\mathcal { W } ^ { \\bot } ) ] - \\theta ^ { \\dagger } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,679,879,771]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The last two results show that, regardless of whether we calculate the expectation of the SSRV with respect to joint of the applied measurement and scale model (i.e., treating $\\mathcal { W }$ as a random variable and $\\mathcal { V }$ as a fixed observed quantity) or with respect to the target measurement model (e.g.， treating $\\mathcal { V }$ as the random variable and $\\mathcal { W }$ as fixed） we get the same form for the asymptotic bias ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[305,785,687,810]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0620/657acef651a54b00b47efe474c4f00ed2d15746c9449ba26eb49baa4dc7e874c.jpg","list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { l i m } _ { n \\to \\infty } \\operatorname { b i a s } ( { \\overline { { \\theta } } } ) = \\operatorname* { l i m } _ { n \\to \\infty } \\operatorname { b i a s } _ { p ^ { \\dag } } ( { \\overline { { \\theta } } } ) = \\operatorname* { l i m } _ { n \\to \\infty } \\operatorname { b i a s } _ { p ^ { ( a ) } } ( { \\overline { { \\theta } } } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"12 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,651,875,667]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,83,893,161]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"43","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where we now use $b i a s ( \\theta )$ , i.e., without subscript,to refer to either ${ \\mathrm { b i a s } } _ { p ^ { \\dagger } } ( \\theta )$ or ${ \\mathrm { b i a s } } _ { p ^ { ( a ) } } ( { \\overline { { \\theta } } } )$ To introduce the next theorem let us generalize this notation. We now write $\\theta ^ { \\dagger }$ as $f ( W ^ { \\parallel \\dagger } , W ^ { \\perp \\dagger } )$ and also to highlight that this value for $\\theta ^ { \\dagger }$ is with respect to the true value of the system scale with the notation bias $( \\theta | W ^ { \\perp \\dagger } )$ ： ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[236,175,756,203]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0843/4e23527fee3300b06c4ae251b126e7a6e61466b88420c5016cada1d428119199.jpg","list_items":null,"page_idx":"43","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname * { l i m } _ { n \\to \\infty } \\mathrm { b i a s } ( \\overline { { \\theta } } | W ^ { \\perp \\dagger } ) = \\mathbb { E } _ { p ^ { ( a ) } ( \\mathcal { W } ^ { \\parallel } | W ^ { \\parallel \\dagger } ) } [ f ( W ^ { \\parallel \\dagger } , \\mathcal { W } ^ { \\perp } ) ] - f ( W ^ { \\parallel \\dagger } , W ^ { \\perp \\dagger } ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,213,877,250]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"43","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem C.3. The asymptotic variance of a faithful scale simulation random variable under regularity conditions provided in Definition $\\it { C . 1 }$ is: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[266,263,727,291]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0601/4498704f2d9d92767f61d87b2e4193eb2f59ee5717a6e7823696dad35bef12bf.jpg","list_items":null,"page_idx":"43","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { l i m } _ { n \\to \\infty } \\operatorname { v a r } _ { p ^ { ( a ) } ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } | Y _ { n } ) } ( \\theta ) = \\mathbb { E } _ { p ^ { ( a ) } ( \\mathcal { W } ^ { \\perp } | W ^ { \\parallel \\dagger } ) } \\left[ b i a s ( \\bar { \\theta } | \\mathcal { W } ^ { \\perp } ) ^ { 2 } \\right] .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,303,874,342]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"43","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof. As notational shorthands,let us define $e ( \\mathcal { W } ^ { \\| } ) \\equiv \\mathbb { E } _ { p ^ { ( a ) } ( \\mathcal { W } ^ { \\bot } | \\mathcal { W } ^ { \\parallel } ) } \\left[ f ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\bot } ) \\right]$ By the law of total variance: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[274,356,720,377]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0325/8562dc762ea02c3e441ad6778fb201f10332272089e93764c51acabae49a5611.jpg","list_items":null,"page_idx":"43","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { \\operatorname { v a r } _ { p ^ { ( a ) } } ( \\theta ) = \\mathbb { E } _ { p ^ { ( a ) } } [ \\operatorname { v a r } _ { p ^ { ( a ) } } ( \\theta \\vert \\mathcal { W } ^ { \\parallel } ) ] + \\operatorname { v a r } _ { p ^ { ( a ) } } [ \\mathbb { E } _ { p ^ { ( a ) } } ( \\theta \\vert \\mathcal { W } ^ { \\parallel } ) ] . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,390,383,407]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"43","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We start with the second term. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[184,417,810,529]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0521/30add5e014c8db1b631b24e2724f2bdd28cd376c4e02d9d9f7e8fdfe5090779c.jpg","list_items":null,"page_idx":"43","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } { \\underset { n \\to \\infty } { \\operatorname* { l i m } } \\mathrm { v a r } _ { p ^ { ( a ) } } [ \\mathbb { E } _ { p ^ { ( a ) } } ( \\theta | \\mathcal { W } ^ { \\| } ) ] = \\underset { n \\to \\infty } { \\operatorname* { l i m } } \\mathrm { v a r } _ { p ^ { ( a ) } } \\left[ \\int \\theta p ^ { ( a ) } ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\| } ) d \\mathcal { W } ^ { \\perp } \\right] } & { } \\\\ { = \\underset { n \\to \\infty } { \\operatorname* { l i m } } \\mathrm { v a r } _ { p ^ { ( a ) } } [ e ( \\mathcal { W } ^ { \\| } ) ] } & { } \\\\ { } & { = \\underset { n \\to \\infty } { \\operatorname* { l i m } } \\int \\left( e ( \\mathcal { W } ^ { \\perp } ) - \\mathbb { E } _ { p ^ { ( a ) } } e ( \\mathcal { W } ^ { \\| } ) \\right) ^ { 2 } p ^ { ( a ) } ( \\mathcal { W } ^ { \\| } | Y _ { n } ) d \\mathcal { W } ^ { \\| } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,536,877,574]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"43","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Using our regularity conditions to apply the dominated convergence theorem the limit can be moved inside the integral leading to ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[194,587,799,623]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0874/0e19b3678b88f62d07dd2080bedd6723266797e9c5faa825d78d697034afa2b2.jpg","list_items":null,"page_idx":"43","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { l i m } _ { n \\to \\infty } \\operatorname { v a r } _ { p ^ { ( a ) } } [ \\mathbb { E } ( \\theta | \\mathcal { W } ^ { \\| } ) ] = \\int \\operatorname* { l i m } _ { n \\to \\infty } \\big ( e ( \\mathcal { W } ^ { \\perp } ) - \\mathbb { E } _ { p ^ { ( a ) } } e ( \\mathcal { W } ^ { \\| } ) \\big ) ^ { 2 } p ^ { ( a ) } ( \\mathcal { W } ^ { \\| } | Y _ { n } ) d \\mathcal { W } ^ { \\| } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,635,753,654]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"43","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Using the fact that $\\theta$ is a faithful scale simulation random variable leads to ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[228,666,764,724]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0950/ae121f246692c2d478c241d081f9f3422d3603b6f8c7755a05f97d295ea94b0c.jpg","list_items":null,"page_idx":"43","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l r } { { \\operatorname* { l i m } _ { n \\to \\infty } \\operatorname { v a r } _ { p ^ { ( a ) } } [ \\mathbb { E } ( \\theta | \\mathcal { W } ^ { \\parallel } ) ] = \\int \\operatorname* { l i m } _ { n \\to \\infty } \\big ( e ( \\mathcal { W } ^ { \\perp } ) - \\mathbb { E } _ { p ^ { ( a ) } } e ( \\mathcal { W } ^ { \\parallel } ) \\big ) ^ { 2 } \\delta _ { W ^ { \\parallel \\dagger } } d \\mathcal { W } ^ { \\parallel } } } \\\\ & { } & { = 0 \\quad } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"43","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"13 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,503,104]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We are therefore left with only the first term: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[158,116,816,271]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0230/3f6e0a44556b114dcf382d7ea0c7baa04dfed868c9a1ec6f383cfad6e28a304d.jpg","list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { l } { \\displaystyle \\operatorname { v a r } _ { p ^ { ( \\alpha ) } } ( \\theta ) = \\underset { n \\to \\infty } { \\operatorname* { l i m } } \\mathbb { E } _ { p ^ { ( \\alpha ) } } \\big [ \\nabla \\mathrm { a r } _ { p ^ { ( \\alpha ) } } \\big ( \\theta | \\mathcal { W } ^ { \\parallel } \\big ) \\big ] } \\\\ { \\displaystyle \\quad = \\operatorname* { l i m } _ { n \\to \\infty } \\mathbb { E } _ { p ^ { ( \\alpha ) } } [ \\int \\big ( \\theta - \\mathbb { E } _ { p ^ { ( \\alpha ) } } \\big ( \\theta \\big | \\mathcal { W } ^ { \\parallel } \\big ) \\big ) ^ { 2 } p ^ { ( \\alpha ) } \\big ( \\mathcal { W } ^ { \\perp } \\big | \\mathcal { W } ^ { \\parallel } \\big ) d \\mathcal { W } ^ { \\perp } \\big | \\mathcal { W } ^ { \\parallel } ] } \\\\ { \\displaystyle \\quad = \\underset { n \\to \\infty } { \\operatorname* { l i m } } \\mathbb { E } _ { p ^ { ( \\alpha ) } } [ \\int \\big ( f ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } ) - e ( \\mathcal { W } ^ { \\parallel } ) \\big ) ^ { 2 } p ^ { ( \\alpha ) } \\big ( \\mathcal { W } ^ { \\perp } \\big | \\mathcal { W } ^ { \\parallel } \\big ) d \\mathcal { W } ^ { \\perp } \\big | \\mathcal { W } ^ { \\parallel } ] } \\\\ { \\displaystyle \\quad = \\underset { n \\to \\infty } { \\operatorname* { l i m } } \\int \\{ \\int \\big ( f ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } ) - e ( \\mathcal { W } ^ { \\parallel } ) \\big ) ^ { 2 } p ^ { ( \\alpha ) } \\big ( \\mathcal { W } ^ { \\perp } \\big | \\mathcal { W } ^ { \\parallel } \\big ) d \\mathcal { W } ^ { \\perp } \\} p ^ { ( \\alpha ) } \\big ( \\mathcal { W } ^ { \\parallel } \\big | \\mathcal { Y } \\big | } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,279,874,315]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Again,applying the dominated convergence theorem and using the fact that this is a faithful scale simulation random variable yields: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[181,325,825,470]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0563/afc30070872b333437cecd2c25c752a1e94f20681f5619caec32b3d3a1b61fdf.jpg","list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { l } { \\displaystyle \\operatorname* { l i m } _ { \\to \\infty } \\operatorname { v a r } _ { p ^ { ( \\alpha ) } } ( \\theta ) = \\int \\left\\{ \\int \\left( f ( \\mathcal { W } ^ { \\parallel } , \\mathcal { W } ^ { \\perp } ) - e ( \\mathcal { W } ^ { \\parallel } ) \\right) ^ { 2 } p ^ { ( \\alpha ) } ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\parallel } ) d \\mathcal { W } ^ { \\perp } \\right\\} \\delta _ { \\mathcal { W } ^ { \\parallel } \\uparrow } d \\mathcal { W } ^ { \\parallel } } \\\\ { \\displaystyle \\quad = \\int \\left( f ( W ^ { \\parallel \\dagger } , \\mathcal { W } ^ { \\perp } ) - e ( W ^ { \\parallel \\dagger } ) \\right) ^ { 2 } p ^ { ( \\alpha ) } ( \\mathcal { W } ^ { \\perp } | W ^ { \\parallel \\dagger } ) d \\mathcal { W } ^ { \\perp } } \\\\ { \\displaystyle \\quad = \\int \\left[ \\mathrm { b i a s } ^ { * } ( \\theta | \\mathcal { W } ^ { \\perp } ) \\right] ^ { 2 } p ^ { ( \\alpha ) } ( \\mathcal { W } ^ { \\perp } | W ^ { \\parallel \\dagger } ) d \\mathcal { W } ^ { \\perp } } \\\\ { \\displaystyle \\quad = \\mathbb { E } _ { p ^ { ( \\alpha ) } ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\parallel \\dagger } ) } \\left[ \\mathrm { b i a s } ( \\bar { \\theta } | \\mathcal { W } ^ { \\perp } ) ^ { 2 } \\right] . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,507,877,655]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This result establishes that the asymptotic variance of SSRVs acknowledges their bias. The asymptotic variance of an SSRV is precisely the expected squared bias with expectation taken with respect to our uncertainty in $\\mathcal { W } ^ { \\perp }$ conditioned on the true value of $W ^ { \\parallel \\dagger }$ which is learned from the data. This also implies that, so long as the scale model has non-zero variance,if the model has non-zero bias then it will be matched by a non-zero variance. This is a critical feature as it helps to avoid spurious conclusions caused by being overly confident in an incorrect answer. These desirable features are not present in many popular models used to analyze compositional surveys. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,681,352,707]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"D Applications ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,720,547,743]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"D.1Simulation Details and Results ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,752,875,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For all simulations, we simulated a data set with 20 taxa across two conditions using multinomial sampling. Sequencing depth was set to 5,0o0, and, unless otherwise specified,sample size was set to 50( $n = 5 0$ ). ALDEx2 and DESeq2 were fit using their respective Bioconductor packages. For the simulations, default settings were used for each. For ALDEx2, 128 Monte Carlo samples were used, and the t-test with the Benjamini-Hochberg correction were used to find significance. For DESeq2, the Wald test was used, and Benjamini-Hochberg p-values were used to determine significance. Unless otherwise stated, $\\alpha = 0 . 0 5$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"14 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,479,875,496]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[263,85,723,262]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 1: Recreating ALDEx2 using SSRVs. To do so, we used a scale model that was a degenerate point mass centered on the geometric mean. Signifcance was assessd using the testing procedure of ALDEx2 at $\\\\alpha = 0 . 0 5$ . \"]","image_footnote":"[]","img_path":"images/0281/2a693362c036bcd2245890a34f281cf36ffe1fc004a3c5453773c3b8429bcebd.jpg","list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[120,353,875,444]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,445,875,517]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For our scale simulation approach, we assessed significance using the same approach as ALDEx2 with $\\alpha = 0 . 0 5$ . This procedure, along with a degenerate total model centered on the geometric mean, can capture the results of the base ALDEx2 model. Results to this effect are shown in Figure 1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,518,877,627]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Further， we provide several more figures to support claims made in the main text. Shown in Supplementary Figure 2a, we show that small amounts of uncertainty in the scale ( $\\alpha = 0 . 5$ ） can greatly reduce the number of false positives while still retaining three out of four true positives. This is in direct contrast to the results of ALDEx2. Shown in Supplementary Figure 2b, we find that inferred results can vary greatly as a function of the uncertainty in the scale model as expressed through $\\alpha$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[122,650,722,671]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"D.2Artificial Gut Application Details and Results ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,680,325,699]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bioreactor Culturing ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,709,877,799]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The bioreactor run consisted of 8 continuous-flow vessels (Multifors 2， Infors） cultured over 25 total days according to previously established protocols (Silverman et al., 2018; Letourneau et al., 2021). Briefly, vessels were inoculated with identical inoculum derived from a single healthy stool donor. Vessels were maintained anaerobic under nitrogen, kept at $3 7 ^ { \\mathrm { ~ o ~ } }$ C, continuously stirred at 1O0 rpm using magnetic impeller stir-shafts, and the total volume of media (40O mL modified Gifu Anaerobic Medium) was turned over once per 24 hours. Samples for sequencing and fow cytometry were collcted daily at 4pm by syringe via a sterilized sampling port. Since the volume of each vessel remained fixed at 400 mL, concentration of cells is directly proportional to absolute abundance. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[488,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"15 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[233,104,758,622]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 2: (a) Modeling results for DESeq2, ALDEx2,and three separate SSRVs. The $\\\\it C L R$ model uses a scale model centered on the geometric mean with $\\\\alpha = 1 e \\\\mathrm { ~ - ~ } 3$ .Note its correspondence with ALDEx2. The Relaxed scale model uses the same mean, but sets $\\\\alpha = 0 . 5$ . The $\\\\it C o D A$ scale model also uses the same mean, but sets $\\\\alpha = 1 0$ . Note that a small amount of noise reduces the number of false positives greatly. (b) Sensitivity analysis showing the estimated SSRV effect size for each taxa under the Relaxed model as a function of $\\\\alpha$ . A black line signifies that the estimated effect size was statistically different from zero at $9 5 \\\\%$ confidence. \"]","image_footnote":"[]","img_path":"images/0081/a1bf8dc72ea2658553208ecd7464fd1f729642ea4ca17dab585fc76277bffd1c.jpg","list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[488,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"16 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,85,877,159]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,179,500,199]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cell Quantification by Flow Cytometry ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,208,875,299]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Samples were serially diluted ten-fold for a fnal dilution of 1:1Ooo. Serial dilutions were performed in PBS, with the final dilution performed into wels of a 96-wel plate containing PBS with 1 μL SYBR green/DMSO (1:499 SYBR:DMSO) per 180 μL total volume. The final 20O μL mixture including sample and SYBR green was incubated 15 min at 37 $\\cup$ C (Vandeputte et al., 2017). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,300,875,372]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Flow cytometry was performed using a MACSQuant Flow Cytometer (Miltenyi Biotec). The 96-well plate containing samples was maintained on a chill rack for the duration of the run. 10 μL was taken for measurement. Forward scatter, side scatter, and SYBR fluorescence (using channel B1, log5, 274 V） were recorded. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,373,877,482]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Flow cytometry data was analyzed in FlowJo using a two-step gating strategy. A single sample containing cells but no SYBR was used to develop gates which were then applied to all samples. First, using a graph of forward scatter vs. side scatter, we roughly selected for the center of point density, omitting extreme values (i.e. possibly representing cellular debris and/or particle aggregates). Second, examining the SYBR channel, we excluded counts that fell within the range of the no-SYBR control. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[122,503,591,521]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"16S rRNA-Gene Encoding Amplicon Sequencing ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,531,877,659]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"16S rRNA gene amplicon sequencing was performed as previously described using custom barcoded primers targeting the V4 region of the gene, resulting in 150 bp paired-end reads sequenced by Ilumina MiniSeq according to previously published protocols (Silverman et al., 2018; Letourneau et al., 2021). DADA2 was used to identify and quantify sequence variants (SVs) in our dataset， using version 123 of the Silva database. Taxa that had counts of less than 2,5O0 across all samples were fltered and amalgamated into a separate category. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,680,258,698]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Data Analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,708,875,780]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For our analysis, we restricted ourselves to days 1 and 14. Furthermore, we used data from vessels 3, 4, 5, 6,7,and 8 only due to limitations in the data. We noted a difference between flow cytometry concentrations between days regardless of vessel. See Figure 3 for details. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[488,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"17 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,877,176]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We tested a model of differential expression including indicator terms for each vessel and day. For ALDEx2, we used the generalized linear model test with 250 Monte Carlo Samples. We assessed differential expressed taxa based on the p-values ( $\\alpha = 0 . 0 5$ ) for day alone. For DESeq2, we employed the likelihood ratio test with a reduced model containing only the vessel and assessed differential abundance using the same criteria as ALDEx2. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,178,877,304]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For the SSRV based on the Multinomial Log Normal model, we set our prior parameters as follows: $\\nu = D + 3$ and $\\Xi = I _ { D }$ reflecting our weak prior knowledge that the taxa were independent; Γ = BlockDiagonal $( 1 0 I _ { 6 } , 1 )$ reflecting our assumption that the some taxa were more abundant than others at baseline and assuming that the signal-to-noise ratio with respect to differential abundance was most likely on the order of 1 (Silverman et al., 2018); and finally $M = 0$ which reflected that we did not have α priori knowledge differentiating the taxa. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,306,870,342]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"31 different sequence variants were declared significant by at least one method. Taxonomic information for these significant sequence variants is presented in Table 1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,361,671,574]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 3: Variation of fow cytometry data by day. Note that there was a higher concentration of cells on day 14 regardless of vessel. This has biological validity based on the experimental goals of the study. \"]","image_footnote":"[]","img_path":"images/0880/d943be76a29af524357920020bbd163df5d3bf6c29e398813db0a1e831e16c90.jpg","list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[488,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"18 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[294,145,687,747]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0726/4fd567fdeb6eb4ca7124764f6184024a901e7aaaab911ef48718d5bc564e6c7d.jpg","list_items":null,"page_idx":"49","sub_type":null,"table_body":"<table><tr><td>Sequence</td><td>Taxonomic Information</td></tr><tr><td>1</td><td>Bacteroides spp.</td></tr><tr><td>2</td><td>Escherichia/Shigella spp.</td></tr><tr><td>3</td><td>Bacteroides uniformis</td></tr><tr><td>4</td><td>Peptoniphilus spp.</td></tr><tr><td>5</td><td>Pyramidobacter piscolens</td></tr><tr><td>6</td><td>Enterococcus spp.</td></tr><tr><td>7</td><td>Veillonella spp.</td></tr><tr><td>8</td><td>Bacteroides spp.</td></tr><tr><td>9</td><td>Streptococcus spp.</td></tr><tr><td>10</td><td>Anaerosalibacter bizertensis</td></tr><tr><td>11</td><td>Bacteroides stercoris</td></tr><tr><td>12</td><td>Bacteroides caccae</td></tr><tr><td>13</td><td>Megasphaera spp.</td></tr><tr><td>14</td><td>Peptoniphilus lacrimalis</td></tr><tr><td>15</td><td>Lachnoclostridium spp.</td></tr><tr><td>16</td><td>Dorea formicigenerans</td></tr><tr><td>17</td><td>Phascolarctobacterium faecium</td></tr><tr><td>18</td><td>Veillonella spp.</td></tr><tr><td>19</td><td>Lachnoclostridium spp.</td></tr><tr><td>20</td><td>Clostridium sensu stricto spp.</td></tr><tr><td>21</td><td>Veillonella spp.</td></tr><tr><td>22</td><td>Peptoclostridium difficile</td></tr><tr><td>23</td><td>Erysipelatoclostridium ramosum</td></tr><tr><td>24</td><td>Hungatella spp.</td></tr><tr><td>25</td><td>Flavonifractor plautii</td></tr><tr><td>26</td><td>Bifidobacterium spp.</td></tr><tr><td>27</td><td>Parabacteroides merdae</td></tr><tr><td>28</td><td>Anaerococcus vaginalis</td></tr><tr><td>29</td><td>Blautia spp.</td></tr><tr><td>30</td><td>Eubacterium spp.</td></tr><tr><td>31</td><td>Anaerostipes hadrus</td></tr></table>","table_caption":"[\"Table 2: Taxonomic information of significant sequence variants identified in the Real Data Analysis. \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[488,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"49","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"19 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[294,85,699,362]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 4: FDR by method (ALDEx2, DESeq2, SSRV - Design) with increasing sample sizes. Additional samples were drawn using bootstrap resampling. Dashed horizontal line denotes the worst possible FDR given the number of true positives in this study. \"]","image_footnote":"[]","img_path":"images/0086/4f0c473aafe3231986e1cfef1cb9d386a54d0fe586b7a5d55db8d93d2c5d10df.jpg","list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[120,450,870,473]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"D.3Deriving the Multinomial Logistic Normal Model as a SSRV ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,483,872,518]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section, we show how the multinomial logistic-normal model for sequence count data presented in the manuscript can be inferred as a SSRV. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,540,598,558]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"D.3.1 Derivation of Conditional Relative Model ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,568,385,585]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recall that the target model is: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[419,616,576,757]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0894/6f3597cc4fbf6633ff7f339e42e2c9a87537f68b57922516e7f87f226fc0a86d.jpg","list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { Y _ { j } \\sim \\mathrm { M u l t } ( \\mathcal { W } _ { j } ^ { \\parallel } ) } \\\\ & { \\mathcal { W } _ { j } = \\mathcal { W } _ { j } ^ { \\perp } \\mathcal { W } _ { j } ^ { \\parallel } } \\\\ & { \\mathcal { W } _ { j } = \\exp ( \\psi _ { j } ) } \\\\ & { \\psi _ { j } \\sim N ( B X , \\Omega , I ) } \\\\ & { B \\sim N ( M , \\Omega , \\Gamma ) } \\\\ & { \\Omega \\sim I W ( \\nu , \\Xi ) } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,768,424,787]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"First, we defined $G$ as a linear map: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"20 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[457,99,537,138]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"1009/46f6a375a285a501d6df5f7900bdbd071e5f97068b5b5bb5994ffed66aa90b50.jpg","list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nG = { \\binom { F } { H } }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,143,875,180]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $F = [ I _ { D - 1 } , - \\mathbf { 1 } _ { D - 1 } ]$ and $H = \\mathbf { 1 } _ { D }$ . Note that for any arbitrary vector $\\theta$ and matrix $\\Sigma$ ， we have: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[289,205,705,291]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0868/f27d099124d816436d82f1c08f139e5a2148317f209dc654bec5ef310d5a582e.jpg","list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n{ \\begin{array} { r l } & { G \\theta = \\left[ F \\theta \\right] = \\left[ \\theta ^ { \\parallel } \\right] } \\\\ & { G \\theta } \\end{array} } \n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,299,602,318]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where the $\\parallel$ and $\\perp$ notation follows the rest of the paper. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,319,640,337]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Applying $G$ to $\\psi$ from the model defined in Equations 2-7: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[446,347,547,387]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0280/b1453dce80577146363ed015bd85a2aaf4493f3c19f836d6951f3bc76350bc12.jpg","list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nG \\psi = \\left[ \\psi ^ { \\parallel } \\right]\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,397,875,434]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since it is an affne transformation, we get (Theorem 2.3.10, Gupta and Nagar (2000), page 64): ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[338,445,656,574]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0058/e3ca7cc64fdbc5b823245efc19186776a8b9e3e24ef240537f234c1fc4d54257.jpg","list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\left[ \\psi ^ { \\parallel } \\right] \\sim N ( G B X , G \\Omega G ^ { T } , I ) } \\\\ & { \\left[ \\psi ^ { \\parallel } \\right] \\sim N \\left( \\left[ \\begin{array} { l } { B ^ { \\parallel } } \\\\ { B ^ { \\perp } } \\end{array} \\right] X , \\left[ \\begin{array} { l l } { \\Omega ^ { \\parallel } } & { \\Omega ^ { + T } } \\\\ { \\Omega ^ { + } } & { \\Omega ^ { \\perp } } \\end{array} \\right] , I \\right) } \\\\ & { \\left[ \\psi ^ { \\parallel } \\right] \\sim N \\left( \\left[ \\begin{array} { l } { B ^ { \\parallel } } \\\\ { B ^ { \\perp } } \\end{array} \\right] X , \\Omega ^ { * } , I \\right) } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,579,872,616]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"from the properties discussed before. Now, we need to marginalize out $B$ and $\\Omega$ . Starting with $B$ , we can write Equation 14 as: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[323,626,674,666]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0875/c230c2cf287daaab9c814c6f45d3ff752d095a448fc7ae957c391b24dd894d6f.jpg","list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\left[ \\psi ^ { \\parallel } \\right] = \\left[ B ^ { \\parallel } \\right] X + E ^ { B } , E ^ { B } \\sim N ( 0 , \\Omega ^ { * } , I )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,676,877,712]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Similarly, since $B \\sim N ( M , \\Omega , I )$ ,by Theorem 2.3.10 (Gupta and Nagar (2000), page 64), we get: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[352,710,642,752]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0654/c77efa5067ec7bfb49e4794bc73b8caf369c8fb26e90b8b8e42bd90240ff6b84.jpg","list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nG B = \\left[ \\frac { B ^ { \\parallel } } { B ^ { \\perp } } \\right] \\sim N \\left( \\left[ M ^ { \\parallel } \\right] , \\Omega ^ { * } , \\Gamma \\right)\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,756,289,772]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Then, we can write: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[330,771,666,811]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0702/7de936a245b8bf9b87085bb0dffd8a994fabb9b60f76f97bac8fb222c1c9a737.jpg","list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\left[ B ^ { \\parallel } \\right] = \\left[ M ^ { \\parallel } \\right] + E ^ { * } , \\ E ^ { * } \\sim N ( 0 , \\Omega ^ { * } , \\Gamma ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"21 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,266,103]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This implies that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[361,113,635,199]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0959/5423743485c48b648419df22c4229f1c97fcf079fb15aa4326d2430e8e3e7c33.jpg","list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { \\left[ \\psi ^ { \\parallel } \\right] = \\left( \\left[ M ^ { \\parallel } \\right] + E ^ { * } \\right) X + E ^ { B } } \\\\ { = \\left[ { M ^ { \\parallel } } \\right] X + E ^ { * } X + E ^ { B } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,205,872,242]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $E ^ { B }$ and $E ^ { * }$ are defined as before. Using Theorem 2.3.10 (Gupta and Nagar (2000), page 64), the distribution of $E ^ { * } X$ is: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[393,252,601,273]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0039/cf8a73181f24a60f18541a0e52c6f31e9d2c829309e47efbdaeb25ffbce81df6.jpg","list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nE ^ { * } X \\sim N ( 0 , \\Omega ^ { * } , X ^ { T } \\Gamma X )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,284,563,304]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Then, by Gupta and Nagar (2000) (page 82), we get ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[334,313,663,353]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0407/cd09e7ed80df5c9390b45af91acb34c12f554d00880e30feaffc9bc5bdec7ba2.jpg","list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\left[ \\psi ^ { \\parallel } \\right] \\sim N \\left( \\left[ M \\parallel \\right] X , \\Omega ^ { * } , I + X ^ { T } \\Gamma X \\right)\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,362,563,381]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To marginalize out $\\Omega ^ { * }$ ， we start by applying $G$ to $\\Omega$ ： ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[392,390,604,411]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0206/8cf9056382364401519d84100f63d5969547b8bef844730854a06154f4268376.jpg","list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nG \\Omega G ^ { T } = \\Omega ^ { * } \\sim \\mathrm { I W } ( \\nu , \\Xi ^ { * } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,420,875,479]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\Xi ^ { * } = G \\Xi G ^ { T } = \\left[ \\Xi ^ { \\parallel } \\quad \\Xi ^ { + ^ { T } } \\right]$ Let $\\hat { E } \\sim N ( 0 , I , I + X ^ { T } \\Gamma X )$ . This implies that (Theorem 4.2.1 of Gupta and Nagar (2000), page 134) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[277,488,717,529]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0454/98d4d7b1102881672e77b0e863bbaf45fe4782699cd30f4ee69143b30c356807.jpg","list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\Omega ^ { - 1 } \\hat { E } + \\left[ M ^ { \\parallel } \\right] X \\sim T \\left( \\nu , \\left[ M ^ { \\parallel } \\right] X , \\Xi ^ { * } , I + X ^ { T } \\Gamma X \\right)\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,537,236,555]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"which implies ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,553,674,593]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0062/bc2cdb4b9d5cc780f681ebb75dbc6aa8f97eb2f35c6dfe8ec324af42b73b9235.jpg","list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\left[ \\psi ^ { \\parallel } \\right] \\sim T \\left( \\nu , \\left[ M ^ { \\parallel } \\right] X , \\Xi ^ { * } , I + X ^ { T } \\Gamma X \\right) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[122,598,771,618]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From Gupta and Nagar (2000) (page 163), the marginal distribution of $\\psi ^ { \\parallel }$ is: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[362,627,632,648]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0668/5e120e4975106c7349e692ce0d2aa8dc8fc58085b2e6e1ff71b2cdbf823acc1e.jpg","list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\psi ^ { \\parallel } \\sim T ( \\nu , M ^ { \\parallel } X , \\Xi ^ { \\parallel } , I + X ^ { T } \\Gamma X )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,659,428,678]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"which implies the model for $\\psi ^ { \\parallel } | Y$ is: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[359,688,635,762]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0303/b0832641d9c2d2b4aa29f98565a49a84b9c18ad5261e07854050041f165c1796.jpg","list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { Y _ { j } \\sim \\mathrm { M u l t } ( \\mathcal { W } _ { j } ^ { \\parallel } ) } \\\\ & { \\psi ^ { \\parallel } \\sim \\mathrm { A L R } _ { D } ( \\mathcal { W } ^ { \\parallel } ) } \\\\ & { \\psi ^ { \\parallel } \\sim T ( \\nu , M ^ { \\parallel } X , \\Xi ^ { \\parallel } , I + X ^ { T } \\Gamma X ) . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,770,877,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $A L R ( \\cdot )$ denotes the additive log-ratio transform. Note that this is exactly the collapse sampler from Silverman et al. (2022). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,828,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"22 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,85,704,104]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"D.3.2Derivation of Implied Scale Model for Bayesian PIM ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,114,398,132]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recall from the previous section: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,145,673,185]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0183/d6c09b184d903551f300ffa5efe80f50f03093474978e672d71c2df013b89b86.jpg","list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\left[ \\psi ^ { \\parallel } \\right] \\sim T \\left( \\nu , \\left[ M ^ { \\parallel } \\right] X , \\Xi ^ { * } , I + X ^ { T } \\Gamma X \\right) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,199,877,236]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Then, from Gupta and Nagar (2000) (page 138),the distribution of $\\psi ^ { \\perp } | \\psi ^ { \\parallel }$ is a multivariate T-distribution: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[158,265,810,315]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0586/c7642ade581c9326b047f84f84533b70e2851ac1041ba34fd58912ded44c6530.jpg","list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\nu ^ { \\bot } | \\psi | ^ { \\rbrack } \\sim T [ \\nu + D , M ^ { \\bot } X + ( \\Xi ^ { + } ) ^ { T } ( \\Xi ^ { \\parallel } ) ^ { - 1 } ( \\psi ^ { \\parallel } - M ^ { \\parallel } X ) , } \\\\ & { \\qquad \\Xi _ { 2 2 \\cdot 1 } , ( I + X ^ { T } \\Gamma X ) ( I + ( I + X ^ { T } \\Gamma X ) ^ { - 1 } ( \\psi ^ { \\parallel } - M ^ { \\parallel } X ) ^ { T } ( \\Xi ^ { \\parallel } ) ^ { - 1 } ( \\psi ^ { \\parallel } - M ^ { \\parallel } X ) ) ] } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,327,439,348]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\Xi _ { 2 2 \\cdot 1 } = \\Xi ^ { \\perp } - ( \\Xi ^ { + } ) ( \\Xi ^ { \\parallel } ) ^ { - 1 } ( \\Xi ^ { + } ) ^ { T }$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,367,558,387]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"D.3.3Collapse-Uncollapse Sampling Details ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,396,877,486]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"While conceptually straightforward, scale simulation requires efficient sampling for three separate stages: a compositional model, a scale model, and an inferential model. For the case of the multinomial logistic normal model, our sampling details closely mimic the Collapse-Uncollapse (CU) sampler of Silverman et al. (2022). Intuitively, the basic sampler follows three steps: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[142,488,707,542]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Sample from the partially identified base scale model $P ( \\mathcal { W } ^ { \\parallel } | \\mathcal { V } )$ \n2. Sample from the scale model $P ( \\mathcal { W } ^ { \\perp } | \\mathcal { W } ^ { \\parallel } )$ \n3. Compute the target estimand from the samples of $\\mathcal { W } ^ { \\parallel }$ and $\\mathcal { W } ^ { \\perp }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,544,877,705]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sampling from the scale model is typically effcient as it is low-dimension and flexible. To date, efficient routines for these models are already implemented in R. However, sampling from Step 1 is cumbersome but were recently made much more effcient and scalable by Silverman et al. (2022). Their approach relies on rewriting the above multinomial logisticnormal model as a marginally latent matrix-t process. Then, they develop efficient inference for this class of models using a Collapse-Uncollapse (CU） sampling approach. Such an approach vastly improves computational effciency, increasing the usability of these models to much larger sequence count data sets. See Silverman et al. (2O22) for complete details on implementation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,707,874,743]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We utilize the logic behind the CU sampler to infer Step 1. These routines are efficiently implemented in the R package fido (Silverman et al., 2022). Formal implementation details ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"23 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[140,128,816,279]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0716/c2f53215236bc2db042324c7ee62ca70bab16ab2fd76328018258b723618339f.jpg","list_items":null,"page_idx":"54","sub_type":null,"table_body":"<table><tr><td>Data: Y Result: S samples of the form {B(s), Ω(s)}</td><td></td></tr></table>","table_caption":"[\"Algorithm 3: Modified Collapse-Uncollapse Sampler for Scale Simulation \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[119,301,816,327]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"EExample of Scale Reliant Inference in Ecology ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[119,340,877,431]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A common task in ecology is to estimate the rate at which an animal species abundance changes between two time-points (Udevitz and Pollock,1992). To avoid needlessly complicating this example, we consider only a single species whose abundance is denoted at time $t$ by $W _ { t }$ . In this case,the relative abundance at time $t$ is trivial $W _ { t } ^ { \\parallel } = 1$ implying that $W _ { t } = W _ { t } ^ { \\perp }$ . Ecologists often want to estimate the value of the logarithmic rate of change: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[421,443,571,527]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"1013/c63d3dd8c480b21fd508480878860744fd79a00d22bc8e46acd436fae05de745.jpg","list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { r = \\log \\left( \\frac { W _ { t + 1 } } { W _ { t } } \\right) } \\\\ { = \\log \\left( \\frac { W _ { t + 1 } ^ { \\perp } } { W _ { t } ^ { \\perp } } \\right) . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,534,591,551]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is clear that this target estimand is fully scale reliant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,554,877,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Yet many ecological surveys that attempt to estimate changes in species abundance over time are limited to using observational data where the scale, e.g., $W _ { t } ^ { \\perp }$ cannot be directly measured. Instead ecologists often use some variation on mark-recapture studies (Krebs, 1999).In a mark-recapture study, animals are captured and released twice. During the first capture,animals are marked prior to release, the total number of animals captured during are recorded ( $Y ^ { 1 }$ ). The second capture not only records the total number of captured animals $Y ^ { 2 t }$ but also the number of those captured that had marks indicating that they had been recaptured $Y ^ { 2 m }$ . By performing independent mark-recapture studies both at timepoint $t$ and $t + 1$ ecologists hope to collect sufficient information to estimate $r$ . We can think of these data as being derived from a compositional survey. Albeit a trivial compositional survey since the relative abundance in this simple 1 species example is trivially identifiable as it is always equal to 1. Below we illustrate a use of mark-recapture coupled with a technique calld the change-in-ratios method (Udevitz and Pollock,1992) that is sometimes used to estimate $r$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"24 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,87,524,103]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for scale simulation are outlined in Algorithm 3. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[119,85,879,195]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Suppose two separate mark-recapture experiments are undertaken at times $t$ and $t +$ 1.Let $\\{ Y _ { t } ^ { 1 } , Y _ { t } ^ { 2 m } , Y _ { t } ^ { 2 t } \\}$ denote the number of initially-marked individuals, the number of recaptured marked individuals,and the total number of observed individuals during the recapture survey at time $t$ Wedenoteananalogous vector $\\{ Y _ { t + 1 } ^ { 1 } , Y _ { t + 1 } ^ { 2 m } , Y _ { t + 1 } ^ { 2 t } \\}$ for time $t + 1$ These data are often used to estimate the population size (species abundance) at each time-point: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[426,209,568,292]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0631/4612eadec75cfd5ca7dd6912de93da82d523259e011452978b99153192713d7c.jpg","list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { c } { { \\hat { W } _ { t } = \\displaystyle { \\frac { Y _ { t } ^ { 1 } Y _ { t } ^ { 2 t } } { Y _ { t } ^ { 2 m } } } } } \\\\ { { \\hat { W } _ { t + 1 } = \\displaystyle { \\frac { Y _ { t + 1 } ^ { 1 } Y _ { t + 1 } ^ { 2 t } } { Y _ { t + 1 } ^ { 2 m } } } } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,303,787,320]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the change-of-ratios methods these estimates are then used to estimate $r$ as: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[354,332,640,428]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0957/bfbdb3103c1a7602b7a7da2dd211184010486d8a12f48f0abda1d44d22356b15.jpg","list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\hat { r } = \\log \\left( \\frac { \\hat { W } _ { t } } { \\hat { W } _ { t + 1 } } \\right) } \\\\ & { \\quad = \\log \\left( \\frac { Y _ { t + 1 } ^ { 1 } } { Y _ { t } ^ { 1 } } \\right) \\log \\left( \\frac { Y _ { t + 1 } ^ { 2 t } / Y _ { t + 1 } ^ { 2 m } } { Y _ { t } ^ { 2 t } / Y _ { t } ^ { 2 m } } \\right) . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[122,435,406,454]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This is an applied estimator for $r$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,454,877,746]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The framework of SRI provides useful concepts to help catalyze discussions about this type of analysis. Does this constitute fully scale reliant， partially scale reliant, or scale invariant inference? Of course, as we have not explicitly specified a measurement model is it not possible to make any formal claim. Still, thinking through the mechanics of a mark-recapture study it seems reasonable to assume that these studies do not identify the system composition. Consider that implicit in the above applied estimator is an assumption of homogeneous mixing and random sampling: there is an assumption that the marked an-imals homogeneously mix into the population such that the recapture probability $Y _ { t } ^ { 2 m } / Y ^ { 2 t }$ is equal to the proportion of marked animals in the entire population. This assumption is clearly not exactly true leaving fundamental uncertainty in the system scale despite the observed data. It therefore seems reasonable to assume that this is a partial compositional survey. In contrast, we expect that the degree of uncertainty is likely far less than what is present in sequence count data and it is likely unreasonable to model these data as coming from a fully compositional survey. In either case,given that the target estimand is oneto-one with the ratio of scales $W _ { t - 1 } ^ { \\perp } / W _ { t } ^ { \\perp }$ it seems clear that inference of $r$ must be scale reliant. We therefore expect that strict frequentist consistency is likely not possible. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,747,875,801]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Overall, this discussion is merely meant to illustrate how the language of SRI can be applied to different felds. Further work is needed if SRI is to provide a comprehensive framework for understanding the role of scale in mark-recapture based ecological studies. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"25 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,80,271,104]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[117,118,874,156]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Gabrielsen, A. (1978). Consistency and identifiability. Journal of Econometrics, 8(2):261- 263. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,166,877,224]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Grantham, N. S., Guan, Y., Reich, B. J., Borer, E. T., and Gross, K. (2020). Mimix: A bayesian mixed-effects model for microbiome data from designed experiments. Journal of the American Statistical Association, 115(530):599-609. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,233,861,255]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Gupta, A. and Nagar, D. (20oo). Matrix Variate Distributions. Chapman & Hall/CRC. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,265,872,303]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Krebs, C. J. (1999). Ecological methodology. Benjamin/Cummings, Menlo Park, Calif, 2nd ed edition. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,314,877,371]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Letourneau, J.， Holmes, Z. C.， Dallow, E. P., Durand, H. K., Jiang, S., Gupta, S. K, Mincey, A. C., Muehlbauer, M. J.， Bain, J. R., and David, L. A. (2021).Ecological memory of prior nutrient exposure in the human gut microbiome. bioRxiv. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,382,877,439]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Props,R., Kerckhof, F.-M., Rubbens, P., De Vrieze, J.， Hernandez Sanabria, E., Waege-man,W.， Monsieurs, P., Hammes, F. and Boon, N. (2017). Absolute quantification of microbial taxon abundances. The ISME journal, 11(2):584-587. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,449,707,469]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Shao, J. (2003). Mathematical statistics. Springer Texts in Statistics. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,479,879,536]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Silverman, J. D., Bloom, R. J., Jiang, S., Durand, H. K., Dallow, E., Mukherjee, S., and David, L. A. (2021). Measuring and mitigating pcr bias in microbiota datasets. PLoS Computational Biology, 17(7):e1009113. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,546,877,604]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Silverman, J. D., Durand, H. K., Bloom, R. J., Mukherjee, S., and David, L. A. (2018). Dynamic linear models guide design and analysis of microbiota studies within artificial human guts. Microbiome, 6(1):1-20. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,614,875,671]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Silverman, J. D.，Roche,K.， Holmes, Z. C.，David,L.A.，and Mukherjee, S. (2022). Bayesian multinomial logistic normal models through marginally latent matrix-t processes. Journal of Machine Learning Research, 23(7):1-42. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,683,870,722]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Udevitz, M. S. and Pollock, K. H. (1992). Change-in-ratio methods for estimating population size. In Wildlife 2001: populations, pages 90-101. Springer. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,731,877,787]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Vandeputte, D., Kathagen, G., D'hoe, K., Vieira-Silva, S., Valles-Colomer, M., Sabino, J., Wang, J., Tito, R. Y., De Commer, L., Darzi, Y., et al. (2017). Quantitative microbiome profiling links gut community variation to microbial load. Nature, 551(7681):507-511. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[486,829,508,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"26 ","text_format":null,"text_level":null,"type":"discarded"}],"string":"[\n {\n \"bbox\": \"[305,148,691,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Scale Reliant Inference \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[166,198,831,243]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Michelle Pistner Nixon1, Jeffrey Letourneau’ Lawrence A. David $_ 3$ 4 \\nNicole A. Lazar; Sayan Mukherjee $^ { 4 6 } ,$ and Justin D. Silverman12478 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[459,281,537,297]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[166,304,828,659]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This article develops new tools and new statistical theory for a statistical problem we call Scale Reliant Inference (SRI). Many scientific fields collect multivariate data that lack scale: where the size, sum, or total of each measurement is arbitrary and is not representative of the scale of the underlying system being measured. For example, in the analysis of high-throughput sequencing data, it is well known that the number of sequencing reads (the sequencing depth） varies substantially due non-biological (technical) factors. This article develops a formal problem statement for SRI which unifies problems seen in multiple scientific fields. Informally,we define SRI as an estimation problem in which an estimand of interest cannot be uniquely identified due to the lack of scale information in the observed data. This problem statement represents a reformulation of the related field of Compositional Data Analysis and allows us to prove fundamental limits on SRI. For example, we prove that inferential criteria such as consistency, calibration,and bias are unattainable for common SRI tasks. Moreover，we show that common methods often applied to SRI implicitly assume infinite knowledge of the system scale and can lead to a troubling phenomena termed unacknowledged bias. Counter-intuitively, we show that this problem worsens with more data and can lead to substantially elevated Type-I and Type-II error rates. Still, we show that rigorous statistical inference is possible so long as models acknowledge the fundamental uncertainty in the system scale. We introduce a class of models we call Scale Simulation Random Variables (SSRVs) as flexible, rigorous, and computationally efficient approach to SRI. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,671,877,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1Equal contributions 2Collge of Information Sciences and Technology, Pennsylvania State University 3Department of Molecular Genetics and Microbiology, Duke University 4Center for Genomic and Computational Biology, Duke University 5Department of Statistics, Pennsylvania State University $_ 6$ Departments of Statistical Science,Mathematics, Computer Science, Biostatistics & Bioinformatics, Duke University 7Department of Medicine, Pennsylvania State University $^ { \\\\mathrm { ~ 8 ~ } }$ Corresponding author: JustinSilverman@psu.edu \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[22,256,62,722]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[491,829,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[491,829,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,85,890,122]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords: Multivariate Analysis, Identifiability, Uncertainty Quantification, Sequence Count Data \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[122,80,343,104]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,119,877,375]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This article develops new theory and tools for a statistical problem we call Scale Reliant Inference (SRI). Consider an unobserved physical system that can be represented as a positive-valued $D \\\\times M$ matrix $W$ with elements $W _ { d m }$ representing the amount of entity $d$ present in configuration $m$ .For example, in macro-ecology, $W _ { d m }$ might represent the abundance of species $d \\\\in \\\\{ 1 , \\\\dots , D \\\\}$ in forest $m \\\\in \\\\{ 1 , \\\\ldots , M \\\\}$ .We call $W$ a scaled system as it can be decomposed uniquely into scale and compositional components: $W _ { d m } =$ $W _ { d m } ^ { \\\\parallel } W _ { m } ^ { \\\\perp }$ where $W _ { d m } ^ { \\\\parallel }$ denotesthesi $\\\\begin{array} { r } { W _ { d m } ^ { \\\\parallel } = W _ { d m } / \\\\sum _ { d = 1 } ^ { D } W _ { d m } ) } \\\\end{array}$ of entities in configuration $m$ that are of type $d$ and $W _ { m } ^ { \\\\perp }$ denotes the scale (i.e.， sum total, $\\\\begin{array} { r } { W ^ { \\\\perp } = \\\\sum _ { d = 1 } ^ { D } W _ { d m } ) } \\\\end{array}$ of configuration $m$ .A scaled system is not directly observable but instead must be measured (e.g., sampled or surveyed). Yet due to limitations of the measurement process, these measurements often do not provide information about the system scale $W ^ { \\\\perp } = ( W _ { 1 } ^ { \\\\perp } , \\\\ldots , W _ { M } ^ { \\\\perp } )$ . Informally, SRI is an estimation problem in which an estimand of interest cannot be uniquely identified due to the lack of scale information in the observed data. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,376,877,702]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A major contribution of this work is the formulation of SRI which provides a unified representation of inferential problems seen in numerous scientific fields. In macro-ecology researchers often want to infer population dynamics of one or more animal species yet only have access to survey data that lacks scale information due to an inability to exhaustively survey all animals in a forest (e.g.， Murphy et al. (2016)). In demography, researchers may want to estimate the size of difficult to measure populations yet be limited to survey methods that do not exuastively survey the population leading to confounded or biased estimates (Rossi et al.， 1987). In the study of human gut microbiota, researchers may want to estimate pairwise covariances between each of $D$ bacterial species yet only be able to measure these communities using 16S rRNA sequencing where the total number of sequences measured in each sample is unrepresentative of total biological load in the sampled microbial communities (e.g., Paulson et al. (20l3)). In all three of these cases, the inferential problem can be framed as an example of SRI: as an estimation problem where the estimand of interest cannot be uniquely determined from the observed data due to the lack of scale information.By providing a formal definition of this problem framed in terms of probabilistic identifiability, we are able to provide new theoretical results on the limits of SRI. We show that for many common SRI tasks Frequentist criteria such as consistency, unbiasedness,and even calibration are impossible. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,704,875,795]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A second contribution of this work is to provide a unified language,and key theoretical constructs,to facilitate discussion of the many modeling assumptions commonly used to circumvent the challenges of SRI. These modeling assumptions are often implicit and poorly understood. As an example, consider the use of data normalizations common in the analysis of microbiome 16S rRNA sequencing data. Recognizing that the total number of 16S rRNA sequences in each sequenced sample (the sequencing depth) varies due to non-biological (i.e., technical) factors, many researchers choose to apply various data transformations intended to “normalize” the data and make “make the microbial abundances comparable across samples” Srinivasan et al. (2021). Often, the transformed data is then analyzed as if the measurements reflect direct measurements of microbial abundances: as if it contained both scale and compositional information. Our formulation of SRI allows us to provide new insights into the implications of these approaches, the assumptions behind them,and to study the sensitivity of study conclusions to errors in these assumptions. For example, we find that the common approach of data normalization in sequence count data analysis fails to acknowledge uncertainty in the system scale and that this lack of uncertainty leads to a troubling phenomena called unacknowledged bias which in turn can lead to substantially elevated Type-I error rates in practice Gustafson (2015). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[491,829,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,877,304]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,306,877,523]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Beyond studying modeling assumptions and proving fundamental limits on SRI, our problem formulation allows us to develop new statistical tools that respect those limits and produce statistically rigorous statistical inference that has both theoretical and practical advantages compared to existing methods. By framing SRI as a problem of identifiability, we are able to leverage existing literature on Bayesian Partially Identified Models (Bayesian PIMs) to develop models that acknowledge their own biases, reduce Type-I error rates. To address computational problems with posterior inference of Bayesian PIMs,we introduce a related class of models we term Scale Simulation Random Variables (SSRVs) which can often by sampled using direct Monte-Carlo simulation without the need to establish a Markov-Chain. In addition, SSRVs allow researchers to incorporate additional information about scale into analyses in ways that are possible with many existing tools. We show examples of how this additional information can even reduce Type-II error rates. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,525,877,705]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The structure of this work is as follows. In Section 2，we provide the definitions and basic results needed to formally state the problem of scale reliant inference and contrast this problem with problems studied in the related field of Compositional Data Analysis. In Section 3, we use the preceding problem formulation to prove fundamental limits on SRI in terms of consistency, calibration,and bias. In Section 4, we define Scale Simulation Random Variables (SSRVs). In Sections 4.1, 4.2, and 4.3, we provide key properties of SSRVs which illustrate their utility in the setting of SRI. In Section 5,we turn our attention to empirical and theoretical studies that contrast SSRVs with existing methods used for SRI in the context of both simulated and real data. Finally, we close with a discussion in Section 6 including a brief discussion of potential connections between our work and Causal Inference. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[491,829,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,80,465,104]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2Problem Formulation \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,118,877,301]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We define Scale Reliant Inference (SRI) as an estimation problem where our ability to uniquely determine the estimand of interest is limited due to a lack of scale information in the observed data. To build this definition，we first need to define two relationships between three constructs (Section 2.1): 1. the relationship between the system we are studying (first construct） and the quantity we wish to estimate (second construct） and 2. the relationship between the system we are studying and our measurements of the system (third construct). We call these three constructs the scaled system, target estimand, and compositional survey respectively. Together these two relationships then allow us to link our measurements to our estimand of interest and define scale reliant inference (Section 2.3). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,325,370,345]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.1 Core Constructs \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,356,875,410]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A scaled system is a representation of the physical system of scientifc interest. It is central to our problem formulation as it provides the context for the questions: “What is being measured?” and “What is being estimated?\\\". \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,422,877,478]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 1 (Scaled System). A scaled system $W \\\\in \\\\mathbf { W }$ isa $D$ -dimensional positive-valued quantity defined over a set of configurations $m \\\\in \\\\mathbf { M }$ such that $W _ { d m }$ denotes a scientifically meaningful amount of entity d in configuration $m$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,489,877,671]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To reflect the diverse goals of different studies in different fields of science, we purposefully leave the definitions of configurations, entities, and amounts flexible. While we use notation $W _ { d m }$ for elements of a scaled system,we do not even require that $W$ is a finite dimensional matrix. For concreteness,consider a researcher interested in human-microbe interactions who is studying how the abundance of different bacterial taxa at the epithelial surface of the colon change over time. Supposing this researcher will ultimately want to em-ploy a continuous-time model of these abundances, they might choose to represent the set of configurations $M$ as a continuous interval of time over which the study was performed, entities as a set of $D$ -different bacterial genera, and amounts as the average concentration per square centimeter of bacterial cells at the colonic epithelial surface. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,672,875,746]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Regardless of the chosen definitions of configuration，entity, and amount, $W$ can be uniquely decomposed into in terms of its scale (summed amounts) and composition (pro-portional amounts). The scale of the system ( $W ^ { \\\\perp } = ( W _ { 1 } ^ { \\\\perp } , \\\\ldots W _ { M } ^ { \\\\perp } ) )$ is an $M$ -vector repre-senting the summed amount of each entity in each configuration $m$ ： \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[429,758,565,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0141/9733d75de895ede0ac510a4c7dfa69b231ce56dcf0e9b7d43c6b0cb279029d43.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nW _ { m } ^ { \\\\perp } = \\\\sum _ { i = 1 } ^ { D } W _ { d m } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[491,829,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,875,122]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The composition of the system $( W ^ { \\\\parallel } )$ is a $D \\\\times M$ -dimensional quantity, defined by a $\\\\boldsymbol { D }$ -vector at each $m \\\\in \\\\mathbf { M }$ , representing the proportional amounts of each entity in each configuratin: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[382,128,614,169]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0392/c64bf2f0ad4a0b3b263f243bf2655c695dc85f0fa699483a1861161505819755.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nW _ { \\\\cdot m } ^ { \\\\parallel } = \\\\left( \\\\frac { W _ { 1 m } } { W _ { m } ^ { \\\\perp } } , \\\\ldots , \\\\frac { W _ { D m } } { W _ { m } ^ { \\\\perp } } \\\\right) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[122,172,875,208]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Together these definitions define the set of $D \\\\times M$ dimensional scaled systems as a Cartesian product space: $\\\\mathbf { W } = \\\\mathbf { W } ^ { \\\\parallel } \\\\times \\\\mathbf { W } ^ { \\\\perp }$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,210,875,300]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The scale of a system plays a special role in scientific inference. Note that the compo-sition of a system is a unit-less quantity, all the information defining the units of $W$ (the definition of amount) is contained in the system scale. The scale of the system therefore is intricately tied to the scientific goals of a study in a way that the composition is not. This feature also helps motivate our study of scale reliant inference. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,301,875,356]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The target estimand is the second construct of SRI and is the answer to the question “What is being estimated?\\\". It is defined through its relationship to the scaled system, a relationship called the target estimator. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,361,877,454]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 2 (Target Estimand and Estimator). Let $\\\\theta \\\\in \\\\Theta$ denote a target estimand: a quantity of scientific interest to a researcher studying a scaled system $W$ . The target estimand is an expression of the scaled system along with potential covariates $X$ such that it can be written as $\\\\theta = f ( W , X )$ where $f$ is a function $f : \\\\mathbf { W } \\\\times \\\\mathbf { X } \\\\to \\\\Theta$ called the target estimator. For notational simplicity, we will often exclude $X$ and simply write $\\\\theta = f ( W )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,460,872,497]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As an example, consider the following two target estimands which reflect common estimation problems in the analysis of sequence count data: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[313,505,681,556]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0420/aa14ebd5e47082bac781c1340fc5abdd335bd6648d025b15cc6c5b73420d29ab.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { \\\\begin{array} { l } { \\\\theta _ { i } = \\\\underset { m : x _ { m } = 1 } { \\\\mathrm { m e a n } } \\\\left( \\\\log W _ { d m } \\\\right) - \\\\underset { m : x _ { m } = 0 } { \\\\mathrm { m e a n } } ( \\\\log W _ { d m } ) } \\\\\\\\ { \\\\Sigma _ { d _ { 1 } , d _ { 2 } } = c o v ( \\\\log W _ { d _ { 1 } \\\\cdot } , \\\\log W _ { d _ { 2 } \\\\cdot } ) . } \\\\end{array} } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,560,877,671]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The first (Equation 3) can be equated to the Log-Fold-Change (LFC) of an entity (i.e., gene or microbe） between two biological conditions $x \\\\in \\\\{ 0 , 1 \\\\}$ . Estimation of LFCs is a central problem in gene expression and microbiome studies and is often called: differential abundance or differential expression analysis Fernandes et al. (2014). The second is the log-scale covariation between two entities which is a common statistic of interest in microecology Friedman and Alm (2012). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,671,872,707]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally, a compositional survey is the measurement process used to study a scaled system. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,713,879,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3 (Compositional Survey). Let $\\\\mathcal { V } \\\\in \\\\mathbf { Y }$ denote a random variable describing the measurement of a scaled system $W$ such that we can denote by ${ \\\\mathcal { N } } _ { m }$ as a stochastic measurement of the system W at the configuration $m \\\\in \\\\mathbf { M }$ and $P _ { W } ( \\\\mathcal { V } )$ as the measurement model. V is a compositional survey if the re-parameterized model $P _ { W ^ { \\\\perp } , W \\\\parallel } ( \\\\mathcal { V } )$ identifies the composition of the system ( $W ^ { \\\\parallel }$ ） but not the scale $( W ^ { \\\\perp } )$ · \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[490,829,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,877,250]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In discussing compositional surveys, we will often refer to them as being scale limited, this term refer to the fact that they do not identify the system scale. We also distinguish between two types of compositional surveys: partial and full compositional surveys. For a full compositional surveys, the set $W ^ { \\\\perp }$ is fully unidentified by the measurement model: any value of $W ^ { \\\\perp } \\\\in \\\\mathbf { W } ^ { \\\\perp }$ could result in the same measurement model. For a partial compositional survey, while $W ^ { \\\\parallel }$ is not uniquely identified, there exists at least one value of $W ^ { \\\\perp } \\\\in \\\\mathbf { W } ^ { \\\\perp }$ that could not result in the same measurement model. In this article we primarily discuss fully compositional surveys and leave partial compositional surveys as a topic for future research. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,272,743,295]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2Invertibility as a Generalization of Identifiability \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,304,877,540]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our goal is to define Scale Reliant Inference (SRI) as an estimation problem where our ability to uniquely determine the estimand of interest is limited due to a lack of scale information in the observed data. The prior subsection already provided the relationship between a compositional survey and a scaled system (the measurement model, $P _ { W ^ { \\\\perp } , W \\\\parallel } ( \\\\mathcal { V } ) )$ and the relationship between a scaled system and a target estimand (the target estimator, $\\\\theta \\\\ = \\\\ f ( W ^ { \\\\perp } , W ^ { \\\\parallel } ) )$ .Moreover, the prior section has already expressed scale limitations of $\\\\mathcal { V }$ in terms of model identifiability. It may therefore seem natural to again use the concept identifiability to define the phrase “our ability to uniquely determine the estimand is limited\\\". For example, some readers may consider defining SRI as problem of inferring $\\\\theta$ when $\\\\theta$ is unidentifiable in a reparameterized model $P _ { f ^ { - 1 } ( \\\\theta ) } ( \\\\mathcal { V } )$ .Unfortunately, this approach is problematic. Many target estimands (e.g., Equations 3 and 4) are functions of $W ^ { \\\\parallel }$ and $W ^ { \\\\perp }$ without a unique inverse. Instead, we need to use to a more general notion of identifiability which we call invertibility. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,541,877,758]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A family of probability models $P _ { W } ( \\\\mathcal { V } )$ is a function $g$ linking a parameter space $\\\\mathbf { W }$ to a set of measures $\\\\mathbf { P }$ defined over the set $\\\\mathbf { Y }$ ： $P _ { W } ( \\\\mathcal { V } ) \\\\equiv g : \\\\mathbf { W } \\\\to \\\\mathbf { P }$ . From this perspective, the identifiability (or unidentifability) of a particular model $P \\\\in \\\\mathbf { P }$ is a statement on the size of the pre-image of the function $g$ . We denote the pre-image of $g$ as $g ^ { - 1 }$ If $g ^ { - 1 } ( P )$ is a singleton set，we say that the model $P$ (or colloquially the model parameter $W$ ）is identifiable. Otherwise, it is unidentifiable. The distinction between fully and partially unidentifiable depends on whether $g ^ { - 1 } ( P ) = \\\\mathbf { W }$ (fully unidentifiable） or $g ^ { - 1 } ( P ) \\\\subsetneq \\\\mathbf { W }$ (partially unidentifable). In fact, this pre-image is precisely the identification region of the model (Basse and Bojinov, 202O). Yet,by expressing identifability in this form,it is now natural to generalize this concept to arbitrary maps $h : \\\\mathbf { A } \\\\mathbf { B }$ . In the following definition, for a map $h : \\\\mathbf { A } \\\\mathbf { B }$ we use the notation $h ^ { - 1 } ( \\\\mathbf { b } )$ to represent the pre-image of the map at the set $\\\\mathbf { b } \\\\subset \\\\mathbf { B }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[125,770,872,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 4 (Invertibility). Consider a map $h : \\\\mathbf { A } \\\\mathbf { B }$ and a subset $\\\\mathbf { b } \\\\subseteq \\\\mathbf { B }$ ， we say that: 1. h is invertible at b if $h ^ { - 1 } ( \\\\mathbf { b } )$ is a singleton set. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[491,829,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[142,85,836,122]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. h is partially uninvertible at b if $h ^ { - 1 } ( \\\\mathbf { b } )$ is not a singleton set and $h ^ { - 1 } ( \\\\mathbf { b } ) \\\\subset \\\\mathbf { A }$ \\n3. h is fully uninvertible at $\\\\mathbf { b }$ if $h ^ { - 1 } ( \\\\mathbf { b } )$ is not a singleton set and $h ^ { - 1 } ( \\\\mathbf { b } ) = \\\\mathbf { A }$ · \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,136,875,190]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Identifiability is therefore a special case of invertibility where the domain is a set of parameters, the co-domain is a set of probability measures, $h$ is a function, and our set $\\\\mathbf { b }$ is a singleton element. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,212,658,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.3Scale Reliant and Scale Reliant Inference \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,243,872,280]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We are finally in a position to define SRI as the problem of uniquely determining the target estimand given the scale limitations of a compositional survey. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,281,877,371]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From our three constructs we have two maps: the target estimand ( $f : \\\\mathbf { W } ^ { \\\\parallel } \\\\times \\\\mathbf { W } ^ { \\\\perp } \\\\to \\\\Theta$ ） and the map representation of the measurement model ( $g : \\\\mathbf { W } ^ { \\\\parallel } \\\\times \\\\mathbf { W } ^ { \\\\perp } \\\\to \\\\mathbf { P }$ where $\\\\mathbf { P }$ denotes a set of probability measures over the set $\\\\mathbf { Y }$ ). We define scale reliance in terms of the invertibility of the composite map $h = g \\\\circ f ^ { - 1 } : \\\\Theta \\\\to \\\\mathbf { P }$ where $f ^ { - 1 }$ denotes the pre-image map of $f$ · \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,383,877,438]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 5 (Scale Reliant Inference). Let $f$ and $g$ represent the maps defined by the target estimator and measurement model respectively and $h = g \\\\circ f ^ { - 1 }$ the composite map. Given a compositional survey $\\\\mathcal { V }$ with distribution $P \\\\in \\\\mathbf { P }$ ， we define inference of $\\\\theta \\\\in \\\\Theta$ as: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[143,440,812,493]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Scale Invariant Inference if $h$ is invertible at $P$ \\n2. Partially Scale Reliant Inference if $h$ is partially uninvertible at $P$ \\n3. Fully Scale Reliant Inference if $h$ is fully uninvertible at $P$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,506,877,616]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An important and unique feature of our definition of SRI is its reliance on the scientific question being asked. Quite intuitively，whether or not an inferential problem is scale reliant or scale invariant depends not only on the available data (through the term $g$ ）but also on the scientific question (through the term $f ^ { - 1 }$ ). As we discuss in Section 2.5, this feature is absent from the related but distinct feld of Compositional Data Analysis which can lead to problems when evaluating inference. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,638,486,660]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.4Tools for Establishing SRI \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,667,877,779]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The prior section defined SRI as the lack of invertibility of the composite map $h = g \\\\circ f ^ { - 1 }$ . However,it can be challenging to establish if a given estimation problem is scale reliant inference using that definition directly. Instead, this section introduces a simpler characterization of target estimands (whether the estimand is scale reliant or scale invariant) and then shows how this characterization can be used to easily establish when an estimation problem constitutes SRI. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[490,829,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"8 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,877,140]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Quite simply, we say a target estimand (or the target estimator) is scale reliant if fixing the value of $W ^ { \\\\parallel }$ is insufficient to uniquely determine $\\\\theta$ . The following definitions formalizes this notion. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,152,879,224]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 6 (Scale Reliance of Target Estimands and Estimators). For a target estimand and estimator $\\\\theta \\\\ : = \\\\ : f ( W ^ { \\\\parallel } , W ^ { \\\\perp } )$ ， $f _ { W ^ { \\\\parallel } }$ denote a restriction of the target estimator to the domain $W ^ { \\\\parallel } \\\\times \\\\mathbf { W } ^ { \\\\perp }$ for a particular value of $W ^ { \\\\| } \\\\in \\\\mathbf { W } ^ { \\\\| }$ . The following three statements are equivalent: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,226,879,373]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. $f$ is a scale invariant target estimator and $\\\\theta$ is a scale invariant target estimand. 2. $f _ { W ^ { \\\\parallel } }$ is invertible at $W ^ { \\\\parallel }$ 3. $A t \\\\ W ^ { \\\\parallel }$ there exists a single unique value of $\\\\theta \\\\in \\\\Theta$ that satisfies $\\\\theta \\\\ : = \\\\ : f ( W ^ { \\\\parallel } , W ^ { \\\\perp } )$ regardless of the value of $W ^ { \\\\perp } \\\\in { \\\\bf W } ^ { \\\\perp }$ \\nIf $f$ (and $\\\\theta$ ) are not scale invariant, then they are scale reliant. There are two types of \\nscale reliant target estimators (and estimands). ·If $f _ { W ^ { \\\\parallel } }$ is partially uninvertible then $f$ (and $\\\\theta$ ） are partially scale reliant. · If $f _ { W ^ { \\\\parallel } }$ is fully uninvertible then $f$ (and $\\\\theta$ ) are fully scale reliant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,385,877,530]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Using this definition, Table 1 summaries a number of propositions and proofs provided in Supplementary Section B.2 which categorize inference as scale invariant, partially scale reliant, or fully scale reliant, depending on whether the data comes from a partial or full compositional survey and whether the target estimand is scale invariant， partially scale reliant,or fully scale reliant. The combination of partially compositional surveys with a partially or fully scale reliant target estimand is substantially more complicated and can induce a composite map $h$ that is invertible,partially uninvertible,or fully uninvertable depending on the specific target estimand and the specific measurement model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,531,877,622]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As examples, note that the target estimands (LFC and Covariance estimands) provided in Equations 3 and 4 are fully and partially scale reliant respectively (proofs provided in Supplementary Section B.2). For example, for LFC estimation we can place no restrictions on the set of possible values of $\\\\theta \\\\in ( - \\\\infty , \\\\infty )$ using knowledge of the system compositional alone.In contrast consider that a target estimator of the form \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[401,633,591,674]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0340/92c56002ae1660a421a54e5f3f715f46bd1c0a025404196a5c25c2f8da0b8c9f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\zeta _ { i } = \\\\operatorname* { m e a n } _ { m } \\\\left( \\\\log \\\\frac { W _ { d m } } { W _ { j m } } \\\\right)\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,688,446,722]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is scale invariant as $\\\\begin{array} { r } { \\\\log { \\\\frac { W _ { d m } } { W _ { j m } } } = \\\\log { \\\\frac { W _ { d m } ^ { \\\\parallel } } { W _ { j m } ^ { \\\\parallel } } } } \\\\end{array}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,739,704,762]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.5 Connections to Compositional Data Analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,770,875,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"SRI can be understood as a reformulation and refocusing of Compositional Data Analysis (CoDA). Both SRI and CoDA focus on the challenge of analyzing of multivariate data in the absence of scale information yet they define this focus in very different ways. The prior section shows how SRI defnes this challenge through two relationships, both centered around the system being studied: the target estimator that defines the scientific goal, and the measurement model that describes the observed data. In contrast, the core of CoDA does not have any notion of the system being studied. Rather than centering on the system, CoDA is centered on the observed data which it defines mathematically as simplicial vectors (vectors $x \\\\in \\\\mathbb { R } ^ { D + }$ that sum to a constant value). In fact，while early work on CoDA discussed the notion of a base-scale, a construct that is conceptually similar to SRI's scaled systems, the base-scale is introduced largely as an special-topic and is neither a central part of CoDA theory nor is part of the central problem formulation of CoDA (Aitchison,1986, Chapter 9). Without the emphasis on the scaled system, it is also not possible for CoDA to defne a construct analogous to SRI's target estimands and estimators: the scientific goal is not an integral part of CoDA theory. Instead, the definition of the estimation task is typically discussed in the context of an applied model $P _ { \\\\theta } ( Y )$ or an applied estimator ${ \\\\hat { \\\\theta } } = r ( Y )$ : constructs that directly relate and estimand with the data. As a result, applied estimators (or models) typically serve two duties in CoDA: they both define the estimation task and define an approach to estimation. In contrast, in SRI we distinguish between the estimation task (the target estimand and estimator) and the approach to estimating the target estimand (the applied estimator). The separation of these two concepts is necessary in order to establish the limits on inference and to properly evaluate applied estimators. In fact, challenges arise when trying to evaluate applied estimators from a CoDA perspective. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[490,829,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"9 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,83,892,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0400/38ded8f9154a52ef9093a119ddf2a27639cd75ec05db378edb40010653a59898.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td></td><td>PropositionCompositional Survey (V)</td><td>Target Estimand (0)</td><td>Inference of 0 from observations of Y</td></tr><tr><td>D.3</td><td>Full</td><td>Scale Invariant</td><td>Scale Invariant</td></tr><tr><td>D.6</td><td>Full</td><td></td><td>Partially Scale ReliantPartially Scale Reliant</td></tr><tr><td>D.5</td><td>Full</td><td>Fully Scale Reliant</td><td>Fully Scale Reliant</td></tr><tr><td>D.3</td><td>Partial</td><td>Scale Invariant</td><td>Scale Invariant</td></tr><tr><td></td><td>Partial</td><td>Partially Scale Reliant</td><td></td></tr><tr><td></td><td>Partial</td><td>Fully Scale Reliant</td><td></td></tr></table>\",\n \"table_caption\": \"[]\",\n \"table_footnote\": \"[\\\"Table 1: Inference of $\\\\\\\\theta$ depending on the type of the data (partial or full compositional survey) and target estimand (scale invariant or scale reliant). \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[119,306,877,689]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,690,875,799]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In SRI, the target estimand gives context to the question “what is being estimated?\\\" and thereby forms a foundation on which an applied estimand can be evaluated. Without an analogous construct, CoDA instead proposes three axioms (or principles) that it argues any applied estimator of compositional data should satisfy (Pawlowsky-Glahn et al., 2015). The most important of these three is the principle of scale inuariance which is nicely summarized by Lovell et al. (2020): \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[488,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[166,85,828,159]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Scale Invariance is a fundamental principle of CoDA. Put simply, scale invariance means that the functions we use to analyze compositions should return identical results when we scale compositions by multiplying all components by a constant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[130,171,834,189]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"'he following example helps to illustrate the problems with this axiomatic approach. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,190,877,356]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Consider an ecologist studying the effect of an invasive beetle on three different species of plants in 10 different forests (half with, $x _ { m } = 1$ , and half without, $x _ { m } = 0$ , the beetle). Suppose that，due to experimental limitations, the ecologist can only measure the pro-portional abundance of each of the plant species in each forest, leading to measurements $Y _ { \\\\cdot 1 } , \\\\dots , Y _ { \\\\cdot D } \\\\in \\\\mathbb { S } ^ { D }$ constrained to the $D$ -dimensional simplex. For rhetorical simplicity, let us pretend there is no measurement noise such that $Y _ { \\\\cdot m } = W _ { \\\\cdot m } ^ { \\\\parallel }$ .Familiar with the field of CoDA,this ecologist decides to first apply the Centered Log-Ratio (CLR) transform to these measurements Wdm $W _ { d m } ^ { \\\\parallel * } = W _ { d m } ^ { \\\\parallel } / G ( W . m )$ (where $G ( \\\\cdot )$ denotes the geometric mean) and then for each species quantify the effect of the invasive species as a difference in means: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[328,367,666,424]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0054/86defdf1bb4ff4ee047940fafeca9045e625f5831632190efa0dc80def77533f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\hat { \\\\theta } _ { d } = \\\\underset { m : x _ { m } = 1 } { \\\\mathrm { m e a n } } ( \\\\log W _ { d m } ^ { \\\\parallel * } ) - \\\\underset { m : x _ { m } = 0 } { \\\\mathrm { m e a n } } ( \\\\log W _ { d m } ^ { \\\\parallel * } ) } \\\\\\\\ & { \\\\quad = r ( W ^ { \\\\parallel } ) . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,434,877,563]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This applied estimator is scale invariant per the CoDA definition: $r ( Y ) = r ( \\\\alpha Y )$ for any constant $\\\\alpha \\\\in \\\\mathsf { \\\\Gamma } ( 0 , \\\\infty )$ . In fact, this estimator even abides by CoDAs two other defining principles (permutation invariance and sub-compositional coherence） (Pawlowsky-Glahn et al., 2015). Yet,this is not sufficient to argue that this is a principled or rigorous analysis of these data. Evaluating this applied estimator (i.e., determining if it is principled, rigorous, or useful) must depend on the precise goals of the researcher: what precisely does the researcher want to estimate? The language of SRI allows us to discuss this issue clearly. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,564,877,763]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Suppose the researcher's target estimand is given by Equation 6: they want to under-stand how the invasive beetle changes the mean CLR normalized abundance of each species. Then researchers applied estimator is clearly a principled analysis of these data as the ap-plied methods are guaranteed to exactly output the target estimand. Suppose instead that the researcher's target estimand is given by Equation 3: they care about how the actual abundance of each species changes in forests with or without the beetle and not the change in CLR normalized abundance. In this case, the target estimand is (fully) scale reliant yet somehow the applied estimator is scale invariant.1 In fact, the assumption is that what is being estimated by the applied estimator is answering the scientifc question implied by the target estimand, that $\\\\theta _ { d } = \\\\hat { \\\\theta } _ { d }$ . With minimal algebra this assumption can be reduced to an assumption that the diference in mean log geometric means of the compositions is exactly equal to the difference in mean log scales: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,775,872,805]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1other than the qualifier /fully/ which is unique to SRI, this sentence holds from the perspective of the SRI definition of scale reliance/invariance of an estimator or the CoDA definition. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[488,829,506,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,537,104]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[160,117,816,145]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0042/e6cede6b370a5931f3e5bebca423794d38ddf39125f3cf71919da62289e908d1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { m e a n } _ { m : x _ { m } = 1 } ( \\\\log W _ { d m } ^ { \\\\perp } ) - \\\\operatorname* { m e a n } _ { m : x _ { m } = 0 } ( \\\\log W _ { d m } ^ { \\\\perp } ) = \\\\operatorname* { m e a n } _ { m : x _ { m } = 1 } ( g ( \\\\log W _ { \\\\cdot m } ^ { \\\\parallel } ) ) - \\\\operatorname* { m e a n } _ { m : x _ { m } = 0 } ( g ( \\\\log W _ { \\\\cdot m } ^ { \\\\parallel } ) ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[122,159,877,194]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We call this the CLR assumption and will return to it later in Section 5.2, when we show this same assumption is at play in popular tools used to analyze Sequence Count Data. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,195,877,323]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This example illustrates how SRI allows for a degree of subtlety, a consideration of the scientific question, not present in CoDAs axiomatic approach. The next section emphasizes this subtlety and shows how more well established concepts such as consistency, calibration, and bias can be used within SRI to establish the limits of inference and evaluate estimators instead of CoDAs axioms. Moreover, in later sections we show how SRI can even provide a framework for showing the functional consequences of violating modeling assumptions and using applied estimators that may be inconsistent with the scientific goal. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,348,620,373]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3Limit on Scale Reliant Inference \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,388,875,424]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"With our problem formulation, we can now state and prove key results on the limits on scale reliant inference in terms of consistency, calibration, and bias. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,425,877,570]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $\\\\theta = f ( W )$ denote a target estimand and estimator pairing, and let $P _ { W } ( \\\\mathcal { V } )$ denote a measurement model for a compositional survey $\\\\mathcal { V }$ . The following results are stated in terms of the composite map $h = g \\\\circ f ^ { - 1 }$ where $g$ denotes the mapping $g \\\\equiv P _ { W } ( \\\\mathcal { V } ) : \\\\mathbf { W } \\\\to \\\\mathbf { P }$ .In Supplementary Section B.3, we prove the following statements: (1） that invertibility is a necessary condition for consistency, (2) that invertibility is a necessary condition for unbi-asedness,and (3) that full uninvertability is incompatable with the existence of calibrated confidence sets. Furthermore, we show that those three results imply the following three limits on scale reliant inference. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,582,874,619]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 1 (Consistency). For partially or fully scale reliant inference， there does not exist a sequence of estimators for θ that is consistent. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,631,870,667]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2 (Calibration). For fully scale reliant inference, there does not exist a calibratec confidence set for $\\\\theta$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,680,867,699]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3 (Bias). For partially or fully scale reliant inference, any estimator is biased. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,712,875,766]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"These results highlight the difficulty of scale reliant inference. In fact, they show that Frequentist ideals such as consistency， calibration， and unbiasedness are fundamentally impossible for fully scale reliant inference. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[488,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"12 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,82,787,107]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4Scale Simulation Random Variables (SSRVs) \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,121,877,212]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The previous section established limits on Frequentist ideals such as unbiasedness, consistency, and calibration in scale reliant inference. In this section, we develop Scale Simulation Random Variables (SSRVs) which operate under a Bayesian inference paradigm. We show that these random variables satisfy weaker Bayesian notions of calibration and are closely related to a class of models called Bayesian Partially Identified Models (Bayesian PIMs). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,213,877,449]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In moving towards a Bayesian viewpoint, we now treat $W$ as a random quantity (which we emphasize with script notation $\\\\mathcal { W }$ ). Similarly, we wil sometimes discuss a compositional survey as a fixed (i.e.， observed, denoted $Y$ ）rather than random (denoted $\\\\mathcal { V }$ ）quantity. Following this shift in viewpoint，we will switch from using notation such as $P _ { W } ( \\\\mathcal { V } )$ to instead using conditional probability or sampling notation $P ( \\\\mathcal { P } | W )$ or $\\\\mathcal { V } \\\\sim P ( W )$ ，either of which are more common in the Bayesian literature. We will start to denote probability densities or probability mass functions with a lowercase $p$ as opposed to the uppercase $P$ which we reserve for measures. Finally, we will denote with an $( a )$ (e.g, $p ^ { ( a ) }$ ）probability models specified by an analyst and contrast that to the true distribution of a random quantity which we denote with a dagger (e.g., $p ^ { \\\\dagger }$ ). In particular,we call $p ^ { \\\\dagger } ( \\\\mathcal { V } | W )$ the target measurement model which represents the true relationship between the distribution of $\\\\mathcal { V }$ and the value of $W$ . When we wish to explicitly refer to the true distribution of $\\\\mathcal { V }$ ， which is conditioned on the true value of $W$ ， we will denote this as $p ^ { \\\\dagger } ( \\\\mathcal { V } | W ^ { \\\\dagger } )$ · \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,460,879,570]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 7 (Scale Simulation Random Variables). Let $\\\\theta = f ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } )$ denote a tar-get estimand and estimator pairing. Let $p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\parallel } )$ denote an applied scale model and $p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\parallel } | Y )$ denote an applied measurement model for an observed compositional survey $Y$ . We define a scale simulation random variable $\\\\theta$ as the transformation of the random variable $( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } )$ via the target estimator. This can be written in hierarchical notation $a s$ ： \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[419,582,576,656]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0582/23d58efb6c9303f7af3cd12da0680e0f860b325aadcc1372c836bc4414627672.jpg\",\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { c } { { \\\\theta = f ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } ) } } \\\\\\\\ { { \\\\mathcal { W } ^ { \\\\perp } \\\\sim p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\parallel } ) } } \\\\\\\\ { { \\\\mathcal { W } ^ { \\\\parallel } \\\\sim p ^ { ( a ) } ( Y ) } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,667,663,686]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"or alternatively as an integral transformation for any set $\\\\mathbf { A } \\\\subset \\\\Theta$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[212,698,782,738]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0130/c14e97008cef98ccdb47afb5e5b7bbc82d348312c48d1abee14dfa3346394e38.jpg\",\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nP ^ { ( a ) } ( \\\\theta \\\\in \\\\mathbf { A } | Y ) = \\\\int _ { ( \\\\mathcal { W } ^ { \\\\bot } , \\\\mathcal { W } ^ { \\\\| } ) \\\\in f ^ { - 1 } ( \\\\mathbf { A } ) } p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\bot } | \\\\mathcal { W } ^ { \\\\| } ) p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\| } | Y ) d \\\\mathcal { W } ^ { \\\\| } d \\\\mathcal { W } ^ { \\\\bot } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,770,874,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Less formally, Scale Simulation Random Variables (SSRVs) are random variables defined over the same space as a target estimand that capture our uncertainty in both the system scale and the system composition. The name scale simulation random variables comes from the fact that they can be readily simulated by Monte-Carlo Simulation without the need to establish a Markov Chain (See Supplementary Algorithm 1). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[488,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"13 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,875,140]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,142,875,213]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A special quantity that we will refer to throughout this work is the Scale-Conditional Random Variable (SCRV） which can be thought of as an SSRV conditional on $\\\\mathcal { W } ^ { \\\\perp }$ ,i.e., without incorporating uncertainty from the applied measurement model. Using the above notation, the SCRV can be defined as: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[235,227,754,267]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0160/6e781f0c5486a2e574588e6ac14dc88a295170d282864deae5877fc7d4d65362.jpg\",\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nP ^ { ( a ) } ( \\\\theta \\\\in \\\\mathbf { A } | Y , W ^ { \\\\perp } ) = \\\\int _ { ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } ) \\\\in f ^ { - 1 } ( \\\\mathbf { A } ) : \\\\mathcal { W } ^ { \\\\perp } = W ^ { \\\\perp } } p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\parallel } | Y ) d \\\\mathcal { W } ^ { \\\\parallel } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,279,874,314]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Later in this work we will show how existing applied estimators such as ALDEx2 can be related to the framework of SSRVs by equating them to SCRVs. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,315,870,348]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The mean of SSRVs, which we will denote as $\\\\bar { \\\\theta }$ have an appealingly simple and intuitive form: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[331,349,665,372]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0248/ee41a7ce9a260f13a9f5b620429a37a8c7f9e1d1aaef89c20cd0c871e1ef27fa.jpg\",\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { \\\\overline { { \\\\theta } } = \\\\mathbb { E } _ { p ( \\\\mathcal { W } ^ { \\\\perp } \\\\mid \\\\mathcal { W } ^ { \\\\parallel } ) } \\\\left[ \\\\mathbb { E } _ { p ( \\\\mathcal { W } ^ { \\\\parallel } \\\\mid Y ) } \\\\left[ f ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } ) \\\\right] \\\\right] . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,377,875,433]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This is simply the expectation of the target estimand with respect to the measurement and scale models. Moreover, $\\\\tilde { \\\\theta }$ can be readily estimated as the mean of Monte-Carlo samples obtained from Supplementary Algorithm 1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,455,787,477]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4.1Relationship to Bayesian Partially Identified Models \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,484,877,651]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bayesian Partially Identified Models (Bayesian PIMs) are a special class of Bayesian hierarchical models where the likelihood identifies only a subset of the parameters that the prior is defined over (Gustafson, 2O15). In the present context, the likelihood is defined with respect to parameters $\\\\begin{array} { r } { \\\\mathcal { W } ^ { \\\\| } \\\\left( p ( \\\\boldsymbol { Y } | \\\\mathcal { W } ^ { \\\\| } ) \\\\right) } \\\\end{array}$ while the prior is defined over the possible values of both $\\\\mathcal { W } ^ { \\\\parallel }$ and $\\\\mathcal { W } ^ { \\\\parallel }$ ， $( p ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } ) )$ .These models are remarkable in that they maintain their calibration properties even in the setting of partially identified inference,a feature that many other models lack (Gustafson, 2015). The following proposition establishes that SSRVs can be seen as the transformed posterior distribution of a Bayesian PIMs (proof provided in Supplementary Section C.1). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,661,879,791]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 1. Let $p ( \\\\mathcal { W } ^ { \\\\perp } , \\\\mathcal { W } ^ { \\\\parallel } | Y )$ denote the posterior of a Bayesian Partially Identified model with likelihood $p ( Y | \\\\mathcal { W } ^ { \\\\| } )$ and prior $p ( \\\\mathcal { W } ^ { \\\\perp } , \\\\mathcal { W } ^ { \\\\parallel } )$ Let $\\\\theta = f ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } )$ denote a tar-get estimand and estimator pairing. The transformation of the posterior distribution for $( W ^ { \\\\parallel } , W ^ { \\\\perp } )$ via the target estimator is a scale simulation random variable with the measure-ment model corresponding to the marginal posterior distribution $p ( W ^ { \\\\parallel } | Y )$ ， and scale model corresponding to the conditional prior $p ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\parallel } )$ which is equal to the conditional posterior $p ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\parallel } , Y )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[488,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"14 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,877,323]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Rather than introducing SSRVs as transformations of Bayesian PIMs, we instead introduce them as in Definition 7 for two reasons. First, as we show in Supplementary Algorithm 1, the modular specifcation of SSRVs tends to produce models that are readily amenable to Monte-Carlo simulation-based inference without the need to establish a Markov Chain. This is in sharp contrast to Bayesian PIMs which do not generally permit direct Monte-Carlo simulation. This computational challenge is often exacerbated by highly anisotropic posteriors that arise due to the only partial updating of the prior (Gustafson, 2O15). Second,we find it simpler to specify these estimators modularly. Since the data $Y$ provides little to no information about the scale of the system, we believe that specifying uncertainty over $\\\\mathcal { W } ^ { \\\\perp }$ should occupy a dedicated part of the model design process. Such modularity also enhances reproducibility as all assumptions about $\\\\mathcal { W } ^ { \\\\perp }$ are clearly stated in the scale model $p ( \\\\mathcal { W } ^ { \\\\perp } | W ^ { \\\\parallel } )$ . In contrast, specifying a prior over $p ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } )$ may involve less obvious assumptions about the conditional relation between $\\\\mathcal { W } ^ { \\\\parallel }$ and $\\\\mathcal { W } ^ { \\\\perp }$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,345,387,367]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4.2SSRV Calibration \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,377,877,686]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"While we have already shown that Frequentist calibration is unattainable for fully scale reliant inference, the relationship between SSRVs and Bayesian PIMs implies that SSRVs enjoy the same Bayesian calibration properties as Bayesian PIMs. Beyond the notion that a Bayesian $1 - \\\\alpha$ credible interval is calibrated in a subjective probability sense, Bayesian credible intervals have a type of calibration that connects to Frequentist concepts of coverage. For example, Gustafson (2015) (pg. 41) shows that for any Bayesian $( 1 - \\\\alpha )$ posterior credible interval, the expected probability that the true parameter value exists in the interval is exactly equal to $( 1 - \\\\alpha )$ . While this Bayesian calibration is comparable to the notion of Frequentist coverage, it is weaker: this coverage is only guaranteed on average where the averaging is weighted according the prior distribution. Connecting this back to SSRVs, we can say that any $( 1 - \\\\alpha )$ probability interval of an SSRV enjoys the typical notion of Bayesian calibration in terms of subjective probability as well as this weaker form of Frequentist calibration. Still， while these calibration properties are appealing, they are not our primary motivation for developing SSRVs (or alternatively introducing Bayesian PIMs) for scale reliant inference: that motivation is best highlighted through the asymptotic properties of SSRVs which show how they can help mitigate inferential errors despite the inevitability of a large-sample bias. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,710,560,731]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4.3Asymptotic Properties of SSRVs \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,741,875,795]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Classically, the variance of an estimator (including the variance of a Bayesian posterior) decreases monotonically with increasing data, approaching zero in the asymptotic limit. Yet on an intuitive level, such behavior is troubling in the setting of scale reliant inference where limitations of the data place fundamental limits on our knowledge. Especially considering that any estimator for a target estimand $\\\\theta$ in the setting of scale reliant inference must be biased (Theorem 3),an asymptotic variance that tends to zero despite a non-zero bias clearly points to a substantial risk for Type I errors. This combination of non-zero bias in the setting of zero asymptotic variance has been called an “unacknowledged bias” in the Bayesian PIM literature (Gustafson, 2015, pg. 35). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[488,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"15 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,877,195]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,196,877,321]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section, we show that,unlike classical models, SSRVs do not suffer from such an unacknowledged bias: so long as the scale model is not degenerate, the asymptotic variance of a SSRV is non-zero reflecting the limits on inference in the setting of uninvertability. In fact, we show that the scale model of an SSRV can be thought of as providing a model for the bias of the SSRV mean and lead to an asymptotic variance that is directly tied to the modeled bias. This leads us to say that SSRVs demonstrate an acknowledged rather than unacknowledged bias. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,325,877,578]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The following results should come as no surprise to readers familiar with the study of Bayesian PIMs where nearly identical properties are wel established. In fact, our treatment of the asymptotic properties of SSRVs is modeled after the discussion of unacknowledged versus acknowledged bias in Gustafson (2015). Still, we include this presentation to highlight these properties and to provide conditions under which identical conditions hold for SSRVs. In particular, to ensure the correspondence between Bayesian PIMs and SSRVs we must placed mild regularity conditions on the target estimator. In brief,we assume that target estimator is bounded and that its expectation with respect to the joint distribution $P ( Y , W ^ { \\\\parallel } , W ^ { \\\\perp } )$ exists (See Supplementary Section C.2 for formal statement). With these conditions,we can state the asymptotic properties in a concise form analogous to the results presented in Gustafson (2015) (pg. 52). In addition, to avoid muddling our presentation with non-scale-related sources of bias and unidentifiability, we assume that the measurement model is faithful such that in the asymptotic limit it converges to a point mass about the true value of the system composition. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,580,875,689]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We use the following short-hand notations to state our results concisely. We denote the true value of the target estimand as $\\\\theta ^ { \\\\dagger } \\\\equiv f ( W ^ { \\\\parallel \\\\dagger } , W ^ { \\\\perp \\\\dagger } )$ , the true measurement model $p ^ { \\\\dagger } \\\\equiv p ^ { \\\\dagger } ( \\\\mathcal { V } | W ^ { \\\\parallel \\\\dagger } , W ^ { \\\\perp \\\\dagger } )$ ， and the joint of the applied measurement and scale models $p ^ { ( a ) } \\\\equiv$ $p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { V } ) = p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\parallel } ) p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\parallel } | \\\\mathcal { V } )$ . Finally, we will use the following notation for bias and variance which emphasizes the model with which expectations are taken: for an estimator $\\\\hat { \\\\phi }$ and a model $P$ we define $\\\\mathrm { b i a s } _ { P } ( \\\\hat { \\\\phi } ) = \\\\mathbb { E } _ { P } [ \\\\hat { \\\\phi } ] - \\\\phi ^ { \\\\dagger }$ and $\\\\mathrm { v a r } _ { P } ( \\\\hat { \\\\phi } ) = \\\\mathbb { E } _ { P } [ ( \\\\hat { \\\\phi } - \\\\mathbb { E } _ { p } ( \\\\hat { \\\\phi } ) ) ^ { 2 } ]$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,689,875,743]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We first show that the asymptotic bias with respect to the true measurement model $( p ^ { \\\\dagger } )$ (the Frequentist bias)is equal to the bias calculated with respect to the posterior $p ^ { ( a ) }$ (the Bayesian bias). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,752,875,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 4. Let $\\\\theta = f ( \\\\mathcal { W } )$ denote a target estimand and estimator pairing. Let $\\\\theta$ denote a faithful scale simulation estimator that satisfies the regularity conditions provided in Definition E.2. Then the asymptotic bias of $\\\\overline { { \\\\theta } }$ with respect to the target measurement model $p ^ { \\\\dagger }$ or with respect to the applied measurement and scale models $p ^ { ( a ) }$ are equal \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[488,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"16 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[117,85,745,104]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[284,117,714,145]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0029/d86c0a5c7db5a1fbd19532ef64ad558220d4033aa4f01440d25b56b4cbd93e61.jpg\",\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\operatorname { b i a s } ( { \\\\overline { { \\\\theta } } } | W ^ { \\\\perp \\\\dagger } ) = \\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\operatorname { b i a s } _ { p ^ { \\\\dagger } } ( { \\\\overline { { \\\\theta } } } ) = \\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\operatorname { b i a s } _ { p ^ { ( a ) } } ( { \\\\overline { { \\\\theta } } } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,157,230,175]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and given by \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[236,188,761,217]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0732/fc06c512bddbbbf3c6263ad3ef9ef2561b9283dfca6d43ab4d301374657cca8b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { l i m } _ { n \\\\to \\\\infty } b i a s ( \\\\overline { { { \\\\theta } } } | W ^ { \\\\perp \\\\dagger } ) = \\\\mathbb { E } _ { p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\parallel } | W ^ { \\\\parallel \\\\dagger } ) } [ f ( W ^ { \\\\parallel \\\\dagger } , \\\\mathcal { W } ^ { \\\\perp } ) ] - f ( W ^ { \\\\parallel \\\\dagger } , W ^ { \\\\perp \\\\dagger } ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,229,875,304]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our notation $\\\\scriptstyle \\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\operatorname { b i a s } ( \\\\theta | W ^ { \\\\perp \\\\dagger } )$ introduced the condition on $W ^ { \\\\perp \\\\dagger }$ in order to emphasize which parts of this bias required knowledge of the true value of the system scale to calculate. Next, we show that the scale model provides a model for this bias and dictates the asymptotic variance. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,315,874,353]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 5. The asymptotic variance of a faithful scale simulation random variable under regularity conditions provided in Definition E.2 is: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[267,366,730,393]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0279/917bb660860d4a4dda69eca903297ec7a960566a0963ae3fe6ea73169f3c45c1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\operatorname { v a r } _ { p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } | Y _ { n } ) } ( \\\\theta ) = \\\\mathbb { E } _ { p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\perp } | W ^ { \\\\parallel \\\\dagger } ) } \\\\left[ \\\\operatorname { b i a s } ( \\\\bar { \\\\theta } | \\\\mathcal { W } ^ { \\\\perp } ) ^ { 2 } \\\\right] .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,405,877,534]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This result establishes what we mean when we say that SSRVs acknowledge their own bias: while the mean of an SSRV is a biased point estimator, the variance of an SSRV depends on the bias. As the true value of $W ^ { \\\\perp }$ is not known, the SSRV models its own bias based on beliefs about the system scale encoded in the scale model. So long as the scale model has non-zero variance (is not degenerate), the asymptotic variance of an SSRV will be non-zero - acknowledging its bias. Next, we show that common models used to analyze sequence count data do not acknowledge their bias. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,559,875,613]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5Empirical Studies of SSRVs applied to Sequence Counting \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,626,877,790]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section, we show that common models used in the analysis of sequence count data can demonstrate elevated rates of Type-1 errors due to unacknowledged bias. We then focus our attention on a single model, ALDEx2, in order to provide intuition behind these results and explain why they should be expected from many currently used methods. To strengthen our interpretation of these results, we show how the ALDEx2 model can be viewed as a SSRV and modified to acknowledge its bias and reduce false positives. In fact, through applications to both real and simulated data, we show that the flexibility provided by SSRVs even allows us to incorporate additional knowledge about the system scale that can reduce Type-II errors as well. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[488,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,875,140]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"17\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For concreteness and brevity, in this section we present al results and discusson relative to a population-level (as opposed to sample-level） version of the LFC estimand given in Equation 3: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[330,141,666,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0382/a033ede5dfad437ebe464724bf4358e36cd101bb62113537c238c078320d0fb4.jpg\",\n \"list_items\": null,\n \"page_idx\": \"17\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\theta _ { d } = \\\\mathbb { E } [ \\\\log W _ { d m : x _ { m } = 1 } ] - \\\\mathbb { E } [ \\\\log W _ { d m : x _ { m } = 0 } ] .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,167,874,204]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"17\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We switch to a population based estimator to be consistent with the ALDEx2 model definition (presented below). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,226,723,247]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"17\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5.1Unacknowledged Bias in ALDEx2 and DESeq2 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,256,877,366]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"17\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As described by Gustafson (2015) (pg. 35)，a hallmark of unacknowledged bias is an increasing rate of Type-1 errors with increasing data. This phenomena is troubling as it defies standard logic: typically， we think of inference improving with more data, not worsening. Figure 1 illustrates the intuition behind this phenomena. As we show next, two popular methods for differential abundance and differential expression analysis of sequence count data display this hallmark of unacknowledged bias. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[316,378,679,569]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 1: The relationship between unacknowledged bias and sample size can be understood as a symp-tom of model misspecification. Typical model-based estimators demonstrate decreasing uncertainty (e.g., decreasing variance)with increasing data,approaching zero in the asymptotic limit. Now consider a model that is misspecified such that a corresponding estimator displays a large-sample bias.When the sample size is small,the uncertainty can be large and encapsulate the bias: large uncertainty arising from even a misspecified model may bufer against the inferential effects of the bias.Yet with increasing sample size the model becomes increasingly certain in a biased estimate; in turn the impact of the bias becomes magnified. In the present context, model misspecification refers to models failing to incorporate uncertainty from the lack of scale information in the observed data. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0840/3a46d23df54c705c062e4b629d756865a8142d8a4ef92f4ea8751149a8471f09.jpg\",\n \"list_items\": null,\n \"page_idx\": \"17\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[119,734,875,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"17\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To demonstrate this phenomena we simulated a scaled system with $D = 2 0$ entities defined over $M$ configurations equally split between two conditions. For concreteness, let us imagine this is a microbiome system consisting of $D$ taxa in the gut of $M$ individuals half of which were randomly assigned to a narrow-spectrum antibiotic treatment and the other half to placebo. For simplicity,we simulate $W _ { d m } \\\\sim \\\\mathrm { P o i s s o n } ( \\\\lambda _ { i x _ { m } } )$ . Of the 20 taxa, we set $\\\\lambda _ { d 0 } = \\\\lambda _ { d 1 }$ in all but four taxa,reflecting no association between their abundance and treatment. In the remaining four, we set $\\\\lambda _ { d 0 } < \\\\lambda _ { d 1 }$ simulating decreased abundance associated with antibiotic treatment. To simulate the loss of scale information，we then simulated observations $Y _ { \\\\cdot m }$ as a multivariate hyper-geometric draw from the corresponding $W _ { \\\\cdot m }$ . We then applied both the ALDEx2 (Fernandes et al., 2014) and DESeq2 (Love et al., 2014) models to the resampled data. We take as our target estimand $\\\\theta _ { d }$ defined as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[488,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"17\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"18 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,877,214]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[253,227,740,246]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0287/d2c8d3ead102baf24e10f2a061313351aae8c69dc6ed26f82357402e7336ab21.jpg\",\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\theta _ { d } = \\\\log \\\\lambda _ { d 1 } - \\\\log \\\\lambda _ { d 0 } = \\\\log \\\\operatorname { \\\\mathbb { E } } [ W _ { d m : x _ { m } = 1 } ] - \\\\log \\\\operatorname { \\\\mathbb { E } } [ W _ { d m : x _ { m } = 0 } ]\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,257,877,330]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and therefore define a false positive as estimating a taxon $d$ is differentially abundant when $\\\\lambda _ { i 0 } = \\\\lambda _ { i 1 }$ . We repeated this process simulating different numbers of samples, $M \\\\in ( 0 , 2 0 0 ]$ ， to study how the False Discovery Rate (FDR) changes as a function of sample size. Full simulation details can be found in Supplementary Section F.1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,330,877,421]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The results are shown in Figure 2 and demonstrate the hallmark of unacknowledged bias: the FDR increases with increasing sample size. With sample sizes as small as 20, both methods demonstrate false discovery rates greater than 6O%. Moreover, in the asymptotic limit both methods display rates of 80% which corresponds to calling every taxon significantly differentially abundant². \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,443,524,465]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5.2Examining ALDEx2 in Detail \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,474,875,546]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We examine the form of the ALDEx2 model to identify the origin of this unacknowledged bias. The ALDEx2 model is defined procedurally in 3 parts. In the first part,each mul-tivariate observation $Y _ { \\\\cdot m }$ is modeled using an independent Bayesian Multinomial-Dirichlet model: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[390,558,606,606]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0665/eaea922b682aa983aadb2386f4fa04d2244a61ddc2cae3b013e6703aef7b424a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { Y _ { \\\\cdot m } \\\\sim \\\\mathrm { M u l t i n o m i a l } ( W _ { \\\\cdot m } ^ { \\\\parallel } ) } \\\\\\\\ & { W _ { \\\\cdot m } ^ { \\\\parallel } \\\\sim \\\\mathrm { D i r i c h l e t } ( \\\\alpha ) . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,614,877,669]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Posterior samples of $W _ { \\\\cdot m }$ are then normalized using the Centered Log-Ratio (CLR) transform For each transformed posterior sample and for each $i \\\\in \\\\{ 1 , \\\\ldots , D \\\\}$ ，ALDEx2 then uses a t-test to test the null hypothesis that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[369,683,622,702]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0135/48777617c5bad441f4c191cefe008673eb4bd49ed83a3972423986cfe2214954.jpg\",\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n0 = \\\\mathbb { E } [ \\\\eta _ { d m : x _ { m = 1 } } ] - \\\\mathbb { E } [ \\\\eta _ { d m : x _ { m = 0 } } ] . ^ { 3 }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,713,875,751]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In short, ALDEx2 is using the CLR transform as a parameter normalization and rather than testing the null hypothesis that $\\\\theta _ { d } = 0$ (with $\\\\theta _ { d }$ defined in Eq. 18) is instead testing \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,760,875,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2The maximum is not $1 0 0 \\\\%$ due to 4 true positives. 3Ultimately the p-value that is reported by ALDEx2 is the mean of the p-values computed on each posterior sample although this step is not central to our discussion. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[488,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"19 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[145,88,838,353]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 2: False discovery rate versus sample size for ALDEx2 and DESeq2. For each method, false discovery rate increases even in the presence of infinite data. The maximum false discovery rate was 0.80, as four of 2O taxa were truly differentially expressed. Sequencing depth was set to $5 , 0 0 0$ . Simulation details are in Supplementary Section D.1. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0371/ced333e56f0ada8767c46a2dd65bedb52cfb498f1fca2ae128f6866ba29f981d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[119,460,323,477]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"the null hypothesis that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[254,489,736,568]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0614/29e9e77cde5fd6376dab84bb0f0efe00647bbc9085538ba0ba76bbadceed1651.jpg\",\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { 0 = \\\\hat { { \\\\boldsymbol \\\\theta } } _ { d } = \\\\hat { { \\\\boldsymbol \\\\theta } } _ { i } = \\\\mathbb { E } [ \\\\eta _ { d m : x _ { m } = 1 } ] - \\\\mathbb { E } [ \\\\eta _ { d m : x _ { m } = 0 } ] } \\\\\\\\ & { \\\\qquad = \\\\mathbb { E } \\\\left[ \\\\log \\\\frac { W _ { d m : x _ { m } = 1 } ^ { \\\\parallel } } { g ( W _ { \\\\cdot m : x _ { m } = 1 } ^ { \\\\parallel } ) } \\\\right] - \\\\mathbb { E } \\\\left[ \\\\log \\\\frac { w _ { d m : x _ { m } = 0 } ^ { \\\\parallel } } { g ( w _ { \\\\cdot m : x _ { m } = 0 } ^ { \\\\parallel } ) } \\\\right] } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,579,733,598]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To develop intuition about the assumption that $\\\\theta _ { d } = \\\\hat { \\\\theta } _ { d }$ , we approximate \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[393,613,604,632]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0885/e76bbfb06b54f85ac7b637f0162a446a08edf3ee6261d84c780f1b01995e9ad1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\log \\\\mathbb { E } [ W _ { d m } ] \\\\approx \\\\mathbb { E } [ \\\\log W _ { d m } ]\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,646,875,702]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"based on the Taylor approximation given by Teh et al. (2006) which is highly accurate for a Poisson random variable with rate parameter much greater than 1. With this approximation, the assumption $\\\\theta _ { d } = \\\\hat { \\\\theta } _ { d }$ reduces to: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[166,714,795,737]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0248/3a6fba98a0e2ff13ef4bd17e680be7c0a9cd42ebfde6f9dc642c7c5377b63bba.jpg\",\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { \\\\mathbb { E } [ \\\\log W _ { m : x _ { m } = 1 } ^ { \\\\perp } ] - \\\\mathbb { E } [ \\\\log W _ { m : x _ { m } = 0 } ^ { \\\\perp } ] = \\\\mathbb { E } [ \\\\log g ( W _ { \\\\cdot m : x _ { m = 0 } } ^ { \\\\parallel } ) ] - \\\\mathbb { E } [ \\\\log g ( W _ { \\\\cdot m : x _ { m = 1 } } ^ { \\\\parallel } ) ] . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,750,877,804]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In other words, ALDEx2 assumes that expected difference in log scales between conditions can be imputed exactly (without error） from the system composition. This is simply the population-level analogue of the CLR assumption presented in Equation 8. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"20 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,85,400,104]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"20\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5.2.1 Generalizing ALDEx2 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,114,875,167]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"20\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"With the ALDEx2 scale assumption exposed， we are now in a position to generalize ALDEx2 and show that introducing even small amounts of uncertainty into this assumption is sufficient to alleviate the unacknowledged bias and mitigate Type-I errors. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,169,875,321]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"20\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The ALDEx2 model can be easily generalized to a SSRV. In ALDEx2, instead of defining $\\\\eta$ in terms of the CLR transform of the system composition: let us instead define $\\\\eta _ { d m } = \\\\log W _ { d m } ^ { \\\\parallel } + \\\\log W _ { m } ^ { \\\\perp }$ .This modified ALDEx2 represents a scale conditional random variable, conditional because the value of $\\\\log W _ { m } ^ { \\\\perp }$ is left unspecified. This SCRV can then be combined with a scale model to form an SSRV which can be easily sampling using Algorithm C.2. For example, in Supplemental Figure 1 and 2 we show that we can recover the original ALDEx2 model by using the following point-mass as a scale model: $p ( \\\\log W _ { m } ^ { \\\\perp } ) \\\\sim \\\\delta _ { - \\\\log G \\\\in \\\\cal W _ { \\\\cdot m } ^ { \\\\parallel } }$ where $\\\\delta _ { x }$ denotes a point-masscntered on $x$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,323,875,412]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"20\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Viewing ALDEx2 as an SSRV highlights the source of its unacknowledged bias. There is no uncertainty in the system scale: the scale model is degenerate. The system scale is not precisely equal to the negative geometric mean of the composition yet the model is certain that it is. As we showed in Section 4.3,a degenerate scale model essentially guarantees an unacknowledged bias. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,412,875,522]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"20\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Some readers may question whether it is in fact the lack of scale uncertainty in ALDEx2 that is responsible for the unacknowledged bias and not some other factors such as the misspecification of Multinomial-Dirichlet model to our simulation procedure. To address this concern and to further reinforce that scale uncertainty is the root of the elevated Type-1 error rates, we use our ALDEx2 based SSRV with a non-degenerate scale model. We generalize ALDEx2's point-mass scale model as the following log-normal model: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[356,526,640,549]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0854/23b6d5db1d4c8bdf85d47cc1c1753a32c59ef99b179b833abc4cd09a73a849d5.jpg\",\n \"list_items\": null,\n \"page_idx\": \"20\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { \\\\log ( W _ { m } ^ { \\\\perp } ) \\\\sim N ( - \\\\log g \\\\left( W _ { \\\\cdot m } ^ { \\\\parallel } \\\\right) , \\\\alpha ^ { 2 } ) . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,554,877,717]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"20\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As $\\\\alpha 0$ this model reduces to the ALDEx2 point mass model. Yet when $\\\\alpha ^ { 2 } \\\\ > \\\\ 0$ ， this model adds to the ALDEx2 point mass model uncertainty in the system scale. In Supplementary Section D.1, we show that adding even small amounts of uncertainty ( $\\\\alpha >$ 0.2) is sufficient to remove allof the false positives while retaining 3 of the 4 true positives results. Moreover, we show that the optimal window for the choice of $\\\\alpha ^ { 2 }$ is large in that this SSRV continues to identify true positives so long as $\\\\alpha < 1 . 8$ . To provide context for this value, consider that for $\\\\alpha > 1 . 8$ we would have to believe that, for any individual in the study, there is a greater than 5% probability that the CLR Estimate of Scale is in error by more than a factor of $e ^ { 1 . 9 6 \\\\times 1 . 8 ^ { \\\\circ } } \\\\approx 5 7 3$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,739,681,762]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"20\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5.3ALDEx2 and CoDA as Opposing Extremes \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,770,872,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"20\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Before progressing to our study of real data, we briefly point out that the above ALDEx2 SSRV can also allow us to contextualize CoDA principle of scale invariance as an extreme opposite of the assumptions of ALDEx2. Using the log-normal scale model above，we recovered ALDEx2's point mass scale model in the limit as $\\\\alpha 0$ .This limit implies infinite knowledge (zero uncertainty) in the system scale given the system composition. In the other extreme,as $\\\\alpha \\\\to \\\\infty$ we assume zero knowledge (infinite uncertainty) in the system scale. Unsurprisingly, in this extreme,any non-zero credible interval calculated from the corresponding SSRV must cover the entire parameter space - enforcing restricting us to only ask scale invariant questions. While this is technically not the same as CoDA's scale invariance axiom (which enforces scale invariance of an applied estimator not the target estimator), it does echo the common view in CoDA that only questions that can be framed in terms of log-ratio can be answered from data that lack scale Aitchison (1994). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,506,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"20\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"21 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,877,267]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,270,875,342]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Between these two extremes (the point mass model of ALDEx2 and CoDA's infinite uncertainty), SSRVs allow us to use express non-zero yet finite uncertainty in the system scale. We believe this view emphasizes the utility of SSRVs, as a middle ground between these two opposing extremes. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,364,710,386]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5.4Application to a novel 16S rRNA-Seq dataset \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,395,877,631]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Artificial gut systems (bioreactors) are seeded with human feces and often used as a model for host-associated microbiota. Yet， many bioreactors undergo a two week equilibration period in which the inoculated communities diverge from those seen in human hosts and adapt to their new ex vivo environment. To evaluate the limits of these model systems we ran and collected data for 8 bioreactors over the course of the first two weeks after inoculation. Each bioreactor was seeded with 100mL of fecal slurry and maintained at a constant 400mL of growth media over the course of the experiment. Our primary interest was in determining which of the microbial taxa present in the innoculum were able to grow in the bioreactors. We quantify this interest using the differential abundance target estimator given in Equation 3 with $x _ { m } = 0$ denoting day 1 of the experiment and $x _ { m } = 1$ denoting day 14. In this case,we were primarily interested in the de facto total scale, i.e., we wanted to identify taxa that increased or decreased in total cellular number over the two week period. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,633,877,796]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Two features of this experimental system allowed us to obtain high-quality measurements of the total scale in these bioreactors. First,because these vessels were kept at a constant 400mL measuring total cellular concentration allowed us to estimate total cellular number. Second, per protocol (Silverman et al., 2018)， these bioreactors are constantly stirred to produce homogeneous growth conditions throughout the bioreactor; this ensured that estimation of concentration from a sample of material reflected the concentration of the overall bioreactor. For each of the 8 bioreactors,at each of the two time-points,we obtained triplicate flow-cytometry measurements of bacterial cellular concentration. Trip-licate measurements were obtained so that we could estimate uncertainty in the scale due \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"22 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,87,307,103]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"to measurement noise. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,104,877,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For analysis of the resultant data, we compared 5 models: 3 SSRVs, ALDEx2 and DESeq2. To introduce our 3 SSRVs, we frst consider the following multinomial log-normal Bayesian PIM which is an extension of Bayesian multinomial logistic-normal models that have become popular for analysis of microbiota survey data (Silverman et al., 2022): \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[388,184,607,253]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0212/3260d029706c283a7679b396a371f24ed57dc5abae810cf2c010588cd70278f8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { Y _ { \\\\cdot m } \\\\sim \\\\mathrm { M u l t i n o m i a l } ( \\\\mathcal { W } _ { \\\\cdot m } ^ { \\\\parallel } ) } \\\\\\\\ & { \\\\mathcal { W } _ { \\\\cdot m } = e ^ { \\\\psi _ { \\\\cdot m } } } \\\\\\\\ & { \\\\psi _ { \\\\cdot m } \\\\sim N ( B X _ { \\\\cdot m } , \\\\Omega ) } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,258,877,405]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"with priors $B \\\\sim \\\\mathit { N } ( M , \\\\Omega , V )$ and $\\\\Omega \\\\sim I W ( u , \\\\Xi )$ (See Supplementary Section D.2 for prior specifications). This is a Bayesian Partially Identified Model since the multinomial likelihood is parameterized in terms of only the parameter $\\\\mathcal { W } ^ { \\\\parallel }$ whereas the prior is defined over $\\\\mathbf { W } = \\\\mathbf { W } ^ { \\\\parallel } \\\\times \\\\mathbf { W } ^ { \\\\perp }$ . Due to computational limitations, it is infeasible to sample this model directly using standard approaches (e.g., MCMC). In contrast, we show in Supplementary Section D.3, that this model can be decomposed into a measurement model and a scale model and sampled efficiently as an SSRV using Algorithm C.1. This SSRV representation of the above Bayesian PIM is the first of the three SSRVs we consider in this section. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,406,877,515]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our next two SSRVs are based upon the above Bayesian Multinomial log-normal model. When written as an SSRV the above Bayesian Multinomial log-normal model has a Multinomial Logistic Normal measurement model of the form studied in Silverman et al. (2022). We created two more SSRVs using this measurement model with diferent scale models. Letting $m \\\\in \\\\mathbf { M }$ denote a given vessel at a given time-point,we build two scale models around the following family of log-normal models: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[393,522,606,542]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0867/ff69182da905f41f3fc931cafc944f14b63a7f28b3a8a76a1679c0dbc0e0ab91.jpg\",\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\log W _ { m } ^ { \\\\perp } \\\\sim N ( c + \\\\mu _ { m } , \\\\alpha _ { m } ^ { 2 } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,550,877,770]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $c$ is some constant ad $\\\\mu$ is subjectgthe consuraint $\\\\sum _ { m = 1 } ^ { M } \\\\mu _ { m } = 0$ The constraint is not a critical requirement mathematically or statistically, but we include it to simplify the task of specifying the scale model. Consider that for diffrential analysis (e.g., target estimand in Equation 3),we do not actually need to know the abundance of each entity in each configuration but only the relative scale between configurations (i.e., the scale up to a constant factor $e ^ { c }$ ). This simplifies our task as biologically we often have prior knowledge about the relative scales between conditions yet have substantially less information about the absolute scale of any one condition. To illustrate this claim, consider that for a healthy individual, we believe that the total number of microbes in that individuals mouth is likely the same today as it is tomorrow, yet we have little prior knowledge about the total number of microbes on either day in isolation. In short,for differential analysis we need only know a composition over the scale and can simply set $c = 1$ with no loss of generality. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,771,872,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Of the two scale models we pair with the measurement model $p ( \\\\mathcal { W } ^ { \\\\parallel } | Y )$ ， the first is considered our gold standard (our benchmark against which we measure false positives), defined by letting $\\\\mu _ { m }$ and $\\\\alpha _ { m } ^ { 2 }$ equal the mean and variance of our flow cytometry measurements. We call the resultant SSRV the flow model. Our second scale model is built based on our knowledge of the experimental design of our study. Since, we know that each vessel contained 400mL of growth media and the innoculum consisted of $1 0 0 ~ \\\\mathrm { m L }$ of fecal slurry, we expect that the total microbial load would, on average, increase from the day 1 to day 14 by a factor of approximately 4. We encapsulated these beliefs in the scale model \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"23 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,877,195]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[367,210,629,252]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0190/b9517480ba9eb95f9bf3daffdff5ca4b7b4e5794b0328f9626083f1c3ad69c52.jpg\",\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\frac { f ( W _ { j , \\\\mathrm { t = 1 } } ^ { \\\\perp } ) } { f ( W _ { j , \\\\mathrm { t = 1 4 } } ^ { \\\\perp } ) } \\\\sim \\\\mathrm { L o g N o r m a l } ( \\\\bar { d } , \\\\tau ^ { 2 } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[122,265,826,284]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $d = \\\\log ( 1 0 0 / 4 0 0 )$ and $\\\\tau = 0 . 5$ .4 We call the resultant SSRV the design model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,285,877,593]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To summarise, we created 3 SSRVs all of which used the same Multinomial Logistic-Normal measurement model but three different scaled models: (1） a scale model chosen to ensure that the resultant SSRV was identical to the above Multinomial Log-Normal model,(2) A scale model built using the flow cytometry measurements,and (3) a scale model built using only our knowledge of the experimental design. We applied the resultant 3 SSRVs as well as ALDEx2 and DESeq2, the results are summarized in Figure 3. As in Section 5.1, we again find that DESeq2 follows the patterns of errors displayed by ALDEx2. Yet in contrast to Section 5.1, here we find that both ALDEx2 and DESeq2 demonstrate reasonable specificities (66.7% and $1 0 0 \\\\%$ respectively） yet poor sensitivities (14.2% and 17.8% respectively). Notably what differs here from prior sections is that here many taxa are truly changing in abundance, growing to occupy the larger 400mL bioreactor. In addition, here we have small sample sizes ( $M = 1 6$ ） compared to the range of sample sizes considered in Section 5.1 ( $M \\\\in ( 1 0 , 1 0 0 ) )$ . Still, by bootstrap resampling we can simulate how results would change had we used more bioreactors. With 25 bioreactors ( $M = 5 0$ ）， we again see the hallmark increasing Type-I error rate of unacknowledged bias: DESeq2 and ALDEx2 give FDRs on the order of $4 0 \\\\%$ and $4 5 \\\\%$ respectively (Supplementary Figure 4). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,594,875,685]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In contrast, across all tested scenarios (i.e., with 8 vessels or 50), the design-based SSRV remains calibrated and displays better sensitivity, FDR,and specifcity than ALDEx2 or DESeq2: FDR of O%, sensitivity of $1 0 0 \\\\%$ ， and specificity of $1 0 0 \\\\%$ . Similar results were found for the Bayesian PIM based scale model which demonstrated a sensitivity of 92.9%, a specificity of $0 \\\\%$ , and a false discovery rate of $1 0 . 3 \\\\%$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,707,382,724]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5.4.1 Sensitivity Analyses \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,734,875,771]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Whereas measurement models are built around real data and therefore have some degree of robustness to miss-specification, scale models (such as our design based scale model above) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[135,784,705,799]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 $\\\\tau ^ { 2 }$ can be compared to $\\\\alpha ^ { 2 }$ in the flow estimator by the relationship $\\\\tau ^ { 2 } = 2 \\\\alpha ^ { 2 }$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"24 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[140,0,839,284]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 3: Comparison of results from ALDEx2, DESeq2,and 3 SSRVs using real data from an ex vivo artificial gut study.“GS\\\\\\\" denotes the gold-standard model against which true/false positives/negatives are defined.Our goal standard (the SSRV using the Flow scale model) is the only model that could use the replicate flow-cytometry measurements collcted in this study. For brevity,taxa that were not significant with respect to any of the 5 models are not shown. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0052/5ec970e554097d57e9c51d8f581854ce9024f9c050e7031fe2cf4e332da33032.jpg\",\n \"list_items\": null,\n \"page_idx\": \"24\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[119,406,875,515]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"24\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"can be purely expressions of subjective belief. It is therefore natural to question whether inferences drawn from an SSRV are sensitive to the chosen scale model: how robust are our claims about sensitivity and specificity of the design model above if we had picked a different value of $\\\\tau ^ { 2 }$ ? When working with SSRVs specified as a target estimand, a measurement model and a scale model, the structure of SSRVs often allows for computationally efficient global sensitivity analyses. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,517,875,626]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"24\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We expect that the most computationally limiting part of Algorithm D.1 will be sampling from the measurement model which is often high-dimensional and conditioned on observed data. Yet to study how the distribution of an SSRV changes with different scale models, the samples from the measurement model need only be obtained once and can be reused for each measurement model. This type of memoization algorithm allows for scalable global sensitivity analyses. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,626,875,752]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"24\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In Figure 4, we show that our results from the design-based SSRV are largely insensitive to our choice of the parameter $\\\\tau$ . So long as $\\\\tau$ is greater than O.35, the specificity of the design-based SSRV will be equal to $1 0 0 \\\\%$ . Moreover， so long as we choose $\\\\tau$ less than 6, then the sensitivity of the design-based SSRV will be greater than that of either DESeq2 or ALDEx2. Consider, that a value of $\\\\tau \\\\geq 6$ would require the unreasonable belief that there is a greater than $5 \\\\%$ chance that the number of microbial cells at day 14 was greater than or equal to $4 e ^ { 1 . 9 6 * 7 ^ { 2 } } \\\\approx 2 \\\\times 1 0 ^ { 4 2 }$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,753,870,789]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"24\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For completeness, we also demonstrate a sensitivity analysis about the parameter $\\\\alpha$ used in SSRV analysis of the simulated dataset from Section 5.2. Those results are provided and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"24\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"25 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[261,85,730,286]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 4: Sensitivity analysis of design-based total model as a function of $\\\\\\\\tau$ . The estimated effect size as a function of $\\\\\\\\tau$ is shown for each taxa in the artificial gut study. The effect size is colored black if the corresponding credible set excluded zero（\\\\\\\"significant\\\\\\\"） and grey otherwise (\\\\\\\"not significant\\\\\\\"). As an illstration, we highlight the 5 taxa that are the most clearly diffrentially abundant in the sense that their credible sets exclude zero over the largest range of $\\\\\\\\tau$ . For comparison,we depict the value of $\\\\\\\\tau$ used in Figure 3 ( $\\\\\\\\tau = 0 . 5$ ）as a black dashed line. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0587/4e20f84b3c7e7f6834de611ba7080a04dee53a1b96d03cffc5b15740d61ebe4d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[120,425,459,443]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"discussed in Supplementary Section D.1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,465,527,488]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5.5Generalizing beyond ALDEx2 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,496,877,787]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"So far,we have focused on providing intuition explaining the origin of ALDEx2's unacknowledged bias. Yet from both Sections 5.1 and 5.4 it is clear that other models such as DESeq2 sufer from unacknowledged bias as well. Unfortunately, whereas the form of the ALDEx2 model permitted in depth dissection, the form of DESeq2 is more complex. For example,whereas the CLR normalization used by ALDEx2 can be expressed as a function of $\\\\mathcal { W } ^ { \\\\perp }$ ， DESeq's median-of-ratios method represents a normalization based on the observed count data（ $Y$ ). It is unclear how or even if this could be represented as a closed form scale model. Still our theoretical results from Section 3 can be applied. DESeq2 is a standard parametric (specifically a Negative Binomial empirical Bayesian model) and has a posterior variance that will collapse to zero in the asymptotic limit. In other words, DE-Seq2 is clearly lacking any uncertainty in the system scale that act against this asymptotic certainty: whatever the scale model used by DESeq2, it is likely degenerate. Moreover, given that any point estimator is biased (Theorem 3), it follows that any model/estimator that aims to perform scale reliant inference yet has a variance that collapses to zero in the asymptotic limit will display an unacknowledged bias. We expect that most models currently used for scale reliant inference fallinto this category. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[137,790,874,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Even beyond sequence counting, we expect unacknowledged bias to be prevalent. For example, in mark-recapture studies it is often assumed (without error) that allindividuals have the same probability of being captured in the second sample as the first (Krebs,1999). We expect that this lack of uncertainty in a critical assumption that can be related to the system scale equates to a degenerate scale model when viewed through the lens of SSRVs. We provided an extended discusson and an example of SRI in the context of ecology in Supplementary Section E. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"26 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,877,195]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,222,313,247]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6Discussion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,261,877,406]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This article has introduced a formal statement of the problem of scale reliant inference. This problem statement was then used to highlight the critical role of modeling assumptions in scale reliant inference. Most notably, for the sub-class of fully scale reliant inference, we show that Frequentist criteria such as consistency， calibration，and unbiasedness are unattainable and even Bayesian notions of calibration require specialized models in order to avoid unacknowledged bias. Still, we showed that there scale reliant inference is possible and that attractive approaches (such as the use of SSRVs) may help avoid the pitfalls of current methods. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,409,875,497]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There are numerous avenues for future expansion and refinement of this work, we re-strict our discussion to four potential future directions: (1) exploring connections to other statistical fields, (2) studying partially scale reliant or biased compositional surveys, (3) developing new tools to quantify the impact of unacknowledged bias, and (4) reevaluating standard attitudes towards experimental design and sample size calculations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,500,877,790]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this article, we have shown how our problem statement represents a reformulation of compositional data analysis and represents an application of Bayesian Partially Identified Models. Still, there are other important connections between scale reliant inference and other statistical fields that should be explored. Perhaps the most notable is the connection between our work and the Potential Outcomes Formulation (POF） of Causal Inference as discussed by Imbens and Rubin (2015). We notice the following connections. Both the POF and scale reliant inference center around clear statements of the scientific question being asked in the form of an estimand (called a causal estimand in the POF and a target estimand here). In both problem formulations,the central challenge is a lack of requisite information (inability to observe both potential outcomes in the POF, lack of knowledge of the system scale here). In both problem formulations,key methods attempt to directly model and quantify uncertainty in assumptions used to address the lack of knowledge (Bayesian model based methods in the POF and Bayesian PIMs or SSRVs here). Beyond the potential for sharing ideas and solutions between the POF and scale reliant inference, we expect that some studies will ask questions that fall at the boundary between these two fields. For example, a researcher studying differential abundance may want to use an estimand that is both a causal estimand and a target estimand: “How would the abundance of taxa $d$ change if that patient had been given the drug versus if they had not been given the drug?” Further work is necessary to answer such questions. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"27 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,87,875,140]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,142,877,431]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A core motivation in this work has been to provide discussion on the limits of scale reliant inference and to relate this back to applied problems in bioinformatics, ecology, and social sciences. While we have discussed limits on scale reliant inference, the relation back to the analysis of real data is non-trivial. In reality, compositional surveys are simply a model and cannot hope to explain the many intricacies and complexities of real data. For example, in practice, we expect that any survey data provides at least minuscule amount of information about the system scale making partially compositional surveys the norm and fully compositional surveys an extreme. Yet with partially compositional surveys conclu-sions about whether inference is fully or partially scale reliant cannot be solely based on the question being asked but requires a more careful consideration of the precise question and the precise information about scale present in the observed data. Beyond this, our definition of compositional surveys assumed that the measurement process identified the system composition. In practice, we expect that measurement bias, such as PCR bias in 16S rRNA sequencing (Silverman et al., 2021), implies that there are other sources of non-invertibility or non-identifiability at play beyond the lack of scale information. Exploring how these other factors impact scale reliant inference would surely be a worthwhile endeavor. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,434,877,650]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A critical limitation of the present work is our inability to adequately answer the ques-tion: “When will unacknowledged bias due to scale limitations lead to problems in an analysis?” Especially in the analysis of sequence count data, this remains and outstanding problem. While there has been some work showing that commonly used methods display elevated Type-1 error rates in a wide variety of real and simulated datasets (Roche and Mukherjee, 2022)， these analyses have all been dependent on the particular simulation model or on assumed fidelity of qPCR or flow cytometry measurements without consideration of potential error in those measurements themselves. Still, the sensitivity analyses we discuss in Section 5.4.1 may find use here. Given a real dataset, these sensitivity analyses allow an analyst to quantify the degree to which their conclusions would change when there is error in their assumptions. Overall, we expect this is a fruitful avenue for future work and development. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,652,877,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Scale reliant inference is a remarkable problem that can defy standard statistical logic. Suppose that a research decides to use ALDEx2 or DESeq2 despite the unacknowledged bias. Standard statistical attitudes towards the value of data might argue “that's probably fine so long as that researcher has sufficient data.” Yet this is the exact opposite of what we have shown: the effect of the bias problems will get worse with more data. Key questions remain: are there instances where having a small number of samples will allow researchers to ignore scale uncertainty and simply using existing tools like DESeq2 and ALDEx2? Again, we expect sensitivity analyses such as those described in prior sections to be helpful in identifying these instances. More broadly, the impact of sample size on experimental design is well described in the Bayesian PIM literature (e.g., see Chapter 3 and Section 7.2 or Gustafson (2015)). Interestingly, in that literature, authors discuss diminishing returns that occur with increasing data: since the model uncertainty asymptotically remains non-zero at a certain point, collcting more data cannot appreciably decrease uncertainty beyond the limiting value. Overall, we expect that further consideration of how scale uncertainty and unacknowledged bias interact with sample size will provide actionable insights. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"28 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,877,214]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,239,669,265]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acknowledgments and Funding Sources \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,279,877,424]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We thank Rachel Silverman， Juan Jose Egozcue, Vera Pawlosky-Glahn, Gregory Gloor, Michael Love,and James Morton for their manuscript comments. LAD acknowledges support from the the Global Probiotics Council, a Searle Scholars Award, the Hartwell Foundation,an Alfred P. Sloan Research Fellowship,the Translational Research Institute through Cooperative Agreement NNX16AO69A, the Damon Runyon Cancer Research Foundation, and NIH 1R01DK116187-01. SM would like to acknowledge the support of grants NSF IIS-1546331, NSF DMS-1418261， NSF IIS-1320357, NSF DMS-1045153,and NSF DMS-1613261. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,452,604,477]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data Access and Code Availability \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,491,870,526]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"All data and R scripts necessary to replicate all analyses in the manuscript and supplement are located at https://github.com/michellepistner/ScaleSimulation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,554,269,578]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,592,874,628]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Aitchison, J. (1986). The Statistical Analysis of Compositional Data. Chapman and Hall, New York. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,641,733,660]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Aitchison, J. (1994). Principles of compositional data analysis. 24:73-81. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,671,872,708]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Basse,G. and Bojinov, I. (202O). A general theory of identification. arXiv preprint arXiv:2002.06041. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,720,877,795]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Fernandes, A. D., Reid, J. N., Macklaim, J. M., McMurrough, T. A., Edgel, D.R., and Gloor，G. B. (2014). Unifying the analysis of high-throughput sequencing datasets: characterizing RNA-seq, 16S rRNA gene sequencing and selective growth experiments by compositional data analysis. Microbiome, 2(1):1-13. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"29 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[112,55,880,804]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"29\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Friedman, J. and Alm, E. J. (2012). Inferring correlation networks from genomic survey data. PLoS Computational Biology, 8(9):e1002687. \\nGustafson， P. (2O15). Bayesian inference for partially identified models: Exploring the limits of limited data, volume 140. CRC Press. \\nImbens, G. W. and Rubin, D.B. (2015). Causal inference in statistics, social, and biomedical sciences. Cambridge University Press. \\nKrebs, C. J. (1999). Ecological methodology. Benjamin/Cummings, Menlo Park, Calif, 2nd ed edition. \\nLove,M. I., Huber, W.,and Anders, S. (2014). Moderated estimation of fold change and dispersion for RNA-seq data with DESeq2. Genome Biology, 15(12):1-21. \\nLovell,D. R., Chua, X.-Y.， and McGrath, A. (2020). Counts: an outstanding challenge for log-ratio analysis of compositional data in the molecular biosciences. NAR Genomics and Bioinformatics, 2(2):lqaa040. \\nMurphy， S. M.， Cox, J. J.，Augustine, B. C.， Hast, J. T.，Guthrie, J. M.， Wright, J., McDermott, J.， Maehr, S. C.，and Plaxico, J. H. (2016). Characterizing recolonization by a reintroduced bear population using genetic spatial capture-recapture. The Journal of Wildlife Management, 80(8):1390-1407. \\nPaulson, J. N.， Stine, O. C.， Bravo, H. C.，and Pop,M. (2013). Diferential abundance analysis for microbial marker-gene surveys. Nature methods,10(12):1200-1202. \\nPawlowsky-Glahn， V.， Egozcue, J. J.，and Tolosana-Delgado， R. (2015). Modeling and analysis of compositional data. John Wiley & Sons. \\nRoche,K. E. and Mukherjee, S. (2022). The accuracy of absolute differential abundance analysis from relative count data. PLOs Computational Biology, 18(7):el010284. \\nRossi,P. H.， Wright, J. D., Fisher,G. A.，and Wilis, G. (1987). The urban homeless: estimating composition and size. Science, 235(4794):1336-1341. \\nSilverman, J. D., Bloom, R. J., Jiang, S., Durand, H. K., Dallow, E., Mukherjee, S., and David,L. A. (2021). Measuring and mitigating pcr bias in microbiota datasets. PLoS Computational Biology, 17(7):e1009113. \\nSilverman, J. D., Durand, H. K., Bloom, R. J., Mukherjee, S.，and David, L. A. (2018). Dynamic linear models guide design and analysis of microbiota studies within artificial human guts. Microbiome, 6(1):1-20. \\nSilverman， J. D.， Roche,K.， Holmes, Z. C.， David,L.A.，and Mukherjee, S. (2022). Bayesian multinomial logistic normal models through marginally latent matrix-t processes. J. Mach. Learn. Res., 23:7-1. \\nSrinivasan，A.， Xue, L.，and Zhan， X. (2021). Compositional knockoff filter for highdimensional regression analysis of microbiome data. Biometrics, 77(3):984-995. \\nTeh, Y., Newman, D.,and Welling, M. (2006). A Collapsed Variational Bayesian Inference Algorithm for Latent Dirichlet Allocation. In Advances in Neural Information Processing Systems, volume 19. MIT Press. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"29\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"30 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[116,85,879,257]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,506,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"31 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[161,154,833,183]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Supplement to “Scale Reliant Inference\\\" \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[173,215,823,260]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Michelle Pistner Nixon, Jefrey Letourneau, Lawrence A. David, Nicole A. Lazar, Sayan Mukherjee, and Justin D. Silverman \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,321,877,376]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this supplement,we present clarifying remarks and the proofs to our main results and additional claims made throughout the paper. In addition, we show additional details and results of our simulations and data analyses. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[124,404,857,428]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"AClarifying Comments on the Importance of Scale \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,441,875,513]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Here, we provide an extended discussion of the following two questions that we have received frequently in the development of this work: (1)\\\"Isn't this problem solved by [field-specific technology]?” and (2) “Why should I care about the totals anyways [in this field]? The totals may not be any more relevant than the proportional abundances.\\\". \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,516,877,733]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Before addressing either question, let us provide an example related to human gut microbiota ilustrating that there can be multiple scientifically meaningful definitions of scale. It can be context-dependent: one researcher may want $W _ { d m }$ to represent the concentration of bacterial taxon $d$ in the colon of human host $m$ , yet another may want $W _ { d m }$ to represent the number of bacterial cells of taxon $d$ in the colon of host $m$ . Either choice implies different scales, i.e., $W _ { m } ^ { \\\\perp }$ represents either the total number or total concentration of bacterial cels in the colon. Yet a third researcher may find both of these scales irrelevant noting the following two trivialities: larger people likely have more bacteria in their gut and the amount of insoluble fiber someone ate the day before (e.g.，wheat bran) can add to dry mass in the colon and therefore impact bacterial concentration. This third researcher may want to use some more esoteric notion of scale such as the total number of bacteria relative to the colonic surface area. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,734,875,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We believe this example helps to address both Questions 1 and 2. With respect to Question 1,we believe its fairly clear that whatever technology is used for measuring to-tal bacterial or total 16S rRNA concentration (e.g.， flow cytometry or qPCR） is unlikely to provide information about all possibly scientifically relevant scales (e.g., total bacterial number or total bacterial number relative to colonic surface area). Moreover, even when the question is stated in terms of a scale that can be measured (e.g., total bacterial concen-tration), the measurements may be noisy or biased (Props et al., 2017) again introducing uncertainty in the system scale. The tools we develop and our discussion can apply equally well to cases where scale measurements are relevant but sufficiently noisy to introduce uncertainty in the estimand of interest ( $\\\\theta$ ） \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[22,257,62,724]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[491,829,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,877,195]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,196,875,287]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"With respect to the second question, note that our work does not advocate for any one scale. In particular， we encourage the reader to not conclude that we are arguing that default total scale (the total number of entities in a configuration) is the scale of interest. Instead, what we develop here is theory and tools for working with any scale of scientific interest - whatever that scale might be. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,314,266,338]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"BProofs \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,352,718,371]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section, we present all of the proofs referenced in the main text. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,371,874,406]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $h = g \\\\circ f ^ { - 1 }$ denote the composite map formed by the measurement model and estimand as discussed in the main text. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,429,745,450]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B.1Proofs of Scale Reliance for Selected Estimators \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,460,856,478]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section, let $W$ denote a $\\\\boldsymbol { D }$ -dimensional scaled system over a set of M conditions. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,500,390,518]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B.1.1 Differential Analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,527,872,564]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We begin by defining differential analysis. Then, we prove that differential analysis is fully scale reliant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,575,877,632]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition B.1 (Differential Analysis). Let $x _ { m } \\\\in \\\\{ 0 , 1 \\\\}$ denote a binary covariate associated with each condition $m \\\\in \\\\mathbf { M }$ . For a given $i \\\\in \\\\{ 1 , \\\\ldots , D \\\\}$ ， differential analysis is defined as a target estimand of the form \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[315,643,679,689]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0233/966604da5d825774f5d464fa032ea05d014de3e4356ff0d825592a63dfe32dec.jpg\",\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\theta _ { i } = \\\\frac { 1 } { n _ { 1 } } \\\\sum _ { m : x _ { m } = 1 } \\\\log W _ { d m } - \\\\frac { 1 } { n _ { 0 } } \\\\sum _ { m : x _ { m } = 0 } \\\\log W _ { d m }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,699,879,736]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $n _ { 0 }$ and $n _ { 1 }$ denote the number of conditions $m \\\\in \\\\mathbf { M }$ such that $x _ { m } = 1$ or $x _ { m } = 0$ respectively. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,747,633,767]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition B.1. Differential Analysis is fully scale reliant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[491,829,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,84,879,122]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Prof.Bythelearityftemeanadusigthation $\\\\log W _ { d m } = \\\\log W _ { d m } ^ { \\\\parallel } + \\\\log W _ { m } ^ { \\\\perp }$ it follows that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[202,135,790,190]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0646/8bd0f9e433159191114237623cd386e43cdae0ba00f130e371c4295486814c5e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\theta _ { d } = \\\\underset { x _ { m } = 1 } { \\\\mathrm { m e a n } } ( \\\\mathrm { l o g } W _ { d } ^ { \\\\parallel } ) - \\\\underset { x _ { m } = 0 } { \\\\mathrm { m e a n } } ( \\\\mathrm { l o g } W _ { d } ^ { \\\\parallel } ) + \\\\underset { x _ { m } = 1 } { \\\\mathrm { m e a n } } ( \\\\mathrm { l o g } W ^ { \\\\perp } ) - \\\\underset { x _ { m } = 0 } { \\\\mathrm { m e a n } } ( \\\\mathrm { l o g } W ^ { \\\\perp } ) } \\\\\\\\ & { \\\\quad = \\\\theta _ { d } ^ { \\\\parallel } + \\\\theta ^ { \\\\perp } . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,204,877,265]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note that $\\\\theta _ { d } ^ { \\\\parallel }$ is solely a function of $\\\\boldsymbol { W } _ { d } ^ { \\\\parallel }$ and $\\\\theta ^ { \\\\perp }$ is solely a function of $W ^ { \\\\perp }$ . In this form it is clear that for any fixed value of $\\\\theta _ { d } ^ { \\\\parallel }$ , we can always select a value of $\\\\theta ^ { \\\\perp } \\\\in \\\\mathbb { R }$ such that $\\\\theta _ { i }$ can take on any value in $\\\\mathbb { R }$ (i.e., in $\\\\Theta$ ). Thus differential analysis is fully scale reliant. $\\\\sqsubset$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,285,393,304]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B.1.2 Covariation Analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,313,879,348]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition B.2 (Covariation Analysis). Define the Covariation analysis target estimand$a s$ ：\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[370,353,624,369]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0330/448a899e720ccab87d6afeef3b97f4717211f11aac34595efeac088020a3370b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\Sigma _ { d _ { 1 } , d _ { 2 } } = c o v ( \\\\log W _ { d _ { 1 } \\\\cdot } , \\\\log W _ { d _ { 2 } \\\\cdot } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,380,741,401]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition B.2. Covariation Analysis $\\\\left( \\\\Sigma _ { d _ { 1 } , d _ { 2 } } \\\\right)$ is partially scale reliant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,410,877,450]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof. For notational convenience, let $\\\\overline { W }$ be a shorthand for $\\\\log W$ . We reparameterize $\\\\Sigma _ { d _ { 1 } d _ { 2 } }$ in terms of $\\\\overline { W } ^ { \\\\| }$ and $\\\\overline { W } ^ { \\\\perp }$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[189,463,808,521]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0291/b19cf83e11247fa4915e6bd956a420714579f5079846dbc055d3047bfe75787f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\Sigma _ { d _ { 1 } d _ { 2 } } = \\\\mathrm { c o v } ( \\\\overline { W } _ { d _ { 1 } } ^ { \\\\parallel } + \\\\overline { W } ^ { \\\\perp } , \\\\overline { W } _ { d _ { 2 } } ^ { \\\\parallel } + \\\\overline { W } ^ { \\\\perp } ) } \\\\\\\\ & { \\\\qquad = \\\\mathrm { c o v } ( \\\\overline { W } _ { d _ { 1 } } ^ { \\\\parallel } , \\\\overline { W } _ { d _ { 2 } } ^ { \\\\parallel } ) + \\\\mathrm { c o v } ( \\\\overline { W } _ { d _ { 1 } } ^ { \\\\parallel } , \\\\overline { W } ^ { \\\\perp } ) + \\\\mathrm { c o v } ( \\\\overline { W } ^ { \\\\perp } , \\\\overline { W } _ { d _ { 2 } } ^ { \\\\parallel } ) + \\\\mathrm { c o v } ( \\\\overline { W } ^ { \\\\perp } , \\\\overline { W } ^ { \\\\perp } ) } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,531,877,570]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We denote the standard deviation of a vector $x$ as $\\\\sigma _ { x } = { \\\\sqrt { \\\\operatorname { v a r } ( x ) } }$ and the covariation of two vectors $x$ and $y$ as $\\\\rho _ { x y } = c o r r ( x , y )$ . We can then write: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[160,583,816,643]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0733/a2dfefaa1ba24578af91d0ef04c826bf85f9c9ae46f5a492d31b7a39ae79209f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { { _ { 1 } d _ { 2 } } = \\\\sigma _ { \\\\overline { { W } } _ { d _ { 1 } } ^ { \\\\parallel } } \\\\sigma _ { \\\\overline { { W } } _ { d _ { 2 } } ^ { \\\\parallel } } \\\\rho ( \\\\overline { { W } } _ { d _ { 1 } } ^ { \\\\parallel } , \\\\overline { { W } } _ { d _ { 2 } } ^ { \\\\parallel } ) + \\\\sigma _ { \\\\overline { { W } } _ { d _ { 1 } } ^ { \\\\parallel } } \\\\sigma _ { \\\\overline { { W } } ^ { \\\\perp } } \\\\rho ( \\\\overline { { W } } _ { d _ { 1 } } ^ { \\\\parallel } , \\\\overline { { W } } ^ { \\\\perp } ) + \\\\sigma _ { \\\\overline { { W } } _ { d _ { 2 } } ^ { \\\\parallel } } \\\\sigma _ { \\\\overline { { W } } ^ { \\\\perp } } \\\\rho ( \\\\overline { { W } } _ { d _ { 2 } } ^ { \\\\parallel } , \\\\overline { { W } } ^ { \\\\perp } ) + } \\\\\\\\ & { \\\\quad = \\\\alpha + \\\\sigma _ { \\\\overline { { W } } ^ { \\\\perp } } \\\\beta + \\\\sigma _ { \\\\overline { { W } } ^ { \\\\perp } } \\\\gamma + \\\\sigma _ { \\\\overline { { W } } ^ { \\\\perp } } ^ { 2 } . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,654,877,715]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For a fixed value of $\\\\alpha$ , the minimum value that $\\\\Sigma _ { d _ { 1 } d _ { 2 } }$ can attain is $\\\\alpha - ( \\\\beta + \\\\gamma ) ^ { 2 } / 4$ which occurs when o2 $\\\\sigma _ { \\\\overline { { W } } ^ { \\\\perp } } ^ { 2 } = ( \\\\beta + \\\\gamma ) / 2$ .If $c < 0$ then the minimum value is $\\\\alpha - c$ which occurs at $\\\\sigma _ { W ^ { \\\\bot } } ^ { 2 } = 1$ . Covariation analysis is therefore partially scale reliant. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,734,467,756]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B.2Proofs of Scale Reliance \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,765,875,803]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section， we provide proofs related to the scale reliance of variaous compositional survey and target estimand pairings. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[490,828,504,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,131,890,284]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0443/0cddb78fa98e8fa9a476296cdd7b522b8ff7f7e052b9803fa2ed0db1ef7e90ac.jpg\",\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td></td><td>Proposition Compositional Survey (V)</td><td>Target Estimand (0)</td><td>Inference of θ from observations of Y</td></tr><tr><td>D.3</td><td>Full</td><td>Scale Invariant</td><td>Scale Invariant</td></tr><tr><td>D.6</td><td>Full</td><td>Partially Scale ReliantPartially Scale Reliant</td><td></td></tr><tr><td>D.5</td><td>Full</td><td>Fully Scale Reliant</td><td>Fully Scale Reliant</td></tr><tr><td>D.3</td><td>Partial</td><td>Scale Invariant</td><td>Scale Invariant</td></tr><tr><td></td><td>Partial</td><td>Partially Scale Reliant</td><td></td></tr><tr><td></td><td>Partial</td><td>Fully Scale Reliant</td><td></td></tr></table>\",\n \"table_caption\": \"[\\\"Table 1: Inference of $\\\\\\\\theta$ depending on the type of the data (partial or full compositional survey) and target estimand (scale invariant or scale reliant). \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[119,303,879,358]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition B.3. If $\\\\mathcal { V }$ is a full or partial compositional survey of a scaled system $W$ and $\\\\theta = f ( W )$ a scale invariant target estimand, then $h$ is invertible at $P$ (i.e., scale invariant inference). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,369,877,444]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof. By definition, if $\\\\mathcal { V }$ is either a partial or full compositional survey then there is a single value of $W ^ { \\\\parallel }$ in the pre-image $g ^ { - 1 } ( P )$ . Since the target estimand is scale invariant, then it is uniquely defined by $W ^ { \\\\parallel }$ . It follows that there exists a single element $\\\\theta \\\\in \\\\Theta$ that is in the pre-image $h ^ { - 1 } ( P )$ . Hence $h$ is invertible at $P$ □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,455,877,511]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition B.4. If $\\\\mathcal { V }$ is a full compositional survey of a scaled system $W$ and $\\\\theta = f ( W )$ is a scale reliant target estimand at $W ^ { \\\\parallel }$ ，then $h$ is uninvertible at $P$ (i.e.， scale reliant inference). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,523,877,559]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof. Since $\\\\theta$ is scale reliant,it follows that there must be at least two $W _ { 1 } ^ { \\\\perp } , W _ { 2 } ^ { \\\\perp } \\\\in { \\\\bf W } ^ { \\\\perp }$ such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[428,571,568,643]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0120/46914c8ead8fdd8c240070156809aadf62b7ccc4098f51da14a59365e21a0c71.jpg\",\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { f ( W ^ { \\\\parallel } , W _ { 1 } ^ { \\\\perp } ) = \\\\theta _ { 1 } } \\\\\\\\ & { f ( W ^ { \\\\parallel } , W _ { 2 } ^ { \\\\perp } ) = \\\\theta _ { 2 } } \\\\\\\\ & { \\\\theta _ { 1 } \\\\neq \\\\theta _ { 2 } . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,654,877,709]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since $\\\\mathcal { V }$ is a full compositional survey then both $W _ { 1 } ^ { \\\\perp }$ and $W _ { 2 } ^ { \\\\perp }$ must be in the pre-image $g ^ { - 1 } ( P )$ . It follows that both values $\\\\theta _ { 1 }$ and $\\\\theta _ { 2 }$ must be in the pre-image $h ^ { - 1 } ( P )$ . Thus, $h$ is uninvertible. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,722,877,777]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition B.5. If Y is a full compositional survey of a scaled system $W$ and $\\\\theta = f ( W )$ is a fully scale reliant target estimand at $W ^ { \\\\parallel }$ ，then $h$ is fully uninvertible at $P$ (i.e.,， fully scale reliant inference). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[491,829,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,875,140]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof. Building on PropositionB.4, we know that $h$ is uninvertible. It suffices to show that with the additional constraint that $\\\\theta$ is fully scale reliant that all possible values of $\\\\theta$ must be in the pre-image $h ^ { - 1 } ( P )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,142,875,231]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Building on PropositionB.4,and using the fact that $\\\\theta$ is fully scale reliant at $W ^ { \\\\parallel }$ it follows that for any value of $\\\\theta \\\\in \\\\Theta$ ，there must exist some $W ^ { \\\\perp } \\\\in \\\\mathbf { W } ^ { \\\\perp }$ such that $f ( W ^ { \\\\parallel } , W ^ { \\\\perp } ) = \\\\theta$ . Again, since $\\\\mathcal { V }$ is a full compositional survey it follows that every such pair of $W ^ { \\\\parallel }$ and $W ^ { \\\\perp }$ must be in the pre-image $g ^ { - 1 } ( P )$ . Thus $h ^ { - 1 } ( P ( \\\\mathcal { V } ) ) = \\\\Theta$ which is the definition of fully uninvertible. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,243,877,299]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition B.6. If Y is a full compositional survey of a scaled system $W$ and $\\\\theta = f ( W )$ is a partially scale reliant target estimand, then $h$ is partially uninvertible at $P$ (i.e.,， partially scale reliant inference). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,310,875,366]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof. Building on Proposition B.4, we know that $h$ is uninvertible. It suffices to show that with the additional constraint that $\\\\theta$ is partially scale reliant that there must exist some $\\\\theta \\\\in \\\\Theta$ that is not in the pre-image $h ^ { - 1 } ( P ( \\\\mathcal { V } ) )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,367,875,455]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Building on Proposition B.4, and using the fact that $\\\\theta$ is partially scale reliant it follows that at $W ^ { \\\\parallel }$ ， there must be some value $\\\\theta \\\\in \\\\Theta$ such that there does not exist a value of $W ^ { \\\\perp } \\\\in { \\\\bf W } ^ { \\\\perp }$ that satisfies $\\\\theta = f ( W ^ { \\\\parallel } , W ^ { \\\\perp } )$ . It therefore follows that this value of $\\\\theta$ cannot be in the pre-image $h ^ { - 1 } ( P ( \\\\mathcal { V } ) )$ and therefore that $h ^ { - 1 } ( P ( \\\\mathcal { V } ) ) \\\\subsetneq \\\\Theta$ . Therefore $h$ is partially uninvertible. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,468,877,559]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally, we do not provide any formal statements about the situation of a partially compositional survey and a partially or fully scale reliant target estimator. We simply note that these situations are substantially more complicated and that the composite map $h$ can be invertible, partially uninvertible, or fully uninvertible depending on the specific target estimand $\\\\theta = f ( W )$ and the specific measurement model $P$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,582,823,603]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B.3Proofs Related to the Limits of Scale Reliant Inference \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,612,875,685]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We will use the following declarations throughout this section. Let $x$ denote a random variable with distribution $P \\\\in \\\\textbf { P }$ Let $\\\\phi \\\\in \\\\Phi$ denote a quantity related to a random variable through a map $h : \\\\Phi { \\\\bf P }$ .Let $\\\\phi ^ { \\\\dagger }$ denote the true value of $\\\\phi$ .Finally, let $\\\\{ \\\\hat { \\\\phi } _ { n } \\\\}$ denote a sequence of estimators. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,686,874,739]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We use the following axiom which serves to exclude trivial estimators defined tautologically with respect to the true value $\\\\phi ^ { \\\\dagger }$ , e.g., to exclude the potential for an omniscient analyst. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,751,875,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition B.3 (Axiom of Learning). Our ability to learn the value of $\\\\phi \\\\in \\\\Phi$ is limited to the pre-image of h such that, were we to know $P$ and know $h$ ， then we must not be able to decide which of the values of $\\\\phi \\\\in h ^ { - 1 } ( P )$ is the true value. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[491,829,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,877,195]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We believe that this axiom reflects a common assumption that is often unstated in statistical theory. Letting $h$ denote a family of probability models such that $\\\\phi$ represents a parameter indexing the family, this axiom implies that our ability to learn about $\\\\phi$ is limited to the identification region of the model family at $P$ . Practically, this also implies that we cannot chose an estimator $\\\\hat { \\\\phi }$ that magically converges to the correct value $\\\\phi ^ { \\\\dagger }$ when $h$ is uninvertible. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,217,308,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B.3.1 Consistency \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,243,748,262]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our definition of consistency is standard, still we state it for completeness. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,270,877,309]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition B.4 (Consistent Estimator). $\\\\{ \\\\hat { \\\\phi } _ { n } \\\\}$ is a consistent sequence of estimators for $\\\\phi$ if and only if, for all $\\\\epsilon > 0$ ， \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[385,320,607,347]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0154/021b765316dcb078aa9e544c840ab71f1b17a595777c546df890ca6b015b2aea.jpg\",\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { l i m } _ { n \\\\infty } P ( | | \\\\hat { \\\\phi } _ { n } - \\\\phi | | < \\\\epsilon ) = 1 .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,354,877,429]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Next,we prove that no estimator can be consistent if $h$ is not invertible. This result and proof represent an extension of the proof by Gabrielsen (1978) that identifiability is a necessary condition for consistency to the more general notion of invertibility rather than identifiability. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,436,877,474]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem B.1. If $h$ is uninvertible at $P$ then $\\\\left\\\\{ { \\\\hat { \\\\phi } } _ { n } \\\\right\\\\}$ cannot be a consistent sequence of estimators. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,484,877,559]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof. By contradiction， suppose that $h$ is uninvertible at $P$ then $h ^ { - 1 } ( P )$ must contain at least two distinct elements $\\\\phi _ { 1 } , \\\\phi _ { 2 } \\\\in h ^ { - 1 } ( P )$ . By the axiom of learning, the sequence of estimators cannot converge to $\\\\phi ^ { \\\\dagger }$ alone.In order to be consistent, it must therefore converge in probability to both $\\\\phi _ { 1 }$ and $\\\\phi _ { 2 }$ which is a contradiction. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,568,875,607]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary B.1. Let $\\\\theta$ be a Target estimand in scale reliant inference. There does not exist a consistent sequence of estimators $\\\\{ \\\\hat { \\\\phi } _ { n } \\\\}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,614,875,651]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof. The proof follows immediately from the definition of scale reliant inference in terms of uninvertability and Proposition B.4. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,671,303,690]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B.3.2Calibration \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,699,880,736]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition B.5 (Calibration). A set estimator $S \\\\in \\\\Phi$ is a confidence set calibrated at level $\\\\beta$ i \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[397,738,598,765]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0667/e5243bc85b1fddea1ea8b5176ce3626e5b2dc58374e078c7f4671ada51b2a2d3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { i n f } _ { \\\\phi \\\\in \\\\Phi , P \\\\in h ( \\\\phi ) } P ( \\\\phi \\\\in S ) \\\\ge \\\\beta\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,770,875,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and if the following two additional conditions hold: (1) $\\\\beta \\\\geq 0$ and (2) for any $\\\\phi \\\\in \\\\Phi$ there exist for each $P \\\\in h ( \\\\phi )$ ， some other $\\\\phi ^ { * } \\\\in \\\\Phi$ such $P ( \\\\phi ^ { * } \\\\notin S ) = 1$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[490,829,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,877,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note that our definition is nearly identical to the standard definition but includes the additional term $P \\\\in h ( \\\\phi )$ in the infimum to address situations in $h ( \\\\phi )$ is not a singleton set. Beyond this modification,our definition emphasizes two additional conditions that are often assumed but not stated. The first conditions, $\\\\beta > 0$ , is included to avoid trivially calling a set estimator calibrated if $\\\\beta = 0$ . The second conditions is added to avoid trivial set estimators that may be calibrated at level $\\\\beta > 0$ by returning the entire set $\\\\Phi$ . Essentially, this second conditions simply states that the set estimator must exclude at least one value of $\\\\phi ^ { * } \\\\in \\\\Phi$ from consideration in order to be called a calibrated confidence set. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,242,870,279]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem B.2.If $h$ is fully uninvertible for some $P \\\\in \\\\textbf { P }$ ， then there does not exist d calibrated confidence set for $\\\\phi$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,290,877,435]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof. By contradiction, suppose that a set estimator $S$ is a calibrated confidence set for $\\\\phi$ and that $h$ is fully uninvertible for some $P \\\\in \\\\mathbf { P }$ . Since $S$ is calibrated, there must then be some $\\\\phi \\\\in h ^ { - 1 } ( P )$ and some other $\\\\phi ^ { * } \\\\in \\\\Phi$ such that, $P ( \\\\phi ^ { * } \\\\notin S ) = 1$ .However，by definition, since $h$ is fully uninvertible at $P$ every value of $\\\\phi \\\\in \\\\Phi$ , including $\\\\phi ^ { * }$ , is in the set $h ^ { - 1 } ( P )$ .Therefore, if $\\\\phi ^ { * }$ is not in $S$ then $P ( \\\\phi ^ { * } \\\\in S ) = 0$ and the infimum becomes zero requiring $\\\\beta = 0$ . However, since we require that $\\\\beta > 0$ for $S$ to be calibrated, and by the axiom of learning, our supposition that $S$ is a calibrated confidence set implies that $\\\\phi ^ { * }$ must be in the set $S$ which is a contradiction. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,446,875,483]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary B.2. Let $\\\\theta$ denote a target estimator in fully scale reliant inference. then there does not exist a calibrated confidence set for $\\\\theta$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,493,872,531]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof. The proof follws immediately from the definition of fully scale reliant inference in terms of full uninvertability and Proposition B.5. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,551,369,569]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B.3.3 Bias and Variance \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,579,779,598]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition B.6 (Bias and Variance). We define the bias of an estimator $\\\\hat { \\\\phi }$ as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[400,609,593,630]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0238/76645de9b92939bdc82e40b72b767574ef7e50809070167afcd9c0e0d998f61c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathrm { b i a s } _ { P } ( \\\\hat { \\\\phi } ) = \\\\mathbb { E } _ { P } [ \\\\hat { \\\\phi } ] - \\\\phi ^ { \\\\dagger } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[122,642,477,660]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We define the variance of an estimator as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[379,672,614,693]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0409/570bf949d07db65d4f27635ae77bbd2074d816ac9c0ddbdff21fce5192864b04.jpg\",\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname { v a r } _ { P } ( \\\\hat { \\\\phi } ) = \\\\mathbb { E } _ { P } [ ( \\\\hat { \\\\phi } - \\\\mathbb { E } _ { P } [ \\\\hat { \\\\phi } ] ) ^ { 2 } ] .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,704,758,723]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem B.3. If $h$ is uninvertible at $P$ ， then any estimator of $\\\\phi$ is biased. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,732,877,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof. For any probability model $P$ ， the expectation $\\\\mathbb { E } _ { P } [ \\\\hat { \\\\phi } ]$ must be a singleton set. If $h$ is uninvertible then $h ^ { - 1 } ( P )$ must contain at least two distinct elements. By the axiom of Learning, the estimator $\\\\hat { \\\\phi }$ cannot select the correct element of the set $h ^ { - 1 } ( P )$ and must therefore be biased. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[491,828,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,877,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary B.3. Let $\\\\theta$ denote a target estimand in scale reliant inference. Let $\\\\hat { \\\\theta }$ denote an estimator for $\\\\theta$ . Then $b i a s _ { P _ { W } } ( \\\\hat { \\\\theta } ) > 0$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,135,874,171]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof. The proof follws immediately from Proposition B.4 and the definition of scale reliance in terms of uninvertability. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,196,671,223]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CScale Simulation Random Variables \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,237,877,455]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section, the notation introduced in Section 4 of the manuscript applies. Briefly, this involves switching from using notation such as $P _ { W } ( \\\\mathcal { V } )$ to conditional probability notation $P ( \\\\mathcal { P } | W )$ which is more common in the Bayesian literature. This also involves specifying models with respect to probability densities or probability mass functions which we will denote with a lowercase $p$ as opposed to an upper case $P$ (which we reserve for probability measures). Finally, we will denote with a $( a )$ (e.g., $p ^ { ( a ) }$ or $P ^ { ( a ) }$ ）probability models that are specified by an analyst (applied models) and contrast this to the true distribution of a random quantity which we denote with a dagger (e.g., $p ^ { \\\\dagger }$ ，or $P ^ { \\\\dagger }$ ). In particular we call $P ^ { \\\\dagger } ( \\\\mathcal { V } | W )$ the target measurement model which represents the true relationship between the distribution of $\\\\mathcal { V }$ and the value of $W$ . When we wish to explicitly refer to the true distribution of $\\\\mathcal { V }$ , i.e.,the target measurement model at the true value of $W ^ { \\\\dagger }$ ，we will explicitly write $P ^ { \\\\dagger } ( \\\\mathcal { V } | W ^ { \\\\dagger } )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,457,875,511]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As suggested in the manuscript, the name scale simulation random variables highlights that this form is readily amenable to evaluation by Monte-Carlo Simulation (See Algorithm 1). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,513,872,683]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/1017/c795c9db184d8e134f881a56fcb76b9ffecacbc305671116d8f1b62d3c5467b7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Algorithm 1: Monte-Carlo Algorithm for Simulating Scale Simulation Random Variables</td></tr><tr><td>Data: Y</td></tr><tr><td>Result: S samples of 0</td></tr><tr><td>for s in {1,..., S} do</td></tr><tr><td>Sample W(s) ~ p(@)(Y)</td></tr><tr><td>Sample W⊥(s) ~ p(@)(W(s))</td></tr><tr><td>Calculate 0(s) = f(W(s),W⊥(s))</td></tr><tr><td>end</td></tr></table>\",\n \"table_caption\": \"[]\",\n \"table_footnote\": \"[\\\"This algorithm can also be adapted for use with Scale Conditional Random Variables (SCRVs） with density $p ^ { ( a ) } ( \\\\\\\\theta ~ \\\\\\\\vert ~ Y , W ^ { \\\\\\\\perp } )$ in combination with a scale model of the form \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[490,829,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"8 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,108,867,257]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0501/1c01eb5389206a87e206b52be4300818106f065328cbb5ed247d582758499622.jpg\",\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Algorithm 2: Monte-Carlo Algorithm for Simulating Scale Simulation Random Variables from an SCRV</td></tr><tr><td>Data: Y</td></tr><tr><td>Result: S samples of 0</td></tr><tr><td>for s in {1,..., S} do</td></tr><tr><td>Sample W⊥(s) ~ p(@)(·)</td></tr><tr><td>Sample 0(s) ~ p(Y,W⊥(s))</td></tr><tr><td>end</td></tr></table>\",\n \"table_caption\": \"[]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[117,260,877,352]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We will often assume the following regularity conditions on SSRVs. This regularity conditions are introduced so that we will be able to apply the dominated convergence theorem in proofs of the bias and variance of SSRVs. In essence, these regularity conditions serve to formally exclude from consideration pathological pairings of target estimands and applied and target models that we expect would be unlikely to be encountered in practice. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,361,879,455]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition C.1 (Regularity Conditions on Scale Simulation Estimators). Let $\\\\mathcal { V }$ denote a compositional survey of a scaled system $\\\\mathcal { W }$ with target measurement model $P ^ { \\\\dagger } ( \\\\mathcal { V } | W ^ { \\\\dagger } )$ . Let $\\\\mathcal { V } _ { n }$ denote a sequence of random variables that converges in probability to $\\\\mathcal { V }$ Let $\\\\theta ( \\\\mathcal { V } _ { n } )$ denote a scale simulation random variable. Let $P ( \\\\mathcal { V } , \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } )$ denote the joint probability distribution defined by: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[245,469,748,488]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0325/5d30f054b6c715e7b6d9607e6ca1c6663c4acb22d428fd4a070471ea893c2162.jpg\",\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nP ( \\\\mathcal { V } , \\\\mathcal { W } ^ { \\\\| } , \\\\mathcal { W } ^ { \\\\bot } ) = P ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\bot } | \\\\mathcal { W } ^ { \\\\| } ) P ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\| } | \\\\mathcal { V } ) P ^ { \\\\dagger } ( \\\\mathcal { V } | \\\\mathcal { W } ^ { \\\\| \\\\dagger } , \\\\mathcal { W } ^ { \\\\bot \\\\dagger } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,501,877,556]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $P ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\parallel } ) P ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\parallel } | \\\\mathcal { V } )$ denote the applied measurement and scale model used in the scale simulation estimand. We assume the following three regularity conditions on scale simulation estimators. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[137,558,874,595]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. That each element of the sequence $\\\\{ \\\\theta ( \\\\mathcal { V } _ { n } ) \\\\}$ is integrable with respect to the joint distribution $P ( \\\\mathcal { V } _ { n } , \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[140,611,803,632]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. Under this joint, that the sequence $\\\\{ \\\\theta ( \\\\mathcal { V } _ { n } ) \\\\}$ converges in probability to $\\\\theta ( \\\\mathcal { V } )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[142,647,879,686]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3. That there exists a random variable $Z$ that is integrable with respect to the this joint which satisfies: $| \\\\theta ( \\\\mathcal { V } _ { n } ) | \\\\leq Z$ for all $n$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,696,875,751]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note that it follows from the dominated convergence theorem (See Theorem 1.1 in Shao (2003)） that under these regularity conditions we may interchange limits and integration, e.g., \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[232,750,761,787]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0247/4416c0886a01be91cbae97a8058cebc3f8898e43e29c2f6b9399b251835be5eb.jpg\",\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\mathbb { E } _ { P ( \\\\mathcal { V } , \\\\mathcal { W } ^ { \\\\| } , \\\\mathcal { W } ^ { \\\\bot } ) } [ \\\\theta ( \\\\mathcal { V } _ { n } ) ] = \\\\int \\\\operatorname* { l i m } _ { n \\\\to \\\\infty } f ( \\\\mathcal { W } ^ { \\\\| } , \\\\mathcal { W } ^ { \\\\bot } ) d P ( \\\\mathcal { V } _ { n } , \\\\mathcal { W } ^ { \\\\| } , \\\\mathcal { W } ^ { \\\\bot } ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[490,829,503,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"9 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[117,87,555,104]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$p ^ { ( a ) } ( W ^ { \\\\perp } , \\\\cdot )$ where · denotes known hyperparameters \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,82,754,106]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C.1 Relationship between SSRVs and Bayesian PIMs \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,114,877,151]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Scale Simulation Random Variables, can alternatively be thought of as a transformation of the posterior distribution of a Bayesian Partially Identified Model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,159,877,217]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition C.1. Let $p ( \\\\mathcal { W } ^ { \\\\perp } , \\\\mathcal { W } ^ { \\\\parallel } | Y )$ denote the posterior of a Bayesian Partially Identified model with likelihood $p ( Y | \\\\mathcal { W } ^ { \\\\| } )$ and prior $p ( \\\\mathcal { W } ^ { \\\\perp } , \\\\mathcal { W } ^ { \\\\parallel } )$ Let $\\\\theta = f ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } )$ denote a target estimand and estimator pairing. The posterior of $\\\\theta$ is a scale simulation random variable. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,226,444,244]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof. Let us first rewrite the prior as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[361,257,632,276]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0855/13163306161807a6ec13729328e249e4a4659e02516e57b2b8307b10b27388a3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\np ( \\\\mathcal { W } ^ { \\\\perp } , \\\\mathcal { W } ^ { \\\\| } ) = p ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\| } ) p ( \\\\mathcal { W } ^ { \\\\| } ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,287,852,308]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $\\\\mathbf { A } \\\\subset \\\\Theta$ . The posterior under the transform $\\\\theta = f ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } )$ can then be written as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[274,319,720,359]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0424/ed96c962d1d49a9cf8534269e5f636ffe4a3e66e351876475f09f461e00969bf.jpg\",\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nP ( \\\\theta \\\\in \\\\mathbf { A } | Y ) = \\\\int _ { \\\\mathcal { W } ^ { \\\\perp } , \\\\mathcal { W } \\\\parallel \\\\in f ^ { - 1 } ( \\\\mathbf { A } ) } p ( \\\\mathcal { W } ^ { \\\\perp } , \\\\mathcal { W } ^ { \\\\parallel } | Y ) d \\\\mathcal { W } ^ { \\\\parallel } d \\\\mathcal { W } ^ { \\\\perp }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,369,872,409]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since in the likelihood, $Y$ is conditionally independent of $\\\\mathcal { W } ^ { \\\\perp }$ the posterior $p ( \\\\mathcal { W } ^ { \\\\perp } , \\\\mathcal { W } ^ { \\\\parallel } | Y )$ can be factored as $p ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\parallel } ) p ( \\\\mathcal { W } ^ { \\\\parallel } | Y )$ leading to \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[246,417,748,458]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0132/71c2dcf778183cad1e2ea08c4ffefe3d0aeb7d4018d617c3c5e97289b021abcc.jpg\",\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nP ( \\\\theta \\\\in \\\\mathbf { A } | Y ) = \\\\int _ { \\\\mathcal { W } ^ { \\\\perp } , \\\\mathcal { W } \\\\parallel \\\\in f ^ { - 1 } ( \\\\mathbf { A } ) } p ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\parallel } ) p ( \\\\mathcal { W } ^ { \\\\parallel } | Y ) d \\\\mathcal { W } ^ { \\\\parallel } d \\\\mathcal { W } ^ { \\\\perp } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,507,872,551]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C.2Asymptotic Properties of Bayesian Scale Simulation Random Variables \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,560,877,671]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Like any model, there are numerous ways in which scale simulation estimators can be misspecified. Still, our goal here is to focus on issues related to the lack of information about the system scale provided by compositional surveys. The definition of Faithful SSRVs serves to focus our attention on that goal and to deemphasizes issues with out ability to learn $W ^ { \\\\parallel }$ ， a topic that is both studied elsewhere and largely domain specific (Silverman et al.,2021, 2022; Grantham et al., 2020). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,680,879,773]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition C.2 (Faithful Scale Simulation Random Variables). Let $\\\\theta$ be a scale simulation random variables related to a target estimand $\\\\theta = f ( \\\\mathcal { W } )$ . We say that $\\\\theta$ is a faithful scale simulation random variable if the measurement model converges to the true value of the composition of the scaled system. That is, if for a sequence of random variables $\\\\{ \\\\mathcal { { D } } _ { n } \\\\}$ that converge in probability to $\\\\mathcal { V }$ that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[392,785,599,811]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0176/86ee9b6e6597916430e9dda489b659f2fc8bf3965f5debfd1165e4138ffb24b8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { l i m } _ { n \\\\to \\\\infty } p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\parallel } | \\\\mathcal { V } _ { n } ) \\\\to \\\\delta _ { W ^ { \\\\parallel } } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[488,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,468,877,484]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,84,877,141]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\delta _ { W ^ { \\\\parallel \\\\dagger } }$ denotes a point mass centered on the true value of $\\\\mathcal { W } ^ { \\\\parallel }$ . Alternatively for a sequence of observations $\\\\left\\\\{ Y _ { n } \\\\right\\\\}$ of a random variable $\\\\mathcal { V }$ ， a faithful scale simulation random variable is one that satisfies: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[362,155,629,183]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0657/9c36650e6c0a71d433a176f3ea559f84ada50a59d9c3dc8e5b5e5e6a8a84cbfb.jpg\",\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { l i m } _ { n \\\\to \\\\infty } p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\parallel } | Y _ { 1 } , \\\\ldots , Y _ { n } ) \\\\to \\\\delta _ { W ^ { \\\\parallel } }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,193,877,289]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In what follows, we provide results about the asymptotic expectation and variance of scale simulation random variables. For notational convenience, we will denote the expecta-tion of a scale simulation random variable as $\\\\overline { { { \\\\theta } } } = \\\\mathbb { E } _ { P ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } | Y ) } [ f ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } ) ]$ . In addition, let us use the following short-hand notations: $p ^ { \\\\dagger } \\\\equiv p ^ { \\\\dagger } ( \\\\mathcal { V } | W ^ { \\\\parallel \\\\dagger } , W ^ { \\\\perp \\\\dagger } )$ ， $\\\\theta ^ { \\\\dagger } \\\\equiv f ( W ^ { \\\\parallel \\\\dagger } , W ^ { \\\\perp \\\\dagger } )$ ， and $p ^ { ( a ) } \\\\equiv p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { V } ) = p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\parallel } ) p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\parallel } | \\\\mathcal { V } )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,297,877,354]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem C.1. Let $\\\\theta \\\\ : = \\\\ : f ( \\\\mathcal { W } )$ denote a target estimand and estimator pairing. Let $\\\\theta$ denote a faithful scale simulation estimator that satisfies the regularity conditions provided in Definition C.1. Then the asymptotic bias of $\\\\bar { \\\\theta }$ is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[300,367,699,395]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0739/952a435c8e82d1e5c782d4736c3c2c1d8ad250a02724c8b1473e3ddfc9a3ab73.jpg\",\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\mathrm { b i a s } _ { p ^ { \\\\dag } } ( \\\\overline { { \\\\theta } } ) = \\\\mathbb { E } _ { p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\perp } | W ^ { | \\\\mathfrak { t } } ) } [ f ( W ^ { | \\\\mathfrak { t } | } , \\\\mathcal { W } ^ { \\\\perp } ) ] - \\\\theta ^ { \\\\dag }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,406,490,425]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof. By the definition of asymptotic bias: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[359,439,633,465]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0235/ed4b7f8c8b1e51e23c0aa7584719dc463378535d3f3725f88f7f882ac5ef6d15.jpg\",\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\operatorname { b i a s } _ { p ^ { \\\\dag } } ( \\\\overline { { \\\\theta } } ) = \\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\mathbb { E } _ { p ^ { \\\\dag } } ( \\\\overline { { \\\\theta } } ) - \\\\theta ^ { \\\\dag } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,477,874,513]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By the dominated convergence theorem in combination with our regularity conditions, we have that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[181,523,839,561]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0942/7c5e4a77919e5d840d443fd5256ae9fb4ef777034b8c7833d444a35752046b7d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n_ { p ^ { \\\\prime } } ( \\\\overline { { { \\\\theta } } } ) = \\\\int \\\\operatorname* { l i m } _ { n \\\\infty } f ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } ) p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\parallel } ) p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\parallel } | \\\\mathcal { V } _ { n } ) p ^ { \\\\dagger } ( \\\\mathcal { V } _ { n } | \\\\mathcal { W } ^ { \\\\parallel \\\\dagger } , \\\\mathcal { W } ^ { \\\\perp \\\\dagger } ) d \\\\mathcal { W } ^ { \\\\parallel } d \\\\mathcal { W } ^ { \\\\perp } d \\\\mathcal { W } ^ { \\\\perp } d \\\\mathcal { W } ^ { \\\\perp } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,571,633,590]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since $\\\\theta$ is a faithful scale simulation estimator it follows that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[184,599,812,703]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0229/c736cb61fe551a7cf3a6abade25ee17d0f01f5e300528ab08501e75e5631e192.jpg\",\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { l } { \\\\displaystyle \\\\operatorname* { l i m } _ { n \\\\to \\\\infty } E _ { p ^ { \\\\dagger } } ( \\\\overline { { \\\\theta } } ) = \\\\int f ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } ) p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\parallel } ) \\\\delta _ { W ^ { \\\\parallel \\\\dagger } } p ^ { \\\\dagger } ( \\\\mathcal { V } | \\\\mathcal { W } ^ { \\\\parallel \\\\dagger } , \\\\mathcal { W } ^ { \\\\perp \\\\dagger } ) d \\\\mathcal { W } ^ { \\\\parallel } d \\\\mathcal { W } ^ { \\\\perp } d \\\\mathcal { Y } } \\\\\\\\ { \\\\displaystyle \\\\quad \\\\quad = \\\\int f ( \\\\mathcal { W } ^ { \\\\parallel \\\\dagger } , \\\\mathcal { W } ^ { \\\\perp } ) p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\parallel \\\\dagger } ) d \\\\mathcal { W } ^ { \\\\perp } \\\\int p ^ { ( a ) } ( \\\\mathcal { V } | \\\\mathcal { W } ^ { \\\\parallel \\\\dagger } , \\\\mathcal { W } ^ { \\\\perp \\\\dagger } ) d \\\\mathcal { V } } \\\\\\\\ { \\\\displaystyle \\\\quad = \\\\mathbb { E } _ { p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\parallel \\\\dagger } ) } \\\\left[ f ( \\\\mathcal { W } ^ { \\\\parallel \\\\dagger } , \\\\mathcal { W } ^ { \\\\perp } ) \\\\right] . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,713,241,731]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It follows that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[292,729,700,757]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0122/00b052c7c4dbe300730c528e77e14d62b91e888ebb69ac3cf0d9c2391ca1134f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\operatorname { b i a s } _ { p ^ { \\\\dag } } ( \\\\overline { { \\\\theta } } ) = \\\\mathbb { E } _ { p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\bot } | W ^ { | \\\\dag } ) } \\\\left[ f ( W ^ { | \\\\dag } , \\\\mathcal { W } ^ { \\\\bot } ) \\\\right] - \\\\theta ^ { \\\\dag }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,762,877,777]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"□ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,877,214]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The above result should come as no surprise given Theorem B.3, in the setting of scale reliant inference (i.e., when $h$ is not invertible), the mean of SSRVs are a biased estimator. Still, there are at least two remarkable facts that tie into this result.First，we will show that the bias is identical when we shift to a Bayesian perspective and calculate bias with respect to the posterior $\\\\operatorname* { l i m } _ { n \\\\to \\\\infty }$ ${ \\\\mathrm { b 1 a s } } _ { p ^ { ( a ) } ( { \\\\mathcal { W } } \\\\parallel , { \\\\mathcal { W } } ^ { \\\\perp } | Y _ { n } ) }$ . Second, we will show that the asymptotic variance of the SSRV actually includes a model for this bias and remains non-zero even in the asymptotic limit. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,226,877,281]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem C.2. Let $\\\\theta \\\\ : = \\\\ : f ( \\\\mathcal { W } )$ denote a target estimand and estimator pairing. Let $\\\\theta$ denote a faithful scale simulation estimator that satisfies the regularity conditions provided in Definition C.1. Then the asymptotic bias of $\\\\bar { \\\\theta }$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[294,295,700,320]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0997/ac3f4ea5a59193dfc9aedfad8e53ee979c308c2498b4a9e49b4f5c33f7b8cec1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\mathrm { b i a s } _ { p ^ { ( a ) } } ( \\\\overline { { \\\\theta } } ) = \\\\mathbb { E } _ { p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\bot } | W ^ { | | \\\\dagger } ) } [ f ( W ^ { | | \\\\dagger } , \\\\mathcal { W } ^ { \\\\bot } ) ] - \\\\theta ^ { \\\\dagger }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,332,392,351]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof. By the definition of bias, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[184,361,810,433]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0536/634eef011cf17fa6f659983416df9a3d3e9155cf2e8b9cdd32c3753d5ba77589.jpg\",\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\underset { n \\\\infty } { \\\\operatorname* { l i m } } \\\\mathrm { b i a s } _ { p ^ { ( a ) } } ( \\\\overline { { \\\\theta } } ) = \\\\underset { n \\\\infty } { \\\\operatorname* { l i m } } \\\\mathbb { E } _ { p ^ { ( a ) } } [ f ( \\\\mathcal { W } ^ { \\\\| } , \\\\mathcal { W } ^ { \\\\perp } ) ] - \\\\theta ^ { \\\\dagger } } \\\\\\\\ & { \\\\mathbb { E } _ { p ^ { ( a ) } } [ f ( \\\\mathcal { W } ^ { \\\\| } , \\\\mathcal { W } ^ { \\\\perp } ) ] = \\\\underset { n \\\\infty } { \\\\operatorname* { l i m } } \\\\displaystyle \\\\int f ( \\\\mathcal { W } ^ { \\\\| } , \\\\mathcal { W } ^ { \\\\perp } ) p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\| } ) p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\| } | Y _ { n } ) d \\\\mathcal { W } ^ { \\\\| } d \\\\mathcal { W } ^ { \\\\perp } . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,440,874,479]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By the dominated convergence theorem (applicable through regularity conditions） and using the fact that $\\\\theta$ is a faithful SSRV, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[186,489,808,593]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0426/a025acae85654ffa4131356d7d2d0508c7356929539e39702d57fe5521b539e1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\mathbb { E } _ { p ^ { ( a ) } } [ f ( \\\\mathcal { W } ^ { \\\\| } , \\\\mathcal { W } ^ { \\\\bot } ) ] = \\\\displaystyle \\\\int \\\\operatorname* { l i m } _ { n \\\\to \\\\infty } f ( \\\\mathcal { W } ^ { \\\\| } , \\\\mathcal { W } ^ { \\\\bot } ) p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\bot } | \\\\mathcal { W } ^ { \\\\| } ) p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\| } | Y _ { n } ) d \\\\mathcal { W } ^ { \\\\| } d \\\\mathcal { W } ^ { \\\\bot } } \\\\\\\\ & { \\\\quad \\\\quad \\\\quad \\\\quad = \\\\displaystyle \\\\int f ( W ^ { \\\\| \\\\dag } , \\\\mathcal { W } ^ { \\\\bot } ) p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\bot } | W ^ { \\\\| \\\\dag } ) d \\\\mathcal { W } ^ { \\\\bot } } \\\\\\\\ & { \\\\quad \\\\quad \\\\quad = \\\\mathbb { E } _ { p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\bot } | W ^ { \\\\| \\\\dag } ) } [ f ( W ^ { \\\\| \\\\dag } , \\\\mathcal { W } ^ { \\\\bot } ) ] . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,603,207,621]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Therefore, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[290,619,702,647]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0485/dc436aeeee7a6a6b2bf0a57541208c9ab14c1639ad14c9a6e23f33df4db72e6b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\mathrm { b i a s } _ { p ^ { ( a ) } } ( \\\\overline { { \\\\theta } } ) = \\\\mathbb { E } _ p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\bot } | W ^ { | \\\\} ) } [ f ( W ^ { | | \\\\dagger } , \\\\mathcal { W } ^ { \\\\bot } ) ] - \\\\theta ^ { \\\\dagger } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,679,879,771]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The last two results show that, regardless of whether we calculate the expectation of the SSRV with respect to joint of the applied measurement and scale model (i.e., treating $\\\\mathcal { W }$ as a random variable and $\\\\mathcal { V }$ as a fixed observed quantity) or with respect to the target measurement model (e.g.， treating $\\\\mathcal { V }$ as the random variable and $\\\\mathcal { W }$ as fixed） we get the same form for the asymptotic bias \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[305,785,687,810]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0620/657acef651a54b00b47efe474c4f00ed2d15746c9449ba26eb49baa4dc7e874c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\operatorname { b i a s } ( { \\\\overline { { \\\\theta } } } ) = \\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\operatorname { b i a s } _ { p ^ { \\\\dag } } ( { \\\\overline { { \\\\theta } } } ) = \\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\operatorname { b i a s } _ { p ^ { ( a ) } } ( { \\\\overline { { \\\\theta } } } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"12 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,651,875,667]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,83,893,161]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"43\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where we now use $b i a s ( \\\\theta )$ , i.e., without subscript,to refer to either ${ \\\\mathrm { b i a s } } _ { p ^ { \\\\dagger } } ( \\\\theta )$ or ${ \\\\mathrm { b i a s } } _ { p ^ { ( a ) } } ( { \\\\overline { { \\\\theta } } } )$ To introduce the next theorem let us generalize this notation. We now write $\\\\theta ^ { \\\\dagger }$ as $f ( W ^ { \\\\parallel \\\\dagger } , W ^ { \\\\perp \\\\dagger } )$ and also to highlight that this value for $\\\\theta ^ { \\\\dagger }$ is with respect to the true value of the system scale with the notation bias $( \\\\theta | W ^ { \\\\perp \\\\dagger } )$ ： \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[236,175,756,203]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0843/4e23527fee3300b06c4ae251b126e7a6e61466b88420c5016cada1d428119199.jpg\",\n \"list_items\": null,\n \"page_idx\": \"43\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname * { l i m } _ { n \\\\to \\\\infty } \\\\mathrm { b i a s } ( \\\\overline { { \\\\theta } } | W ^ { \\\\perp \\\\dagger } ) = \\\\mathbb { E } _ { p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\parallel } | W ^ { \\\\parallel \\\\dagger } ) } [ f ( W ^ { \\\\parallel \\\\dagger } , \\\\mathcal { W } ^ { \\\\perp } ) ] - f ( W ^ { \\\\parallel \\\\dagger } , W ^ { \\\\perp \\\\dagger } ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,213,877,250]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"43\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem C.3. The asymptotic variance of a faithful scale simulation random variable under regularity conditions provided in Definition $\\\\it { C . 1 }$ is: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[266,263,727,291]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0601/4498704f2d9d92767f61d87b2e4193eb2f59ee5717a6e7823696dad35bef12bf.jpg\",\n \"list_items\": null,\n \"page_idx\": \"43\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\operatorname { v a r } _ { p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } | Y _ { n } ) } ( \\\\theta ) = \\\\mathbb { E } _ { p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\perp } | W ^ { \\\\parallel \\\\dagger } ) } \\\\left[ b i a s ( \\\\bar { \\\\theta } | \\\\mathcal { W } ^ { \\\\perp } ) ^ { 2 } \\\\right] .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,303,874,342]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"43\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof. As notational shorthands,let us define $e ( \\\\mathcal { W } ^ { \\\\| } ) \\\\equiv \\\\mathbb { E } _ { p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\bot } | \\\\mathcal { W } ^ { \\\\parallel } ) } \\\\left[ f ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\bot } ) \\\\right]$ By the law of total variance: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[274,356,720,377]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0325/8562dc762ea02c3e441ad6778fb201f10332272089e93764c51acabae49a5611.jpg\",\n \"list_items\": null,\n \"page_idx\": \"43\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { \\\\operatorname { v a r } _ { p ^ { ( a ) } } ( \\\\theta ) = \\\\mathbb { E } _ { p ^ { ( a ) } } [ \\\\operatorname { v a r } _ { p ^ { ( a ) } } ( \\\\theta \\\\vert \\\\mathcal { W } ^ { \\\\parallel } ) ] + \\\\operatorname { v a r } _ { p ^ { ( a ) } } [ \\\\mathbb { E } _ { p ^ { ( a ) } } ( \\\\theta \\\\vert \\\\mathcal { W } ^ { \\\\parallel } ) ] . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,390,383,407]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"43\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We start with the second term. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[184,417,810,529]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0521/30add5e014c8db1b631b24e2724f2bdd28cd376c4e02d9d9f7e8fdfe5090779c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"43\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } { \\\\underset { n \\\\to \\\\infty } { \\\\operatorname* { l i m } } \\\\mathrm { v a r } _ { p ^ { ( a ) } } [ \\\\mathbb { E } _ { p ^ { ( a ) } } ( \\\\theta | \\\\mathcal { W } ^ { \\\\| } ) ] = \\\\underset { n \\\\to \\\\infty } { \\\\operatorname* { l i m } } \\\\mathrm { v a r } _ { p ^ { ( a ) } } \\\\left[ \\\\int \\\\theta p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\| } ) d \\\\mathcal { W } ^ { \\\\perp } \\\\right] } & { } \\\\\\\\ { = \\\\underset { n \\\\to \\\\infty } { \\\\operatorname* { l i m } } \\\\mathrm { v a r } _ { p ^ { ( a ) } } [ e ( \\\\mathcal { W } ^ { \\\\| } ) ] } & { } \\\\\\\\ { } & { = \\\\underset { n \\\\to \\\\infty } { \\\\operatorname* { l i m } } \\\\int \\\\left( e ( \\\\mathcal { W } ^ { \\\\perp } ) - \\\\mathbb { E } _ { p ^ { ( a ) } } e ( \\\\mathcal { W } ^ { \\\\| } ) \\\\right) ^ { 2 } p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\| } | Y _ { n } ) d \\\\mathcal { W } ^ { \\\\| } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,536,877,574]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"43\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Using our regularity conditions to apply the dominated convergence theorem the limit can be moved inside the integral leading to \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[194,587,799,623]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0874/0e19b3678b88f62d07dd2080bedd6723266797e9c5faa825d78d697034afa2b2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"43\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\operatorname { v a r } _ { p ^ { ( a ) } } [ \\\\mathbb { E } ( \\\\theta | \\\\mathcal { W } ^ { \\\\| } ) ] = \\\\int \\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\big ( e ( \\\\mathcal { W } ^ { \\\\perp } ) - \\\\mathbb { E } _ { p ^ { ( a ) } } e ( \\\\mathcal { W } ^ { \\\\| } ) \\\\big ) ^ { 2 } p ^ { ( a ) } ( \\\\mathcal { W } ^ { \\\\| } | Y _ { n } ) d \\\\mathcal { W } ^ { \\\\| } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,635,753,654]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"43\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Using the fact that $\\\\theta$ is a faithful scale simulation random variable leads to \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[228,666,764,724]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0950/ae121f246692c2d478c241d081f9f3422d3603b6f8c7755a05f97d295ea94b0c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"43\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l r } { { \\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\operatorname { v a r } _ { p ^ { ( a ) } } [ \\\\mathbb { E } ( \\\\theta | \\\\mathcal { W } ^ { \\\\parallel } ) ] = \\\\int \\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\big ( e ( \\\\mathcal { W } ^ { \\\\perp } ) - \\\\mathbb { E } _ { p ^ { ( a ) } } e ( \\\\mathcal { W } ^ { \\\\parallel } ) \\\\big ) ^ { 2 } \\\\delta _ { W ^ { \\\\parallel \\\\dagger } } d \\\\mathcal { W } ^ { \\\\parallel } } } \\\\\\\\ & { } & { = 0 \\\\quad } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"43\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"13 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,503,104]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We are therefore left with only the first term: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[158,116,816,271]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0230/3f6e0a44556b114dcf382d7ea0c7baa04dfed868c9a1ec6f383cfad6e28a304d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { l } { \\\\displaystyle \\\\operatorname { v a r } _ { p ^ { ( \\\\alpha ) } } ( \\\\theta ) = \\\\underset { n \\\\to \\\\infty } { \\\\operatorname* { l i m } } \\\\mathbb { E } _ { p ^ { ( \\\\alpha ) } } \\\\big [ \\\\nabla \\\\mathrm { a r } _ { p ^ { ( \\\\alpha ) } } \\\\big ( \\\\theta | \\\\mathcal { W } ^ { \\\\parallel } \\\\big ) \\\\big ] } \\\\\\\\ { \\\\displaystyle \\\\quad = \\\\operatorname* { l i m } _ { n \\\\to \\\\infty } \\\\mathbb { E } _ { p ^ { ( \\\\alpha ) } } [ \\\\int \\\\big ( \\\\theta - \\\\mathbb { E } _ { p ^ { ( \\\\alpha ) } } \\\\big ( \\\\theta \\\\big | \\\\mathcal { W } ^ { \\\\parallel } \\\\big ) \\\\big ) ^ { 2 } p ^ { ( \\\\alpha ) } \\\\big ( \\\\mathcal { W } ^ { \\\\perp } \\\\big | \\\\mathcal { W } ^ { \\\\parallel } \\\\big ) d \\\\mathcal { W } ^ { \\\\perp } \\\\big | \\\\mathcal { W } ^ { \\\\parallel } ] } \\\\\\\\ { \\\\displaystyle \\\\quad = \\\\underset { n \\\\to \\\\infty } { \\\\operatorname* { l i m } } \\\\mathbb { E } _ { p ^ { ( \\\\alpha ) } } [ \\\\int \\\\big ( f ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } ) - e ( \\\\mathcal { W } ^ { \\\\parallel } ) \\\\big ) ^ { 2 } p ^ { ( \\\\alpha ) } \\\\big ( \\\\mathcal { W } ^ { \\\\perp } \\\\big | \\\\mathcal { W } ^ { \\\\parallel } \\\\big ) d \\\\mathcal { W } ^ { \\\\perp } \\\\big | \\\\mathcal { W } ^ { \\\\parallel } ] } \\\\\\\\ { \\\\displaystyle \\\\quad = \\\\underset { n \\\\to \\\\infty } { \\\\operatorname* { l i m } } \\\\int \\\\{ \\\\int \\\\big ( f ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } ) - e ( \\\\mathcal { W } ^ { \\\\parallel } ) \\\\big ) ^ { 2 } p ^ { ( \\\\alpha ) } \\\\big ( \\\\mathcal { W } ^ { \\\\perp } \\\\big | \\\\mathcal { W } ^ { \\\\parallel } \\\\big ) d \\\\mathcal { W } ^ { \\\\perp } \\\\} p ^ { ( \\\\alpha ) } \\\\big ( \\\\mathcal { W } ^ { \\\\parallel } \\\\big | \\\\mathcal { Y } \\\\big | } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,279,874,315]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Again,applying the dominated convergence theorem and using the fact that this is a faithful scale simulation random variable yields: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[181,325,825,470]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0563/afc30070872b333437cecd2c25c752a1e94f20681f5619caec32b3d3a1b61fdf.jpg\",\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { l } { \\\\displaystyle \\\\operatorname* { l i m } _ { \\\\to \\\\infty } \\\\operatorname { v a r } _ { p ^ { ( \\\\alpha ) } } ( \\\\theta ) = \\\\int \\\\left\\\\{ \\\\int \\\\left( f ( \\\\mathcal { W } ^ { \\\\parallel } , \\\\mathcal { W } ^ { \\\\perp } ) - e ( \\\\mathcal { W } ^ { \\\\parallel } ) \\\\right) ^ { 2 } p ^ { ( \\\\alpha ) } ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\parallel } ) d \\\\mathcal { W } ^ { \\\\perp } \\\\right\\\\} \\\\delta _ { \\\\mathcal { W } ^ { \\\\parallel } \\\\uparrow } d \\\\mathcal { W } ^ { \\\\parallel } } \\\\\\\\ { \\\\displaystyle \\\\quad = \\\\int \\\\left( f ( W ^ { \\\\parallel \\\\dagger } , \\\\mathcal { W } ^ { \\\\perp } ) - e ( W ^ { \\\\parallel \\\\dagger } ) \\\\right) ^ { 2 } p ^ { ( \\\\alpha ) } ( \\\\mathcal { W } ^ { \\\\perp } | W ^ { \\\\parallel \\\\dagger } ) d \\\\mathcal { W } ^ { \\\\perp } } \\\\\\\\ { \\\\displaystyle \\\\quad = \\\\int \\\\left[ \\\\mathrm { b i a s } ^ { * } ( \\\\theta | \\\\mathcal { W } ^ { \\\\perp } ) \\\\right] ^ { 2 } p ^ { ( \\\\alpha ) } ( \\\\mathcal { W } ^ { \\\\perp } | W ^ { \\\\parallel \\\\dagger } ) d \\\\mathcal { W } ^ { \\\\perp } } \\\\\\\\ { \\\\displaystyle \\\\quad = \\\\mathbb { E } _ { p ^ { ( \\\\alpha ) } ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\parallel \\\\dagger } ) } \\\\left[ \\\\mathrm { b i a s } ( \\\\bar { \\\\theta } | \\\\mathcal { W } ^ { \\\\perp } ) ^ { 2 } \\\\right] . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,507,877,655]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This result establishes that the asymptotic variance of SSRVs acknowledges their bias. The asymptotic variance of an SSRV is precisely the expected squared bias with expectation taken with respect to our uncertainty in $\\\\mathcal { W } ^ { \\\\perp }$ conditioned on the true value of $W ^ { \\\\parallel \\\\dagger }$ which is learned from the data. This also implies that, so long as the scale model has non-zero variance,if the model has non-zero bias then it will be matched by a non-zero variance. This is a critical feature as it helps to avoid spurious conclusions caused by being overly confident in an incorrect answer. These desirable features are not present in many popular models used to analyze compositional surveys. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,681,352,707]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"D Applications \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,720,547,743]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"D.1Simulation Details and Results \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,752,875,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For all simulations, we simulated a data set with 20 taxa across two conditions using multinomial sampling. Sequencing depth was set to 5,0o0, and, unless otherwise specified,sample size was set to 50( $n = 5 0$ ). ALDEx2 and DESeq2 were fit using their respective Bioconductor packages. For the simulations, default settings were used for each. For ALDEx2, 128 Monte Carlo samples were used, and the t-test with the Benjamini-Hochberg correction were used to find significance. For DESeq2, the Wald test was used, and Benjamini-Hochberg p-values were used to determine significance. Unless otherwise stated, $\\\\alpha = 0 . 0 5$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"14 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,479,875,496]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[263,85,723,262]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 1: Recreating ALDEx2 using SSRVs. To do so, we used a scale model that was a degenerate point mass centered on the geometric mean. Signifcance was assessd using the testing procedure of ALDEx2 at $\\\\\\\\alpha = 0 . 0 5$ . \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0281/2a693362c036bcd2245890a34f281cf36ffe1fc004a3c5453773c3b8429bcebd.jpg\",\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[120,353,875,444]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,445,875,517]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For our scale simulation approach, we assessed significance using the same approach as ALDEx2 with $\\\\alpha = 0 . 0 5$ . This procedure, along with a degenerate total model centered on the geometric mean, can capture the results of the base ALDEx2 model. Results to this effect are shown in Figure 1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,518,877,627]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Further， we provide several more figures to support claims made in the main text. Shown in Supplementary Figure 2a, we show that small amounts of uncertainty in the scale ( $\\\\alpha = 0 . 5$ ） can greatly reduce the number of false positives while still retaining three out of four true positives. This is in direct contrast to the results of ALDEx2. Shown in Supplementary Figure 2b, we find that inferred results can vary greatly as a function of the uncertainty in the scale model as expressed through $\\\\alpha$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,650,722,671]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"D.2Artificial Gut Application Details and Results \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,680,325,699]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bioreactor Culturing \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,709,877,799]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The bioreactor run consisted of 8 continuous-flow vessels (Multifors 2， Infors） cultured over 25 total days according to previously established protocols (Silverman et al., 2018; Letourneau et al., 2021). Briefly, vessels were inoculated with identical inoculum derived from a single healthy stool donor. Vessels were maintained anaerobic under nitrogen, kept at $3 7 ^ { \\\\mathrm { ~ o ~ } }$ C, continuously stirred at 1O0 rpm using magnetic impeller stir-shafts, and the total volume of media (40O mL modified Gifu Anaerobic Medium) was turned over once per 24 hours. Samples for sequencing and fow cytometry were collcted daily at 4pm by syringe via a sterilized sampling port. Since the volume of each vessel remained fixed at 400 mL, concentration of cells is directly proportional to absolute abundance. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[488,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"15 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[233,104,758,622]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 2: (a) Modeling results for DESeq2, ALDEx2,and three separate SSRVs. The $\\\\\\\\it C L R$ model uses a scale model centered on the geometric mean with $\\\\\\\\alpha = 1 e \\\\\\\\mathrm { ~ - ~ } 3$ .Note its correspondence with ALDEx2. The Relaxed scale model uses the same mean, but sets $\\\\\\\\alpha = 0 . 5$ . The $\\\\\\\\it C o D A$ scale model also uses the same mean, but sets $\\\\\\\\alpha = 1 0$ . Note that a small amount of noise reduces the number of false positives greatly. (b) Sensitivity analysis showing the estimated SSRV effect size for each taxa under the Relaxed model as a function of $\\\\\\\\alpha$ . A black line signifies that the estimated effect size was statistically different from zero at $9 5 \\\\\\\\%$ confidence. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0081/a1bf8dc72ea2658553208ecd7464fd1f729642ea4ca17dab585fc76277bffd1c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[488,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"16 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,85,877,159]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,179,500,199]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cell Quantification by Flow Cytometry \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,208,875,299]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Samples were serially diluted ten-fold for a fnal dilution of 1:1Ooo. Serial dilutions were performed in PBS, with the final dilution performed into wels of a 96-wel plate containing PBS with 1 μL SYBR green/DMSO (1:499 SYBR:DMSO) per 180 μL total volume. The final 20O μL mixture including sample and SYBR green was incubated 15 min at 37 $\\\\cup$ C (Vandeputte et al., 2017). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,300,875,372]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Flow cytometry was performed using a MACSQuant Flow Cytometer (Miltenyi Biotec). The 96-well plate containing samples was maintained on a chill rack for the duration of the run. 10 μL was taken for measurement. Forward scatter, side scatter, and SYBR fluorescence (using channel B1, log5, 274 V） were recorded. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,373,877,482]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Flow cytometry data was analyzed in FlowJo using a two-step gating strategy. A single sample containing cells but no SYBR was used to develop gates which were then applied to all samples. First, using a graph of forward scatter vs. side scatter, we roughly selected for the center of point density, omitting extreme values (i.e. possibly representing cellular debris and/or particle aggregates). Second, examining the SYBR channel, we excluded counts that fell within the range of the no-SYBR control. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[122,503,591,521]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"16S rRNA-Gene Encoding Amplicon Sequencing \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,531,877,659]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"16S rRNA gene amplicon sequencing was performed as previously described using custom barcoded primers targeting the V4 region of the gene, resulting in 150 bp paired-end reads sequenced by Ilumina MiniSeq according to previously published protocols (Silverman et al., 2018; Letourneau et al., 2021). DADA2 was used to identify and quantify sequence variants (SVs) in our dataset， using version 123 of the Silva database. Taxa that had counts of less than 2,5O0 across all samples were fltered and amalgamated into a separate category. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,680,258,698]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data Analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,708,875,780]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For our analysis, we restricted ourselves to days 1 and 14. Furthermore, we used data from vessels 3, 4, 5, 6,7,and 8 only due to limitations in the data. We noted a difference between flow cytometry concentrations between days regardless of vessel. See Figure 3 for details. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[488,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"17 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,877,176]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We tested a model of differential expression including indicator terms for each vessel and day. For ALDEx2, we used the generalized linear model test with 250 Monte Carlo Samples. We assessed differential expressed taxa based on the p-values ( $\\\\alpha = 0 . 0 5$ ) for day alone. For DESeq2, we employed the likelihood ratio test with a reduced model containing only the vessel and assessed differential abundance using the same criteria as ALDEx2. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,178,877,304]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For the SSRV based on the Multinomial Log Normal model, we set our prior parameters as follows: $\\\\nu = D + 3$ and $\\\\Xi = I _ { D }$ reflecting our weak prior knowledge that the taxa were independent; Γ = BlockDiagonal $( 1 0 I _ { 6 } , 1 )$ reflecting our assumption that the some taxa were more abundant than others at baseline and assuming that the signal-to-noise ratio with respect to differential abundance was most likely on the order of 1 (Silverman et al., 2018); and finally $M = 0$ which reflected that we did not have α priori knowledge differentiating the taxa. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,306,870,342]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"31 different sequence variants were declared significant by at least one method. Taxonomic information for these significant sequence variants is presented in Table 1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,361,671,574]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 3: Variation of fow cytometry data by day. Note that there was a higher concentration of cells on day 14 regardless of vessel. This has biological validity based on the experimental goals of the study. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0880/d943be76a29af524357920020bbd163df5d3bf6c29e398813db0a1e831e16c90.jpg\",\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[488,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"18 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[294,145,687,747]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0726/4fd567fdeb6eb4ca7124764f6184024a901e7aaaab911ef48718d5bc564e6c7d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"49\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Sequence</td><td>Taxonomic Information</td></tr><tr><td>1</td><td>Bacteroides spp.</td></tr><tr><td>2</td><td>Escherichia/Shigella spp.</td></tr><tr><td>3</td><td>Bacteroides uniformis</td></tr><tr><td>4</td><td>Peptoniphilus spp.</td></tr><tr><td>5</td><td>Pyramidobacter piscolens</td></tr><tr><td>6</td><td>Enterococcus spp.</td></tr><tr><td>7</td><td>Veillonella spp.</td></tr><tr><td>8</td><td>Bacteroides spp.</td></tr><tr><td>9</td><td>Streptococcus spp.</td></tr><tr><td>10</td><td>Anaerosalibacter bizertensis</td></tr><tr><td>11</td><td>Bacteroides stercoris</td></tr><tr><td>12</td><td>Bacteroides caccae</td></tr><tr><td>13</td><td>Megasphaera spp.</td></tr><tr><td>14</td><td>Peptoniphilus lacrimalis</td></tr><tr><td>15</td><td>Lachnoclostridium spp.</td></tr><tr><td>16</td><td>Dorea formicigenerans</td></tr><tr><td>17</td><td>Phascolarctobacterium faecium</td></tr><tr><td>18</td><td>Veillonella spp.</td></tr><tr><td>19</td><td>Lachnoclostridium spp.</td></tr><tr><td>20</td><td>Clostridium sensu stricto spp.</td></tr><tr><td>21</td><td>Veillonella spp.</td></tr><tr><td>22</td><td>Peptoclostridium difficile</td></tr><tr><td>23</td><td>Erysipelatoclostridium ramosum</td></tr><tr><td>24</td><td>Hungatella spp.</td></tr><tr><td>25</td><td>Flavonifractor plautii</td></tr><tr><td>26</td><td>Bifidobacterium spp.</td></tr><tr><td>27</td><td>Parabacteroides merdae</td></tr><tr><td>28</td><td>Anaerococcus vaginalis</td></tr><tr><td>29</td><td>Blautia spp.</td></tr><tr><td>30</td><td>Eubacterium spp.</td></tr><tr><td>31</td><td>Anaerostipes hadrus</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 2: Taxonomic information of significant sequence variants identified in the Real Data Analysis. \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[488,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"49\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"19 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[294,85,699,362]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 4: FDR by method (ALDEx2, DESeq2, SSRV - Design) with increasing sample sizes. Additional samples were drawn using bootstrap resampling. Dashed horizontal line denotes the worst possible FDR given the number of true positives in this study. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0086/4f0c473aafe3231986e1cfef1cb9d386a54d0fe586b7a5d55db8d93d2c5d10df.jpg\",\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[120,450,870,473]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"D.3Deriving the Multinomial Logistic Normal Model as a SSRV \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,483,872,518]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section, we show how the multinomial logistic-normal model for sequence count data presented in the manuscript can be inferred as a SSRV. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,540,598,558]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"D.3.1 Derivation of Conditional Relative Model \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,568,385,585]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recall that the target model is: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[419,616,576,757]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0894/6f3597cc4fbf6633ff7f339e42e2c9a87537f68b57922516e7f87f226fc0a86d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { Y _ { j } \\\\sim \\\\mathrm { M u l t } ( \\\\mathcal { W } _ { j } ^ { \\\\parallel } ) } \\\\\\\\ & { \\\\mathcal { W } _ { j } = \\\\mathcal { W } _ { j } ^ { \\\\perp } \\\\mathcal { W } _ { j } ^ { \\\\parallel } } \\\\\\\\ & { \\\\mathcal { W } _ { j } = \\\\exp ( \\\\psi _ { j } ) } \\\\\\\\ & { \\\\psi _ { j } \\\\sim N ( B X , \\\\Omega , I ) } \\\\\\\\ & { B \\\\sim N ( M , \\\\Omega , \\\\Gamma ) } \\\\\\\\ & { \\\\Omega \\\\sim I W ( \\\\nu , \\\\Xi ) } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,768,424,787]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"First, we defined $G$ as a linear map: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"20 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[457,99,537,138]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"1009/46f6a375a285a501d6df5f7900bdbd071e5f97068b5b5bb5994ffed66aa90b50.jpg\",\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nG = { \\\\binom { F } { H } }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,143,875,180]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $F = [ I _ { D - 1 } , - \\\\mathbf { 1 } _ { D - 1 } ]$ and $H = \\\\mathbf { 1 } _ { D }$ . Note that for any arbitrary vector $\\\\theta$ and matrix $\\\\Sigma$ ， we have: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[289,205,705,291]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0868/f27d099124d816436d82f1c08f139e5a2148317f209dc654bec5ef310d5a582e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n{ \\\\begin{array} { r l } & { G \\\\theta = \\\\left[ F \\\\theta \\\\right] = \\\\left[ \\\\theta ^ { \\\\parallel } \\\\right] } \\\\\\\\ & { G \\\\theta } \\\\end{array} } \\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,299,602,318]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where the $\\\\parallel$ and $\\\\perp$ notation follows the rest of the paper. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,319,640,337]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Applying $G$ to $\\\\psi$ from the model defined in Equations 2-7: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[446,347,547,387]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0280/b1453dce80577146363ed015bd85a2aaf4493f3c19f836d6951f3bc76350bc12.jpg\",\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nG \\\\psi = \\\\left[ \\\\psi ^ { \\\\parallel } \\\\right]\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,397,875,434]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since it is an affne transformation, we get (Theorem 2.3.10, Gupta and Nagar (2000), page 64): \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[338,445,656,574]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0058/e3ca7cc64fdbc5b823245efc19186776a8b9e3e24ef240537f234c1fc4d54257.jpg\",\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\left[ \\\\psi ^ { \\\\parallel } \\\\right] \\\\sim N ( G B X , G \\\\Omega G ^ { T } , I ) } \\\\\\\\ & { \\\\left[ \\\\psi ^ { \\\\parallel } \\\\right] \\\\sim N \\\\left( \\\\left[ \\\\begin{array} { l } { B ^ { \\\\parallel } } \\\\\\\\ { B ^ { \\\\perp } } \\\\end{array} \\\\right] X , \\\\left[ \\\\begin{array} { l l } { \\\\Omega ^ { \\\\parallel } } & { \\\\Omega ^ { + T } } \\\\\\\\ { \\\\Omega ^ { + } } & { \\\\Omega ^ { \\\\perp } } \\\\end{array} \\\\right] , I \\\\right) } \\\\\\\\ & { \\\\left[ \\\\psi ^ { \\\\parallel } \\\\right] \\\\sim N \\\\left( \\\\left[ \\\\begin{array} { l } { B ^ { \\\\parallel } } \\\\\\\\ { B ^ { \\\\perp } } \\\\end{array} \\\\right] X , \\\\Omega ^ { * } , I \\\\right) } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,579,872,616]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"from the properties discussed before. Now, we need to marginalize out $B$ and $\\\\Omega$ . Starting with $B$ , we can write Equation 14 as: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[323,626,674,666]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0875/c230c2cf287daaab9c814c6f45d3ff752d095a448fc7ae957c391b24dd894d6f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left[ \\\\psi ^ { \\\\parallel } \\\\right] = \\\\left[ B ^ { \\\\parallel } \\\\right] X + E ^ { B } , E ^ { B } \\\\sim N ( 0 , \\\\Omega ^ { * } , I )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,676,877,712]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Similarly, since $B \\\\sim N ( M , \\\\Omega , I )$ ,by Theorem 2.3.10 (Gupta and Nagar (2000), page 64), we get: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[352,710,642,752]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0654/c77efa5067ec7bfb49e4794bc73b8caf369c8fb26e90b8b8e42bd90240ff6b84.jpg\",\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nG B = \\\\left[ \\\\frac { B ^ { \\\\parallel } } { B ^ { \\\\perp } } \\\\right] \\\\sim N \\\\left( \\\\left[ M ^ { \\\\parallel } \\\\right] , \\\\Omega ^ { * } , \\\\Gamma \\\\right)\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,756,289,772]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Then, we can write: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[330,771,666,811]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0702/7de936a245b8bf9b87085bb0dffd8a994fabb9b60f76f97bac8fb222c1c9a737.jpg\",\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left[ B ^ { \\\\parallel } \\\\right] = \\\\left[ M ^ { \\\\parallel } \\\\right] + E ^ { * } , \\\\ E ^ { * } \\\\sim N ( 0 , \\\\Omega ^ { * } , \\\\Gamma ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"21 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,266,103]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This implies that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[361,113,635,199]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0959/5423743485c48b648419df22c4229f1c97fcf079fb15aa4326d2430e8e3e7c33.jpg\",\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { \\\\left[ \\\\psi ^ { \\\\parallel } \\\\right] = \\\\left( \\\\left[ M ^ { \\\\parallel } \\\\right] + E ^ { * } \\\\right) X + E ^ { B } } \\\\\\\\ { = \\\\left[ { M ^ { \\\\parallel } } \\\\right] X + E ^ { * } X + E ^ { B } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,205,872,242]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $E ^ { B }$ and $E ^ { * }$ are defined as before. Using Theorem 2.3.10 (Gupta and Nagar (2000), page 64), the distribution of $E ^ { * } X$ is: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[393,252,601,273]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0039/cf8a73181f24a60f18541a0e52c6f31e9d2c829309e47efbdaeb25ffbce81df6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nE ^ { * } X \\\\sim N ( 0 , \\\\Omega ^ { * } , X ^ { T } \\\\Gamma X )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,284,563,304]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Then, by Gupta and Nagar (2000) (page 82), we get \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[334,313,663,353]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0407/cd09e7ed80df5c9390b45af91acb34c12f554d00880e30feaffc9bc5bdec7ba2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left[ \\\\psi ^ { \\\\parallel } \\\\right] \\\\sim N \\\\left( \\\\left[ M \\\\parallel \\\\right] X , \\\\Omega ^ { * } , I + X ^ { T } \\\\Gamma X \\\\right)\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,362,563,381]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To marginalize out $\\\\Omega ^ { * }$ ， we start by applying $G$ to $\\\\Omega$ ： \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[392,390,604,411]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0206/8cf9056382364401519d84100f63d5969547b8bef844730854a06154f4268376.jpg\",\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nG \\\\Omega G ^ { T } = \\\\Omega ^ { * } \\\\sim \\\\mathrm { I W } ( \\\\nu , \\\\Xi ^ { * } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,420,875,479]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\Xi ^ { * } = G \\\\Xi G ^ { T } = \\\\left[ \\\\Xi ^ { \\\\parallel } \\\\quad \\\\Xi ^ { + ^ { T } } \\\\right]$ Let $\\\\hat { E } \\\\sim N ( 0 , I , I + X ^ { T } \\\\Gamma X )$ . This implies that (Theorem 4.2.1 of Gupta and Nagar (2000), page 134) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[277,488,717,529]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0454/98d4d7b1102881672e77b0e863bbaf45fe4782699cd30f4ee69143b30c356807.jpg\",\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\Omega ^ { - 1 } \\\\hat { E } + \\\\left[ M ^ { \\\\parallel } \\\\right] X \\\\sim T \\\\left( \\\\nu , \\\\left[ M ^ { \\\\parallel } \\\\right] X , \\\\Xi ^ { * } , I + X ^ { T } \\\\Gamma X \\\\right)\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,537,236,555]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"which implies \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,553,674,593]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0062/bc2cdb4b9d5cc780f681ebb75dbc6aa8f97eb2f35c6dfe8ec324af42b73b9235.jpg\",\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left[ \\\\psi ^ { \\\\parallel } \\\\right] \\\\sim T \\\\left( \\\\nu , \\\\left[ M ^ { \\\\parallel } \\\\right] X , \\\\Xi ^ { * } , I + X ^ { T } \\\\Gamma X \\\\right) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[122,598,771,618]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From Gupta and Nagar (2000) (page 163), the marginal distribution of $\\\\psi ^ { \\\\parallel }$ is: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[362,627,632,648]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0668/5e120e4975106c7349e692ce0d2aa8dc8fc58085b2e6e1ff71b2cdbf823acc1e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\psi ^ { \\\\parallel } \\\\sim T ( \\\\nu , M ^ { \\\\parallel } X , \\\\Xi ^ { \\\\parallel } , I + X ^ { T } \\\\Gamma X )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,659,428,678]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"which implies the model for $\\\\psi ^ { \\\\parallel } | Y$ is: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[359,688,635,762]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0303/b0832641d9c2d2b4aa29f98565a49a84b9c18ad5261e07854050041f165c1796.jpg\",\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { Y _ { j } \\\\sim \\\\mathrm { M u l t } ( \\\\mathcal { W } _ { j } ^ { \\\\parallel } ) } \\\\\\\\ & { \\\\psi ^ { \\\\parallel } \\\\sim \\\\mathrm { A L R } _ { D } ( \\\\mathcal { W } ^ { \\\\parallel } ) } \\\\\\\\ & { \\\\psi ^ { \\\\parallel } \\\\sim T ( \\\\nu , M ^ { \\\\parallel } X , \\\\Xi ^ { \\\\parallel } , I + X ^ { T } \\\\Gamma X ) . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,770,877,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $A L R ( \\\\cdot )$ denotes the additive log-ratio transform. Note that this is exactly the collapse sampler from Silverman et al. (2022). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,828,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"22 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,85,704,104]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"D.3.2Derivation of Implied Scale Model for Bayesian PIM \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,114,398,132]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recall from the previous section: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,145,673,185]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0183/d6c09b184d903551f300ffa5efe80f50f03093474978e672d71c2df013b89b86.jpg\",\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left[ \\\\psi ^ { \\\\parallel } \\\\right] \\\\sim T \\\\left( \\\\nu , \\\\left[ M ^ { \\\\parallel } \\\\right] X , \\\\Xi ^ { * } , I + X ^ { T } \\\\Gamma X \\\\right) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,199,877,236]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Then, from Gupta and Nagar (2000) (page 138),the distribution of $\\\\psi ^ { \\\\perp } | \\\\psi ^ { \\\\parallel }$ is a multivariate T-distribution: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[158,265,810,315]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0586/c7642ade581c9326b047f84f84533b70e2851ac1041ba34fd58912ded44c6530.jpg\",\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\nu ^ { \\\\bot } | \\\\psi | ^ { \\\\rbrack } \\\\sim T [ \\\\nu + D , M ^ { \\\\bot } X + ( \\\\Xi ^ { + } ) ^ { T } ( \\\\Xi ^ { \\\\parallel } ) ^ { - 1 } ( \\\\psi ^ { \\\\parallel } - M ^ { \\\\parallel } X ) , } \\\\\\\\ & { \\\\qquad \\\\Xi _ { 2 2 \\\\cdot 1 } , ( I + X ^ { T } \\\\Gamma X ) ( I + ( I + X ^ { T } \\\\Gamma X ) ^ { - 1 } ( \\\\psi ^ { \\\\parallel } - M ^ { \\\\parallel } X ) ^ { T } ( \\\\Xi ^ { \\\\parallel } ) ^ { - 1 } ( \\\\psi ^ { \\\\parallel } - M ^ { \\\\parallel } X ) ) ] } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,327,439,348]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\Xi _ { 2 2 \\\\cdot 1 } = \\\\Xi ^ { \\\\perp } - ( \\\\Xi ^ { + } ) ( \\\\Xi ^ { \\\\parallel } ) ^ { - 1 } ( \\\\Xi ^ { + } ) ^ { T }$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,367,558,387]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"D.3.3Collapse-Uncollapse Sampling Details \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,396,877,486]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"While conceptually straightforward, scale simulation requires efficient sampling for three separate stages: a compositional model, a scale model, and an inferential model. For the case of the multinomial logistic normal model, our sampling details closely mimic the Collapse-Uncollapse (CU) sampler of Silverman et al. (2022). Intuitively, the basic sampler follows three steps: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[142,488,707,542]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Sample from the partially identified base scale model $P ( \\\\mathcal { W } ^ { \\\\parallel } | \\\\mathcal { V } )$ \\n2. Sample from the scale model $P ( \\\\mathcal { W } ^ { \\\\perp } | \\\\mathcal { W } ^ { \\\\parallel } )$ \\n3. Compute the target estimand from the samples of $\\\\mathcal { W } ^ { \\\\parallel }$ and $\\\\mathcal { W } ^ { \\\\perp }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,544,877,705]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sampling from the scale model is typically effcient as it is low-dimension and flexible. To date, efficient routines for these models are already implemented in R. However, sampling from Step 1 is cumbersome but were recently made much more effcient and scalable by Silverman et al. (2022). Their approach relies on rewriting the above multinomial logisticnormal model as a marginally latent matrix-t process. Then, they develop efficient inference for this class of models using a Collapse-Uncollapse (CU） sampling approach. Such an approach vastly improves computational effciency, increasing the usability of these models to much larger sequence count data sets. See Silverman et al. (2O22) for complete details on implementation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,707,874,743]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We utilize the logic behind the CU sampler to infer Step 1. These routines are efficiently implemented in the R package fido (Silverman et al., 2022). Formal implementation details \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"23 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[140,128,816,279]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0716/c2f53215236bc2db042324c7ee62ca70bab16ab2fd76328018258b723618339f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Data: Y Result: S samples of the form {B(s), Ω(s)}</td><td></td></tr></table>\",\n \"table_caption\": \"[\\\"Algorithm 3: Modified Collapse-Uncollapse Sampler for Scale Simulation \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[119,301,816,327]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"EExample of Scale Reliant Inference in Ecology \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,340,877,431]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A common task in ecology is to estimate the rate at which an animal species abundance changes between two time-points (Udevitz and Pollock,1992). To avoid needlessly complicating this example, we consider only a single species whose abundance is denoted at time $t$ by $W _ { t }$ . In this case,the relative abundance at time $t$ is trivial $W _ { t } ^ { \\\\parallel } = 1$ implying that $W _ { t } = W _ { t } ^ { \\\\perp }$ . Ecologists often want to estimate the value of the logarithmic rate of change: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[421,443,571,527]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"1013/c63d3dd8c480b21fd508480878860744fd79a00d22bc8e46acd436fae05de745.jpg\",\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { r = \\\\log \\\\left( \\\\frac { W _ { t + 1 } } { W _ { t } } \\\\right) } \\\\\\\\ { = \\\\log \\\\left( \\\\frac { W _ { t + 1 } ^ { \\\\perp } } { W _ { t } ^ { \\\\perp } } \\\\right) . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,534,591,551]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is clear that this target estimand is fully scale reliant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,554,877,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Yet many ecological surveys that attempt to estimate changes in species abundance over time are limited to using observational data where the scale, e.g., $W _ { t } ^ { \\\\perp }$ cannot be directly measured. Instead ecologists often use some variation on mark-recapture studies (Krebs, 1999).In a mark-recapture study, animals are captured and released twice. During the first capture,animals are marked prior to release, the total number of animals captured during are recorded ( $Y ^ { 1 }$ ). The second capture not only records the total number of captured animals $Y ^ { 2 t }$ but also the number of those captured that had marks indicating that they had been recaptured $Y ^ { 2 m }$ . By performing independent mark-recapture studies both at timepoint $t$ and $t + 1$ ecologists hope to collect sufficient information to estimate $r$ . We can think of these data as being derived from a compositional survey. Albeit a trivial compositional survey since the relative abundance in this simple 1 species example is trivially identifiable as it is always equal to 1. Below we illustrate a use of mark-recapture coupled with a technique calld the change-in-ratios method (Udevitz and Pollock,1992) that is sometimes used to estimate $r$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"24 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,87,524,103]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for scale simulation are outlined in Algorithm 3. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[119,85,879,195]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Suppose two separate mark-recapture experiments are undertaken at times $t$ and $t +$ 1.Let $\\\\{ Y _ { t } ^ { 1 } , Y _ { t } ^ { 2 m } , Y _ { t } ^ { 2 t } \\\\}$ denote the number of initially-marked individuals, the number of recaptured marked individuals,and the total number of observed individuals during the recapture survey at time $t$ Wedenoteananalogous vector $\\\\{ Y _ { t + 1 } ^ { 1 } , Y _ { t + 1 } ^ { 2 m } , Y _ { t + 1 } ^ { 2 t } \\\\}$ for time $t + 1$ These data are often used to estimate the population size (species abundance) at each time-point: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[426,209,568,292]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0631/4612eadec75cfd5ca7dd6912de93da82d523259e011452978b99153192713d7c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { c } { { \\\\hat { W } _ { t } = \\\\displaystyle { \\\\frac { Y _ { t } ^ { 1 } Y _ { t } ^ { 2 t } } { Y _ { t } ^ { 2 m } } } } } \\\\\\\\ { { \\\\hat { W } _ { t + 1 } = \\\\displaystyle { \\\\frac { Y _ { t + 1 } ^ { 1 } Y _ { t + 1 } ^ { 2 t } } { Y _ { t + 1 } ^ { 2 m } } } } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,303,787,320]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the change-of-ratios methods these estimates are then used to estimate $r$ as: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[354,332,640,428]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0957/bfbdb3103c1a7602b7a7da2dd211184010486d8a12f48f0abda1d44d22356b15.jpg\",\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\hat { r } = \\\\log \\\\left( \\\\frac { \\\\hat { W } _ { t } } { \\\\hat { W } _ { t + 1 } } \\\\right) } \\\\\\\\ & { \\\\quad = \\\\log \\\\left( \\\\frac { Y _ { t + 1 } ^ { 1 } } { Y _ { t } ^ { 1 } } \\\\right) \\\\log \\\\left( \\\\frac { Y _ { t + 1 } ^ { 2 t } / Y _ { t + 1 } ^ { 2 m } } { Y _ { t } ^ { 2 t } / Y _ { t } ^ { 2 m } } \\\\right) . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[122,435,406,454]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This is an applied estimator for $r$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,454,877,746]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The framework of SRI provides useful concepts to help catalyze discussions about this type of analysis. Does this constitute fully scale reliant， partially scale reliant, or scale invariant inference? Of course, as we have not explicitly specified a measurement model is it not possible to make any formal claim. Still, thinking through the mechanics of a mark-recapture study it seems reasonable to assume that these studies do not identify the system composition. Consider that implicit in the above applied estimator is an assumption of homogeneous mixing and random sampling: there is an assumption that the marked an-imals homogeneously mix into the population such that the recapture probability $Y _ { t } ^ { 2 m } / Y ^ { 2 t }$ is equal to the proportion of marked animals in the entire population. This assumption is clearly not exactly true leaving fundamental uncertainty in the system scale despite the observed data. It therefore seems reasonable to assume that this is a partial compositional survey. In contrast, we expect that the degree of uncertainty is likely far less than what is present in sequence count data and it is likely unreasonable to model these data as coming from a fully compositional survey. In either case,given that the target estimand is oneto-one with the ratio of scales $W _ { t - 1 } ^ { \\\\perp } / W _ { t } ^ { \\\\perp }$ it seems clear that inference of $r$ must be scale reliant. We therefore expect that strict frequentist consistency is likely not possible. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,747,875,801]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Overall, this discussion is merely meant to illustrate how the language of SRI can be applied to different felds. Further work is needed if SRI is to provide a comprehensive framework for understanding the role of scale in mark-recapture based ecological studies. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"25 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,80,271,104]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,118,874,156]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Gabrielsen, A. (1978). Consistency and identifiability. Journal of Econometrics, 8(2):261- 263. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,166,877,224]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Grantham, N. S., Guan, Y., Reich, B. J., Borer, E. T., and Gross, K. (2020). Mimix: A bayesian mixed-effects model for microbiome data from designed experiments. Journal of the American Statistical Association, 115(530):599-609. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,233,861,255]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Gupta, A. and Nagar, D. (20oo). Matrix Variate Distributions. Chapman & Hall/CRC. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,265,872,303]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Krebs, C. J. (1999). Ecological methodology. Benjamin/Cummings, Menlo Park, Calif, 2nd ed edition. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,314,877,371]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Letourneau, J.， Holmes, Z. C.， Dallow, E. P., Durand, H. K., Jiang, S., Gupta, S. K, Mincey, A. C., Muehlbauer, M. J.， Bain, J. R., and David, L. A. (2021).Ecological memory of prior nutrient exposure in the human gut microbiome. bioRxiv. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,382,877,439]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Props,R., Kerckhof, F.-M., Rubbens, P., De Vrieze, J.， Hernandez Sanabria, E., Waege-man,W.， Monsieurs, P., Hammes, F. and Boon, N. (2017). Absolute quantification of microbial taxon abundances. The ISME journal, 11(2):584-587. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,449,707,469]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Shao, J. (2003). Mathematical statistics. Springer Texts in Statistics. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,479,879,536]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Silverman, J. D., Bloom, R. J., Jiang, S., Durand, H. K., Dallow, E., Mukherjee, S., and David, L. A. (2021). Measuring and mitigating pcr bias in microbiota datasets. PLoS Computational Biology, 17(7):e1009113. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,546,877,604]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Silverman, J. D., Durand, H. K., Bloom, R. J., Mukherjee, S., and David, L. A. (2018). Dynamic linear models guide design and analysis of microbiota studies within artificial human guts. Microbiome, 6(1):1-20. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,614,875,671]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Silverman, J. D.，Roche,K.， Holmes, Z. C.，David,L.A.，and Mukherjee, S. (2022). Bayesian multinomial logistic normal models through marginally latent matrix-t processes. Journal of Machine Learning Research, 23(7):1-42. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,683,870,722]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Udevitz, M. S. and Pollock, K. H. (1992). Change-in-ratio methods for estimating population size. In Wildlife 2001: populations, pages 90-101. Springer. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,731,877,787]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Vandeputte, D., Kathagen, G., D'hoe, K., Vieira-Silva, S., Valles-Colomer, M., Sabino, J., Wang, J., Tito, R. Y., De Commer, L., Darzi, Y., et al. (2017). Quantitative microbiome profiling links gut community variation to microbial load. Nature, 551(7681):507-511. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[486,829,508,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"26 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.48550/arxiv.2201.03616"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Mathematics and Computational Science"},"sha256":{"kind":"string","value":"1e30c5c25ba1526d99c1371d2bbee48fd7f186a284b4b7814eb868e6877871df"},"title":{"kind":"string","value":"Scale Reliant Inference"}}},{"rowIdx":51,"cells":{"abstract":{"kind":"string","value":"Classification of cardinal numerals is an important step in studying Chinese numerals. The study is carried out in the framework of the functional, structural (morphemic), and functional-structural (combined) classification approaches, as well as in accordance with the integrity of numerals and the accuracy of their quantifying properties. Chinese numerals are considered from the perspective of the first three classifications. The results obtained reveal the nomenclature volatility of Chinese cardinal numerals with the same classification group having several names to denote it. Hence, the article attempts to streamline the respective terminology.\r\nThe functional classification divides cardinal numerals into coefficient and digit-position numerals. The cases when digit-position numerals may function as coefficient numerals are analyzed.\r\nIn the framework of the structural classification, it is suggested to divide cardinal numerals into simple (coefficient) and compound ones, the latter being broken into additive (elementary and miscellaneous) and multiplicative. The article also focuses on answering the question of whether compound numerals should be regarded as words or phrases. The conclusion is made that they should be perceived as words.\r\nThe functional-structural classification makes a substantial contribution to the study of Chinese cardinal numerals. The classification has not been introduced to the Russian sinology yet. This fact entails the necessity to translate the respective terms and expressions, the choice of which is explained in this article. The variations of the combined functional-structural classification introduced by Zhu Dexi, Guan Yanqìng, Xiao Guozheng, and Li Yingzhe are discussed. Based on the analysis of these variations, an understanding of the combined functional-structural division of Chinese cardinal numerals is suggested."},"author":{"kind":"string","value":"L.L. 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","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,146,811,256]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IIOIO6HOH HeoIHOpOINOCbH aHaJIH3npyeMoJ YacTH peuH N BbI3BaHa IOTpe6-HocTB B ee rpaMMaTHuecKOM HccJIeIOBAHH. IIO cJIOBaM ΓyaHb JHBIIHa (关彦庆 Guan Yanqing), «KpaIHe Heo6XoINMO IIPOBeCTN IOJIHOe H BCEo6bEmJIIOIIee OIIHCaHne IIPIHIINOB CTpykTpyrpoBaHH KHTaIckHX YHCSJIHTeJIbHBIX》² [6, c. 3]. Mbl Je yTOUHNM, YTO NaATb TaKyIO pa60Ty CJIeDyET HMeHHO C KOJIInCeTBeHHbIX YHCSJIHTeJIbHBIX KAK 6a-3OBbIX, OT KOTOpbIX o6pa3OBaHH IIopRAIKOBbIX YHCSJIHTeJIbHBIX HaXOJNTcB IIPrMoH 3aBHCmOcTH [7, c. 313]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[173,259,811,556]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Kpome TOrO, paccMOrpeTB KOJIHueCTBHeHHbIe YHcJIHTeJIbHbIe IIO6yJdaeT OTCYtCTBHe Hx cHCTeMHOrO oIHcAHn. XOTa HeKOToPbIE IPOBeJeHHbIe paHee HccJIeIOBaHNr OITINuHOTcRgIy6HnOH aHaJIIn3a, HaIIpHMeP: «HcJIeIOBaHne cTpyKtpybI cOBpeMeHHbIX KHTaIckHX KOJIHueCTBHeHHbIX YHcJIHTeJIbHbIX» (现代汉语基数词的结构研究) ΓyaHb JHBiHa [6], «OIIHCaHHe HCCJIeIOBaHNr KJIaCCNΦHKaIINr YHcJIHTeJIbHbIX» (数词分类研究述论) JIo PInN (刘苹 Liu Ping) [8], «UHcJIHTeJIbHbIe N IHX cTpyKtypa» (数词和数词结构) Chy DiCh (朱德熙 Zhū Déxi) [9], «CNCTeMHa OpraHHaIry, oCo6eHHocTH IIprMeHEnHr N EBOJIHOIIIG KHTaIckHX OIIpeJEJIeHHbIX YHcJIHTeJIbHbIX» (汉语确数词的系统构成、使用特点和历史演进) CNo TOnKoHa N JIn INuKo3 (萧国政 Xiao Guózheng, 李英哲 Li Yingzhe) [10], «IcSeIOBaHne YHcJIHTeJIbHbIX B COBpeMeHHOM KHTaIckOM Ja3bIKe» (现代汉语数词研究) ChkoY IaJIyHb (周彩莲 Zhōu Cāilián) [11], «I3UyeHne BOIpocOB, CBRAHHbIX C KOJIHueCTBOM N UHCJOM B KHTaIckOM Ja3bIKe» (汉语涉数问题研究) FIO HAry (郭攀 Guo Pan) [12], B YacTHN BBIIIE- IpeHuCJIeHHbIX pa6OT KOJIHueCTBHeHHbIe YHcJIHTeJIbHbIe paCCMaTpNBaOTc C ODHORO paKypca - B pAMkax ΦyHKIHOHaJIbHO-CTpyKtpyHO KJIaCCNΦHKaIINr [6; 9]. IomHMo 3TOrO, OTDeJIbHbIe BblBnRaembe B IaHHbIX Tpydax IIOJooKeHHra (CM. [8; 10-12]) BBI- 3bIBaOH TBOIIPOCSBI (TJIaBHbIM H3 KOTOpBX RaBJYeTcRA paIPpeJeJIeHHe YHcJIHTeJIbHbIX BHyTPN KJIaCCNΦHKaIINr) H Tpe6ByOT KpHTNHecKOrO aHaJIIN3a. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,557,811,651]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Болбшннсво ще Лингьнстческнх HCCJIeIOBaHиЧСЛNTeЛьНБИ XKТансКOrOЯзka HOCaT pa3po3HeHHbI xapaKTeРи ПpeДCTaBJIaHOT co6oN obocOJIeHHbIe 6JIOKNHΦopMaIIHN, KOTOpbIe Mbl IOCTaBnIH IeJIbIO CO6paTB, O6OJIHTb N KOMIIJIeKCHOIIpoHaJIIN3HpOBaTB, IIpeДIOXKNB cBOH peIIeHnI CIIOPHBIX BOIIPOcOB.Прп этOM BaЖ-HOH 3aДaчeн HabToIe pa6OtBI YIopЯIOUHBAHNe TepMHHOJIOrHYeCKORO aIInapaTa. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[168,652,811,733]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"По опpeДeJIeHио B.И. Гореява, КОЛичecТВeHьie чсJInteЛьныe (基数词ji shuci) оБОЗнayaOT «OTBЛeueHьie чсЛЯ Илк OКЛичecТВОpyeДмETOB» [13, c. 43].ЭTO co3ByuHO KHTaIckHM oрpeДeJIeHЯM: «КОЛичecТВeHьie чсJInteЛьныe оБОЗнayaOT BeJINuHHyЧсЛЯ» [14, c. 28]; «KOLINчecТВeHьie чсJInteЛьныe уka3bIbaIOTHa 3NaueHHe ChsJa I O6OЗнayaeMoE ИМ КОЛичecТВО» [15, c. 115]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,733,811,812]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"KHTaicn J3bIKoBeJ IIO IIIN IIIIIEeT O TOM, YTO cyIeCTByeT Dba Bua KJIaccNcKaIuIyIeJIbHbIX: yHKIIOHaJIbHaN CtpyKtypHaN [8, c. 123]. OHaKO Mbl CHTaem, YTO BILJeIAYB BVOTJeJIbHbIN BND CJJeIyET N CMEIIaHHYIO yHKIIOHaJIbHO-CTpyKtPyHyIO KJIaCCNΦHKaIIHIO. DaHHbIe TpN BnJa KJIaCCNΦHKaIIHOHHbIX IOXIOOB, BOCHOBY KOToPbIX IIOJOKeH MeToI DeIyKIIHN [16; 17, c. 255], 6yIyT paccMOrpeHb IN HAcTOnIIe ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[173,77,201,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"176 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[430,76,549,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"JI.JI.BAHKOBA ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[201,832,507,847]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 3Jecb Hn daJIe HepeBoH Ha ypcckn IaIbIK haII. -J.I.B. 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","text_format":null,"text_level":null,"type":"text"},{"bbox":"[359,177,650,193]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Функуновальна Класснkaцья ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[183,198,826,382]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CorJIacHO TpaIINIOHHOJIaCCNΦHKaIINI IO FOyHKIINM, KJIacc KOJIInueCTBeHHbIX YHCJIInTeJIbHbIX DeJIInTCra Ha IBa IIOKJIaCCa: KO3ΦΦHIIeHTHBIE (系数 xi shu)и pa3- pIaIbIe (位数 wei shu).K KO3ΦΦHIIeHTHBIM YHCJIInTeJIbHbIM,KaK IIpaBnJO,OTHOcT 零 ling 'HyIb', 一yi 'OiIN', 二 er 'IbA', 两 liang 'IbA', 三 san 'TpN', 四 si 'YeIbpe', 五 w'u 'IpytB', 六 liu 'IIEcTB', 七 qi 'CEmb', 八 ba 'BOceMb', 九 jiU 'JEBytB', 十 shi 'IdecTb' [15, c. 115; 18, c. 133-135]. Pa3pIINbIMN YHCJIInTeJIbHbIMN CHTaIOTc+ shi 'IdecTb', 百 bai 'CTo', 千 qian 'TbICya', 万 wan 'IeCSrTB TbcIaY', 亿 yi 'CTOMMHJINIOHOB' N.T.Д. [6;9].KaK BnIHNO, KOJIInueCTBO IINΦp B KO3ΦΦHIIeHTHom IIOKJIacce KOHeHNO IpaBHO IBeHaIuaTH, IIO3TOMy eRO MOxHO IprH3HaTb 3aKpbITbIM. KoJIInueCTBO IINΦp B pa3pIINOM IIOKJIacce CTpeMHTCR K 6ecKOHeUHocTH, B CBzH C YEm OH MoXeT CHTaTbCRA OTKpbITbIM.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[183,385,826,549]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"HeKoTOpbIe HCCJIeIOBAteJIH BKJIIOUaIOT B IIOJIKJIacC pa3pIINbIX YNCJIInTeJIbHbIX TaKKe 十万 shiwan 'cTO TBICaY', 百万 baiwan 'MHJIIOH', 千万 qianwan 'DecaTb MHIJIHOHOB', 万亿 wanyi 'TpNIIOH' [15, c. 115].MbI He corJIacHbI c TaKoI IIOHNIIeH, H IIpuHHoN TOMy ABJIaETcA YeTBIpexIO3NIIOHbI XapaKTep KHTaIckOc HcTeMbI cHCJIeHNH. B HeI peIepHBIMN TOUkAMN KaK pa3 I YBJIIOHTC pa3PdIHbIE YNCJIInTeJIbHbIe WAN 'DecaTb TBICaY', 亿 yi 'cTO MHJIIOHOB' N T.Д. O6O3HaIoIIne IpoMeKyuToHbIe pa3pIbI, 十万 shiwan 'cTO TBICaY', 百万 baiwan 'MHJIIOH', 千万 qianwan 'DecaTb MHIJIIOHOB' B TAkOM CJUYae He YBJIIOCTcA MoCtOaTEJIbHbIMN pa3pIINbIMN YNCJIInTeJIbHbIMN H IIpeIcTaBJIAOT co60I KOHCTpyKtBi, COCTOJIuHE H3 coYeTaHHMOPfem.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[183,551,826,715]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C KO3ΦHIIHeHTHBIM IOIKJIaCCOM KOJIHueCTBHeHHbIX YHCJITeJIbHbIX C83Ha Hpo-6JIeMa BkJIIOUeHnB HeIg HJIA. TaK, JIN IOHMHN (李宇明 LiYuming) HCKJIIOuaET HyJB N3 YHCJa KO3ΦHIIHeHTHBIX YHCJITeJIbHbIX, OTHOCE rO K MaTeMaTHueCKHM 3HaKaM B HePOrJIHΦHueCKo 3aIINc³ [19, c. 7-8]. C HNM coJIHApHbI C80 TOnJxH N JIn IHHx [10, c. 34]. OJHaKO Mbl IOJIaRaEM, YTO 零 ling 'HyJB' c IOJIbIM IIpaBOM MOKeT cHTaTbcR YNCJIteJIbHbIM, IOscKJIbKy o6JIaAet BCeMH eRO KaYeCTBaMn. BoIIepBbIX, OH o6O3HaHaet OTCyTCTBHe BeJIHcHHbI NIN CJIyKHT OTIIpaBHOI TOOKI IIpr IIpeYncJIeHHN IIpeIMeTOB. Bo-BTOPbIX, 零 ling 'HyJB' JABJIErTc3HaMeHaTEJIbHbIM CJO-BOM 6JIaIqIapr cIIOco6HoCTn CaMOCTeJIbHO φyHKIIHOHpOBaTB B IIpeJIIOJKeHHN (结果等于零 Jiéguo dengyu ling 'Pe3yJIbTaT paBeH HyJIIO'). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[183,717,826,814]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CpeH KHTaCKHX 3bIKOBeIOB Ha6JIHOJaOTcpa3HOJAcN TaKKe IIO IOBOy OIIpeJIeHHKJIaCCnHkauHNOHOrO cTaTuCa YncJIInTeJIbHOrO + shi 'IecTb' B paMKax fHyHKIIHOHaJIbHO KJIaCCnHkauHHKoiJIueCTBeHHbIX YncJIInTeJIbHbIX. He BCE HCCJIeIOBaTeJIN IIpIePJKHBaOTc MHeHHN, YTO OHO YbJIeTc Ko3ΦΦHnEHTbIM, HApnpMep CraO TOnK3H N JIn INuK [10, c. 34]. CyIeCTByET n Dpyra ToUka 3peHHN, cToPOHHKamN KOtOpOH MoJHo Ha3BaTb Yxu I3cn, YkaH BnH (张斌 Zhang Bin) n JIo IOxya ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[351,76,657,89]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ФУHKIиОHAЛьнА,СТPyКТYPHА... ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[796,77,825,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"177 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[216,832,722,847]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 ΘOT BapHaHT KJIaCCHΦHKaUHH KOJIInueCTBeHNbIX YIcJIHTeJIbHbIX 6yIET paccMOrpeH HNKe. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[169,116,811,181]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(刘月华Liu Yuehua), cHTaHOIIHX, YTO + shi 'DecaTb' OINOBpeMeHHO RAJIeTcH IN KO-ΦΦHIIHeHTHBIM I pa3PraHbIM YHCJIHTeJIbHbIM, IOTOMy YTO OHO 3aHNMaET o6e II03Iu: 十万 shiwan 'cTo TBicay' N 二十 ershi 'IbAuaTb' [15, c. 115; 18, c. 133-135; 20, c. 45]. DaHHa TOnKa 3peHnIpeCTaBJIeTcH aM HAnboJIee IIprHemJIoMoi. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,181,811,249]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CraOToJHnJIHnHnKoIJIaIaIOT,HTOHeKOToPbIe pa3pIaIIbIe YncJIInTeJIbHbIE MOrYt BbICTyIaT BpoJIIN KO3ΦΦHIIHeHTHBIX,3aHHMaI IO3HIIIO HepeI万wan‘DecrTBbICr'N亿yi‘CTO MmJIInHOB':万亿wanyi‘TpHJIInOH',十万shiwan‘cTO TbICr' [10,c.37]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,250,811,456]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B cBOIO oupeIb, uHcJIHTeJIbHOe千 qian 'TbICa' H B HeKOToPbIX cJIyAix YHcJIHTeJIbHOe 万 wan 'DecaTb TbcIa' He MOryT IIOABJIaTbcR B KOaΦoHIIHeHTHOI IO3HIIHN. Te coUeTaHHa HEPoJIHΦOB, B KOTOpBX 千 qian 'TbICa' HaxOHTcRA Ha IepBOM MeCTe, He JBIaIOTc YHCIOBBIM 3aINCSaMn (HaIPHMep, 千万 $x =$ 千 $\\mathbf{x} +$ 万 $x)$ . B Tex CJYuaX, KOrIa 万 wan 'DecaTb TbcIa' 3aHIMaeT KOaΦoHIIeHTHyIO IO3HIIIO IpeEi万 wan 'DecaTb TbcIa' B pa3pIHOI IO3HIIHN HJIM MeHBIM pa3pIIBIM YHCJIHTeJIbHBIM (HaIPHMep, 千 qian 'TbICa'), 3TO TaKKe He JBIaETc YHCIOBoI 3aINCSbIO. B CBA3N C OTHM CNo TQWKH IN JIn IHHK3 HCKIOAHOT IN 3 pIAOB YHCJIHTeJIbHBIX TAkne cIOBa, KaK 千万 qianwan 'HeIpeMeHHO', 万万 wanwan 'BO Yo TO HN cTAJO' N万千 wanqian 'Macca, MHOJceTBO; Heo6buaHIO pa3HOO6pa3HbI, KpaIHe H3MeHUNBbI' [10, c. 37-38]. Tem He MeHee UHCJIHTeJIbHOe 万 wan 'DecaTb TbcIa' BCE-TaKHIOHHPOBaTb KaK KOaΦoHIIeHTHOe, 3a HCKIOueHHem BblIEOIIHCANHbIX cIJyaeB.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,457,811,504]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"HeoH03HaHbI IOxOJ K paIpeJIeHHIO YHcJIHTeJIbHbIX IIO yHKIIHM IIpeJIIOXJI JIN IOIMHN. CoJIaCHO erO KJIacchΦHkaIHH, BCE KOJIInYeCTBeHHbIe YHCJIH-TeJIbHbIE DeJIATcR Ha TPh IIOKJIacca: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,505,811,678]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"- Πифпь (数码 shùma): 一 yī ‘оДин’, 二 èr ‘Два’, 三 san ‘Три’……九 jiǔ ‘Дe-BЯть’, 十 shí ‘Дecять’;\",\"pa3pIbI (数位 shiwei): 十shi‘IecTb', 百bai‘cTo', 千qian‘TbICHa', 万 wan 'IecTb TbcIy', 亿yi‘cTO MIIJIHOHOB';\",\"MaTeMaTHUeCKHe 3HaKN B HepoRJIHΦHueCKo3aIIHcN (数符 shifu): 零 ling 'HyIb'; 分 $f e n$ 'uaCTb,ДоЛЯ'и cTpyKtypHaJy auctHua之zhī (cIyJkaTДЯ HEPoRJIHΦHueCKo 3aIIHcN obbyHoiДpo6h, coeINHЯЧИСЛHTeJIb И 3HaMeHaTeJIb); 又you‘H,Дa' (cIyJKTДЯ HEPoRJIHΦHueCKO 3aIIHcN cMeIIaHHoДpo6h, coeINHЯпeIyUo И дpo6-HyU OuaCTH), 点di'an'TOчka'(В DeecrTNUHoiДpo6h), IIpeΦHKc IopraIkoBOrO YHcJIH-TeJIbHOrO第di[19,c.7-8]. \"]","page_idx":"3","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[169,681,811,743]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Данньий пождов Вьзбьает рял замechи. Bo-первьх, Инфраим можно Ha3Baыт He ToЛько пereчincJIeHHbIe ЛИ IOйMHOMВ первом пдКласс КоэфншечСЛHTeBJHbIe, HO IPAЗрДнБIEЧСЛNTeJIbHbIe, KOТОрБIу Hero Ha3bIBaIOTсЯ pa3рДaMH (cM. Второй пождд). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,744,811,847]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bo-BTOPbIX, IocJIeHNII IOKJIacc (MaTeMaTHueCKHe 3HaKN B HepoRJIHΦHueCKO3aIINCH) BBNdy cBOe pa3HOpOJHOCTn IIpeIcTaBJIaETCscomHHTeJIbHBIM. K Hemy JINIOHMNH OTHOCHT N IpeΦHKC IopraIKOBORO YHCJInTEJBHOrO 第 di, IN KOJIInueCTBeHHOE YNCJIInTEJBHOe 零 ling 'HyJI', IN co6CTBeHHO 3aIIHCAHHbIe HepoRJIHΦaMH MaTeMaTHueCKHe 3HaKN, CJlyXaIIHe dJIa3aIIHcN Ipo6HBIX YHcEJI (分 fen 'YacTB, I0JI', 点 di'an 'ToUka', cTpyKtypHaJyAcTHNuIa之zhī), IN CO1O3 又 yòu 'H, Ia'. IocJIeHN II BCTrpeHaetc","text_format":null,"text_level":null,"type":"text"},{"bbox":"[173,77,201,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"178 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[430,76,549,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"JI.JI.BAHKOBA ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[179,116,826,244]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"TOJIbKO B HeporJINHΦHueCKO 3aIINc CMeIIaHHo Ipo6n (三又二分之一 san you er fen zhi yi 'Trn IeJIbIX (H) OJHa BTOpaJ), HO He oTpaKaetcB 3aIINCH IIHΦpamN (3½), BCJIeICTBHe Yero He MoKET cHTaTBc MaTeMaTHUeCKHM 3HaKOM. Mbl IIOJaAeM, YTO BbIeJIaTb IIOJIaIacc «MaTeMaTHUeCKHe 3HaKN B HeporJIHΦHueCKO 3aIINcH» HeIeJIeCOO6pa3HO, IOTOMy YTO IpeHkC第 di He ABJIaETCa MaTeMaTHUeCKHM 3HaKOM, 零 ling 'HyJIb' cJIeIyET OTHeCTH K IIOJIaCCy «IINΦpbI» B IpeJIlaRaEMOM KJIaCCNΦHKaIIHOHOMpacIIpeJIeHHN, a cJOBa 分 fen 'YacTb, IIOJIa', 之 zhi, 又 you 'H, Ja', 点 dian 'ToUka' YHCJIInTEJIbHBIMH He ABJIaHOTc4'.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[183,247,826,310]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"JIIOIINOnpeJJIa KJIacchHcHKaIHNO JIN IOIMnHa KaK cTpykTypHyO (MOPeMHyIO) [8, c. 124]. OndaKo cTpykTypHa KJIacchHcHKaIHNa DeJIHT YNCJIHTeJIbHbIe IIO KOJIInueCTBy BXOJIIHX B HNX MOpΦeM, a B DaHHOM DeJIeHHN peYb HJeT O ΦyHKIIHX YNCJIHTeJIbHbIX, IIO3TOMy MbI cHTaEM ee ΦyHKIIHOHaJIbHoI. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,325,687,341]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CtpyKtpyHn (MopfemHa) KJIacchfKaun ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[184,347,826,734]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CoJIaCHO cTpyKTHpyHOH (MOPΦeMHoH) KJIaCCNΦHKaIHH (HNOrga ee Ha3bIBaOT cJIOBOO6pa3OBaTEJIbHOH [20, c. 204]), B OCHOBY KOToPOH IOJIooKeH NO KoJIInueCTBY MOpΦEM IIOPIAky HX IOJIIOKeHNA, KJIACC KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX IIOpa3JeJIeTcRA Na DBA IOJIaACCA: IIOcTBie5 (简单数词 jiandan shuci) IN CJOxHbIE (复合数词 fuhé shuci). IIOKJIaCC IIOcTBIX KOJIInueCTBeHHbIX YNCJIInTeJIbHbIMN (系数词 xi shuci); IOIJKLIACC CJOxHbIX KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX COCTOHT N3 DBYX rpyIIIN: KO3ΦΦHnEHTBIX IN pa3-pyIHbIX (系数词 xi shuci + 位数词 wei shuci) [9; 15; 21]. ΘTo 6a3OBoE cTpyKTHpyHOe DeJIeHne KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX. B KJIaCCNΦHKaIHHX KNTaIcKHX HCCJeIOBaTeJIe H E yUHTbIAeTcR IIOpIIOK CJIeIOBaHHa pa3pIIOB B CJOxHbIX YNCJIInTeJIbHbIX, XOTB H NEKOToPbIX CJIyAax (ot 11 do 19) MoKET 6bITb H HAO6OpOT: pa3pIINa H KOIInueCTBeHHa JAcTb + shi 'Iecr' IIpeIIeCTByET KO3ΦΦHnEHTHOH (位数词 wei shuci + 系数词 xi shuci)6. B KaueCTBE IIprMEpa IIprHEm KJIaCCNΦHKaIHHO YNCJIInTeJIbHbIX YJXao IIIHKaI (赵世开 Zhao Shikai), B pAMkax KOToPOH ON IIpeIIOxHI KJIaCC KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX DEJIHTB Ha IBA IOJIKlaccs: IIIOcTBie KOJIInueCTBeHHbIX (简单基数词 jiandan jI shuci) IN CJOxHbIE KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX (复合基数词 fuhé jI shuci). IIOKJIaCC IIIOcTBIX KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX c ToUKN 3peHN HfYHKIIHN DEJIHTCRA Ha IBE rpyIIIN: KO3ΦΦHnEHTBIX (系数词 xi shuci) IN pa3pIINHie (位数词 wei shuci) (IIprMEpbI cM. BblIe B φyHKIIHOHaJIbHO KJIaCCNΦHKaIHH). B IOJIKlaccce CJOxHbIX KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX TAKKe coCTOAT N3 3JIeMeHTOB IByX rpyIIIN - KO3ΦΦHnEHTBIX IN pa3-pyIHbIX [22, c. 180] - 6e3 yka3aHHa IIOpaIKA IN CJOeIOBaHHa (十三 shisan 'TrpHa-ⅡaTb', 三十 sānshi 'TprNiTaTb' IN T.П.). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[351,76,657,89]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ФУHKIиОHAЛьнА,СТPyКТYPHА... ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[796,77,825,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"179 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[184,760,825,785]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4B DAHHOM cIyae 3TN MMOI03NaHbIe cIIOBA IpeIcTAbJIHOt CO6Oi cyIeCTBHTeJIbHOe 分 fen \"auctb,doJIa\", cIJyKe6HOe cIIOBO (ctpykTyprHaa cIaTHIIa atPb6yTbHbOH IIpPrOdb)之 zhi,coHO3 又you'H,Ia', cyIeCTBHTeJIbHOe 点 dian 'TOUka'.","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[220,785,615,799]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 B ypcckom 33bke BCTpeaetc TepMHN «IeepBoo6pa3HbIe» [20, c. 193]. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[188,799,823,822]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 Bo3MOxHO, 3TO C8Ba3aHO CO CBOHCTBeHHbIM KHTaHCKOMY 3bIKy OIIyUHeHHEM BO MHOrNX KOHTeKCTAX YHCJIH-TeJbHORO - yI 'OINH', YTO BIAEETcRA peaJIH3aIHNe IPNHIIHa 3KOHOOMHN B 3bIke. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[186,822,823,846]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7 B ΘΟΙ Κλαςηφικαίουν ηειλήτελβόν έρορασελεγούτερ Ακόληχειδεν Ενρόμακονθειν Στέληθειν [22, c. 180]. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[169,115,815,404]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CooTHoIIeHHe KO3ΦΦHIIeHTHO H pa3pIHOJ yAcTeB CLOXHbIX YHcJIHTeJIbHbIX yHtBIAeTcB MOpΦeMHo KJIaCCnΦKauHH B.I. TopeJIOba, KUla IIO6abJIeHo DeJIeHne IIOKJIacca CLOXHbIX KOJIueCTBeHHbIX YNCJIHTeJIbHbIX Ha aIITNBHyO IMyJIbTIINIKaTHBHyO rpyIINb. B ee pAMkax IPOCTbIMN cHTaIOTcOJHomOPΦeMHbIE YHCJIHTeJIbHbIe, O603HaayAOIIe YNCJIa OT OJHO TIO DecrTH. CLOXHbIe YNCJIHTeJIbHbIe OT 11 Do 19 o6pa3yIOTc SLOXeHHem IPOCTbIX 3JIeMeHTOB (DecrTH I OJHOmOPΦeMHoro IPOCTOrO) H Na3bIBaOTcA JINTHBHBM, IIOCKOJIbKY B 3aINCH IIΦpaMn IpeCTaBJIaOT cyMMY YNCJIOBbIX 3NaueHNN: 十三 shisān 'TrpHaIaTb' - 3TO 10 + 3. CLOXHbIe YNCJIHTeJIbHbIe OT 20 Do 90 YBLIAOTc MYJbTIINIKaTHBbIMN, IOTOMY TTO IH 3aINCBIpeCTaBJIaET pe3yJIbTaT yMHOKeHH YNCJIOBbIX 3NaueHNN (B J3bIke - IPOCTbIX MOpΦEM), OJHO N3 KOTOpBX BcerTa paBHO DecrTH: 三十 sānshi 'TrpHIaTb' - 3TO 3 × 10 [13, c. 43]. OJHaKO IIO6aBM, YTO MYJbTIINIKaTHBbIMN MOrYT 6bITb HE TOJIbKO YNCJIHTeJIbHbIe OT 20 Do 90 (KaK y B.I. TopeJIOba), HO I JIO6bIe IpyTHe, KpaTHbe IdecTn, IJIA IINΦpOBoH 3aINCH YNCJIOBbIX BbyPaJaeHH KOTOpBX Hc-IOJIb3YeTcToljBO yMHOKeHH. HaIPmEp, IINΦpOByO φOpMy 3aINCH YNCJIHTeJIbHOr三百 sanbai 'TrpHcTa'MoJHO IPOCTaBHTB BnIe BbyPajKeHH 300 = 3 × 10 × 10, KOTopoe IpeCIaTbJIaER c6oB IIPOHN3BeJeHH UHCJIOBbIX 3NaueHH, BbyPajKeHHbIX B J3bIke IPOCTbIMN MOpΦeMaMn. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[173,405,813,764]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Heo6xoIIMO 6paTHTB BHHMaHHe Ha TO o6cTOJTeJIbCTBO, YTO B KJIaaccnΦHKaIIHN B.I. TopeIOBa He OIppeJeIeK KJIaCCNΦHKaIIHN CtaTyc TaKHX CJOXHBIX YHCJIINTEJBHbIX, B IINΦpOBoI 3aIIHcN UHCIOBBIX BBIPAKeHNI KOTOpBX IIpNCYTCBYOT N CJOxKHeHne, H yMHOxKHeHNE: HApPIMep, 三百五十二 sanbaiwushi'er 'TpncTa IYTBDecaTBA' -3To $(3\\times 100) + (5\\times 10) + 2$ .KHTaIckAJIINHBrHcTNUeCKAraPiHIIINOTHOCHNT HX KAaIITNBHomU TIIpy [6; 24, c. 191]. OHaKO, HxCxOJa I3 ToRTO, YTO IOIO6HbIE YHCJIINTEJBHbIE 6pa3yIOTcIIOpeCTBOM CIOxKeHNI EJIeMeHTOB, IINΦpOBAI 3aIIHcB KOTOpBX, B CBOIO OuepeIb, IpeIcTaBJIeHa C IIOMOIIbU yMHoXeHHN, HX Heo6xoIIMO IIpoTHBOIOCTABHTTEM aIINTNBHbIM YHCJIHTeJBHbIM, EJIeMeHTBI KOTOpBX IIPOCTBI IN Hepa3IOJIKMbI (十三 shisan 'TrnHaIATb' = 10 + 3). IIOaTOMy IpeJIlaRaem rpyIINy CIOxKbIX AIDHTNBbIX KOJIuYeCTBeHHbIX YHCJIHTeJBHbIX IIOpa3IeJIaTb HA IIOIpyIIIY EJIeMeHTAPbIX aIINTNBHbIX IIOIpyIIIY CMEIIAHbIX aIINTNBHbIX (cm. Ta6JI. 1). YTOHNMI pIH NTOM, YTO B ycIOBHX HaJIuHry yMHOxKeHNI B IOIpyIIIcMEIIAHbIX aIINTNBHbIX YHCJIHTeJBHbIX IINOPHTEHTBI CTAtyc IIOJIyaeT HMeHHO CIOxKeHNe HA TOM OCHOBAHNI, YTO IIPN pa36HeHH N IOIO6bIX CIOXHbIX YHCJIHTeJBHbIX HA KOMIOHeBTs CHaHaJIa BVIDeJIaOTcTc Te, KOTOpBIE IIOJIyHeBI IITcMeCIOxKeHHN, a 3aTeM yJke Te, KOTOpBIE IIOJIyHeBI B pe3yJIbTaTe yMHoXeHHN. 3TO XopoIIIO BnIHNO IIpn OKpyJIeHHN YnCeJI, KOrDa ONyckaETcTa Yaactb, KOTopAR B 3aIIHcN IIOΦpAMN IIpNCoeIHHeHa K IIpeBdIyIe NOCpeICTBOM CIOxKeHHN. HanpIMep, YHCJIHTeJBHoe 三百五十二 sanbaiwushi'er 'TpncTa IYTBDecaTIBa'B pe3yJIbTaTe OKpyJIeHHN MOKeT IIOJIyauT BnD 四百 sibAI 'YeTbipeCTa HIn三百多 sanbai duO '6oJIee Tpexcot' HIn三百五十多 sanbaiwushi duo '6oJIee Tpexcot IYTHdeCteTNaHa OCHOBAHNN IINΦpOBoW 3aIIHcN $(3\\times 100) + (5\\times 10) + 2$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[173,77,201,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"180 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[430,76,549,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"JI.JI.BAHKOBA ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[169,787,810,822]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"8 CoJIaCHO HcCJIeIOBaHHIO, IPOBEJIeHHOMy B. KOMPn, B cHCTeMaX cHcJIeHHN 172 a3bIKOB H3 npOHaJIIN3HpoBAHHbIX 196 (TO coCTABJIeT IPrH6JIIN3HTeJIbHO 88%) Iprn o6pa3OBAHHI YHCJIInTeJIbHbIX OJHOBpeMeHHo 3aJIeCTBOBAHbI H MUYJIbTHIIINKATNBHbI N aAJITNBHbI CIIco6bI [23]. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[169,822,810,846]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"9 3decb MoKHO IPOBecTH aHaIIOHHo CO CJIOAMH B 6yKBeHHO 3aIHNC, IIpH pa3JIOKeHHN KOtOpbIX Ha coCTabHbIe YAcTH ChauJIa IIpoHCXoIHT DeJeHHe HA CJOH, A NOTOM Ha 6yKBbl. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[206,151,806,288]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0863/becc927f7fe248492f89215e38b12ebfce5a767058aceb456a95a1439290cc49.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":"<table><tr><td>Пресные</td><td colspan=\"3\">Сложные</td></tr><tr><td rowspan=\"2\">Коэффишиентные\n一 yī ‘ODINH’, \n二 èr ‘ДBA’, \n三 sān ‘Трн’…….\n九 jiǔ ‘ДЕВÀТ’, \n十 shí ‘ДECÀТ’</td><td colspan=\"2\">Адптуньibles</td><td rowspan=\"2\">Мульшлилкатоньibles\n二十 èrshí ‘ДВаДцаТ’, \n三十 сānshi ‘Трпд-\nцать’…….\n八百 babαι ‘ВOCмь-\ncot’, \n九千 jiǔqian ‘ДЕВÀТь\nТысяч’</td></tr><tr><td>Элем entapны\nадптуньibles\not 11 до 19\n十一 shíyī ‘ODINH-\nнадцату’…….\n十九 shíjiǔ ‘ДЕВÀТ-\nнадцату’</td><td>Сmeшаны\nадптуньibles\n九十三 jiǔshísān\n‘ДЕВЯносту Трп'\n二百二十二\nèrbαι ‘Èrshi’ èr ‘ДBE-\nсту ДВаДцату Два’</td></tr></table>","table_caption":"[\"Ta6JI.1 \",\"CtpyKTypna (MopΦeMHa) KJIacCSHnKaIIHn KOJIInueCTBeHHbIX YHcJIHTeJIbHbIX \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[184,305,826,793]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CyuuctByet TaKke ctpyKtupHa KJIaccnФнKaIIHcIN Φyn (邢福义 Xing Fuyi), H3JIOKeHHa B Tpye «TpncTa BOIPOcoB KHTaIckOg rpaMMaTHKN» (汉语语法三百问) [25]. CoJIacHo eN, MeTaKJIacc CnCJIteJBHbIX DeJIITcH a KJIaccBbI O6IIHX (统数 tong shu) HIOpRAKOBbIX (序数 xshu). CpeIN IepBbIX BbIeJIaOTcRcIeIyIOIIHe YcTeBpe IOLJIACCA: KOJIueCTBeHHbIE (基数 ji shu), KpaTHbIE (倍数 bei shu) (三倍 san bei 'TpexKpaTHbI'), Ipo6HbIE (分数 fen shu) (三分之二 san fen zhi er 'DBe TpeH; 三点四 san dian si 'Trn IeJIbIX (H) YcTeBpe IecrTBIX) H IIpN6JIIN3H TeJbHbe YNCJIteJBHbIE (概数 gai shu) (二三米 er-san mi 'IbA-TPn MeTpa'; 十余次 shi yu ci '6oJIee Idecrtn pa3'). BTObIe IeJIATcH a IBa IIOJIACCA: KOJIueCTBeHHbIE YNCJIteJBHbIE c IpeΦHKcOM 第 di (第 di + 基数 ji shu) (第一 diyI 'IepBbI') H IOpRAKOBbIE YNCJIteJBHbIE BΦOpMe KOJIueCTBeHHbIX YNCJIteJBHbIX (基数形式 jI shu xingshi) (一号 yi hao 'IepBoe YncIoo'). B pamkax 3ToH KJIaccnΦnKaIIHN IIOJIACCC KOJIueCTBeHHbIX YNCJIteJBHbIX, B CBOIO OUpeIb, DeJIITcH a Tprn IpyIbI: YNCJIteJBHbIE pa3PraJa eINHH (个位数 gewei shu), paHROBbie YNCJIteJBHbIE (段位数 duanwéi shu) H CIOBOOCoTeAHN C YNCJIteJBHbIM (数词短语 shuci duanyu) [25, c. 146]. Oco6eHHocTAM N OINCAHHO KLIACCnΦnKaIIHN YBJIeTcR BBOHMAR CINH Φyn HOBAI TepMHHOLOTN BMeCTO o6IIeIPnHToH. HanpImep, IJIA O6O3NaueHNG KoEΦnIHNEHTHBIX YNCJIteJBHbIX NCIOJB3yeTc TepMHN 个位数 gewei shu BMeCTO IPINBvHOrO 系数 xi shu, a IJIA HANMeHOBAHN pa3PraIbIX - 段位数 duanwéi shu BMeCTO 位数词 wei shuci. Kpome TOTO, IIOPTNBOIOCTaBJIeHN He YNCJIteJBHbIX IIOHCXODHT He IIO IpiNHIUY «KOJIueCTBeHHbIE vs IIOPAIKOBbIE), a IIO IpiNHIIIY «O6IIe vs IIOPAIKOBbIE), n TaKIM o6pa3OM BBOITcR eINH KLIACCnΦnHOHH bYPOBeHb. YTO kacaETc R camx KOJIueCTBeHHbIX YNCJIteJBHbIX, TO IpeIIOKeHHa IpyIIHPOBKa B IOIDKIAcc EKOJIueCTBeHHbIX YNCJIteJBHbIX (YNCJIteJBHbIE pa3PraJa eINHH (个位数 gewei shu), paHROBbie YNCJIteJBHbIE (段位数 duanwéi shu), CIOBOOCoTeAHN C YNCJIteJBHbIM (数词短语 shuci duanyu)) HjXdAETcR B DAIBHeIIe IeTAIBHOI IPOPabOTke. IepBbIE ZBE rpyIIbl - He YTO HHOe, KaK IIOPCtBie KOJIueCTBeHHbIE YNCJIteJBHbIE. IIOSeIeHHa rpyIIHa - CIOBOOCoTeAHN C YNCJIteJBHbIM - IpeCIaTaNJEET B3rJlAnd CINH ΦHyN Ha To, YTO B HAcToIe pa6Ote TpaKTyETcKak CIOXHBeK OJOUeCTBeHHbIE YNCJIteJBHbIE.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[184,796,828,843]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"OHa H3 IeHTpaJIbHbIX IIpo6JIeM H3UyeHnKoJIHueCTBeHHbIX YIcJIHTeJIbHbIX 3a-KJIIOuaETcB OIIpeJIeHnTOrO, YEm JBIJHOITcCIOXHbIE YIcJIHTeJIbHbIE (HaIIpHMeP, 三十 sanshi 'TpNIaTb', 十三 shisan 'TpINaIaTb', 五百五十二 wUbaiwushi'er ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[351,76,658,89]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ФУHKIиОHAJIbHA,СТPyКТУРHA... ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[796,77,821,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"181 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[169,116,813,263]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"「IITbcOT IITbIdecT IBa）：cIOBAMH（复合数词fuhé shuci）HIN cIOBOOCOeTAHHMn (数词词组shuci ciü).CTopoHHNKamn IIprH3HaHHN IOIO6HbIX YHCJIHTeJIbHbIX CIOJX-HbIMN cIOBAMn JBIHOCTC SoI GouK3HN JIN INHJX3, OIIpeJIeJIHOIIne Hx KaK «cIOJX-HbIE cIOBA, IIOCTPOEHHbIe IIO aTpH6yTHBHOJ MOJIIN》(偏正复合词pianzheng fuheci)[10,c.37]10.HcxOJa H3 IIprHHIIIOB pa3JIHyEHHN cIOB H cIOBOOCoTeAHN B KNTaIKCKOM 3bIke, H3IOJKeHHbIX B.M. H.B. CoJIHIIeBBIMN [26,c.51-84], Mbl TaKKe cHtAem, YTO 3TO cIOBa, a He cIOBOOCoTeAHN11.O6OCHOBaHHem JBJIeTcra To, YTO KaJXIOe TaKoe cIOBO COcTOHT H3 ODNHOJbIX 3JIeMeHTOB, CNHTaKChueckn $\\phi$ yHKIIHOHHpyIOHx KaK eDHHOE IeIOe.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,263,811,384]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Дя сраьння: Дрбные coуетаня, тakhe kaK三分之二 san fen zhi er ‘Дытпгьх’и 五点九 wu dian jiu ‘ПЯТь ΚЕЛБИХ И DeВЯТь DecaTbIX’, coJIepka TJIeMeHTbI 分fen ‘чacTB,doJIЯ’, 点dian‘TOUka’и 之zhī, ПОТOMy MOrY 6bITb OIIpeJeHbI KaK CJIOBOcoTeaHnI, ПOTOMYЧTO КРOME ChIJIteJIbHbIX OHH COCTOrt H3 cyIeCTBHTeJIb-Horo(分fen‘чacTB,doJIЯ); 点dian‘TOUka’)и CTpykTypHOJчaTnIbI(之zhī). BcJIeI 3aД.СИИнаво[20,c.204]ДЯIx obO3HaueHnI 6byem HcIIOL3OBaTb TepMHnCocma6hbIe YucIoobIe bIpaJceHnI. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[294,397,685,413]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ФункионаЛьно-структурна Класснфкадья. ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[169,418,811,678]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B 1982 r.Чжу Дэси 6ылла пеДLOЖЕHA DeТАЛьная KJIaCCNФИКаИЯчСЛNTeЛь-НБИX - ФункIHОHAJIbHO-cTpYKtypHЯ $^{12}$ .В Hei KJIaCC KJIInueCTBeHHbIX ChCJI NTeJIbHbIX DeJIINTCЯ Ha ПОДКJIaCCbI co6cTBeHHO YnCJI NTeJIbHbIX (数词shuci)ИЧСЛNTeJIbHbIX KOHcTpyKIIIM (数词结构shuci jiégou).ПерБИй ПОДКJIaCC,В cBOIO OUEpeIb,coCToHT N3 TaKHX rpyIII, KaK: KoФФHIIeHTHBIE YnCJI NTeJIbHbIE (系数词xi shuci),pa3PraIIbIe YnCJI NTeJIbHbIE (位数词wei shuci)ИпriбЛЗNTeJIbHbIE YnCJI NTeJIbHbIE (概数词gai shuci) (ПрIMeРысm. ВИПЕВ CTpyKtypHо KJIaCCNФИКaIIIN).Bo BTOPOI ПОДКЛaCC BКЛIOUeHBI rpyIIIbI:1)KoФФHIIeHTHO-pa3PraIIbIe cTpYKtypbl (系位构造xiwéigouzao)(В cTpyKtypHо KJIaCCNФИКaIIIN-MyJbTHIIINKAATHBHbIe YnCJI NTeJIbHbIe:二十ershi‘ДВaIITb',五万wuwan‘ПЯТьДecrT TbcIcY);2)KoФФHIIeHTHO-pa3PraIIbIe KOM6HHaIIIN (系位组合xiwéizuhe)(ИЛN CMEIIAHHbIe aДЛNTNBHbIe YnCJI NTeJIbHbIe:三千五百 sanqianwubai「TPn TBICaI INIITbcOT);3)cIoxHbIe KOrФФHIIeHTHO-pa3PraIIbIe cTpYKtypbl (复杂的系位构造fuzá de xiwéigouzao)(coCTO7R H3 IIprNo3NTeJIbHBIX YnCJI NTeJIbHbIX,KoФФHIIeHTHO-pa3PraIIbIX cTpYKtypИЛN KOM6HHaIIIN:两三千 liangsānqian ‘ДBE-ТРи TBICaIIN’,二十万ershiwan‘ДВecSTN TBICaY)[21,c.45-47]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,678,811,774]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"KpnTHKy B3rJlIbIy Jky Iocn, JIO PInH PIIHIEET, YTO $\\ll$ KJIacchNfHKaIIHn Ha KO3-phiHHeHTbIe H pa3pIINbIe YHCJIHTeJIbHbIe IIPOBOITcC IIO3HnI rpaMMaTHKN, B TO BpeM KaB bIJeHne IIpi6JIIN3HTeJIbHbIX YNCJIHTeJIbHbIX IIPOHCXOINT C IOHOHTNHOfYHKUHOHaJIbHOI TOKN 3peHn. OChOBaHHN KJIacchNfHKaIIHn OTJIuHaOTc, YTO He IIO3BOJIeT cTaBHTb Hx Ha OINH yPoBeHb》 [8, c. 123]. Mbl He MoJem coTJIacHTbcRc C JIO IIHN, BO-IIepBbIX, B TOM, KaK OHa B DaHHOM cJlyue oIpTeJIeT KJIaccNfHKaIIHn. 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IocKOJIbky BCE 3JIeMeHTb paBHbI, TO OTHIOIIeHHM MeKJy HmH KONJIYTHBHe, CJeIOBaTeJIbHO, CIOOCoOM CIOBOO6pa3OBaHHA JBJIETcra CIOBOCIOJOKeHne. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[205,821,620,834]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"11 B aHJIINKOM J3bike, HAnpoTHB, 3TO cJIOBOOCOteHaHHa [27, c. 15; 28; 29]. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[203,834,741,847]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"12 ΘTo 6bIa IepeCmOTpeHna Bepcna eTo KJIaccnФнkaIHH YIcJIHTeJIbHbIX 1958 r. [9, c. 185-187]. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[178,151,833,479]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0180/e6410af9e4d84a43a112169b46c2db9d2d85e7cf983bba9a3114c3462240c2bc.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":"<table><tr><td colspan=\"5\">Количес委书记нычеслистовы (基数词ji shùci)</td></tr><tr><td colspan=\"2\">Өлемэнтарные количес委书记нычеслистовы (基本基数词jiběn ji shùci)</td><td colspan=\"3\">ВарнаимBERые количес委书记нычеслистовы (变化基数词bianhuà ji shùci)</td></tr><tr><td rowspan=\"2\">Коэфшиcent-hычеслистовы (系数词xi shùci) 一yi'один', 二 èr'лва', 三 san'tprn'....... 十shi'deсять'</td><td rowspan=\"2\">Раразднычес-лпеслистовы (位数词wèi shùci) 十shi'deсять', 百bái'ctо', 千qiān'tbíсая', 万wan'tdeсять тьсч', 亿yi'cto мнлл-jионов'</td><td>КоэфшиcentHo-pažядные мульштолкатнвные КOLIчес委书记нычеслистовы (系位构造基数词xiwèi gòuzào ji shùci)</td><td colspan=\"2\">КоэфшиcentHo-pažядные aДЛТНВье кOLIчес委书记нычеслистовы (系位组合基数词xiwèi zuhé ji shùci)</td></tr><tr><td>Осычные коэфшиcentHTo-раздные мульштолка-ТВВье КOLIчес委书记нычеслистовы (特殊系位构造tèshū xiwèi gòuzào) 八十bāshi 'BOCEMbDESCYT', 八万亿bāwanyi 'BOCEMb TpNi-лнонов'</td><td>Спецima-bные коэфшиcentHTo-рazдные aД-ДЛТНВье КOLIчес委书记нычеслистовы (一般系位组合yībān xiwèi zuhé) 八十八bāshibā 'BOCEMbDESCYT BOCEMb'</td><td>Спецima-bные коэфшиcentHTo-рazдные aДИ-ТНВье KOLIчес委书记нычеслистov(特殊系位组合tèshū xiwèi zuhé) 一千五六百yīqianwūliubäi 'ОДи Тьсяча ПЯтусOT-ОДи Тьсяча Шость-cot'</td></tr></table>","table_caption":"[\"Ta6II.2 \",\"Функшоньсгтчякша КласнфКацьkoJичecтВeHHbIX YaHbYhBnHa \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[183,491,826,648]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ko3ΦHIIINHeHTBHe I pa3pIbHbIe BnIbI YnCJIInTeJIbHbIX, KaK 6bIIO OIIHcAHO BbIIe, BbIeJIaOTcB pe3yJIbTaTe ΦyHKIIOHaJIbHOrO KJIacchΦHIIuPobAHnA, a IIpN6JIIN3TEJIbHbIE YNCJIInTeJIbHbIe AIBIAOTcpe3yJIbTaTOM KJIaCCNΦHIIuPobAHnHX IO TOUHOCTH, O YempeYb IIOJETB IIPOJOJIkeHH NCCJIeIOBAHnA. Bo-BTOpbIX, He MOxEM corJIacHTbcR, YTO IIpN KJIaCCNΦHIIuPobAHnKo3ΦHIIuNEHTHBIX, pa3pIbHIX I IIpN6JIIN3TEJIbHbIX YNCJIInTeJIbHbIX OCHOBAHnKJIaCCNΦHkaIaIpa3JIuaOTcR. HaIpOTNB, YxJy D'cn HcIOJIb3yETOINO KJIaCCNΦHkaIaIHHOe OCHOBAHNe - cTrpykTypy, KOtOpA y IIpN6JIIN3TEJIbHbIX YNCJIInTeJIbHbIX CIOXHaR. Ha NaI B3rIaI, IIpN paccMOrpeHH Nco3ΦHIIuNEHTHBIX, pa3pIbHIX I IIpN6JIIN3TEJIbHbIX YNCJIInTeJIbHbIX cФyHKIIOHaJIbHO-cTrpykTypHOr ToUKN 3peHHN 6blIO 6blYmecTHO He HcIOJIb3OBaTB TepMHN «IIpN6JIIN3TEJIbHbIE YNCJIInTeJIbHbIe>. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[183,649,826,743]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Tak, ot yIIOTppe6JIeHnJaHHHO TepMHa OTKa3bIbAetcra YaHb JaHbIH H CBOeM BapHaHTe $\\phi$ yHKIIHOHaJIbHO-CTpyKTypHO KJIacchФHKaIIHN KOJIHecTeBHeHHbIX YHCJIHTeJIbHbIX. BaXHBIM erO IOCTHXKeHem MOKHO cHTaTB KaTeROP3aIIIO cJOXHBIX KOJIHueCTBeHHbIX YNCJIHTEJIbHbIX, IIpNBedeHHyIO B BVNe Ta6JIHbI B [6, c. 2], c KOTopoR Mbl CHTaEM YMeCTHBIM IIO3HaKOMITb YHTaTeJI, IIpeJIOxHB PyCCKO83bIHyBIE TepMHHOJIoTHiueCKHe aHaJIOrn H IIpHMepBi (cM. Ta6JI. 2). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[183,744,826,847]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B KJIaccnФнKaIIHryaHbЯHbIINHa IpocTbIe H cIIOJHbIE KOJIHueCTBeHHbIe YHcJIITeJIbHbIe, KOTOpbIe B BbIIIEoIIncAHHO CtpyKtypHO (MOPΦeMHo) KJIaccnФнKaIIHnHa3bIBaJIncb 简单(基)数词jiandan(ji)shuci IIpocTbIe (KOJIHueCTBeHHbIe) YHcJIITeJIbHbIe' n复合(基)数词fuhé(ji)shuci CIOJXbIe (KOJIHueCTBeHHbIe) YHcJIITeJIbHbIe', Ha3bIBaIOTc 基本基数词jibenji shuci 'JIeMeHTapHbIE KOJIHueCTBeHHbIe YHcJIITeJIbHbIe' n变化基数词 bianhua ji shuci'BapNaTHBHbIE KOJIHueCTBeHHbIe YHcJIITeJIbHbIe' ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[351,76,658,89]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ФУHKIиОHAJIbHA,СТPyКТУРHA... ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[796,77,825,89]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"183 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[169,116,810,159]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"COOTBETCTBEHHO. O6bIcHTb 3TO MOxHO TEM, YTO B IepBOI IIape TepMHOB aKIIeHT cJIeJIaH Ha CTpykTpe, a BO BTOPOI - HA $\\phi$ yHKIIHX B COOTBETCTBHN C THIOM KJIacCHHII-KaIIHH. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,162,813,355]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"TyaHb YhbHn OHpaetcHa IpeJIIOKeHHoe B 1982 r. Jxky Ioc HJeHHe IO ctpykType BCEX KOJIInueCTBeHHbIX YncJIHTeJIbHbIX Ha IBe 6OJIbIIHe rpyIINbI: Ko3ΦHNIeHTHO-pa3pIINbIE KOM6HNHaIIH (系位组合 xiwei gouzao) IN KO3ΦHNIeHTHO-pa3pIINbIE cJIeDyHOIIHe pyccKo3bYHbIE aHaJIOTn: KO3ΦHNIeHTHO-pa3pIINbIE MYJBTHIIHKaTINBHe KOJIInueCTBeHHbIE YncJIITeJIbHbIE IN KO3ΦHNIeHTHO-pa3pIINbIE aIINTHB-Hbe KOJIInueCTBeHHbIE YncJIITeJIbHbIE COOTBEtCTBeHHO. O6OCHOBAHHem cJyKHT pa3JIuHne B BVdax MaTeMaTHueckHX ONEpaIH (CIOJKeHH eHH yMHOGKeHHe) C hCIOBbIM 3HaueHnMn B IIOΦPOBbIX φOpMax 3aIIHCn BbIPaKeHH, B BVNe KOToPbIX MOKHO IIpeIcTaBNTb YncJILO, Bep6aJIIN3yeMOe KOJIInueCTBeHHbIM YncJIHTeJIbHBIM. IMeHHO 3TOT fakT akIeHTnpYeTc Pn IpeBoHe Ha3BaHH n IByX paccMaTpNBaEMbIX rpyIII qhcJIIN-TeJIbHbIX IN X IOIgpryIII. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[203,356,657,370]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PaccmOTpHIM IIOIpo6Hee KaJkyo H3 BbIIeYkaaHHbIX rpyII. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,372,811,502]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ko3ΦHnHHeHTHO-pa3pIHBIE MyJBTHIIINKaTHBHBie KOJInueCTBeHHbIe YHCJIHTeJIbHbIe COCTOHT H3 Ko3ΦHnIHHeHTHO n pa3pIHOH qacteH, KOTOpBE B IINΦpoB0I φOpMe 3aIINCH Nx BBipajKeHn yMHOkaOTcI pyr Ha Ipyra. ΓaHb YHbIINHa3bIBaET HX CLOXHbIMN YNCJIHTeJIbHBIMN (复合数词 fuhé shici). B DaHHoI rpyPiNE BO3MOXHO BbIeJIITb IBe IIOIpyIbI: O6bIyHbIe KO3ΦHnIHHeHTHO-pa3pIINbIE MjIBTHIIHKaTHBHBIE KOJInueCTBeHHbIE YNCJIHTeJIbHBIE (一般系位构造 yibān xiwei gouzao) IN cIIeIIHaJIbHBie Ko3ΦHnIHHeHTHO-pa3pIINbIE MyJBTHIIINKaTHBHBIE KOJInueCTBeHHbIE YNCJIHTeJIbHBIE (特殊系位构造 teshui xiwéi gouzao). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,505,811,731]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"O6bIHyBIE Ko3ΦHIIeHTHO-pa3pIHHbIe MyJIbTHIIJHKaTHBbIe KOJIHyEcTBeHHbIe YHCJIHTeJIbHbIe IpeIcTaBIAIOH TnB BapHaTHBbIX KOJIHyEcTBeHHbIX YHCJIHTeJIbHbIX, KOTOpBIE, KaK IIpaBNILO, BKJIHOaIO TpN yPOBbMyJIbTHIIJHKaTHBbIX. B pa6oTe ΓyaHb YHbIINHa OINcHa NIO6HaN TpEXyPObHEBa MoJeIb. OJHOyOBHeBOe MyJIbTHIIJHKaTHBHOe YHCJIHTeJIbHOe, O6O3NaUaEMoe KaK A1, MoJeIHyEETcIIO PPINHIUY Ko3ΦfUuueHMna Yacmb $\\times$ pa3pOdHa Yacmb, HApNMeP, 八百 babi 'BOCEMbCOT' (800 = 8 × 100).ДByxypOBHeBOe Ko3ΦfHIIeHTHO-pa3pIHHOE MyJIbTHIIJHKaTHBHOe KOJIHyEcTBeHHOE YHCJIHTeJIbHOe (A2) MoKeT 6bITb IIpeIcTaBJIeHO KaK A1 $\\times$ pa3pOdHa Yacmb, HApNMeP, 八百万 babiwan 'BOCEMb MIIJINOHOB' (8 000 000 = 800 × 10 000). Ko3ΦfHIIeHTHO-pa3pIHOe MyJIbTHIIJHKaTHBHOe KOJIHyEcTBeHHOE YHCJIHTeJIbHOe c TpoIHoM MyJIbTHIIJHKaTHBHOcCB3bIO (A3) HMe-ET BnA2 $\\times$ pa3pOdHa Yacmb, HApNMeP, 八百万 babiWanyi 'BOCEMbCOT TpHIJIHOHOB' (800 000 000 000 000 = 8 000 000 × 100 000 000). ΘTO TNIHNHbIe MyJIb-THIJIHKaTHBHBie MOJI IN IOCTPoEHNA YNCJIHTeJIbHbIX. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,732,811,794]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B cIeHnJIbHbIX KO3ΦΦHIIHeHTHO-pa3pIINbIX MyJIbTHIIJIHKaTHBbIX KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX IBe KO3ΦΦHIIHeHTbIX YacTH (B IIOpJKe BO3paCTaHH) COBMIIeHbIC OIHHO pa3pIINo H cTpoTcI IO MoJeH (KO3ΦFuuuEHHma YacMb • KO3ΦFuuuEHHma YacMb) × pa3pIaHaY acmb13, rIe cOIOJOKeHHbIE KO3ΦΦHIIHeHTHbIe YacTH ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[173,77,201,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"184 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[430,76,549,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"JI.JI.BAHKOBA ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[203,832,482,846]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"13 3decb cHMOBJO3Haayet coIOJOKeHHe qacte. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[183,116,825,149]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IpeIcTaBJIaHOT co6oB BapHaTnBHyU OacTb: 三四万 sānsiwan ‘TpNiuaTb-copok Tbicu', 两三千 liangsānqian ‘IBe-tpN TbICaHN’. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[183,152,828,231]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ko3ΦHnHeHTHO-pa3pIHBie aIHTNBHbIe KOJIhUecTBeHHbIe YHCJIHTeJIb-HbIe (系位组合基数词 xiwéi zuhé jishuci) B KJIaccnФHKaIIH Igamma HbIHNHa IIpeIcTaBJIHO tco60B TOpYIO rpyIIy BapNaTHBbIX KOJIhUecTBeHHbIX YNCJIHTeJIbHbIX. OTHOIIeHnMaJMyJIbTHIIJIHKaTHBbIMN qacTAMN B Ko3ΦHIIeHTHO-pa3pIHOKOM6HnHaIIHЯBJHOtcaAДNTHBHbIMN[6,c.2]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[183,231,825,278]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"KoΦHHeHTHO-pa3pIHBie aIITHBHbIe KOJIHyEcTBHeHbIe YHCJITeJIbHbIe IpeIcTAbJIeHbI cIeIyIOIIHM NtBIO MoJeIMN, rIe A1 MoJeINpyeTcI IO II pIHIIpy Ko-ΦfquueHmHa yacmb $\\times$ pa3pIaHa yacmb: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[183,279,825,638]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1) A1 + A1 (ВарнаNT: A1 + Коэффншени enthaя част): 八百八 babаibā ‘BOCEMbCOT BOCEMbДecЯT’; 八十八 bashiba ‘BOCEMbДecЯT BOCEMb’. B Данно модел самоe 60Льшоe КОЛичесТВО pa3ряпов paBНо ПЯтп: 八万八千八百八十八 bawanbaqiānbabäibashiba ‘BOCEMbДecЯT BOCEMb ТБИЧ BOCEMbСOT BOCEMbДecЯT BOCEMb’, Ге самын 60Льшои pa3ряД - DeсЯт TBICЯн (万 wan); \",\"2) A1 + A2: 八亿八千万 bayibāqianwan 'BOCEMBcot BOCEMBdeCAY MHIJIHOHB'. B ΘToI MOJIeJIH YacTBIO A1 MOKeT 6bITb TOJIbKO 亿 yi 'CTO MHIJIHOHB', a pa3pIiHOYacTBIO B A2 - ToJIbKO 万 wan 'DEcAeTb TBICaY'; \",\"3) A1 + Hylb + A1 (BapnaHT: A1 + Hylb + pa3pndna qactb): 八万零八(百/十)bawanlingba(bai/shi) 'BOCEMbIeCt TBicayu BOCemb (coTeH/IecrTkoB). HaOJaIaIcIpeIHyIe pa3pndna KOJIueCTBeHHaqactb (A1) IIpeBbIIaet haoJMyoScIocJIe HyJIpa3pndHyIO KOJIueCTBeHHyIO qactb (A1); \",\"4) A2 + HUJIb + A1: 六千万零八百 liuqianwànlingbabai 'IIEcTBIIecrT MHIJIHO-HOB BOCembcot'. B KaueCTBe pa3pJHNoKoJIHueCTBHeHHoJ caTn, o6O3HaayEmoA2, BbICtYIIaOT 亿 yi 'cTO MHIJIHOHOB' N万 wan 'IeCraTb TbICr', KOTOpbIe IIpeBbIIaHT pa3pJHHyO KoJIHueCTBHeHHyO qacTb A1; \",\"5) A1 + BapHaTHBNaJyacTb: 二十一二 ershiy'er 'ДВaIIaTb OINH-ДВaIIaTb IBa', 三十五六 sānshiWuliu 'TpHIIaTb IIaTb-tpHIIaTb IIecTb', 一百七八 yibaiqibā 'ctO cEmbIeCrt-CTO BOCEMbIeCrt', 一千五六（百） yiqianwuliu(bai) 'TbICya IIaTbCOITbICya IIecTbcot', 一万八九（千） yiwanbajiu(qian) 'TbICya BOCEMbCOT-TbICya IIeBraTbcot' [6, c. 7-9].\"]","page_idx":"10","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[183,639,825,669]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"KoJIHueCTBeHHbIe YHcJIHTeJIbHbIe O6JIaIaIOT OIIpeJIeHHbIMN JINHeiHHbIMN xa- paKaTepnCTHKaMn: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[183,671,825,719]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1) rpyiHa Ka03ΦHnHeHTHO-pa3pAINbIX MyJIbTHIIJIHKaTHBbIX KOJIHueCTBeHHbIX YHCJIInTeJIbHbIX: MeHbIiH pa3pI HaxOJNTcBIIpeI, 6JIbIiH pa3pI - IIO3aIi: 五百万亿 wubaiwanyi 'IITbcOT TpHJIINOHOB'15; ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[351,76,657,89]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ФУHKIиОHAЛьнА,СТPyКТYPHА... ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[796,77,825,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"185 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[183,764,825,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":".14B daHHoI rpyiHne YHCJIHTeJIbHbIX IIpr IINΦpOBOI 3aIHCH HX YHCIOBbIX BbIPAKeHHN 3aJeCTBOBaHbI DBA BHJa MaTeMaTHueCckHX OIIepaIcN C YHCIOBbIMN 3HaueHHMn (ymHOJKeHH N cJOKeHHe), YTO, corJIacHO IppeblNyIeI KJIaccHΦHKaIHH, IIO3BOJIeT CHTaTB HX CMEIIAHbIMN aIITNTHBbIMN. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[183,797,825,847]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"15 TOBOPHTb O IOIO6HOH XapakTePCHTKe BO3MOKHO, TOJIbKO ocTaBaCb HA IO3NIOHN HCKJIIOUeHH N3 UHCJa pa3pIbIX YNCJIITHeJIbIX 十万 shiwan 'cTO TBICaY', 百万 baiwan 'MNIJIOH', 千万 qianwan 'DecaTb MNIJIOHOB' H. I. B IPOTHBOM CJIyae, 6yIET JIN6O ODNH pa3pI (KAk B I pHMepe: 百万亿 baiwanyi 'cTO TPHIJIOHOB'), JIN6o pa3pIbIb 6yIyT paioJIaRaTbc OR 60JIbIIIEr K MeHbIIEMy.","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[169,116,811,164]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2) rpyiHa Ko3ΦHHeHTHO-pa3pIHyIx aIITHBbIX KOJIHueCTBeHHbIX YHCJIInTeJIbHbIX: 6OJIbIIHn pa3pI NaXoIHTcB IpeEIN, MeHbIIHn - I03aIIN: 八千八百八十baqianbabaibashi' BOCemTbICyu BOCemBCOT BOCemDeCrt' [6, c. 8; 30, c. 348]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,165,811,225]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IIO cpaBHeHnIO CO CJIOBaMn, OTHOCaIMMncs K IpyTm YacTm peuH, OTJeJIbHbIE BnIbI KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX (KO3ΦΦHIIeHTHO-pa3pIbIe aIINITNBHbIe) XapaKTepn3yOTcra 6OJIbIIHM KOJIInueCTBOM MOpΦem. HaIIpHMeP, KOJIInueCTBeHHoe YNCJIInTeJIbHoe ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[346,227,667,243]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"六亿九千五百二十七万三千一百四十一","text_format":null,"text_level":"1","type":"text"},{"bbox":"[169,244,810,275]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"liuyijiuaqianwubai'ershiqiwansanqianyibaishiyi IIeCTbcOT DeBraHocTo IaTb MIIJIHOHB IBecTH cEmbIeCt TpN TBICAH CTO copOK OINH' BKJIOUaET 17 MOpem. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,277,811,378]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C ΦyHKIIOHOJbHO-cTpkyTpyHOn TOck 3peHHaPaccMaTpHBaOT YNCJIHTeJbHbIE TaKKe Cno TQxK3H N JIn HxK3, HcIOJIb3yprn 3tOM HOByo TepMHHOJIoTHNo. DeJIeHNHe KJIacca YNCSJIHTeJbHbIX IPOIN3BOJNTcR Ha IOJKIaCCbI OIIpeJIeHHbIX (确数 què shù) H IIpN6JIH3HTeJbHbIX (概数 gài shù). OIIpeJIeHHbIe YNCJIHTeJbHbIe (HJIN KOJIInueCTBeHHbIe) IIOJa3IeJIaHOTcR Ha IBe 6OJIbIIHe rpyIIbI: 3JIeMeHTapHbIe (基本数词 jiběn shùci) H CJIOXHbIe (复合数词 fuhé shúci). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,380,811,429]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\Gamma \\rho \\mathbf{y}\\Pi \\Pi$ 3JIeMeHTapHbIX KOJIuHecTBeHHbIX YHcJIHTeJIbHbIX DeJIHTcRA Ha K0eΦHIIeHTHbIe (系数词 $x i s h u c i)$ H pa3pIINbIe (位数词 wei shuci) IOIgpyIIbI, KaK B paMkaX $\\phi \\mathbf{y}\\mathbf{h}\\mathbf{k}\\mathbf{I}\\mathbf{n}\\mathbf{O}\\mathbf{H}\\mathbf{a}\\mathbf{J}\\mathbf{b}\\mathbf{H}\\mathbf{O}$ KJIacchΦHKaIIH. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,430,811,565]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Гуйпа сюжных чслтелых пдраимлгяетс на пдгрупьк кофнichteН-рабдных мльтейнлкатнвых колчесъевных чслтелых (系位构造 xiweigouzao) (二十 ershi 'Двацат',三百 sanbai 'Трпст', 四千 siqian 'Чтбpe Тьсян', 三百万 sanbaiwan 'Трн МИЛиога', 五千四百二十万 wuqiansibai'ershiwan 'ПЯтБесят chebipe мльллона Довсп Тьсян', 四千三百八十一亿 siqiansanbaibashiyi 'ЧтбpecTa Tprnцать BOCem мльллардов СTO МИЛиогов') И диншьх КОЛчесъевных чслтелых (整零构造 zheng ling gouzao) (十六 shiliu 'Шесту), 九十九 jiushjiu 'Девеносто Девяты', 七百六十 qibaliushi 'семсOTИsectbDecaT'). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,568,811,600]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Crao TcHJIN HcKoINcBIAOT TpN MoJeJI, KOTOpbIMM MOyT 6bITb IIpeICTaBJIeHbI aINTHBHbIE KOJIYueCTBeHHbIe YNCJIHTeJIbHbIe (整零构造zheng ling gouzao): ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,601,811,783]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1) MyJIbTHIIJIHKaTHBHaJaCTb + MyJIbTHIIJIHKaTHBHaJaAcTb $^{16}$ (JAcTei MoKeT 6bITb 6oJIbIe IByx): 三百五十 sanbaiwushi 'TpncTa IIpIbDecr', 四万三千五百 siwansanqianwubai 'copok Tprn Tbcryn IIpIbcoT'; \",\"2) MyJIbTHIIJIHKaTHBHaJa YacTb + KoOΦΦHIIHeHTHa JaCtB: 二十二 ershi'er 'IbA-IIaTb IBa', 五千零二十五 wUqianling'ershiwu 'IITb TBICrJ IBaIIaTb IYrTB' (MyJIbTHIIJIHKaTHBbIX YacTei MoKeT 6bITb 6OJIbIe IByX);\",\"3) IecraTb + KO3ΦΦHHeHTHa auctb: 十一 shiyi 'OINHHaIIaTb' [10, c. 38].\",\"B IIpeIcTaBJIeHHbIX Crao TgUxK3HOM H JIn IHHxK3 MoJeJX MyJIbTHIIJIHKaTHBHJa YacTb He pa3JIoxKeHa Ha yPoBHN A1 H A2, KaK 3To 6bIJIO cJIeJIaHO $\\\\Gamma$ yaHb JHBIIHOM pRn OIIHCaHHN MOJIeJ K0eΦHIIeHTHO-pa3PraIHHbIX aLIITINHBbIX KOJIHueCTBeHHbIX YHcJIINTeJIbHbIX. O6bAcHReTcra To TEM, YTO «HOCHTeJIH KHTaIcCKOro Ja3bIka BocIIpHHMaIOT \"]","page_idx":"11","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[173,77,201,89]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"186 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[430,76,549,89]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"JI.JI.BAHKOBA ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[169,809,811,846]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"16 OtCyTCTBHe pa3pHIOH qacTH B daHHo MoJeH IObJcHReT OcyTCTBHe oIpeJIeHnRA $\\langle \\kappa \\mathrm{O}3\\phi \\phi \\mathrm{H}$ HHTHOpa3pHIOHe] npH IepeBOHe Ha pycckn J3bIK TepMHHa 整零构造zheng ling gouzao 'aIINTHBHOe KOJInueCTBeHHoe YHCINTEJIbHOe'. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[183,115,823,181]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"60JIbIIoe YHcJIo KaK IeJIoe, a MaJIeHbKOe - KaK ocTaTOK, KOtOpbIE COOTHOcTcKaK IeJIoe HuaCTb》. IIpn 3OM 整 zheng B TepMHHe 整零构造 zheng ling gouzao cHTaTeTc6 60JIbIIM YHcJIOM, a 零 ling - MaJIeHbKHM IO IIpHHIINy 前项大后项小 qián xiang dà hou xiang xiao '60JIbIIoe - BIIpeEN, MaJIoe - IIO3aIIN' [10, c. 39]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[183,181,826,455]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"I-TaK, B xOide KJIacchΦHIIpOBaHHN KOJIInueCTBeHHbIX YHcJIInTeJbHbIX C ΦyHKIIHOJIbHO-CTpykTpyHOJ TOckn 3peHHN B IaHHOM pa3JeJIe 6blJIN paccMOteHbI DbaIOxOJa. Obe KJIaccHΦHkaIHH OueHb IOXoJN. I ΓayHb YHbIHN, H CToYK3H C JI IN HUJK3 CXODTcB BO B3FJIaIax (B TOM YHcJIe B TepMHNOJIOTHneCKOM IIaHe) Ha 3JIeMeHTapHBie KOJIInueCTBeHHbIE YHcJIIneB. B OTHOIIeHH CJOXHbIX KOJIInueCTBeHHbIX YHcJIIneTbHBIX TAKe Ha6JIHOaETcB COBIIaHEHne TepMHOB IIpr O6O3HaYeHH MUYbTHIIHKaTHBbIX KOJIInueCTBeHHbIX (系位构造 xiWei gouzao 'KO3ΦHIIHeHTHO-pa3PdIbIE MyJBTHIIHKaTHBbIX KOJIInueCTBeHHbIE YHcJIInTeJbHBIX'. OJHaKO B O6O3HaueHH CJOXHbIX KOJIInueCTBeHHbIX YHcJIInTeJbHBIX, O6pa3OBAHHbIX IocpeIcTBOM CJOXeHH N HCJIObIX 3HaueHH B IIΦpOBoB 3aIIHC, Ha6JIHOaETcPa-cXoJDeHne B TepMHIOIOrHH. TaK, B KJIacchΦHkaIHH FyaHb YHbIHNa AKIeHT cDEJAH HA TOM, H3 KaKHX 3JIeMeHTapHBIX qacteON OHn coCToT (系位组合 xiWei zuhé 'KO3ΦHIIHeHTHO-pa3PdIbIE aDIHTHBIE KOJIInueCTBeHHbIE YHcJIInTeJbHBIX'), B TO BpeM KaCra ToxK3H JIN IHuK3 DeJIaOT aIKIeHT Ha pa3Mepe IN POCJIeIOBaTEJbHOCTN coCTABJIHOIIHX Hx qacteN (整零构造 zheng ling gouzao 'OyK8. KOHCtpyKIIHN YHcJIInTeJbHOrO CcoHTaHHem 60JIbIOI H MaIOJ qacte; aDIHTHBHOE KOJIInueCTBeHHOE YHcJIInTeJbHOe'). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[183,457,826,631]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Kpome TOrO, B o6oHb BapHaHTax KJIaccnФHKaIIH aIITNBHbIe YNCJIHTeJIbHbIe He IIOpa3JeJIOTcH Ha 3JIeMeHTapHbIe aIITNBHbIe H cMEIIaHHbIe aIITNBHbIe (cM. cTpykTpyHyo (MopΦemHyO) KJIaccnФHKaIIHO). Te YNCJIHTeJIbHbIe, KOtOpBie Mbl IIpHHMaem 3a 3JIeMeHTapHbIe aIITNBHbIe, B KJIaccnФHKaIIHN ΓyaHb YHBIIINHa coBcEM He ΦIpynpHOT. Y CSo IOTX3Ha H JIn INHxK3 OHN BXOJAT B IIOIpyIIY «aI- IITNBHbIE KOJIHyCEtBeHHbIe YNCJIHTeJIbHbIe》(整零构造zheng ling gouzao). OHaKaC ΦyHKUHOHaJIbHO-cTpykTpyHO TOckN 3peHn A.INTNBHbIE KOJIHyceTBeHHbIE YNCJIHTeJIbHbIe cJIeDyET pa3JeJIaTb Ha 3JIeMeHTapHbIe aIITNBHbIe H cMEIIaHHbIe aIITNBHbIe KaB I IOIpyIIIE o6bUHbIX KO3ΦΦHIIHeHTHO-pa3PraIHbIX aIITNBHbIX KOJIHyceTBeHHbIX YNCJIHTeJIbHbIX, TaK H B IOIpyIIIE cIeIIaJIbHBIX KO3ΦΦHIIHeHTHOpa3PraIHbIX aIITNBHBIX KOJIHyceTBeHHbIX YNCJIHTeJIbHBIX. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[183,632,826,760]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IJIaBHbIM JKe OTJINHHe MBAIEcTc BBeJeHHe IyaHb JHBiHOM KJIacchHkAIIOHbIX IIOIgpyIIcIeIIHaJIbHbIX KO3ΦΦHIIeHTHO-pa3pIbIX KOJIueCTBeHHbIX YHCJIHTeJIbHbIX KaK B MyJIbTHIIJIHKaTHBbIX, TaK H B aIITNTBbIX rpyIIax. 3TN cIeIIaJIbHbIE IIOIgpyIIbI IIpeIcTaBJIeHbI IIpi6JIIN3HTeJIbHbIMN YHCJIInTeJIbHbIMN. HeIOctaTOK KJIaccnHkAIIHN IgammaH JHBiHa BVIIITcR B TOM, YTO IIOIO6Ha CIIeIIHaJIbHaIIOIgpyIIHa IOJIXHa 6bl 6blr N y Ko3ΦΦHIIeHTbIX YHCJIInTeJIbHbIX, IIOCKOJIbKy OHN TaKKe MOryT BbIPAkaTb IIpH6JIIN3HTeJIbHOe YNCJIO, COUeTAcB dpyr c IpyTom (三四 sansi 'TpH-YeTbipe'). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[183,760,823,822]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C yueTOM 3aMeHaHn, BBiCKa3aHHbIX OTHoCHTeJIbHO BblIIepacCMOTpeHHbIX BaPnaHTOB $\\phi$ yHKIIOHaJIbHO-CTpyKTypHOJ KJIaCCNΦHkaIIIN KOJIIneCTBeHHbIX YHCJIInTeJIbHbIX, IIpeJIlaRaem aBTOpCKoe BnJeHHe aHaJIIN3HpyeMOrO KJIaCCNΦHkaIINHOHor DeJIeHn, IIpeIcTabJIeHHoeB Ta6JI.3. 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","text_format":null,"text_level":null,"type":"header"},{"bbox":"[796,77,825,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"187 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[163,151,820,570]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0594/465b3abb8590189a430e162ee9ccde1de384b3a714434a5548349e223b9ec1c7.jpg","list_items":null,"page_idx":"13","sub_type":null,"table_body":"<table><tr><td>Колочастовские ч��нштейпьные\nэдемен�ограды\nЧелеговские ч��нштейпь\nчелеговские ч��нштейпь\nчелеговские ч��нштейпь\nчелеговские ч��нштейпь\nчелеговские ч��нштейпь\nчелеговские ч��нштейпь\nчелеговские ч��нштейпь\nчелеговские ч��нштейпь\nчелergовские ч��нштейпь\nчелеговские ч��нштейпь\nчелеговские ч��нштейпь\nчелеговские ч��нштейпь\nчелеговские ч��нштейпь\nчелеговские ч��нштейпь\nчелеговские ч��нштейпь\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелergовrias\nчелergовrias\nчелergовrias\nчелergовrias\nчелergовrias\nчелergовrias\nчелergовrias\nчелergовrias\nчелergовrias\nчелergовrias\nчелergовrias\nчелergовrias\nчелergовrias\nчелergовrias\nчелergовrias\nчелergовrias\nчелergовrias\nЧелеговrias\nС�лочастовские ч��нштейпь\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговías\nчелеговías\nчелеговías\nчелеговías\nчелеговías\nчелеговías\nчелеговías\nчелеговías\nчелеговías\nчелеговías\nчелеговías\nчелеговías\nчелеговías\nчелеговías\nчелеговías\nчелеговías\nчелеговías\nчелergовías\nчелergовías\nчелergовías\nчелergовías\nчелergовías\nчелergовías\nчелergовías\nчелergовías\nчелergовías\nчелergовías\nчелergовías\nчелergовías\nчелergовías\nчелergовías\nчелergовías\nчелergовías\nчелergовías\nЧелеговrias\nС�лочастовские ч��нштейпь\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелеговrias\nчелегovrias\nчелегovrias\nчелегovrias\nчелегovrias\nчелегovrias\nчелегovrias\nчелегovrias\nчелегovrias\nчелегovrias\nчелегovrias\nчелегovrias\nчелегovrias\nчелегovrias\nчелегovrias\nчелегovrias\nчелегovrias\nчелегovrias\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nЧелеговrias\nС�лочастовские ч��нштейпь\nчелеговrias\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight;\nчlight\nчlight;\nчlight\nчlight;\nчlight\nчlight;\nчlight\nчlight;\nчlight\nчlight;\nчlight\nчlight;\nчlight\nчlight;\nчlight\nчlight;\nчlight\nчlight;\nчlight\nчlight;\nчlight\nчlight;\nчlight\nчlight;\nчlight\nчlight;\nчlight\nчlight;\nчlight\nчlight;\nчlight\nчlight;\nчlight\nчlight;\nЧелеговrias\nС�лочастовские ч��нштейпь\nчелеговrias\nС�лочастовские ч��нштейпь\nчелеговrias\nС�лочастовские ч��нштейпь\nчелеговrias\nС�лочастовские ч��нштейпь\nчелеговrias\nС�лочастовские ч��нштейпь\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;\nчelight;</td></tr></table>","table_caption":"[\"Ta6JI.3 \",\"ΦyHKIIOHaJIbHO-CTpyKTypHa KJIaCCHΦHKaIIHA KOJIInueCTBeHHbIX YHCJINTeJIbHBIX \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[438,585,544,598]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3akJIOUeHne ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[169,606,811,763]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PaccMOTpeHne $\\phi$ yHKIIHOHaJIbHO, cTpyKTypHO n $\\phi$ yHKIIHOHaJIbHO-cTpyKTypHO KJIaCCnΦHKaIIN KOJIInueCTBeHHbIX YHCJIInTeJIbHbIX B KHTaNcKOM Y3bIke BBJBAIO IpO-6JIeMy HOMeHKJIaTypHO HeOJHOpOJDHOCTN, KOJa OJHO I TO JKe IIOHrTne IJIN KJIaCCNΦHKaIINHHa TpyIIa HMEOT HeCKoJIbKO HaHMeHOBaHN, YTO CBHJeTEJIbCTByET O HeycToBIIeMcTepMHNOJIoTHueckOM aIIIapate. Ha $\\phi$ OHe HeKOTopoTepMHNOJIOHueckO HeOIIpeJEJIeHHOCTH B KHTaNcKOM Y3bIKe B OTeHecTBeHHOM KHTaEBeJEHN IIo paccMaTpHbAemomy BOIIPOcy Ha6JIIOJaEtC TOCYtCTBNe 6OJIbIIe qactN TepMHNOJIOrHEcKHX aHaJIIOOB, YTO B IIepByIO oupeIb KaCaetC $\\phi$ yHKIIHOHaJIbHO-cTpyKTypHO KJIaCCnΦHKaIIN.ДЯ TaKHx TepMHHO B HAcTOrIe CTAtBe IIpeIcTaBJIeHbI pyccko-3bIuHbIe BapHaHTbl. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[169,765,811,843]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B xOJIe HCCJIeIOBAHnI IOpa6oTaHbI paHee IIpeJIIOJKeHHbIe KNTaIckHMN JINHHBnCTaMn cTpyKtypHaI N ΦyHKIIHOHaJIbHo-cTpyKtypHaI KJIaCCNΦHKaIHH KOIJIueCTBeHN HbIX YNCJIInTeJIbHbIX. B IepcIIeKTHBe pa60TbI KNTaIckNe KOIJIueCTBeHNbIE YNCJIInTeJIbHbIe 6yUyt KJIaCCNΦHIIpOBaHbI CORJIaCHO TOUHOCTN KOJIInueCTBeHNH O XapaKTepnCTKN BbIPAkaEMOrO HmN YHCJa I eTo IeJIocTHOcTH. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[174,77,201,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"188 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[431,76,549,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"JI.JI.BAHKOBA ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[458,115,552,130]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"JInTepaTypa ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[186,135,826,810]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1.ДрауноA.A.IIcJIeIOBaHnI IO rpaMaTHKe COBpeMeHHOro KHTaIckoroЯ3bIka.-M.;JI.:ИЗД-ВО AH CCCP,1952.-T.1:YacTH peuN.-231 c. \",\"2. Дякоба И.Н. ЭвлioиmaTeMaTeMaTHuecKOrMoIIJIeHnI cTaHOBJeHHe NMeH YHcJIteJIbHbIX kak actnepeu//YUeH.3aII.IIeTpo3aBOJ.roc.yH-Ta.-2018.-No4.-C.49-54.- doi: 10.15393/uchz.art.2018.148. \",\"3. BaIaJIbIkuHa ə.A. YHcJIHTeJIbHbIe KaK oO6aJ qAcTb peuH B pyccKom J3bIke // YueH. 3aII. Ka3aH. yH-Ta. Cep. ΓyMaHHT. HayKn. - 2009. - T. 151, KH. 3. - C. 205-214. \",\"4. CmapukoBa $\\\\Gamma ,H$ . CoBpeMeHHbIe YnCJIHTeJIbHbIe B aCIIeKTe 3bIKOBoi IHHAMKn // BecTH. Tom. roc. yH-Ta. ΦHJIOJIoHra. - 2011. - No 4. - C. 34-40. \",\"5. Buho2pa06 B.B. Имя чилгельhoe // BHHoIpaIOB B.B. Рсckн я3bIK: ГраMaTHueckoe yueHne o cIoBe. - M.: Гoc.yue6.-II.II.N3II-BO, 1972. - C. 233-254. \",\"6. 关彦庆. 现代汉语基数词的结构研究 // 通化师范学院学报. -2012. -第33卷第5期. -页码 $1-10. = \\\\Gamma y a n b$ 。IccJIeIOBaHHe cTpyKtypbI coBpeMeHHbIX KHTaIcknx KoJIHcECTBeHHbIX YHcJIteJIbHbIX // BeCTH. IIe.I. HN-Ta B TyHxya. -2012. -T. 33, № 5. -C. 1-10.\",\"7. Bo2daHOB C.H. YHcJIHTeJIbHoe // BoIaHOB C.I., BoeKoBa M.I. И np. MopΦOJIOrHa coBpeMeHHoro pyccKOrO y3bika. - CII6.: ΦaK. ΦHIOJOrHn H nckycCTB C.-IteP6. roc. yH-Ta, 2009. - C. 301-322. \",\"8. 刘平. 数词分类研究述论 // 重庆邮电大学学报 (社会科学版). -2013. -第25卷第2期. -页码 122-127. =JIo II. OnHcAHN He HccJIeIOBaHnR KJIaCCNΦHKaIIN NcJIHTeJIbHbIX // BecTH. UyHIIHH. yH-Ta cBra3H n TeJIeKOMMyHnKaIIN. Cep. O6IeCTB. HayKn. -2013. -T. 25, № 2. -C. 122-127.\",\"9. 朱德熙. 数词和数词结构 // 中国语文. -1958. -第7卷第4期. -页码185-187. =ЧуД. Исклenteльные ихстуктура// Кнайсяς Фнlioюгия. -1958. -T.7, №4. -С.185-187. \",\"10. 萧国政, 李英哲. 汉语确数词的系统构成、使用特点和历史演进 // 武汉教育学院学报 (哲学社会科学版). - 1997. - 第16卷 第1期. - 页码 34-44. = Cao Γ., Jiu H. CnctemHnaopraHHaIIH, oco6eHHocTH npHMeHeHnI N BOJIHOuN I KHTaIckHX OIIpeJeJIeHHbIX YHcJIHTeJIbHbIX // BeCTH. YXaH. HN-Ta. Cep. ΦHIOcoΦnI N o6IecTB. HayKn. - 1997. - T. 16, No 1. - C. 34-44.\",\"11. 周彩莲. 现代汉语数词研究. -哈尔滨: 黑龙江大学, 2002. - 146页. = Yɔkeoy II. HccJIe-Доване УнсЛиТeЛьНБИХВ COВpeMeHHom KHTaɪСКOMЯЗIKE. - Xap6HH: Xэйлuyнizян. yH-T, 2002. - 146 c. \",\"12. 郭攀. 汉语涉数问题研究. - 北京: 中华书局, 2004. - 277 页 = Φ Π. Изуне Ворpo-cob, Связанных с КОЛичecтбом И чслomВ КНтуckомЯз biome. - ΠЕКИN: KNT. КНГонз-дательстбо, 2004. - 277 c. \",\"13. Tópeño B.H. TeopéTHueckárgaMMaTHKa KHTaɪcKOrO y3bIka. - M.: IIpocBeIIeHne, 1989. - 318 c. \",\"14. 实用汉语语法教程 / 付鸿军主编. - 北京: 北京师范大学出版社, 2015. - 371 页. = Yuchoe IocOBe IIpo IIpaKTHueckoI rpaMMaTHKe KHTaIcKOro J3bIka / IIOД peI. Фу X. - Пекин: ИзД-ВО Пекин. ПeI. уН-Ta, 2015. - 371 c. \",\"15. 实用现代汉语语法/刘月华.-增订本.-北京:商务印书馆,2001.-1005页.=IpaKTHUeCKa rpaMaTHKa coBpeMeHHOro KHTaHcKOro y3bIka / IIO peJ.IHO IO.-IeKNH:KomMepuecka IIpecca,2001.-1005c. \",\"16. Cyb6omun A./JI. KJIaCCnΦHkaIINr // IeHTp rymaHHTapHbIX TeXHOJIOTn.- M.: ΙΦ PAH, 2001.- URL: https://gtmarket.ru/laboratory/basis/3794, cbo6oIhBn. \"]","page_idx":"14","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[351,76,657,89]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ФУHKIиОHAЛьнА,СТPyКТYPHА... ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[796,77,825,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"189 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[173,114,811,634]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"17. Понкун И.В., PeobkaHa A.II. Класснфнадя Кам MeToД hayUHOr HCCJIeIOBaHHa, B YacT-HocTH B IOpHINueckoи HayKe // BeCTH. ПЕм. yH-Ta. IOpHД. hayKn. - 2017. - ByII. 37. - C. 249-259. - doi: 10.17072/1995-4190-2017-37-249-259. \",\"18. 现代汉语描写语法/张斌主编.-北京：商务印书馆，2010.-1306页. =ОпиcaTeЛьнай ГраМатиka coВpeMeHного КИтайсКOrO яЗыкa / Пов peД. Ужан Б.-ПeКин: Komмер-ческа пeccа, 2010.-1306c. \",\"19. 李宇明. 数学语言初见 // 语文教学与研究. -1986. -第17卷第7期. -页码2-15. = JIu IO. ΠερΒαγ BCΤρεγας MaTeMaTHηεκΚοῦ ΙΙΝΗΓΒνιδΤΗΚοῦ // ΠρεπΟДaВΑΗΝΕ έΝΙΥηεΗΝΕ ΑλθΙΚΑ έΝΙΠΕταρατυβΙ. -1986. -T. 17, № 7. -C. 2-15. \",\"20. Cuyuha B. YHcJIHTeJIbHbIe // MaTePHaJIbI K KopIIyCHoI rpaMMaTHKe pyccKOrO y3bIka. - CII6.: HecToP-IVctOpHn, 2018. - BbII. 3: Yactn peuN nJIeKCHKO-rpaMMaTHueckHe KJIacCbI. - C. 193-257. \",\"21. 朱德熙. 语法讲义. - 北京: 商务印书馆, 1982. - 231 页. = UzhvД. Kypc Лжий по гравиматник. - Пекин: Коммереская песca, 1982. - 231 c. \",\"22. 汉英对比语法论集 / 赵世开主编. - 上海: 上海外语教育出版社, 1999. - 349 页. = C6opHnK cTaTei IIO KOMIIapaTHBHOI rpaMMaTHKe KHTaIcKOrO H aHIJIINcKOrO J3bIKOB / IOp peI. YJkao III. - IIaHxaI: IIaHx. H3Д-Bo yYe6. JInT. Ha INHocTp. J3., 1999. - 349 c.\",\"23. Comrie B. Numeral bases // The World Atlas of Language Structures Online. – Munich: Max Planck Digital Libr., 2011. – URL: http://wals.info/features/131,сыобдь. \",\"24. He C. Complex numerals in Mandarin Chinese are constituents // Linguua. - 2015. - V. 164, Pt. A. - P. 189-214. - doi: 10.1016/j.lingua.2015.06.014. \",\"25. 邢福义. 汉语语法三百问. - 北京: 商务印书馆, 2002. - 264 页. = Cun Φ. Трпста Восторов Ктошкоу гравлары. - Пекин: Коммерочая песca, 2002. - 264 c. \",\"26. ColnueB B.M., ColnueBa H.B. TeopeTHueKaI rpaMMaTHka coBpeMeHHoro KHTaNcKOra 3bIka. IIpo6JIeMbI MOpΦOJOΓHn.-M.: IN-H-T BOCTOKOBeJEHHa PAH, BoeHHbYH-T MInHO6OpHbI PΦ, 2016.-164 c. \",\"27. One-Soon Her. Deriving classifier word order typology, or Greenberg's Universal 20A and Universal 20 // Linguistics. - 2017. - V. 55, No 2. - P. 1-31. - doi: 10.1515/ling-2016-0044. \",\"28. Corver N., Zwarts J. Prepositional numerals // Lingua. - 2006. - V. 116, No 6. - P. 811-835. - doi: 10.1016/j.lingua.2005.03.008. \",\"29. Hurford J. The Linguistic Theory of Numerals. – Cambridge: Camb. Univ. Press, 2011. – 308 p. (Cambridge Studies in Linguistics. V. 16). \",\"30. Ionin T., Matushansky O. The composition of complex cardinals // J. Semantics. - 2006. - V. 23, No 4. - P. 315-360. - doi: 10.1093/jos/ffl006. \"]","page_idx":"15","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[653,651,805,675]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IocTuHnla BpeaKiHo 17.08.2021 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[171,689,808,703]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"BaanKobA JIOIMHJa JbBOBa, KaHINaT 0JIIOJIOrHuecknx HayK, IOneHT KafeIpbI KHTaiCKoro A3bKa ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[205,706,547,719]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Mockobckn I ropoIcKo IIeIaIorHueckn yHInBepcHTet ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[228,720,660,731]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2-Ⅱ CeIbckoxO3aIcTBeHHbI IIpoe3I, I. 4, r. 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SERIYA GUMANITARNYE NAUKI ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[339,166,670,179]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(Proceedings of Kazan University. Humanities Series) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[389,185,620,198]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2021, vol. 163, no. 4-5, pp. 175-193 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[184,210,366,223]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ORIGINAL ARTICLE ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[184,229,457,241]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"doi: 10.26907/2541-7738.2021.4-5.175-193 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[324,249,685,275]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Functional, Structural, and Functional-Structural Classifications of Chinese Cardinal Numerals ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[457,284,552,296]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"L.L. Bankova ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[351,300,658,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moscow City University, Moscow, 129226 Russia ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[421,313,589,326]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E-mail: lwittsa@yandex.ru ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[421,332,586,344]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Received August 17, 2021 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[475,351,536,363]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[184,370,823,458]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Classification of cardinal numerals is an important step in studying Chinese numerals. The study is carried out in the framework of the functional, structural (morphemic), and functional-structural (combined) classification approaches, as well as in accordance with the integrity of numerals and the accuracy of their quantifying properties. Chinese numerals are considered from the perspective of the first three classifications. The results obtained reveal the nomenclature volatility of Chinese cardinal numerals with the same classification group having several names to denote it. Hence, the article attempts to streamline the respective terminology. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[184,458,823,482]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The functional classification divides cardinal numerals into coefficient and digit-position numerals. The cases when digit-position numerals may function as coefficient numerals are analyzed. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[184,482,825,531]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the framework of the structural classification, it is suggested to divide cardinal numerals into simple (coefficient) and compound ones, the latter being broken into additive (elementary and miscellaneous) and multiplicative. The article also focuses on answering the question of whether compound numerals should be regarded as words or phrases. The conclusion is made that they should be perceived as words. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[184,531,825,605]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The functional-structural classification makes a substantial contribution to the study of Chinese cardinal numerals. The classification has not been introduced to the Russian sinology yet. This fact entails the necessity to translate the respective terms and expressions, the choice of which is explained in this article. The variations of the combined functional-structural classification introduced by Zhu Dexi, Guan Yanqing, Xiao Guozheng, and Li Yingzhe are discussed. Based on the analysis of these variations, an understanding of the combined functional-structural division of Chinese cardinal numerals is suggested. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[184,608,823,633]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords: Chinese language, Chinese cardinal numerals, functional classification, structural (morphic) classification, functional-structural classification ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[467,646,544,659]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[184,665,826,829]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. Dragunov A.A. Issledovaniya po grammatike sovremennogo kitaiskogo yazyka [Studies on Modern Chinese Grammar]. Vol. 1: Parts of speech. Moscow, Izd. Akad. Nauk SSSR, 1952. 231 p. (In Russian) \",\"2. D'yachkova I.N. The evolution of mathematical thinking and genesis of numerals as parts of speech. Uchenye Zapiski Petrozavodskogo Gosudarstvennogo Universiteta, 2018, vol. 173, no. 4. pp. 49-54. doi: 10.15393/uchz.art.2018.148. (In Russian) \",\"3. Balalykina E.A. Numerals as special category in modern Russian language. Uchenye Zapiski Kazanskogo Universiteta. Seriya Gumanitarnye Nauki, 2009, vol. 151, no. 3, pp. 205-214. (In Russian) \",\"4. Starikova G.N. Current numerals in the aspect of language dynamics. Vestnik Tomskogo Gosudarstvennogo Universiteta, 2011, no. 4, pp. 34-40. (In Russian) \",\"5. Vinogradov V.V. Numerals. In: Vinogradov V.V. Russkii yazyk: Grammaticeskoe uchenie o slove [The Russian Language (A Grammatical Theory of the Word)]. Moscow, Gos. Ucheb.-Ped. Izd., 1972, pp. 233-254. (In Russian) \"]","page_idx":"16","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[351,76,658,89]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ФУHКИПОHAЛьнА,СТPyKТYPHA... ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[796,77,821,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"191 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[171,114,811,796]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"6. Guan Ya. A research on the structure of modern Chinese cardinal numerals. Journal of Tonghua Normal University, 2012, vol. 33, no. 5. pp. 1-10. (In Chinese) \",\"7. Bogdanov S.I. Numerals. In: Bogdanov S.I., Voeikova M.D. et al. Morfologiya sovremennogo russkogo yazyka [Morphology of the Modern Russian Language]. St. Petersburg, Fak. Filol. Iskusstv S.-Peterb. Univ., 2009. pp. 301-322. (In Russian) \",\"8. Liu P. A description of numerals classification research. Journal of Chongqing University of Posts and Telecommunications, 2013, vol. 25, no. 2, pp. 122-127. (In Chinese) \",\"9. Zhu D. Numerals and their structure. Chinese Philology, 1958, vol. 7, no. 4. pp. 185-187. (In Chinese) \",\"10. Xiao G., Li Y. Chinese exact numerals system: Structure, peculiarities of use and evolution. Journal of Wuhan Institute of Education. Philosophy and Social Sciences Series, 1997, vol. 16, no. 1, pp. 34-44. (In Chinese) \",\"11. Zhou C. Research on Modern Chinese Numerals. Harbin, Heilongjiang Univ., 2002. 146 p. (In Chinese) \",\"12. Guo P. A Research on Problems Related to Chinese Numerals. Beijing, Zhonghua Book Co., 2004. 277 p. (In Chinese) \",\"13. Gorelov V.I. Teoreticheskaya grammatika kitaiskogo yazyka [Theoretical Chinese Grammar]. Moscow, Prosveshchenie, 1989. 318 p. (In Russian) \",\"14. Fu H. (Ed.) Practical Chinese Grammar Teaching Materials. Beijing, Beijing Normal Univ. Press, 2015. 371 p. (In Chinese) \",\"15. Liu Y. Practical Modern Chinese Grammar. Beijing, Commer. Press, 2001. 1005 p. (In Chinese) \",\"16. Subbotin A.L. Classification. In: Tsentr gumanitarnykh tekhnologii [Humanitarian Technology Center]. Moscow, IF Ross. Akad. Nauk, 2001. Available at: https://gtmarket.ru/laboratory/basis/3794. (In Russian) \",\"17. Ponkin I.V., Red'kina A.I. Classification as a method of scientific research, particularly in jurisprudence. Vestnik Permskogo Universiteta. Yuridicheskie Nauki, 2017, no. 37, pp. 249-259. doi: 10.17072/1995-4190-2017-37-249-259. (In Russian) \",\"18. Zhang B. (Ed.) Descriptive Grammar of Modern Chinese. Beijing, Commer. Press, 2010. 1306 p. (In Chinese) \",\"19. Li Yu. A first glance at the language of mathematics. Language and Literature Teaching and Studying, 1986, vol. 17, no. 7, pp. 2-15. (In Chinese) \",\"20. Sichinava D.V. Numerals. In: Materialy k korpusnoi grammatike russkogo yazyka [Materials for the Corpus Grammar of the Russian Language]. St. Petersburg, Nestor-Istoriya, 2018, no. 3: Parts of speech and lexico-grammatical classes, pp. 193-257. (In Russian) \",\"21. Zhu D. Lectures on Grammar. Beijing, Commer. Press, 1982. 231 p. (In Chinese) \",\"22. Zhao S. A Collection of Articles on Comparative Grammar of the Chinese and English Languages. Shanghai, SFLEP, 1999. 349 p. (In Chinese) \",\"23. Comrie B. Numeral bases. In: Haspelmath M., Dryer M.S., Gill D., Comrie B. (Eds.) The World Atlas of Language Structures Online. Munich, Max Planck Digital Libr. 2011. Available at: http://wals.info/features/131. \",\"24. He C. Complex numerals in Mandarin Chinese are constituents. *Lingua*, 2015, vol. 164, pt. A, pp. 189–214. doi: 10.1016/j.lingua.2015.06.014. \",\"25. Xing F. Three Hundred Problems of Chinese Grammar. Beijing, Commer. Press, 2002. 264 p. (In Chinese) \",\"26. Solntsev V.M., Solntseva N.V. Teoreticheskaya grammatika sovremennogo kitaiskogo yazyka (Problemy morfologii) [Theoretical Grammar of Modern Chinese (Problems of Morphology)]. Moscow, Inst. Vostokoved. Ross. Akad. Nauk, Voen. Univ. Minoborony RF, 2016. 164 p. (In Russian) \",\"27. One-Soon Her. Deriving classifier word order typology, or Greenberg's Universal 20A and Universal 20. Linguistics, 2017, vol. 55, no. 2, pp. 1-31. doi: 10.1515/ling-2016-0044. \",\"28. Corver N., Zwarts J. Prepositional numerals. Lingua, 2006, vol. 116, no. 6, pp. 811-835. doi: 10.1016/j.lingua.2005.03.008. \"]","page_idx":"17","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[173,77,201,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"17","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"192 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[431,76,549,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"17","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"JI.JI.BAHKOBA ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[186,114,825,168]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"29. Hurford J. The Linguistic Theory of Numerals. Cambridge, Cambridge Univ. Press, 2011. 308 p. (Cambridge Studies in Linguistics. Vol. 16) \",\"30. Ionin T., Matushansky O. The composition of complex cardinals. Journal of Semantics, 2006, vol. 23, no. 4, pp. 315-360. doi: 10.1093/jos/ffl006. \"]","page_idx":"18","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[216,198,791,247]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Дя ушорань: Baнкова Л.Л. Функюнальна, стуктурази и Функюнальностуктурази Классфкаддд Клесчесггдд Кнслтеловьх КНотайсоком ясыke // YueH. 3aII. Ka3aH. yH-Ta. Cep. ГумаHT. Haykn. - 2021. - T. 163, кH. 4-5. - C. 175-193. - doi: 10.26907/2541-7738.2021.4-5.175-193. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[216,262,793,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For citation: Bankova L.L. Functional, structural, and functional-structural classifications of Chinese cardinal numerals. Uchenye Zapiski Kazanskogo Universiteta. Seriya Gumanitarnye Nauki, 2021, vol. 163, no. 4-5, pp. 175-193. doi: 10.26907/2541-7738.2021.4-5.175-193. (In Russian) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[351,76,660,89]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ФУHKIиОHAJIbHA,СТPyKТYPHA... 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XOTa HeKOToPbIE IPOBeJeHHbIe paHee HccJIeIOBaHNr OITINuHOTcRgIy6HnOH aHaJIIn3a, HaIIpHMeP: «HcJIeIOBaHne cTpyKtpybI cOBpeMeHHbIX KHTaIckHX KOJIHueCTBHeHHbIX YHcJIHTeJIbHbIX» (现代汉语基数词的结构研究) ΓyaHb JHBiHa [6], «OIIHCaHHe HCCJIeIOBaHNr KJIaCCNΦHKaIINr YHcJIHTeJIbHbIX» (数词分类研究述论) JIo PInN (刘苹 Liu Ping) [8], «UHcJIHTeJIbHbIe N IHX cTpyKtypa» (数词和数词结构) Chy DiCh (朱德熙 Zhū Déxi) [9], «CNCTeMHa OpraHHaIry, oCo6eHHocTH IIprMeHEnHr N EBOJIHOIIIG KHTaIckHX OIIpeJEJIeHHbIX YHcJIHTeJIbHbIX» (汉语确数词的系统构成、使用特点和历史演进) CNo TOnKoHa N JIn INuKo3 (萧国政 Xiao Guózheng, 李英哲 Li Yingzhe) [10], «IcSeIOBaHne YHcJIHTeJIbHbIX B COBpeMeHHOM KHTaIckOM Ja3bIKe» (现代汉语数词研究) ChkoY IaJIyHb (周彩莲 Zhōu Cāilián) [11], «I3UyeHne BOIpocOB, CBRAHHbIX C KOJIHueCTBOM N UHCJOM B KHTaIckOM Ja3bIKe» (汉语涉数问题研究) FIO HAry (郭攀 Guo Pan) [12], B YacTHN BBIIIE- IpeHuCJIeHHbIX pa6OT KOJIHueCTBHeHHbIe YHcJIHTeJIbHbIe paCCMaTpNBaOTc C ODHORO paKypca - B pAMkax ΦyHKIHOHaJIbHO-CTpyKtpyHO KJIaCCNΦHKaIINr [6; 9]. IomHMo 3TOrO, OTDeJIbHbIe BblBnRaembe B IaHHbIX Tpydax IIOJooKeHHra (CM. [8; 10-12]) BBI- 3bIBaOH TBOIIPOCSBI (TJIaBHbIM H3 KOTOpBX RaBJYeTcRA paIPpeJeJIeHHe YHcJIHTeJIbHbIX BHyTPN KJIaCCNΦHKaIINr) H Tpe6ByOT KpHTNHecKOrO aHaJIIN3a. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,557,811,651]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Болбшннсво ще Лингьнстческнх HCCJIeIOBaHиЧСЛNTeЛьНБИ XKТансКOrOЯзka HOCaT pa3po3HeHHbI xapaKTeРи ПpeДCTaBJIaHOT co6oN obocOJIeHHbIe 6JIOKNHΦopMaIIHN, KOTOpbIe Mbl IOCTaBnIH IeJIbIO CO6paTB, O6OJIHTb N KOMIIJIeKCHOIIpoHaJIIN3HpOBaTB, IIpeДIOXKNB cBOH peIIeHnI CIIOPHBIX BOIIPOcOB.Прп этOM BaЖ-HOH 3aДaчeн HabToIe pa6OtBI YIopЯIOUHBAHNe TepMHHOJIOrHYeCKORO aIInapaTa. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[168,652,811,733]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"По опpeДeJIeHио B.И. Гореява, КОЛичecТВeHьie чсJInteЛьныe (基数词ji shuci) оБОЗнayaOT «OTBЛeueHьie чсЛЯ Илк OКЛичecТВОpyeДмETOB» [13, c. 43].ЭTO co3ByuHO KHTaIckHM oрpeДeJIeHЯM: «КОЛичecТВeHьie чсJInteЛьныe оБОЗнayaOT BeJINuHHyЧсЛЯ» [14, c. 28]; «KOLINчecТВeHьie чсJInteЛьныe уka3bIbaIOTHa 3NaueHHe ChsJa I O6OЗнayaeMoE ИМ КОЛичecТВО» [15, c. 115]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,733,811,812]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"KHTaicn J3bIKoBeJ IIO IIIN IIIIIEeT O TOM, YTO cyIeCTByeT Dba Bua KJIaccNcKaIuIyIeJIbHbIX: yHKIIOHaJIbHaN CtpyKtypHaN [8, c. 123]. OHaKO Mbl CHTaem, YTO BILJeIAYB BVOTJeJIbHbIN BND CJJeIyET N CMEIIaHHYIO yHKIIOHaJIbHO-CTpyKtPyHyIO KJIaCCNΦHKaIIHIO. DaHHbIe TpN BnJa KJIaCCNΦHKaIIHOHHbIX IOXIOOB, BOCHOBY KOToPbIX IIOJOKeH MeToI DeIyKIIHN [16; 17, c. 255], 6yIyT paccMOrpeHb IN HAcTOnIIe \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[173,77,201,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"176 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[430,76,549,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JI.JI.BAHKOBA \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[201,832,507,847]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 3Jecb Hn daJIe HepeBoH Ha ypcckn IaIbIK haII. -J.I.B. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[183,116,825,161]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"cTaTbe. Kpome TOrO, IIpeIcTaBJIeTcra BO3MOXHbIM KJIaCCNΦHIIrpoBaTb KOJIInueCTBeHHbIE YHCSJIHTeJIbHbIe Ha OCHOBAHH TOUHOCTn IX KOJIInueCTBeHHoXapaKTePHeTKN IIEJIOCTHOCTn YHCSJIa, YeMy 6yIeT IIOCBIAHE NOPOJOJIKeHne HCCJIeIOBAHHa. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[359,177,650,193]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Функуновальна Класснkaцья \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[183,198,826,382]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CorJIacHO TpaIINIOHHOJIaCCNΦHKaIINI IO FOyHKIINM, KJIacc KOJIInueCTBeHHbIX YHCJIInTeJIbHbIX DeJIInTCra Ha IBa IIOKJIaCCa: KO3ΦΦHIIeHTHBIE (系数 xi shu)и pa3- pIaIbIe (位数 wei shu).K KO3ΦΦHIIeHTHBIM YHCJIInTeJIbHbIM,KaK IIpaBnJO,OTHOcT 零 ling 'HyIb', 一yi 'OiIN', 二 er 'IbA', 两 liang 'IbA', 三 san 'TpN', 四 si 'YeIbpe', 五 w'u 'IpytB', 六 liu 'IIEcTB', 七 qi 'CEmb', 八 ba 'BOceMb', 九 jiU 'JEBytB', 十 shi 'IdecTb' [15, c. 115; 18, c. 133-135]. Pa3pIINbIMN YHCJIInTeJIbHbIMN CHTaIOTc+ shi 'IdecTb', 百 bai 'CTo', 千 qian 'TbICya', 万 wan 'IeCSrTB TbcIaY', 亿 yi 'CTOMMHJINIOHOB' N.T.Д. [6;9].KaK BnIHNO, KOJIInueCTBO IINΦp B KO3ΦΦHIIeHTHom IIOKJIacce KOHeHNO IpaBHO IBeHaIuaTH, IIO3TOMy eRO MOxHO IprH3HaTb 3aKpbITbIM. KoJIInueCTBO IINΦp B pa3pIINOM IIOKJIacce CTpeMHTCR K 6ecKOHeUHocTH, B CBzH C YEm OH MoXeT CHTaTbCRA OTKpbITbIM.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[183,385,826,549]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"HeKoTOpbIe HCCJIeIOBAteJIH BKJIIOUaIOT B IIOJIKJIacC pa3pIINbIX YNCJIInTeJIbHbIX TaKKe 十万 shiwan 'cTO TBICaY', 百万 baiwan 'MHJIIOH', 千万 qianwan 'DecaTb MHIJIHOHOB', 万亿 wanyi 'TpNIIOH' [15, c. 115].MbI He corJIacHbI c TaKoI IIOHNIIeH, H IIpuHHoN TOMy ABJIaETcA YeTBIpexIO3NIIOHbI XapaKTep KHTaIckOc HcTeMbI cHCJIeHNH. B HeI peIepHBIMN TOUkAMN KaK pa3 I YBJIIOHTC pa3PdIHbIE YNCJIInTeJIbHbIe WAN 'DecaTb TBICaY', 亿 yi 'cTO MHJIIOHOB' N T.Д. O6O3HaIoIIne IpoMeKyuToHbIe pa3pIbI, 十万 shiwan 'cTO TBICaY', 百万 baiwan 'MHJIIOH', 千万 qianwan 'DecaTb MHIJIIOHOB' B TAkOM CJUYae He YBJIIOCTcA MoCtOaTEJIbHbIMN pa3pIINbIMN YNCJIInTeJIbHbIMN H IIpeIcTaBJIAOT co60I KOHCTpyKtBi, COCTOJIuHE H3 coYeTaHHMOPfem.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[183,551,826,715]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C KO3ΦHIIHeHTHBIM IOIKJIaCCOM KOJIHueCTBHeHHbIX YHCJITeJIbHbIX C83Ha Hpo-6JIeMa BkJIIOUeHnB HeIg HJIA. TaK, JIN IOHMHN (李宇明 LiYuming) HCKJIIOuaET HyJB N3 YHCJa KO3ΦHIIHeHTHBIX YHCJITeJIbHbIX, OTHOCE rO K MaTeMaTHueCKHM 3HaKaM B HePOrJIHΦHueCKo 3aIINc³ [19, c. 7-8]. C HNM coJIHApHbI C80 TOnJxH N JIn IHHx [10, c. 34]. OJHaKO Mbl IOJIaRaEM, YTO 零 ling 'HyJB' c IOJIbIM IIpaBOM MOKeT cHTaTbcR YNCJIteJIbHbIM, IOscKJIbKy o6JIaAet BCeMH eRO KaYeCTBaMn. BoIIepBbIX, OH o6O3HaHaet OTCyTCTBHe BeJIHcHHbI NIN CJIyKHT OTIIpaBHOI TOOKI IIpr IIpeYncJIeHHN IIpeIMeTOB. Bo-BTOPbIX, 零 ling 'HyJB' JABJIErTc3HaMeHaTEJIbHbIM CJO-BOM 6JIaIqIapr cIIOco6HoCTn CaMOCTeJIbHO φyHKIIHOHpOBaTB B IIpeJIIOJKeHHN (结果等于零 Jiéguo dengyu ling 'Pe3yJIbTaT paBeH HyJIIO'). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[183,717,826,814]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CpeH KHTaCKHX 3bIKOBeIOB Ha6JIHOJaOTcpa3HOJAcN TaKKe IIO IOBOy OIIpeJIeHHKJIaCCnHkauHNOHOrO cTaTuCa YncJIInTeJIbHOrO + shi 'IecTb' B paMKax fHyHKIIHOHaJIbHO KJIaCCnHkauHHKoiJIueCTBeHHbIX YncJIInTeJIbHbIX. He BCE HCCJIeIOBaTeJIN IIpIePJKHBaOTc MHeHHN, YTO OHO YbJIeTc Ko3ΦΦHnEHTbIM, HApnpMep CraO TOnK3H N JIn INuK [10, c. 34]. CyIeCTByET n Dpyra ToUka 3peHHN, cToPOHHKamN KOtOpOH MoJHo Ha3BaTb Yxu I3cn, YkaH BnH (张斌 Zhang Bin) n JIo IOxya \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[351,76,657,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ФУHKIиОHAЛьнА,СТPyКТYPHА... \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[796,77,825,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"177 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[216,832,722,847]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 ΘOT BapHaHT KJIaCCHΦHKaUHH KOJIInueCTBeHNbIX YIcJIHTeJIbHbIX 6yIET paccMOrpeH HNKe. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[169,116,811,181]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(刘月华Liu Yuehua), cHTaHOIIHX, YTO + shi 'DecaTb' OINOBpeMeHHO RAJIeTcH IN KO-ΦΦHIIHeHTHBIM I pa3PraHbIM YHCJIHTeJIbHbIM, IOTOMy YTO OHO 3aHNMaET o6e II03Iu: 十万 shiwan 'cTo TBicay' N 二十 ershi 'IbAuaTb' [15, c. 115; 18, c. 133-135; 20, c. 45]. DaHHa TOnKa 3peHnIpeCTaBJIeTcH aM HAnboJIee IIprHemJIoMoi. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,181,811,249]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CraOToJHnJIHnHnKoIJIaIaIOT,HTOHeKOToPbIe pa3pIaIIbIe YncJIInTeJIbHbIE MOrYt BbICTyIaT BpoJIIN KO3ΦΦHIIHeHTHBIX,3aHHMaI IO3HIIIO HepeI万wan‘DecrTBbICr'N亿yi‘CTO MmJIInHOB':万亿wanyi‘TpHJIInOH',十万shiwan‘cTO TbICr' [10,c.37]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,250,811,456]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B cBOIO oupeIb, uHcJIHTeJIbHOe千 qian 'TbICa' H B HeKOToPbIX cJIyAix YHcJIHTeJIbHOe 万 wan 'DecaTb TbcIa' He MOryT IIOABJIaTbcR B KOaΦoHIIHeHTHOI IO3HIIHN. Te coUeTaHHa HEPoJIHΦOB, B KOTOpBX 千 qian 'TbICa' HaxOHTcRA Ha IepBOM MeCTe, He JBIaIOTc YHCIOBBIM 3aINCSaMn (HaIPHMep, 千万 $x =$ 千 $\\\\mathbf{x} +$ 万 $x)$ . B Tex CJYuaX, KOrIa 万 wan 'DecaTb TbcIa' 3aHIMaeT KOaΦoHIIeHTHyIO IO3HIIIO IpeEi万 wan 'DecaTb TbcIa' B pa3pIHOI IO3HIIHN HJIM MeHBIM pa3pIIBIM YHCJIHTeJIbHBIM (HaIPHMep, 千 qian 'TbICa'), 3TO TaKKe He JBIaETc YHCIOBoI 3aINCSbIO. B CBA3N C OTHM CNo TQWKH IN JIn IHHK3 HCKIOAHOT IN 3 pIAOB YHCJIHTeJIbHBIX TAkne cIOBa, KaK 千万 qianwan 'HeIpeMeHHO', 万万 wanwan 'BO Yo TO HN cTAJO' N万千 wanqian 'Macca, MHOJceTBO; Heo6buaHIO pa3HOO6pa3HbI, KpaIHe H3MeHUNBbI' [10, c. 37-38]. Tem He MeHee UHCJIHTeJIbHOe 万 wan 'DecaTb TbcIa' BCE-TaKHIOHHPOBaTb KaK KOaΦoHIIeHTHOe, 3a HCKIOueHHem BblIEOIIHCANHbIX cIJyaeB.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,457,811,504]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"HeoH03HaHbI IOxOJ K paIpeJIeHHIO YHcJIHTeJIbHbIX IIO yHKIIHM IIpeJIIOXJI JIN IOIMHN. CoJIaCHO erO KJIacchΦHkaIHH, BCE KOJIInYeCTBeHHbIe YHCJIH-TeJIbHbIE DeJIATcR Ha TPh IIOKJIacca: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,505,811,678]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"- Πифпь (数码 shùma): 一 yī ‘оДин’, 二 èr ‘Два’, 三 san ‘Три’……九 jiǔ ‘Дe-BЯть’, 十 shí ‘Дecять’;\\\",\\\"pa3pIbI (数位 shiwei): 十shi‘IecTb', 百bai‘cTo', 千qian‘TbICHa', 万 wan 'IecTb TbcIy', 亿yi‘cTO MIIJIHOHOB';\\\",\\\"MaTeMaTHUeCKHe 3HaKN B HepoRJIHΦHueCKo3aIIHcN (数符 shifu): 零 ling 'HyIb'; 分 $f e n$ 'uaCTb,ДоЛЯ'и cTpyKtypHaJy auctHua之zhī (cIyJkaTДЯ HEPoRJIHΦHueCKo 3aIIHcN obbyHoiДpo6h, coeINHЯЧИСЛHTeJIb И 3HaMeHaTeJIb); 又you‘H,Дa' (cIyJKTДЯ HEPoRJIHΦHueCKO 3aIIHcN cMeIIaHHoДpo6h, coeINHЯпeIyUo И дpo6-HyU OuaCTH), 点di'an'TOчka'(В DeecrTNUHoiДpo6h), IIpeΦHKc IopraIkoBOrO YHcJIH-TeJIbHOrO第di[19,c.7-8]. \\\"]\",\n \"page_idx\": \"3\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[169,681,811,743]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Данньий пождов Вьзбьает рял замechи. Bo-первьх, Инфраим можно Ha3Baыт He ToЛько пereчincJIeHHbIe ЛИ IOйMHOMВ первом пдКласс КоэфншечСЛHTeBJHbIe, HO IPAЗрДнБIEЧСЛNTeJIbHbIe, KOТОрБIу Hero Ha3bIBaIOTсЯ pa3рДaMH (cM. Второй пождд). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,744,811,847]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bo-BTOPbIX, IocJIeHNII IOKJIacc (MaTeMaTHueCKHe 3HaKN B HepoRJIHΦHueCKO3aIINCH) BBNdy cBOe pa3HOpOJHOCTn IIpeIcTaBJIaETCscomHHTeJIbHBIM. K Hemy JINIOHMNH OTHOCHT N IpeΦHKC IopraIKOBORO YHCJInTEJBHOrO 第 di, IN KOJIInueCTBeHHOE YNCJIInTEJBHOe 零 ling 'HyJI', IN co6CTBeHHO 3aIIHCAHHbIe HepoRJIHΦaMH MaTeMaTHueCKHe 3HaKN, CJlyXaIIHe dJIa3aIIHcN Ipo6HBIX YHcEJI (分 fen 'YacTB, I0JI', 点 di'an 'ToUka', cTpyKtypHaJyAcTHNuIa之zhī), IN CO1O3 又 yòu 'H, Ia'. IocJIeHN II BCTrpeHaetc\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[173,77,201,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"178 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[430,76,549,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JI.JI.BAHKOBA \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[179,116,826,244]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"TOJIbKO B HeporJINHΦHueCKO 3aIINc CMeIIaHHo Ipo6n (三又二分之一 san you er fen zhi yi 'Trn IeJIbIX (H) OJHa BTOpaJ), HO He oTpaKaetcB 3aIINCH IIHΦpamN (3½), BCJIeICTBHe Yero He MoKET cHTaTBc MaTeMaTHUeCKHM 3HaKOM. Mbl IIOJaAeM, YTO BbIeJIaTb IIOJIaIacc «MaTeMaTHUeCKHe 3HaKN B HeporJIHΦHueCKO 3aIINcH» HeIeJIeCOO6pa3HO, IOTOMy YTO IpeHkC第 di He ABJIaETCa MaTeMaTHUeCKHM 3HaKOM, 零 ling 'HyJIb' cJIeIyET OTHeCTH K IIOJIaCCy «IINΦpbI» B IpeJIlaRaEMOM KJIaCCNΦHKaIIHOHOMpacIIpeJIeHHN, a cJOBa 分 fen 'YacTb, IIOJIa', 之 zhi, 又 you 'H, Ja', 点 dian 'ToUka' YHCJIInTEJIbHBIMH He ABJIaHOTc4'.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[183,247,826,310]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JIIOIINOnpeJJIa KJIacchHcHKaIHNO JIN IOIMnHa KaK cTpykTypHyO (MOPeMHyIO) [8, c. 124]. OndaKo cTpykTypHa KJIacchHcHKaIHNa DeJIHT YNCJIHTeJIbHbIe IIO KOJIInueCTBy BXOJIIHX B HNX MOpΦeM, a B DaHHOM DeJIeHHN peYb HJeT O ΦyHKIIHX YNCJIHTeJIbHbIX, IIO3TOMy MbI cHTaEM ee ΦyHKIIHOHaJIbHoI. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,325,687,341]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CtpyKtpyHn (MopfemHa) KJIacchfKaun \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[184,347,826,734]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CoJIaCHO cTpyKTHpyHOH (MOPΦeMHoH) KJIaCCNΦHKaIHH (HNOrga ee Ha3bIBaOT cJIOBOO6pa3OBaTEJIbHOH [20, c. 204]), B OCHOBY KOToPOH IOJIooKeH NO KoJIInueCTBY MOpΦEM IIOPIAky HX IOJIIOKeHNA, KJIACC KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX IIOpa3JeJIeTcRA Na DBA IOJIaACCA: IIOcTBie5 (简单数词 jiandan shuci) IN CJOxHbIE (复合数词 fuhé shuci). IIOKJIaCC IIOcTBIX KOJIInueCTBeHHbIX YNCJIInTeJIbHbIMN (系数词 xi shuci); IOIJKLIACC CJOxHbIX KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX COCTOHT N3 DBYX rpyIIIN: KO3ΦΦHnEHTBIX IN pa3-pyIHbIX (系数词 xi shuci + 位数词 wei shuci) [9; 15; 21]. ΘTo 6a3OBoE cTpyKTHpyHOe DeJIeHne KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX. B KJIaCCNΦHKaIHHX KNTaIcKHX HCCJeIOBaTeJIe H E yUHTbIAeTcR IIOpIIOK CJIeIOBaHHa pa3pIIOB B CJOxHbIX YNCJIInTeJIbHbIX, XOTB H NEKOToPbIX CJIyAax (ot 11 do 19) MoKET 6bITb H HAO6OpOT: pa3pIINa H KOIInueCTBeHHa JAcTb + shi 'Iecr' IIpeIIeCTByET KO3ΦΦHnEHTHOH (位数词 wei shuci + 系数词 xi shuci)6. B KaueCTBE IIprMEpa IIprHEm KJIaCCNΦHKaIHHO YNCJIInTeJIbHbIX YJXao IIIHKaI (赵世开 Zhao Shikai), B pAMkax KOToPOH ON IIpeIIOxHI KJIaCC KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX DEJIHTB Ha IBA IOJIKlaccs: IIIOcTBie KOJIInueCTBeHHbIX (简单基数词 jiandan jI shuci) IN CJOxHbIE KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX (复合基数词 fuhé jI shuci). IIOKJIaCC IIIOcTBIX KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX c ToUKN 3peHN HfYHKIIHN DEJIHTCRA Ha IBE rpyIIIN: KO3ΦΦHnEHTBIX (系数词 xi shuci) IN pa3pIINHie (位数词 wei shuci) (IIprMEpbI cM. BblIe B φyHKIIHOHaJIbHO KJIaCCNΦHKaIHH). B IOJIKlaccce CJOxHbIX KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX TAKKe coCTOAT N3 3JIeMeHTOB IByX rpyIIIN - KO3ΦΦHnEHTBIX IN pa3-pyIHbIX [22, c. 180] - 6e3 yka3aHHa IIOpaIKA IN CJOeIOBaHHa (十三 shisan 'TrpHa-ⅡaTb', 三十 sānshi 'TprNiTaTb' IN T.П.). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[351,76,657,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ФУHKIиОHAЛьнА,СТPyКТYPHА... \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[796,77,825,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"179 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[184,760,825,785]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4B DAHHOM cIyae 3TN MMOI03NaHbIe cIIOBA IpeIcTAbJIHOt CO6Oi cyIeCTBHTeJIbHOe 分 fen \\\"auctb,doJIa\\\", cIJyKe6HOe cIIOBO (ctpykTyprHaa cIaTHIIa atPb6yTbHbOH IIpPrOdb)之 zhi,coHO3 又you'H,Ia', cyIeCTBHTeJIbHOe 点 dian 'TOUka'.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[220,785,615,799]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 B ypcckom 33bke BCTpeaetc TepMHN «IeepBoo6pa3HbIe» [20, c. 193]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[188,799,823,822]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 Bo3MOxHO, 3TO C8Ba3aHO CO CBOHCTBeHHbIM KHTaHCKOMY 3bIKy OIIyUHeHHEM BO MHOrNX KOHTeKCTAX YHCJIH-TeJbHORO - yI 'OINH', YTO BIAEETcRA peaJIH3aIHNe IPNHIIHa 3KOHOOMHN B 3bIke. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[186,822,823,846]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7 B ΘΟΙ Κλαςηφικαίουν ηειλήτελβόν έρορασελεγούτερ Ακόληχειδεν Ενρόμακονθειν Στέληθειν [22, c. 180]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[169,115,815,404]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CooTHoIIeHHe KO3ΦΦHIIeHTHO H pa3pIHOJ yAcTeB CLOXHbIX YHcJIHTeJIbHbIX yHtBIAeTcB MOpΦeMHo KJIaCCnΦKauHH B.I. TopeJIOba, KUla IIO6abJIeHo DeJIeHne IIOKJIacca CLOXHbIX KOJIueCTBeHHbIX YNCJIHTeJIbHbIX Ha aIITNBHyO IMyJIbTIINIKaTHBHyO rpyIINb. B ee pAMkax IPOCTbIMN cHTaIOTcOJHomOPΦeMHbIE YHCJIHTeJIbHbIe, O603HaayAOIIe YNCJIa OT OJHO TIO DecrTH. CLOXHbIe YNCJIHTeJIbHbIe OT 11 Do 19 o6pa3yIOTc SLOXeHHem IPOCTbIX 3JIeMeHTOB (DecrTH I OJHOmOPΦeMHoro IPOCTOrO) H Na3bIBaOTcA JINTHBHBM, IIOCKOJIbKY B 3aINCH IIΦpaMn IpeCTaBJIaOT cyMMY YNCJIOBbIX 3NaueHNN: 十三 shisān 'TrpHaIaTb' - 3TO 10 + 3. CLOXHbIe YNCJIHTeJIbHbIe OT 20 Do 90 YBLIAOTc MYJbTIINIKaTHBbIMN, IOTOMY TTO IH 3aINCBIpeCTaBJIaET pe3yJIbTaT yMHOKeHH YNCJIOBbIX 3NaueHNN (B J3bIke - IPOCTbIX MOpΦEM), OJHO N3 KOTOpBX BcerTa paBHO DecrTH: 三十 sānshi 'TrpHIaTb' - 3TO 3 × 10 [13, c. 43]. OJHaKO IIO6aBM, YTO MYJbTIINIKaTHBbIMN MOrYT 6bITb HE TOJIbKO YNCJIHTeJIbHbIe OT 20 Do 90 (KaK y B.I. TopeJIOba), HO I JIO6bIe IpyTHe, KpaTHbe IdecTn, IJIA IINΦpOBoH 3aINCH YNCJIOBbIX BbyPaJaeHH KOTOpBX Hc-IOJIb3YeTcToljBO yMHOKeHH. HaIPmEp, IINΦpOByO φOpMy 3aINCH YNCJIHTeJIbHOr三百 sanbai 'TrpHcTa'MoJHO IPOCTaBHTB BnIe BbyPajKeHH 300 = 3 × 10 × 10, KOTopoe IpeCIaTbJIaER c6oB IIPOHN3BeJeHH UHCJIOBbIX 3NaueHH, BbyPajKeHHbIX B J3bIke IPOCTbIMN MOpΦeMaMn. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[173,405,813,764]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Heo6xoIIMO 6paTHTB BHHMaHHe Ha TO o6cTOJTeJIbCTBO, YTO B KJIaaccnΦHKaIIHN B.I. TopeIOBa He OIppeJeIeK KJIaCCNΦHKaIIHN CtaTyc TaKHX CJOXHBIX YHCJIINTEJBHbIX, B IINΦpOBoI 3aIIHcN UHCIOBBIX BBIPAKeHNI KOTOpBX IIpNCYTCBYOT N CJOxKHeHne, H yMHOxKHeHNE: HApPIMep, 三百五十二 sanbaiwushi'er 'TpncTa IYTBDecaTBA' -3To $(3\\\\times 100) + (5\\\\times 10) + 2$ .KHTaIckAJIINHBrHcTNUeCKAraPiHIIINOTHOCHNT HX KAaIITNBHomU TIIpy [6; 24, c. 191]. OHaKO, HxCxOJa I3 ToRTO, YTO IOIO6HbIE YHCJIINTEJBHbIE 6pa3yIOTcIIOpeCTBOM CIOxKeHNI EJIeMeHTOB, IINΦpOBAI 3aIIHcB KOTOpBX, B CBOIO OuepeIb, IpeIcTaBJIeHa C IIOMOIIbU yMHoXeHHN, HX Heo6xoIIMO IIpoTHBOIOCTABHTTEM aIINTNBHbIM YHCJIHTeJBHbIM, EJIeMeHTBI KOTOpBX IIPOCTBI IN Hepa3IOJIKMbI (十三 shisan 'TrnHaIATb' = 10 + 3). IIOaTOMy IpeJIlaRaem rpyIINy CIOxKbIX AIDHTNBbIX KOJIuYeCTBeHHbIX YHCJIHTeJBHbIX IIOpa3IeJIaTb HA IIOIpyIIIY EJIeMeHTAPbIX aIINTNBHbIX IIOIpyIIIY CMEIIAHbIX aIINTNBHbIX (cm. Ta6JI. 1). YTOHNMI pIH NTOM, YTO B ycIOBHX HaJIuHry yMHOxKeHNI B IOIpyIIIcMEIIAHbIX aIINTNBHbIX YHCJIHTeJBHbIX IINOPHTEHTBI CTAtyc IIOJIyaeT HMeHHO CIOxKeHNe HA TOM OCHOBAHNI, YTO IIPN pa36HeHH N IOIO6bIX CIOXHbIX YHCJIHTeJBHbIX HA KOMIOHeBTs CHaHaJIa BVIDeJIaOTcTc Te, KOTOpBIE IIOJIyHeBI IITcMeCIOxKeHHN, a 3aTeM yJke Te, KOTOpBIE IIOJIyHeBI B pe3yJIbTaTe yMHoXeHHN. 3TO XopoIIIO BnIHNO IIpn OKpyJIeHHN YnCeJI, KOrDa ONyckaETcTa Yaactb, KOTopAR B 3aIIHcN IIOΦpAMN IIpNCoeIHHeHa K IIpeBdIyIe NOCpeICTBOM CIOxKeHHN. HanpIMep, YHCJIHTeJBHoe 三百五十二 sanbaiwushi'er 'TpncTa IYTBDecaTIBa'B pe3yJIbTaTe OKpyJIeHHN MOKeT IIOJIyauT BnD 四百 sibAI 'YeTbipeCTa HIn三百多 sanbai duO '6oJIee Tpexcot' HIn三百五十多 sanbaiwushi duo '6oJIee Tpexcot IYTHdeCteTNaHa OCHOBAHNN IINΦpOBoW 3aIIHcN $(3\\\\times 100) + (5\\\\times 10) + 2$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[173,77,201,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"180 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[430,76,549,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JI.JI.BAHKOBA \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[169,787,810,822]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"8 CoJIaCHO HcCJIeIOBaHHIO, IPOBEJIeHHOMy B. KOMPn, B cHCTeMaX cHcJIeHHN 172 a3bIKOB H3 npOHaJIIN3HpoBAHHbIX 196 (TO coCTABJIeT IPrH6JIIN3HTeJIbHO 88%) Iprn o6pa3OBAHHI YHCJIInTeJIbHbIX OJHOBpeMeHHo 3aJIeCTBOBAHbI H MUYJIbTHIIINKATNBHbI N aAJITNBHbI CIIco6bI [23]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[169,822,810,846]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"9 3decb MoKHO IPOBecTH aHaIIOHHo CO CJIOAMH B 6yKBeHHO 3aIHNC, IIpH pa3JIOKeHHN KOtOpbIX Ha coCTabHbIe YAcTH ChauJIa IIpoHCXoIHT DeJeHHe HA CJOH, A NOTOM Ha 6yKBbl. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[206,151,806,288]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0863/becc927f7fe248492f89215e38b12ebfce5a767058aceb456a95a1439290cc49.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Пресные</td><td colspan=\\\"3\\\">Сложные</td></tr><tr><td rowspan=\\\"2\\\">Коэффишиентные\\n一 yī ‘ODINH’, \\n二 èr ‘ДBA’, \\n三 sān ‘Трн’…….\\n九 jiǔ ‘ДЕВÀТ’, \\n十 shí ‘ДECÀТ’</td><td colspan=\\\"2\\\">Адптуньibles</td><td rowspan=\\\"2\\\">Мульшлилкатоньibles\\n二十 èrshí ‘ДВаДцаТ’, \\n三十 сānshi ‘Трпд-\\nцать’…….\\n八百 babαι ‘ВOCмь-\\ncot’, \\n九千 jiǔqian ‘ДЕВÀТь\\nТысяч’</td></tr><tr><td>Элем entapны\\nадптуньibles\\not 11 до 19\\n十一 shíyī ‘ODINH-\\nнадцату’…….\\n十九 shíjiǔ ‘ДЕВÀТ-\\nнадцату’</td><td>Сmeшаны\\nадптуньibles\\n九十三 jiǔshísān\\n‘ДЕВЯносту Трп'\\n二百二十二\\nèrbαι ‘Èrshi’ èr ‘ДBE-\\nсту ДВаДцату Два’</td></tr></table>\",\n \"table_caption\": \"[\\\"Ta6JI.1 \\\",\\\"CtpyKTypna (MopΦeMHa) KJIacCSHnKaIIHn KOJIInueCTBeHHbIX YHcJIHTeJIbHbIX \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[184,305,826,793]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CyuuctByet TaKke ctpyKtupHa KJIaccnФнKaIIHcIN Φyn (邢福义 Xing Fuyi), H3JIOKeHHa B Tpye «TpncTa BOIPOcoB KHTaIckOg rpaMMaTHKN» (汉语语法三百问) [25]. CoJIacHo eN, MeTaKJIacc CnCJIteJBHbIX DeJIITcH a KJIaccBbI O6IIHX (统数 tong shu) HIOpRAKOBbIX (序数 xshu). CpeIN IepBbIX BbIeJIaOTcRcIeIyIOIIHe YcTeBpe IOLJIACCA: KOJIueCTBeHHbIE (基数 ji shu), KpaTHbIE (倍数 bei shu) (三倍 san bei 'TpexKpaTHbI'), Ipo6HbIE (分数 fen shu) (三分之二 san fen zhi er 'DBe TpeH; 三点四 san dian si 'Trn IeJIbIX (H) YcTeBpe IecrTBIX) H IIpN6JIIN3H TeJbHbe YNCJIteJBHbIE (概数 gai shu) (二三米 er-san mi 'IbA-TPn MeTpa'; 十余次 shi yu ci '6oJIee Idecrtn pa3'). BTObIe IeJIATcH a IBa IIOJIACCA: KOJIueCTBeHHbIE YNCJIteJBHbIE c IpeΦHKcOM 第 di (第 di + 基数 ji shu) (第一 diyI 'IepBbI') H IOpRAKOBbIE YNCJIteJBHbIE BΦOpMe KOJIueCTBeHHbIX YNCJIteJBHbIX (基数形式 jI shu xingshi) (一号 yi hao 'IepBoe YncIoo'). B pamkax 3ToH KJIaccnΦnKaIIHN IIOJIACCC KOJIueCTBeHHbIX YNCJIteJBHbIX, B CBOIO OUpeIb, DeJIITcH a Tprn IpyIbI: YNCJIteJBHbIE pa3PraJa eINHH (个位数 gewei shu), paHROBbie YNCJIteJBHbIE (段位数 duanwéi shu) H CIOBOOCoTeAHN C YNCJIteJBHbIM (数词短语 shuci duanyu) [25, c. 146]. Oco6eHHocTAM N OINCAHHO KLIACCnΦnKaIIHN YBJIeTcR BBOHMAR CINH Φyn HOBAI TepMHHOLOTN BMeCTO o6IIeIPnHToH. HanpImep, IJIA O6O3NaueHNG KoEΦnIHNEHTHBIX YNCJIteJBHbIX NCIOJB3yeTc TepMHN 个位数 gewei shu BMeCTO IPINBvHOrO 系数 xi shu, a IJIA HANMeHOBAHN pa3PraIbIX - 段位数 duanwéi shu BMeCTO 位数词 wei shuci. Kpome TOTO, IIOPTNBOIOCTaBJIeHN He YNCJIteJBHbIX IIOHCXODHT He IIO IpiNHIUY «KOJIueCTBeHHbIE vs IIOPAIKOBbIE), a IIO IpiNHIIIY «O6IIe vs IIOPAIKOBbIE), n TaKIM o6pa3OM BBOITcR eINH KLIACCnΦnHOHH bYPOBeHb. YTO kacaETc R camx KOJIueCTBeHHbIX YNCJIteJBHbIX, TO IpeIIOKeHHa IpyIIHPOBKa B IOIDKIAcc EKOJIueCTBeHHbIX YNCJIteJBHbIX (YNCJIteJBHbIE pa3PraJa eINHH (个位数 gewei shu), paHROBbie YNCJIteJBHbIE (段位数 duanwéi shu), CIOBOOCoTeAHN C YNCJIteJBHbIM (数词短语 shuci duanyu)) HjXdAETcR B DAIBHeIIe IeTAIBHOI IPOPabOTke. IepBbIE ZBE rpyIIbl - He YTO HHOe, KaK IIOPCtBie KOJIueCTBeHHbIE YNCJIteJBHbIE. IIOSeIeHHa rpyIIHa - CIOBOOCoTeAHN C YNCJIteJBHbIM - IpeCIaTaNJEET B3rJlAnd CINH ΦHyN Ha To, YTO B HAcToIe pa6Ote TpaKTyETcKak CIOXHBeK OJOUeCTBeHHbIE YNCJIteJBHbIE.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[184,796,828,843]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"OHa H3 IeHTpaJIbHbIX IIpo6JIeM H3UyeHnKoJIHueCTBeHHbIX YIcJIHTeJIbHbIX 3a-KJIIOuaETcB OIIpeJIeHnTOrO, YEm JBIJHOITcCIOXHbIE YIcJIHTeJIbHbIE (HaIIpHMeP, 三十 sanshi 'TpNIaTb', 十三 shisan 'TpINaIaTb', 五百五十二 wUbaiwushi'er \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[351,76,658,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ФУHKIиОHAJIbHA,СТPyКТУРHA... \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[796,77,821,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"181 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[169,116,813,263]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"「IITbcOT IITbIdecT IBa）：cIOBAMH（复合数词fuhé shuci）HIN cIOBOOCOeTAHHMn (数词词组shuci ciü).CTopoHHNKamn IIprH3HaHHN IOIO6HbIX YHCJIHTeJIbHbIX CIOJX-HbIMN cIOBAMn JBIHOCTC SoI GouK3HN JIN INHJX3, OIIpeJIeJIHOIIne Hx KaK «cIOJX-HbIE cIOBA, IIOCTPOEHHbIe IIO aTpH6yTHBHOJ MOJIIN》(偏正复合词pianzheng fuheci)[10,c.37]10.HcxOJa H3 IIprHHIIIOB pa3JIHyEHHN cIOB H cIOBOOCoTeAHN B KNTaIKCKOM 3bIke, H3IOJKeHHbIX B.M. H.B. CoJIHIIeBBIMN [26,c.51-84], Mbl TaKKe cHtAem, YTO 3TO cIOBa, a He cIOBOOCoTeAHN11.O6OCHOBaHHem JBJIeTcra To, YTO KaJXIOe TaKoe cIOBO COcTOHT H3 ODNHOJbIX 3JIeMeHTOB, CNHTaKChueckn $\\\\phi$ yHKIIHOHHpyIOHx KaK eDHHOE IeIOe.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,263,811,384]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Дя сраьння: Дрбные coуетаня, тakhe kaK三分之二 san fen zhi er ‘Дытпгьх’и 五点九 wu dian jiu ‘ПЯТь ΚЕЛБИХ И DeВЯТь DecaTbIX’, coJIepka TJIeMeHTbI 分fen ‘чacTB,doJIЯ’, 点dian‘TOUka’и 之zhī, ПОТOMy MOrY 6bITb OIIpeJeHbI KaK CJIOBOcoTeaHnI, ПOTOMYЧTO КРOME ChIJIteJIbHbIX OHH COCTOrt H3 cyIeCTBHTeJIb-Horo(分fen‘чacTB,doJIЯ); 点dian‘TOUka’)и CTpykTypHOJчaTnIbI(之zhī). BcJIeI 3aД.СИИнаво[20,c.204]ДЯIx obO3HaueHnI 6byem HcIIOL3OBaTb TepMHnCocma6hbIe YucIoobIe bIpaJceHnI. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[294,397,685,413]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ФункионаЛьно-структурна Класснфкадья. \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,418,811,678]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B 1982 r.Чжу Дэси 6ылла пеДLOЖЕHA DeТАЛьная KJIaCCNФИКаИЯчСЛNTeЛь-НБИX - ФункIHОHAJIbHO-cTpYKtypHЯ $^{12}$ .В Hei KJIaCC KJIInueCTBeHHbIX ChCJI NTeJIbHbIX DeJIINTCЯ Ha ПОДКJIaCCbI co6cTBeHHO YnCJI NTeJIbHbIX (数词shuci)ИЧСЛNTeJIbHbIX KOHcTpyKIIIM (数词结构shuci jiégou).ПерБИй ПОДКJIaCC,В cBOIO OUEpeIb,coCToHT N3 TaKHX rpyIII, KaK: KoФФHIIeHTHBIE YnCJI NTeJIbHbIE (系数词xi shuci),pa3PraIIbIe YnCJI NTeJIbHbIE (位数词wei shuci)ИпriбЛЗNTeJIbHbIE YnCJI NTeJIbHbIE (概数词gai shuci) (ПрIMeРысm. ВИПЕВ CTpyKtypHо KJIaCCNФИКaIIIN).Bo BTOPOI ПОДКЛaCC BКЛIOUeHBI rpyIIIbI:1)KoФФHIIeHTHO-pa3PraIIbIe cTpYKtypbl (系位构造xiwéigouzao)(В cTpyKtypHо KJIaCCNФИКaIIIN-MyJbTHIIINKAATHBHbIe YnCJI NTeJIbHbIe:二十ershi‘ДВaIITb',五万wuwan‘ПЯТьДecrT TbcIcY);2)KoФФHIIeHTHO-pa3PraIIbIe KOM6HHaIIIN (系位组合xiwéizuhe)(ИЛN CMEIIAHHbIe aДЛNTNBHbIe YnCJI NTeJIbHbIe:三千五百 sanqianwubai「TPn TBICaI INIITbcOT);3)cIoxHbIe KOrФФHIIeHTHO-pa3PraIIbIe cTpYKtypbl (复杂的系位构造fuzá de xiwéigouzao)(coCTO7R H3 IIprNo3NTeJIbHBIX YnCJI NTeJIbHbIX,KoФФHIIeHTHO-pa3PraIIbIX cTpYKtypИЛN KOM6HHaIIIN:两三千 liangsānqian ‘ДBE-ТРи TBICaIIN’,二十万ershiwan‘ДВecSTN TBICaY)[21,c.45-47]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,678,811,774]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"KpnTHKy B3rJlIbIy Jky Iocn, JIO PInH PIIHIEET, YTO $\\\\ll$ KJIacchNfHKaIIHn Ha KO3-phiHHeHTbIe H pa3pIINbIe YHCJIHTeJIbHbIe IIPOBOITcC IIO3HnI rpaMMaTHKN, B TO BpeM KaB bIJeHne IIpi6JIIN3HTeJIbHbIX YNCJIHTeJIbHbIX IIPOHCXOINT C IOHOHTNHOfYHKUHOHaJIbHOI TOKN 3peHn. OChOBaHHN KJIacchNfHKaIIHn OTJIuHaOTc, YTO He IIO3BOJIeT cTaBHTb Hx Ha OINH yPoBeHb》 [8, c. 123]. Mbl He MoJem coTJIacHTbcRc C JIO IIHN, BO-IIepBbIX, B TOM, KaK OHa B DaHHOM cJlyue oIpTeJIeT KJIaccNfHKaIIHn. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[173,77,201,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"182 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[430,76,549,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JI.JI.BAHKOBA \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[169,785,810,821]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10 MbI He MoKem CoTJIacHTbcra CcRo TOnK3HOM N JIn HnUxK B OIIpeJeIeHHN CIOBOO6pa3OBaTeJbHOH MoJeHN KAK aTPh6yTHBHOI. IocKOJIbky BCE 3JIeMeHTb paBHbI, TO OTHIOIIeHHM MeKJy HmH KONJIYTHBHe, CJeIOBaTeJIbHO, CIOOCoOM CIOBOO6pa3OBaHHA JBJIETcra CIOBOCIOJOKeHne. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[205,821,620,834]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"11 B aHJIINKOM J3bike, HAnpoTHB, 3TO cJIOBOOCOteHaHHa [27, c. 15; 28; 29]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[203,834,741,847]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"12 ΘTo 6bIa IepeCmOTpeHna Bepcna eTo KJIaccnФнkaIHH YIcJIHTeJIbHbIX 1958 r. [9, c. 185-187]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[178,151,833,479]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0180/e6410af9e4d84a43a112169b46c2db9d2d85e7cf983bba9a3114c3462240c2bc.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td colspan=\\\"5\\\">Количес委书记нычеслистовы (基数词ji shùci)</td></tr><tr><td colspan=\\\"2\\\">Өлемэнтарные количес委书记нычеслистовы (基本基数词jiběn ji shùci)</td><td colspan=\\\"3\\\">ВарнаимBERые количес委书记нычеслистовы (变化基数词bianhuà ji shùci)</td></tr><tr><td rowspan=\\\"2\\\">Коэфшиcent-hычеслистовы (系数词xi shùci) 一yi'один', 二 èr'лва', 三 san'tprn'....... 十shi'deсять'</td><td rowspan=\\\"2\\\">Раразднычес-лпеслистовы (位数词wèi shùci) 十shi'deсять', 百bái'ctо', 千qiān'tbíсая', 万wan'tdeсять тьсч', 亿yi'cto мнлл-jионов'</td><td>КоэфшиcentHo-pažядные мульштолкатнвные КOLIчес委书记нычеслистовы (系位构造基数词xiwèi gòuzào ji shùci)</td><td colspan=\\\"2\\\">КоэфшиcentHo-pažядные aДЛТНВье кOLIчес委书记нычеслистовы (系位组合基数词xiwèi zuhé ji shùci)</td></tr><tr><td>Осычные коэфшиcentHTo-раздные мульштолка-ТВВье КOLIчес委书记нычеслистовы (特殊系位构造tèshū xiwèi gòuzào) 八十bāshi 'BOCEMbDESCYT', 八万亿bāwanyi 'BOCEMb TpNi-лнонов'</td><td>Спецima-bные коэфшиcentHTo-рazдные aД-ДЛТНВье КOLIчес委书记нычеслистовы (一般系位组合yībān xiwèi zuhé) 八十八bāshibā 'BOCEMbDESCYT BOCEMb'</td><td>Спецima-bные коэфшиcentHTo-рazдные aДИ-ТНВье KOLIчес委书记нычеслистov(特殊系位组合tèshū xiwèi zuhé) 一千五六百yīqianwūliubäi 'ОДи Тьсяча ПЯтусOT-ОДи Тьсяча Шость-cot'</td></tr></table>\",\n \"table_caption\": \"[\\\"Ta6II.2 \\\",\\\"Функшоньсгтчякша КласнфКацьkoJичecтВeHHbIX YaHbYhBnHa \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[183,491,826,648]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ko3ΦHIIINHeHTBHe I pa3pIbHbIe BnIbI YnCJIInTeJIbHbIX, KaK 6bIIO OIIHcAHO BbIIe, BbIeJIaOTcB pe3yJIbTaTe ΦyHKIIOHaJIbHOrO KJIacchΦHIIuPobAHnA, a IIpN6JIIN3TEJIbHbIE YNCJIInTeJIbHbIe AIBIAOTcpe3yJIbTaTOM KJIaCCNΦHIIuPobAHnHX IO TOUHOCTH, O YempeYb IIOJETB IIPOJOJIkeHH NCCJIeIOBAHnA. Bo-BTOpbIX, He MOxEM corJIacHTbcR, YTO IIpN KJIaCCNΦHIIuPobAHnKo3ΦHIIuNEHTHBIX, pa3pIbHIX I IIpN6JIIN3TEJIbHbIX YNCJIInTeJIbHbIX OCHOBAHnKJIaCCNΦHkaIaIpa3JIuaOTcR. HaIpOTNB, YxJy D'cn HcIOJIb3yETOINO KJIaCCNΦHkaIaIHHOe OCHOBAHNe - cTrpykTypy, KOtOpA y IIpN6JIIN3TEJIbHbIX YNCJIInTeJIbHbIX CIOXHaR. Ha NaI B3rIaI, IIpN paccMOrpeHH Nco3ΦHIIuNEHTHBIX, pa3pIbHIX I IIpN6JIIN3TEJIbHbIX YNCJIInTeJIbHbIX cФyHKIIOHaJIbHO-cTrpykTypHOr ToUKN 3peHHN 6blIO 6blYmecTHO He HcIOJIb3OBaTB TepMHN «IIpN6JIIN3TEJIbHbIE YNCJIInTeJIbHbIe>. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[183,649,826,743]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Tak, ot yIIOTppe6JIeHnJaHHHO TepMHa OTKa3bIbAetcra YaHb JaHbIH H CBOeM BapHaHTe $\\\\phi$ yHKIIHOHaJIbHO-CTpyKTypHO KJIacchФHKaIIHN KOJIHecTeBHeHHbIX YHCJIHTeJIbHbIX. BaXHBIM erO IOCTHXKeHem MOKHO cHTaTB KaTeROP3aIIIO cJOXHBIX KOJIHueCTBeHHbIX YNCJIHTEJIbHbIX, IIpNBedeHHyIO B BVNe Ta6JIHbI B [6, c. 2], c KOTopoR Mbl CHTaEM YMeCTHBIM IIO3HaKOMITb YHTaTeJI, IIpeJIOxHB PyCCKO83bIHyBIE TepMHHOJIoTHiueCKHe aHaJIOrn H IIpHMepBi (cM. Ta6JI. 2). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[183,744,826,847]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B KJIaccnФнKaIIHryaHbЯHbIINHa IpocTbIe H cIIOJHbIE KOJIHueCTBeHHbIe YHcJIITeJIbHbIe, KOTOpbIe B BbIIIEoIIncAHHO CtpyKtypHO (MOPΦeMHo) KJIaccnФнKaIIHnHa3bIBaJIncb 简单(基)数词jiandan(ji)shuci IIpocTbIe (KOJIHueCTBeHHbIe) YHcJIITeJIbHbIe' n复合(基)数词fuhé(ji)shuci CIOJXbIe (KOJIHueCTBeHHbIe) YHcJIITeJIbHbIe', Ha3bIBaIOTc 基本基数词jibenji shuci 'JIeMeHTapHbIE KOJIHueCTBeHHbIe YHcJIITeJIbHbIe' n变化基数词 bianhua ji shuci'BapNaTHBHbIE KOJIHueCTBeHHbIe YHcJIITeJIbHbIe' \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[351,76,658,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ФУHKIиОHAJIbHA,СТPyКТУРHA... \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[796,77,825,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"183 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[169,116,810,159]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"COOTBETCTBEHHO. O6bIcHTb 3TO MOxHO TEM, YTO B IepBOI IIape TepMHOB aKIIeHT cJIeJIaH Ha CTpykTpe, a BO BTOPOI - HA $\\\\phi$ yHKIIHX B COOTBETCTBHN C THIOM KJIacCHHII-KaIIHH. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,162,813,355]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"TyaHb YhbHn OHpaetcHa IpeJIIOKeHHoe B 1982 r. Jxky Ioc HJeHHe IO ctpykType BCEX KOJIInueCTBeHHbIX YncJIHTeJIbHbIX Ha IBe 6OJIbIIHe rpyIINbI: Ko3ΦHNIeHTHO-pa3pIINbIE KOM6HNHaIIH (系位组合 xiwei gouzao) IN KO3ΦHNIeHTHO-pa3pIINbIE cJIeDyHOIIHe pyccKo3bYHbIE aHaJIOTn: KO3ΦHNIeHTHO-pa3pIINbIE MYJBTHIIHKaTINBHe KOJIInueCTBeHHbIE YncJIITeJIbHbIE IN KO3ΦHNIeHTHO-pa3pIINbIE aIINTHB-Hbe KOJIInueCTBeHHbIE YncJIITeJIbHbIE COOTBEtCTBeHHO. O6OCHOBAHHem cJyKHT pa3JIuHne B BVdax MaTeMaTHueckHX ONEpaIH (CIOJKeHH eHH yMHOGKeHHe) C hCIOBbIM 3HaueHnMn B IIOΦPOBbIX φOpMax 3aIIHCn BbIPaKeHH, B BVNe KOToPbIX MOKHO IIpeIcTaBNTb YncJILO, Bep6aJIIN3yeMOe KOJIInueCTBeHHbIM YncJIHTeJIbHBIM. 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ΓaHb YHbIINHa3bIBaET HX CLOXHbIMN YNCJIHTeJIbHBIMN (复合数词 fuhé shici). B DaHHoI rpyPiNE BO3MOXHO BbIeJIITb IBe IIOIpyIbI: O6bIyHbIe KO3ΦHnIHHeHTHO-pa3pIINbIE MjIBTHIIHKaTHBHBIE KOJInueCTBeHHbIE YNCJIHTeJIbHBIE (一般系位构造 yibān xiwei gouzao) IN cIIeIIHaJIbHBie Ko3ΦHnIHHeHTHO-pa3pIINbIE MyJBTHIIINKaTHBHBIE KOJInueCTBeHHbIE YNCJIHTeJIbHBIE (特殊系位构造 teshui xiwéi gouzao). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,505,811,731]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"O6bIHyBIE Ko3ΦHIIeHTHO-pa3pIHHbIe MyJIbTHIIJHKaTHBbIe KOJIHyEcTBeHHbIe YHCJIHTeJIbHbIe IpeIcTaBIAIOH TnB BapHaTHBbIX KOJIHyEcTBeHHbIX YHCJIHTeJIbHbIX, KOTOpBIE, KaK IIpaBNILO, BKJIHOaIO TpN yPOBbMyJIbTHIIJHKaTHBbIX. B pa6oTe ΓyaHb YHbIINHa OINcHa NIO6HaN TpEXyPObHEBa MoJeIb. OJHOyOBHeBOe MyJIbTHIIJHKaTHBHOe YHCJIHTeJIbHOe, O6O3NaUaEMoe KaK A1, MoJeIHyEETcIIO PPINHIUY Ko3ΦfUuueHMna Yacmb $\\\\times$ pa3pOdHa Yacmb, HApNMeP, 八百 babi 'BOCEMbCOT' (800 = 8 × 100).ДByxypOBHeBOe Ko3ΦfHIIeHTHO-pa3pIHHOE MyJIbTHIIJHKaTHBHOe KOJIHyEcTBeHHOE YHCJIHTeJIbHOe (A2) MoKeT 6bITb IIpeIcTaBJIeHO KaK A1 $\\\\times$ pa3pOdHa Yacmb, HApNMeP, 八百万 babiwan 'BOCEMb MIIJINOHOB' (8 000 000 = 800 × 10 000). Ko3ΦfHIIeHTHO-pa3pIHOe MyJIbTHIIJHKaTHBHOe KOJIHyEcTBeHHOE YHCJIHTeJIbHOe c TpoIHoM MyJIbTHIIJHKaTHBHOcCB3bIO (A3) HMe-ET BnA2 $\\\\times$ pa3pOdHa Yacmb, HApNMeP, 八百万 babiWanyi 'BOCEMbCOT TpHIJIHOHOB' (800 000 000 000 000 = 8 000 000 × 100 000 000). ΘTO TNIHNHbIe MyJIb-THIJIHKaTHBHBie MOJI IN IOCTPoEHNA YNCJIHTeJIbHbIX. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,732,811,794]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B cIeHnJIbHbIX KO3ΦΦHIIHeHTHO-pa3pIINbIX MyJIbTHIIJIHKaTHBbIX KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX IBe KO3ΦΦHIIHeHTbIX YacTH (B IIOpJKe BO3paCTaHH) COBMIIeHbIC OIHHO pa3pIINo H cTpoTcI IO MoJeH (KO3ΦFuuuEHHma YacMb • KO3ΦFuuuEHHma YacMb) × pa3pIaHaY acmb13, rIe cOIOJOKeHHbIE KO3ΦΦHIIHeHTHbIe YacTH \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[173,77,201,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"184 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[430,76,549,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JI.JI.BAHKOBA \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[203,832,482,846]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"13 3decb cHMOBJO3Haayet coIOJOKeHHe qacte. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[183,116,825,149]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IpeIcTaBJIaHOT co6oB BapHaTnBHyU OacTb: 三四万 sānsiwan ‘TpNiuaTb-copok Tbicu', 两三千 liangsānqian ‘IBe-tpN TbICaHN’. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[183,152,828,231]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ko3ΦHnHeHTHO-pa3pIHBie aIHTNBHbIe KOJIhUecTBeHHbIe YHCJIHTeJIb-HbIe (系位组合基数词 xiwéi zuhé jishuci) B KJIaccnФHKaIIH Igamma HbIHNHa IIpeIcTaBJIHO tco60B TOpYIO rpyIIy BapNaTHBbIX KOJIhUecTBeHHbIX YNCJIHTeJIbHbIX. OTHOIIeHnMaJMyJIbTHIIJIHKaTHBbIMN qacTAMN B Ko3ΦHIIeHTHO-pa3pIHOKOM6HnHaIIHЯBJHOtcaAДNTHBHbIMN[6,c.2]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[183,231,825,278]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"KoΦHHeHTHO-pa3pIHBie aIITHBHbIe KOJIHyEcTBHeHbIe YHCJITeJIbHbIe IpeIcTAbJIeHbI cIeIyIOIIHM NtBIO MoJeIMN, rIe A1 MoJeINpyeTcI IO II pIHIIpy Ko-ΦfquueHmHa yacmb $\\\\times$ pa3pIaHa yacmb: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[183,279,825,638]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1) A1 + A1 (ВарнаNT: A1 + Коэффншени enthaя част): 八百八 babаibā ‘BOCEMbCOT BOCEMbДecЯT’; 八十八 bashiba ‘BOCEMbДecЯT BOCEMb’. B Данно модел самоe 60Льшоe КОЛичесТВО pa3ряпов paBНо ПЯтп: 八万八千八百八十八 bawanbaqiānbabäibashiba ‘BOCEMbДecЯT BOCEMb ТБИЧ BOCEMbСOT BOCEMbДecЯT BOCEMb’, Ге самын 60Льшои pa3ряД - DeсЯт TBICЯн (万 wan); \\\",\\\"2) A1 + A2: 八亿八千万 bayibāqianwan 'BOCEMBcot BOCEMBdeCAY MHIJIHOHB'. B ΘToI MOJIeJIH YacTBIO A1 MOKeT 6bITb TOJIbKO 亿 yi 'CTO MHIJIHOHB', a pa3pIiHOYacTBIO B A2 - ToJIbKO 万 wan 'DEcAeTb TBICaY'; \\\",\\\"3) A1 + Hylb + A1 (BapnaHT: A1 + Hylb + pa3pndna qactb): 八万零八(百/十)bawanlingba(bai/shi) 'BOCEMbIeCt TBicayu BOCemb (coTeH/IecrTkoB). HaOJaIaIcIpeIHyIe pa3pndna KOJIueCTBeHHaqactb (A1) IIpeBbIIaet haoJMyoScIocJIe HyJIpa3pndHyIO KOJIueCTBeHHyIO qactb (A1); \\\",\\\"4) A2 + HUJIb + A1: 六千万零八百 liuqianwànlingbabai 'IIEcTBIIecrT MHIJIHO-HOB BOCembcot'. B KaueCTBe pa3pJHNoKoJIHueCTBHeHHoJ caTn, o6O3HaayEmoA2, BbICtYIIaOT 亿 yi 'cTO MHIJIHOHOB' N万 wan 'IeCraTb TbICr', KOTOpbIe IIpeBbIIaHT pa3pJHHyO KoJIHueCTBHeHHyO qacTb A1; \\\",\\\"5) A1 + BapHaTHBNaJyacTb: 二十一二 ershiy'er 'ДВaIIaTb OINH-ДВaIIaTb IBa', 三十五六 sānshiWuliu 'TpHIIaTb IIaTb-tpHIIaTb IIecTb', 一百七八 yibaiqibā 'ctO cEmbIeCrt-CTO BOCEMbIeCrt', 一千五六（百） yiqianwuliu(bai) 'TbICya IIaTbCOITbICya IIecTbcot', 一万八九（千） yiwanbajiu(qian) 'TbICya BOCEMbCOT-TbICya IIeBraTbcot' [6, c. 7-9].\\\"]\",\n \"page_idx\": \"10\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[183,639,825,669]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"KoJIHueCTBeHHbIe YHcJIHTeJIbHbIe O6JIaIaIOT OIIpeJIeHHbIMN JINHeiHHbIMN xa- paKaTepnCTHKaMn: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[183,671,825,719]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1) rpyiHa Ka03ΦHnHeHTHO-pa3pAINbIX MyJIbTHIIJIHKaTHBbIX KOJIHueCTBeHHbIX YHCJIInTeJIbHbIX: MeHbIiH pa3pI HaxOJNTcBIIpeI, 6JIbIiH pa3pI - IIO3aIi: 五百万亿 wubaiwanyi 'IITbcOT TpHJIINOHOB'15; \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[351,76,657,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ФУHKIиОHAЛьнА,СТPyКТYPHА... \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[796,77,825,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"185 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[183,764,825,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \".14B daHHoI rpyiHne YHCJIHTeJIbHbIX IIpr IINΦpOBOI 3aIHCH HX YHCIOBbIX BbIPAKeHHN 3aJeCTBOBaHbI DBA BHJa MaTeMaTHueCckHX OIIepaIcN C YHCIOBbIMN 3HaueHHMn (ymHOJKeHH N cJOKeHHe), YTO, corJIacHO IppeblNyIeI KJIaccHΦHKaIHH, IIO3BOJIeT CHTaTB HX CMEIIAHbIMN aIITNTHBbIMN. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[183,797,825,847]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"15 TOBOPHTb O IOIO6HOH XapakTePCHTKe BO3MOKHO, TOJIbKO ocTaBaCb HA IO3NIOHN HCKJIIOUeHH N3 UHCJa pa3pIbIX YNCJIITHeJIbIX 十万 shiwan 'cTO TBICaY', 百万 baiwan 'MNIJIOH', 千万 qianwan 'DecaTb MNIJIOHOB' H. I. B IPOTHBOM CJIyae, 6yIET JIN6O ODNH pa3pI (KAk B I pHMepe: 百万亿 baiwanyi 'cTO TPHIJIOHOB'), JIN6o pa3pIbIb 6yIyT paioJIaRaTbc OR 60JIbIIIEr K MeHbIIEMy.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[169,116,811,164]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2) rpyiHa Ko3ΦHHeHTHO-pa3pIHyIx aIITHBbIX KOJIHueCTBeHHbIX YHCJIInTeJIbHbIX: 6OJIbIIHn pa3pI NaXoIHTcB IpeEIN, MeHbIIHn - I03aIIN: 八千八百八十baqianbabaibashi' BOCemTbICyu BOCemBCOT BOCemDeCrt' [6, c. 8; 30, c. 348]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,165,811,225]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IIO cpaBHeHnIO CO CJIOBaMn, OTHOCaIMMncs K IpyTm YacTm peuH, OTJeJIbHbIE BnIbI KOJIInueCTBeHHbIX YNCJIInTeJIbHbIX (KO3ΦΦHIIeHTHO-pa3pIbIe aIINITNBHbIe) XapaKTepn3yOTcra 6OJIbIIHM KOJIInueCTBOM MOpΦem. HaIIpHMeP, KOJIInueCTBeHHoe YNCJIInTeJIbHoe \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[346,227,667,243]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"六亿九千五百二十七万三千一百四十一\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,244,810,275]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"liuyijiuaqianwubai'ershiqiwansanqianyibaishiyi IIeCTbcOT DeBraHocTo IaTb MIIJIHOHB IBecTH cEmbIeCt TpN TBICAH CTO copOK OINH' BKJIOUaET 17 MOpem. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,277,811,378]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C ΦyHKIIOHOJbHO-cTpkyTpyHOn TOck 3peHHaPaccMaTpHBaOT YNCJIHTeJbHbIE TaKKe Cno TQxK3H N JIn HxK3, HcIOJIb3yprn 3tOM HOByo TepMHHOJIoTHNo. DeJIeHNHe KJIacca YNCSJIHTeJbHbIX IPOIN3BOJNTcR Ha IOJKIaCCbI OIIpeJIeHHbIX (确数 què shù) H IIpN6JIH3HTeJbHbIX (概数 gài shù). OIIpeJIeHHbIe YNCJIHTeJbHbIe (HJIN KOJIInueCTBeHHbIe) IIOJa3IeJIaHOTcR Ha IBe 6OJIbIIHe rpyIIbI: 3JIeMeHTapHbIe (基本数词 jiběn shùci) H CJIOXHbIe (复合数词 fuhé shúci). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,380,811,429]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\Gamma \\\\rho \\\\mathbf{y}\\\\Pi \\\\Pi$ 3JIeMeHTapHbIX KOJIuHecTBeHHbIX YHcJIHTeJIbHbIX DeJIHTcRA Ha K0eΦHIIeHTHbIe (系数词 $x i s h u c i)$ H pa3pIINbIe (位数词 wei shuci) IOIgpyIIbI, KaK B paMkaX $\\\\phi \\\\mathbf{y}\\\\mathbf{h}\\\\mathbf{k}\\\\mathbf{I}\\\\mathbf{n}\\\\mathbf{O}\\\\mathbf{H}\\\\mathbf{a}\\\\mathbf{J}\\\\mathbf{b}\\\\mathbf{H}\\\\mathbf{O}$ KJIacchΦHKaIIH. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,430,811,565]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Гуйпа сюжных чслтелых пдраимлгяетс на пдгрупьк кофнichteН-рабдных мльтейнлкатнвых колчесъевных чслтелых (系位构造 xiweigouzao) (二十 ershi 'Двацат',三百 sanbai 'Трпст', 四千 siqian 'Чтбpe Тьсян', 三百万 sanbaiwan 'Трн МИЛиога', 五千四百二十万 wuqiansibai'ershiwan 'ПЯтБесят chebipe мльллона Довсп Тьсян', 四千三百八十一亿 siqiansanbaibashiyi 'ЧтбpecTa Tprnцать BOCem мльллардов СTO МИЛиогов') И диншьх КОЛчесъевных чслтелых (整零构造 zheng ling gouzao) (十六 shiliu 'Шесту), 九十九 jiushjiu 'Девеносто Девяты', 七百六十 qibaliushi 'семсOTИsectbDecaT'). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,568,811,600]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Crao TcHJIN HcKoINcBIAOT TpN MoJeJI, KOTOpbIMM MOyT 6bITb IIpeICTaBJIeHbI aINTHBHbIE KOJIYueCTBeHHbIe YNCJIHTeJIbHbIe (整零构造zheng ling gouzao): \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,601,811,783]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1) MyJIbTHIIJIHKaTHBHaJaCTb + MyJIbTHIIJIHKaTHBHaJaAcTb $^{16}$ (JAcTei MoKeT 6bITb 6oJIbIe IByx): 三百五十 sanbaiwushi 'TpncTa IIpIbDecr', 四万三千五百 siwansanqianwubai 'copok Tprn Tbcryn IIpIbcoT'; \\\",\\\"2) MyJIbTHIIJIHKaTHBHaJa YacTb + KoOΦΦHIIHeHTHa JaCtB: 二十二 ershi'er 'IbA-IIaTb IBa', 五千零二十五 wUqianling'ershiwu 'IITb TBICrJ IBaIIaTb IYrTB' (MyJIbTHIIJIHKaTHBbIX YacTei MoKeT 6bITb 6OJIbIe IByX);\\\",\\\"3) IecraTb + KO3ΦΦHHeHTHa auctb: 十一 shiyi 'OINHHaIIaTb' [10, c. 38].\\\",\\\"B IIpeIcTaBJIeHHbIX Crao TgUxK3HOM H JIn IHHxK3 MoJeJX MyJIbTHIIJIHKaTHBHJa YacTb He pa3JIoxKeHa Ha yPoBHN A1 H A2, KaK 3To 6bIJIO cJIeJIaHO $\\\\\\\\Gamma$ yaHb JHBIIHOM pRn OIIHCaHHN MOJIeJ K0eΦHIIeHTHO-pa3PraIHHbIX aLIITINHBbIX KOJIHueCTBeHHbIX YHcJIINTeJIbHbIX. O6bAcHReTcra To TEM, YTO «HOCHTeJIH KHTaIcCKOro Ja3bIka BocIIpHHMaIOT \\\"]\",\n \"page_idx\": \"11\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[173,77,201,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"186 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[430,76,549,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JI.JI.BAHKOBA \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[169,809,811,846]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"16 OtCyTCTBHe pa3pHIOH qacTH B daHHo MoJeH IObJcHReT OcyTCTBHe oIpeJIeHnRA $\\\\langle \\\\kappa \\\\mathrm{O}3\\\\phi \\\\phi \\\\mathrm{H}$ HHTHOpa3pHIOHe] npH IepeBOHe Ha pycckn J3bIK TepMHHa 整零构造zheng ling gouzao 'aIINTHBHOe KOJInueCTBeHHoe YHCINTEJIbHOe'. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[183,115,823,181]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"60JIbIIoe YHcJIo KaK IeJIoe, a MaJIeHbKOe - KaK ocTaTOK, KOtOpbIE COOTHOcTcKaK IeJIoe HuaCTb》. IIpn 3OM 整 zheng B TepMHHe 整零构造 zheng ling gouzao cHTaTeTc6 60JIbIIM YHcJIOM, a 零 ling - MaJIeHbKHM IO IIpHHIINy 前项大后项小 qián xiang dà hou xiang xiao '60JIbIIoe - BIIpeEN, MaJIoe - IIO3aIIN' [10, c. 39]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[183,181,826,455]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"I-TaK, B xOide KJIacchΦHIIpOBaHHN KOJIInueCTBeHHbIX YHcJIInTeJbHbIX C ΦyHKIIHOJIbHO-CTpykTpyHOJ TOckn 3peHHN B IaHHOM pa3JeJIe 6blJIN paccMOteHbI DbaIOxOJa. Obe KJIaccHΦHkaIHH OueHb IOXoJN. I ΓayHb YHbIHN, H CToYK3H C JI IN HUJK3 CXODTcB BO B3FJIaIax (B TOM YHcJIe B TepMHNOJIOTHneCKOM IIaHe) Ha 3JIeMeHTapHBie KOJIInueCTBeHHbIE YHcJIIneB. B OTHOIIeHH CJOXHbIX KOJIInueCTBeHHbIX YHcJIIneTbHBIX TAKe Ha6JIHOaETcB COBIIaHEHne TepMHOB IIpr O6O3HaYeHH MUYbTHIIHKaTHBbIX KOJIInueCTBeHHbIX (系位构造 xiWei gouzao 'KO3ΦHIIHeHTHO-pa3PdIbIE MyJBTHIIHKaTHBbIX KOJIInueCTBeHHbIE YHcJIInTeJbHBIX'. OJHaKO B O6O3HaueHH CJOXHbIX KOJIInueCTBeHHbIX YHcJIInTeJbHBIX, O6pa3OBAHHbIX IocpeIcTBOM CJOXeHH N HCJIObIX 3HaueHH B IIΦpOBoB 3aIIHC, Ha6JIHOaETcPa-cXoJDeHne B TepMHIOIOrHH. TaK, B KJIacchΦHkaIHH FyaHb YHbIHNa AKIeHT cDEJAH HA TOM, H3 KaKHX 3JIeMeHTapHBIX qacteON OHn coCToT (系位组合 xiWei zuhé 'KO3ΦHIIHeHTHO-pa3PdIbIE aDIHTHBIE KOJIInueCTBeHHbIE YHcJIInTeJbHBIX'), B TO BpeM KaCra ToxK3H JIN IHuK3 DeJIaOT aIKIeHT Ha pa3Mepe IN POCJIeIOBaTEJbHOCTN coCTABJIHOIIHX Hx qacteN (整零构造 zheng ling gouzao 'OyK8. KOHCtpyKIIHN YHcJIInTeJbHOrO CcoHTaHHem 60JIbIOI H MaIOJ qacte; aDIHTHBHOE KOJIInueCTBeHHOE YHcJIInTeJbHOe'). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[183,457,826,631]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Kpome TOrO, B o6oHb BapHaHTax KJIaccnФHKaIIH aIITNBHbIe YNCJIHTeJIbHbIe He IIOpa3JeJIOTcH Ha 3JIeMeHTapHbIe aIITNBHbIe H cMEIIaHHbIe aIITNBHbIe (cM. cTpykTpyHyo (MopΦemHyO) KJIaccnФHKaIIHO). Te YNCJIHTeJIbHbIe, KOtOpBie Mbl IIpHHMaem 3a 3JIeMeHTapHbIe aIITNBHbIe, B KJIaccnФHKaIIHN ΓyaHb YHBIIINHa coBcEM He ΦIpynpHOT. Y CSo IOTX3Ha H JIn INHxK3 OHN BXOJAT B IIOIpyIIY «aI- IITNBHbIE KOJIHyCEtBeHHbIe YNCJIHTeJIbHbIe》(整零构造zheng ling gouzao). OHaKaC ΦyHKUHOHaJIbHO-cTpykTpyHO TOckN 3peHn A.INTNBHbIE KOJIHyceTBeHHbIE YNCJIHTeJIbHbIe cJIeDyET pa3JeJIaTb Ha 3JIeMeHTapHbIe aIITNBHbIe H cMEIIaHHbIe aIITNBHbIe KaB I IOIpyIIIE o6bUHbIX KO3ΦΦHIIHeHTHO-pa3PraIHbIX aIITNBHbIX KOJIHyceTBeHHbIX YNCJIHTeJIbHbIX, TaK H B IOIpyIIIE cIeIIaJIbHBIX KO3ΦΦHIIHeHTHOpa3PraIHbIX aIITNBHBIX KOJIHyceTBeHHbIX YNCJIHTeJIbHBIX. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[183,632,826,760]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IJIaBHbIM JKe OTJINHHe MBAIEcTc BBeJeHHe IyaHb JHBiHOM KJIacchHkAIIOHbIX IIOIgpyIIcIeIIHaJIbHbIX KO3ΦΦHIIeHTHO-pa3pIbIX KOJIueCTBeHHbIX YHCJIHTeJIbHbIX KaK B MyJIbTHIIJIHKaTHBbIX, TaK H B aIITNTBbIX rpyIIax. 3TN cIeIIaJIbHbIE IIOIgpyIIbI IIpeIcTaBJIeHbI IIpi6JIIN3HTeJIbHbIMN YHCJIInTeJIbHbIMN. HeIOctaTOK KJIaccnHkAIIHN IgammaH JHBiHa BVIIITcR B TOM, YTO IIOIO6Ha CIIeIIHaJIbHaIIOIgpyIIHa IOJIXHa 6bl 6blr N y Ko3ΦΦHIIeHTbIX YHCJIInTeJIbHbIX, IIOCKOJIbKy OHN TaKKe MOryT BbIPAkaTb IIpH6JIIN3HTeJIbHOe YNCJIO, COUeTAcB dpyr c IpyTom (三四 sansi 'TpH-YeTbipe'). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[183,760,823,822]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C yueTOM 3aMeHaHn, BBiCKa3aHHbIX OTHoCHTeJIbHO BblIIepacCMOTpeHHbIX BaPnaHTOB $\\\\phi$ yHKIIOHaJIbHO-CTpyKTypHOJ KJIaCCNΦHkaIIIN KOJIIneCTBeHHbIX YHCJIInTeJIbHbIX, IIpeJIlaRaem aBTOpCKoe BnJeHHe aHaJIIN3HpyeMOrO KJIaCCNΦHkaIINHOHor DeJIeHn, IIpeIcTabJIeHHoeB Ta6JI.3. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[351,76,657,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ФУHKIиОHAЛьнА,СТPyКТYPHА... \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[796,77,825,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"187 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[163,151,820,570]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0594/465b3abb8590189a430e162ee9ccde1de384b3a714434a5548349e223b9ec1c7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Колочастовские ч��нштейпьные\\nэдемен�ограды\\nЧелеговские ч��нштейпь\\nчелеговские ч��нштейпь\\nчелеговские ч��нштейпь\\nчелеговские ч��нштейпь\\nчелеговские ч��нштейпь\\nчелеговские ч��нштейпь\\nчелеговские ч��нштейпь\\nчелеговские ч��нштейпь\\nчелergовские ч��нштейпь\\nчелеговские ч��нштейпь\\nчелеговские ч��нштейпь\\nчелеговские ч��нштейпь\\nчелеговские ч��нштейпь\\nчелеговские ч��нштейпь\\nчелеговские ч��нштейпь\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелergовrias\\nчелergовrias\\nчелergовrias\\nчелergовrias\\nчелergовrias\\nчелergовrias\\nчелergовrias\\nчелergовrias\\nчелergовrias\\nчелergовrias\\nчелergовrias\\nчелergовrias\\nчелergовrias\\nчелergовrias\\nчелergовrias\\nчелergовrias\\nчелergовrias\\nЧелеговrias\\nС�лочастовские ч��нштейпь\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговías\\nчелеговías\\nчелеговías\\nчелеговías\\nчелеговías\\nчелеговías\\nчелеговías\\nчелеговías\\nчелеговías\\nчелеговías\\nчелеговías\\nчелеговías\\nчелеговías\\nчелеговías\\nчелеговías\\nчелеговías\\nчелеговías\\nчелergовías\\nчелergовías\\nчелergовías\\nчелergовías\\nчелergовías\\nчелergовías\\nчелergовías\\nчелergовías\\nчелergовías\\nчелergовías\\nчелergовías\\nчелergовías\\nчелergовías\\nчелergовías\\nчелergовías\\nчелergовías\\nчелergовías\\nЧелеговrias\\nС�лочастовские ч��нштейпь\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелеговrias\\nчелегovrias\\nчелегovrias\\nчелегovrias\\nчелегovrias\\nчелегovrias\\nчелегovrias\\nчелегovrias\\nчелегovrias\\nчелегovrias\\nчелегovrias\\nчелегovrias\\nчелегovrias\\nчелегovrias\\nчелегovrias\\nчелегovrias\\nчелегovrias\\nчелегovrias\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nчelight;\\nЧелеговrias\\nС�лочастовские 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yHKIIHOHaJIbHO-cTpyKTypHO KJIaCCnΦHKaIIN KOJIInueCTBeHHbIX YHCJIInTeJIbHbIX B KHTaNcKOM Y3bIke BBJBAIO IpO-6JIeMy HOMeHKJIaTypHO HeOJHOpOJDHOCTN, KOJa OJHO I TO JKe IIOHrTne IJIN KJIaCCNΦHKaIINHHa TpyIIa HMEOT HeCKoJIbKO HaHMeHOBaHN, YTO CBHJeTEJIbCTByET O HeycToBIIeMcTepMHNOJIoTHueckOM aIIIapate. Ha $\\\\phi$ OHe HeKOTopoTepMHNOJIOHueckO HeOIIpeJEJIeHHOCTH B KHTaNcKOM Y3bIKe B OTeHecTBeHHOM KHTaEBeJEHN IIo paccMaTpHbAemomy BOIIPOcy Ha6JIIOJaEtC TOCYtCTBNe 6OJIbIIe qactN TepMHNOJIOrHEcKHX aHaJIIOOB, YTO B IIepByIO oupeIb KaCaetC $\\\\phi$ yHKIIHOHaJIbHO-cTpyKTypHO KJIaCCnΦHKaIIN.ДЯ TaKHx TepMHHO B HAcTOrIe CTAtBe IIpeIcTaBJIeHbI pyccko-3bIuHbIe BapHaHTbl. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[169,765,811,843]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B xOJIe HCCJIeIOBAHnI IOpa6oTaHbI paHee IIpeJIIOJKeHHbIe KNTaIckHMN JINHHBnCTaMn cTpyKtypHaI N ΦyHKIIHOHaJIbHo-cTpyKtypHaI KJIaCCNΦHKaIHH KOIJIueCTBeHN HbIX YNCJIInTeJIbHbIX. B IepcIIeKTHBe pa60TbI KNTaIckNe KOIJIueCTBeHNbIE YNCJIInTeJIbHbIe 6yUyt KJIaCCNΦHIIpOBaHbI CORJIaCHO TOUHOCTN KOJIInueCTBeHNH O XapaKTepnCTKN BbIPAkaEMOrO HmN YHCJa I eTo IeJIocTHOcTH. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[174,77,201,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"188 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[431,76,549,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JI.JI.BAHKOBA \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[458,115,552,130]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JInTepaTypa \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[186,135,826,810]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1.ДрауноA.A.IIcJIeIOBaHnI IO rpaMaTHKe COBpeMeHHOro KHTaIckoroЯ3bIka.-M.;JI.:ИЗД-ВО AH CCCP,1952.-T.1:YacTH peuN.-231 c. \\\",\\\"2. Дякоба И.Н. ЭвлioиmaTeMaTeMaTHuecKOrMoIIJIeHnI cTaHOBJeHHe NMeH YHcJIteJIbHbIX kak actnepeu//YUeH.3aII.IIeTpo3aBOJ.roc.yH-Ta.-2018.-No4.-C.49-54.- doi: 10.15393/uchz.art.2018.148. \\\",\\\"3. BaIaJIbIkuHa ə.A. YHcJIHTeJIbHbIe KaK oO6aJ qAcTb peuH B pyccKom J3bIke // YueH. 3aII. Ka3aH. yH-Ta. Cep. ΓyMaHHT. HayKn. - 2009. - T. 151, KH. 3. - C. 205-214. \\\",\\\"4. CmapukoBa $\\\\\\\\Gamma ,H$ . CoBpeMeHHbIe YnCJIHTeJIbHbIe B aCIIeKTe 3bIKOBoi IHHAMKn // BecTH. Tom. roc. yH-Ta. ΦHJIOJIoHra. - 2011. - No 4. - C. 34-40. \\\",\\\"5. Buho2pa06 B.B. Имя чилгельhoe // BHHoIpaIOB B.B. Рсckн я3bIK: ГраMaTHueckoe yueHne o cIoBe. - M.: Гoc.yue6.-II.II.N3II-BO, 1972. - C. 233-254. \\\",\\\"6. 关彦庆. 现代汉语基数词的结构研究 // 通化师范学院学报. -2012. -第33卷第5期. -页码 $1-10. = \\\\\\\\Gamma y a n b$ 。IccJIeIOBaHHe cTpyKtypbI coBpeMeHHbIX KHTaIcknx KoJIHcECTBeHHbIX YHcJIteJIbHbIX // BeCTH. IIe.I. HN-Ta B TyHxya. -2012. -T. 33, № 5. -C. 1-10.\\\",\\\"7. Bo2daHOB C.H. YHcJIHTeJIbHoe // BoIaHOB C.I., BoeKoBa M.I. И np. MopΦOJIOrHa coBpeMeHHoro pyccKOrO y3bika. - CII6.: ΦaK. ΦHIOJOrHn H nckycCTB C.-IteP6. roc. yH-Ta, 2009. - C. 301-322. \\\",\\\"8. 刘平. 数词分类研究述论 // 重庆邮电大学学报 (社会科学版). -2013. -第25卷第2期. -页码 122-127. =JIo II. OnHcAHN He HccJIeIOBaHnR KJIaCCNΦHKaIIN NcJIHTeJIbHbIX // BecTH. UyHIIHH. yH-Ta cBra3H n TeJIeKOMMyHnKaIIN. Cep. O6IeCTB. HayKn. -2013. -T. 25, № 2. -C. 122-127.\\\",\\\"9. 朱德熙. 数词和数词结构 // 中国语文. -1958. -第7卷第4期. -页码185-187. =ЧуД. Исклenteльные ихстуктура// Кнайсяς Фнlioюгия. -1958. -T.7, №4. -С.185-187. \\\",\\\"10. 萧国政, 李英哲. 汉语确数词的系统构成、使用特点和历史演进 // 武汉教育学院学报 (哲学社会科学版). - 1997. - 第16卷 第1期. - 页码 34-44. = Cao Γ., Jiu H. CnctemHnaopraHHaIIH, oco6eHHocTH npHMeHeHnI N BOJIHOuN I KHTaIckHX OIIpeJeJIeHHbIX YHcJIHTeJIbHbIX // BeCTH. YXaH. HN-Ta. Cep. ΦHIOcoΦnI N o6IecTB. HayKn. - 1997. - T. 16, No 1. - C. 34-44.\\\",\\\"11. 周彩莲. 现代汉语数词研究. -哈尔滨: 黑龙江大学, 2002. - 146页. = Yɔkeoy II. HccJIe-Доване УнсЛиТeЛьНБИХВ COВpeMeHHom KHTaɪСКOMЯЗIKE. - Xap6HH: Xэйлuyнizян. yH-T, 2002. - 146 c. \\\",\\\"12. 郭攀. 汉语涉数问题研究. - 北京: 中华书局, 2004. - 277 页 = Φ Π. Изуне Ворpo-cob, Связанных с КОЛичecтбом И чслomВ КНтуckомЯз biome. - ΠЕКИN: KNT. КНГонз-дательстбо, 2004. - 277 c. \\\",\\\"13. Tópeño B.H. TeopéTHueckárgaMMaTHKa KHTaɪcKOrO y3bIka. - M.: IIpocBeIIeHne, 1989. - 318 c. \\\",\\\"14. 实用汉语语法教程 / 付鸿军主编. - 北京: 北京师范大学出版社, 2015. - 371 页. = Yuchoe IocOBe IIpo IIpaKTHueckoI rpaMMaTHKe KHTaIcKOro J3bIka / IIOД peI. Фу X. - Пекин: ИзД-ВО Пекин. ПeI. уН-Ta, 2015. - 371 c. \\\",\\\"15. 实用现代汉语语法/刘月华.-增订本.-北京:商务印书馆,2001.-1005页.=IpaKTHUeCKa rpaMaTHKa coBpeMeHHOro KHTaHcKOro y3bIka / IIO peJ.IHO IO.-IeKNH:KomMepuecka IIpecca,2001.-1005c. \\\",\\\"16. Cyb6omun A./JI. KJIaCCnΦHkaIINr // IeHTp rymaHHTapHbIX TeXHOJIOTn.- M.: ΙΦ PAH, 2001.- URL: https://gtmarket.ru/laboratory/basis/3794, cbo6oIhBn. \\\"]\",\n \"page_idx\": \"14\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[351,76,657,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ФУHKIиОHAЛьнА,СТPyКТYPHА... \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[796,77,825,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"189 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[173,114,811,634]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"17. Понкун И.В., PeobkaHa A.II. Класснфнадя Кам MeToД hayUHOr HCCJIeIOBaHHa, B YacT-HocTH B IOpHINueckoи HayKe // BeCTH. ПЕм. yH-Ta. IOpHД. hayKn. - 2017. - ByII. 37. - C. 249-259. - doi: 10.17072/1995-4190-2017-37-249-259. \\\",\\\"18. 现代汉语描写语法/张斌主编.-北京：商务印书馆，2010.-1306页. =ОпиcaTeЛьнай ГраМатиka coВpeMeHного КИтайсКOrO яЗыкa / Пов peД. Ужан Б.-ПeКин: Komмер-ческа пeccа, 2010.-1306c. \\\",\\\"19. 李宇明. 数学语言初见 // 语文教学与研究. -1986. -第17卷第7期. -页码2-15. = JIu IO. ΠερΒαγ BCΤρεγας MaTeMaTHηεκΚοῦ ΙΙΝΗΓΒνιδΤΗΚοῦ // ΠρεπΟДaВΑΗΝΕ έΝΙΥηεΗΝΕ ΑλθΙΚΑ έΝΙΠΕταρατυβΙ. -1986. -T. 17, № 7. -C. 2-15. \\\",\\\"20. Cuyuha B. YHcJIHTeJIbHbIe // MaTePHaJIbI K KopIIyCHoI rpaMMaTHKe pyccKOrO y3bIka. - CII6.: HecToP-IVctOpHn, 2018. - BbII. 3: Yactn peuN nJIeKCHKO-rpaMMaTHueckHe KJIacCbI. - C. 193-257. \\\",\\\"21. 朱德熙. 语法讲义. - 北京: 商务印书馆, 1982. - 231 页. = UzhvД. Kypc Лжий по гравиматник. - Пекин: Коммереская песca, 1982. - 231 c. \\\",\\\"22. 汉英对比语法论集 / 赵世开主编. - 上海: 上海外语教育出版社, 1999. - 349 页. = C6opHnK cTaTei IIO KOMIIapaTHBHOI rpaMMaTHKe KHTaIcKOrO H aHIJIINcKOrO J3bIKOB / IOp peI. YJkao III. - IIaHxaI: IIaHx. H3Д-Bo yYe6. JInT. Ha INHocTp. J3., 1999. - 349 c.\\\",\\\"23. Comrie B. Numeral bases // The World Atlas of Language Structures Online. – Munich: Max Planck Digital Libr., 2011. – URL: http://wals.info/features/131,сыобдь. \\\",\\\"24. He C. Complex numerals in Mandarin Chinese are constituents // Linguua. - 2015. - V. 164, Pt. A. - P. 189-214. - doi: 10.1016/j.lingua.2015.06.014. \\\",\\\"25. 邢福义. 汉语语法三百问. - 北京: 商务印书馆, 2002. - 264 页. = Cun Φ. Трпста Восторов Ктошкоу гравлары. - Пекин: Коммерочая песca, 2002. - 264 c. \\\",\\\"26. ColnueB B.M., ColnueBa H.B. TeopeTHueKaI rpaMMaTHka coBpeMeHHoro KHTaNcKOra 3bIka. IIpo6JIeMbI MOpΦOJOΓHn.-M.: IN-H-T BOCTOKOBeJEHHa PAH, BoeHHbYH-T MInHO6OpHbI PΦ, 2016.-164 c. \\\",\\\"27. One-Soon Her. Deriving classifier word order typology, or Greenberg's Universal 20A and Universal 20 // Linguistics. - 2017. - V. 55, No 2. - P. 1-31. - doi: 10.1515/ling-2016-0044. \\\",\\\"28. Corver N., Zwarts J. Prepositional numerals // Lingua. - 2006. - V. 116, No 6. - P. 811-835. - doi: 10.1016/j.lingua.2005.03.008. \\\",\\\"29. Hurford J. The Linguistic Theory of Numerals. – Cambridge: Camb. Univ. Press, 2011. – 308 p. (Cambridge Studies in Linguistics. V. 16). \\\",\\\"30. Ionin T., Matushansky O. The composition of complex cardinals // J. Semantics. - 2006. - V. 23, No 4. - P. 315-360. - doi: 10.1093/jos/ffl006. \\\"]\",\n \"page_idx\": \"15\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[653,651,805,675]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IocTuHnla BpeaKiHo 17.08.2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[171,689,808,703]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"BaanKobA JIOIMHJa JbBOBa, KaHINaT 0JIIOJIOrHuecknx HayK, IOneHT KafeIpbI KHTaiCKoro A3bKa \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[205,706,547,719]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Mockobckn I ropoIcKo IIeIaIorHueckn yHInBepcHTet \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[228,720,660,731]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2-Ⅱ CeIbckoxO3aIcTBeHHbI IIpoe3I, I. 4, r. MockBa, 129226, Pocn \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[205,732,371,744]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E-mail: lwittsa@yandex.ru \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[173,77,201,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"190 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[430,76,549,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JI.JI.BAHKOBA \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[662,118,805,131]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ISSN 2541-7738 (Print) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[663,131,815,143]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ISSN 2500-2171 (Online) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[189,149,821,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"UCHENYE ZAPISKI KAZANSKOGO UNIVERSITETA. SERIYA GUMANITARNYE NAUKI \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[339,166,670,179]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(Proceedings of Kazan University. Humanities Series) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[389,185,620,198]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2021, vol. 163, no. 4-5, pp. 175-193 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[184,210,366,223]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ORIGINAL ARTICLE \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[184,229,457,241]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"doi: 10.26907/2541-7738.2021.4-5.175-193 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[324,249,685,275]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Functional, Structural, and Functional-Structural Classifications of Chinese Cardinal Numerals \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[457,284,552,296]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"L.L. Bankova \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[351,300,658,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moscow City University, Moscow, 129226 Russia \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[421,313,589,326]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E-mail: lwittsa@yandex.ru \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[421,332,586,344]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Received August 17, 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[475,351,536,363]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[184,370,823,458]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Classification of cardinal numerals is an important step in studying Chinese numerals. The study is carried out in the framework of the functional, structural (morphemic), and functional-structural (combined) classification approaches, as well as in accordance with the integrity of numerals and the accuracy of their quantifying properties. Chinese numerals are considered from the perspective of the first three classifications. The results obtained reveal the nomenclature volatility of Chinese cardinal numerals with the same classification group having several names to denote it. Hence, the article attempts to streamline the respective terminology. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[184,458,823,482]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The functional classification divides cardinal numerals into coefficient and digit-position numerals. The cases when digit-position numerals may function as coefficient numerals are analyzed. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[184,482,825,531]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the framework of the structural classification, it is suggested to divide cardinal numerals into simple (coefficient) and compound ones, the latter being broken into additive (elementary and miscellaneous) and multiplicative. The article also focuses on answering the question of whether compound numerals should be regarded as words or phrases. The conclusion is made that they should be perceived as words. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[184,531,825,605]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The functional-structural classification makes a substantial contribution to the study of Chinese cardinal numerals. The classification has not been introduced to the Russian sinology yet. This fact entails the necessity to translate the respective terms and expressions, the choice of which is explained in this article. The variations of the combined functional-structural classification introduced by Zhu Dexi, Guan Yanqing, Xiao Guozheng, and Li Yingzhe are discussed. Based on the analysis of these variations, an understanding of the combined functional-structural division of Chinese cardinal numerals is suggested. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[184,608,823,633]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords: Chinese language, Chinese cardinal numerals, functional classification, structural (morphic) classification, functional-structural classification \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[467,646,544,659]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[184,665,826,829]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. Dragunov A.A. Issledovaniya po grammatike sovremennogo kitaiskogo yazyka [Studies on Modern Chinese Grammar]. Vol. 1: Parts of speech. Moscow, Izd. Akad. Nauk SSSR, 1952. 231 p. (In Russian) \\\",\\\"2. D'yachkova I.N. The evolution of mathematical thinking and genesis of numerals as parts of speech. Uchenye Zapiski Petrozavodskogo Gosudarstvennogo Universiteta, 2018, vol. 173, no. 4. pp. 49-54. doi: 10.15393/uchz.art.2018.148. (In Russian) \\\",\\\"3. Balalykina E.A. Numerals as special category in modern Russian language. Uchenye Zapiski Kazanskogo Universiteta. Seriya Gumanitarnye Nauki, 2009, vol. 151, no. 3, pp. 205-214. (In Russian) \\\",\\\"4. Starikova G.N. Current numerals in the aspect of language dynamics. Vestnik Tomskogo Gosudarstvennogo Universiteta, 2011, no. 4, pp. 34-40. (In Russian) \\\",\\\"5. Vinogradov V.V. Numerals. In: Vinogradov V.V. Russkii yazyk: Grammaticeskoe uchenie o slove [The Russian Language (A Grammatical Theory of the Word)]. Moscow, Gos. Ucheb.-Ped. Izd., 1972, pp. 233-254. (In Russian) \\\"]\",\n \"page_idx\": \"16\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[351,76,658,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ФУHКИПОHAЛьнА,СТPyKТYPHA... \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[796,77,821,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"191 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[171,114,811,796]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"6. Guan Ya. A research on the structure of modern Chinese cardinal numerals. Journal of Tonghua Normal University, 2012, vol. 33, no. 5. pp. 1-10. (In Chinese) \\\",\\\"7. Bogdanov S.I. Numerals. In: Bogdanov S.I., Voeikova M.D. et al. Morfologiya sovremennogo russkogo yazyka [Morphology of the Modern Russian Language]. St. Petersburg, Fak. Filol. Iskusstv S.-Peterb. Univ., 2009. pp. 301-322. (In Russian) \\\",\\\"8. Liu P. A description of numerals classification research. Journal of Chongqing University of Posts and Telecommunications, 2013, vol. 25, no. 2, pp. 122-127. (In Chinese) \\\",\\\"9. Zhu D. Numerals and their structure. Chinese Philology, 1958, vol. 7, no. 4. pp. 185-187. (In Chinese) \\\",\\\"10. Xiao G., Li Y. Chinese exact numerals system: Structure, peculiarities of use and evolution. Journal of Wuhan Institute of Education. Philosophy and Social Sciences Series, 1997, vol. 16, no. 1, pp. 34-44. (In Chinese) \\\",\\\"11. Zhou C. Research on Modern Chinese Numerals. Harbin, Heilongjiang Univ., 2002. 146 p. (In Chinese) \\\",\\\"12. Guo P. A Research on Problems Related to Chinese Numerals. Beijing, Zhonghua Book Co., 2004. 277 p. (In Chinese) \\\",\\\"13. Gorelov V.I. Teoreticheskaya grammatika kitaiskogo yazyka [Theoretical Chinese Grammar]. Moscow, Prosveshchenie, 1989. 318 p. (In Russian) \\\",\\\"14. Fu H. (Ed.) Practical Chinese Grammar Teaching Materials. Beijing, Beijing Normal Univ. Press, 2015. 371 p. (In Chinese) \\\",\\\"15. Liu Y. Practical Modern Chinese Grammar. Beijing, Commer. Press, 2001. 1005 p. (In Chinese) \\\",\\\"16. Subbotin A.L. Classification. In: Tsentr gumanitarnykh tekhnologii [Humanitarian Technology Center]. Moscow, IF Ross. Akad. Nauk, 2001. Available at: https://gtmarket.ru/laboratory/basis/3794. (In Russian) \\\",\\\"17. Ponkin I.V., Red'kina A.I. Classification as a method of scientific research, particularly in jurisprudence. Vestnik Permskogo Universiteta. Yuridicheskie Nauki, 2017, no. 37, pp. 249-259. doi: 10.17072/1995-4190-2017-37-249-259. (In Russian) \\\",\\\"18. Zhang B. (Ed.) Descriptive Grammar of Modern Chinese. Beijing, Commer. Press, 2010. 1306 p. (In Chinese) \\\",\\\"19. Li Yu. A first glance at the language of mathematics. Language and Literature Teaching and Studying, 1986, vol. 17, no. 7, pp. 2-15. (In Chinese) \\\",\\\"20. Sichinava D.V. Numerals. In: Materialy k korpusnoi grammatike russkogo yazyka [Materials for the Corpus Grammar of the Russian Language]. St. Petersburg, Nestor-Istoriya, 2018, no. 3: Parts of speech and lexico-grammatical classes, pp. 193-257. (In Russian) \\\",\\\"21. Zhu D. Lectures on Grammar. Beijing, Commer. Press, 1982. 231 p. (In Chinese) \\\",\\\"22. Zhao S. A Collection of Articles on Comparative Grammar of the Chinese and English Languages. Shanghai, SFLEP, 1999. 349 p. (In Chinese) \\\",\\\"23. Comrie B. Numeral bases. In: Haspelmath M., Dryer M.S., Gill D., Comrie B. (Eds.) The World Atlas of Language Structures Online. Munich, Max Planck Digital Libr. 2011. Available at: http://wals.info/features/131. \\\",\\\"24. He C. Complex numerals in Mandarin Chinese are constituents. *Lingua*, 2015, vol. 164, pt. A, pp. 189–214. doi: 10.1016/j.lingua.2015.06.014. \\\",\\\"25. Xing F. Three Hundred Problems of Chinese Grammar. Beijing, Commer. Press, 2002. 264 p. (In Chinese) \\\",\\\"26. Solntsev V.M., Solntseva N.V. Teoreticheskaya grammatika sovremennogo kitaiskogo yazyka (Problemy morfologii) [Theoretical Grammar of Modern Chinese (Problems of Morphology)]. Moscow, Inst. Vostokoved. Ross. Akad. Nauk, Voen. Univ. Minoborony RF, 2016. 164 p. (In Russian) \\\",\\\"27. One-Soon Her. Deriving classifier word order typology, or Greenberg's Universal 20A and Universal 20. Linguistics, 2017, vol. 55, no. 2, pp. 1-31. doi: 10.1515/ling-2016-0044. \\\",\\\"28. Corver N., Zwarts J. Prepositional numerals. Lingua, 2006, vol. 116, no. 6, pp. 811-835. doi: 10.1016/j.lingua.2005.03.008. \\\"]\",\n \"page_idx\": \"17\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[173,77,201,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"17\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"192 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[431,76,549,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"17\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JI.JI.BAHKOBA \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[186,114,825,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"29. Hurford J. The Linguistic Theory of Numerals. Cambridge, Cambridge Univ. Press, 2011. 308 p. (Cambridge Studies in Linguistics. Vol. 16) \\\",\\\"30. Ionin T., Matushansky O. The composition of complex cardinals. Journal of Semantics, 2006, vol. 23, no. 4, pp. 315-360. doi: 10.1093/jos/ffl006. \\\"]\",\n \"page_idx\": \"18\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[216,198,791,247]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Дя ушорань: Baнкова Л.Л. Функюнальна, стуктурази и Функюнальностуктурази Классфкаддд Клесчесггдд Кнслтеловьх КНотайсоком ясыke // YueH. 3aII. Ka3aH. yH-Ta. Cep. ГумаHT. Haykn. - 2021. - T. 163, кH. 4-5. - C. 175-193. - doi: 10.26907/2541-7738.2021.4-5.175-193. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[216,262,793,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For citation: Bankova L.L. Functional, structural, and functional-structural classifications of Chinese cardinal numerals. Uchenye Zapiski Kazanskogo Universiteta. Seriya Gumanitarnye Nauki, 2021, vol. 163, no. 4-5, pp. 175-193. doi: 10.26907/2541-7738.2021.4-5.175-193. (In Russian) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[351,76,660,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ФУHKIиОHAJIbHA,СТPyKТYPHA... \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[796,77,825,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"193 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.26907/2541-7738.2021.4-5.175-193"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Physics/Mathematics and Computational Science"},"sha256":{"kind":"string","value":"8f5d40bf33ca719aec2295b1e02ad9917f742061cec44b571b0131efddf934ba"},"title":{"kind":"string","value":"Functional, Structural, and Functional-Structural Classifications of Chinese Cardinal Numerals"}}},{"rowIdx":52,"cells":{"abstract":{"kind":"string","value":""},"author":{"kind":"string","value":"Pozzo, Federico|Forestieri, Gabriela|Ianna, Giulia|Vit, Filippo|Tissino, Erika|Bittolo, Tamara|Papotti, Robel|Terzi DI Bergamo, Lodovico|Steffan, Agostino|Laureana, Roberta|Tafuri, Agostino|Chiarenza, Annalisa|DI Raimondo, Francesco|Olivieri, Jacopo|Zaja, Francesco|Laurenti, Luca|Ilaria DEL Principe, Maria|Bomben, Riccardo|Zucchetto, Antonella|Rossi, Davide|Gattei, Valter"},"content_list":{"kind":"list like","value":[{"bbox":"[62,115,872,150]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"P594 THE CXCR4-LOW/CD5-HIGH PROLIFERATIVE FRACTION IS ENRICHED IN BTK MUTATIONS AND ANTICIPATES RELAPSE IN IBRUTINIB-TREATED CLL ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[62,161,616,176]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Topic: 5. Chronic lymphocytic leukemia and related disorders - Biology & Translational Research ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,187,934,236]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Federico Pozzo\\*1, Gabriela Forestieri2, Giulia Ianna1, Filippo Vit1, Erika Tissino1, Tamara Bittolo1, Robel Papotti1, Lodovico Terzi Di Bergamo2, Agostino Steffan1, Roberta Laureana3, Agostino Tafuri4, Annalisa Chiarenza5, Francesco Di Raimondo5, Jacopo Olivieri6, Francesco Zaja7, Luca Laurenti8, Maria Ilaria Del Principe3, Riccardo Bomben1, Antonella Zucchetto1, Davide Rossi2, Valter Gattei1 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[58,249,924,315]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"<sup>1</sup>Centro Di Riferimento Oncologico Di Aviano (Cro), Irccs, Aviano, Italy; <sup>2</sup>Eoc Istituto Oncologico Della Svizzera Italiana, Bellinzona, Switzerland; <sup>3</sup>Hematology, Dept Of Biomedicine And Prevention, Universita' Tor Vergata, Roma, Italy; <sup>4</sup>Azienda Ospedaliero - Universitaria Sant'Andrea, Italy; <sup>5</sup>Presidio Ospedaliero Ferrarotto, Catania, Italy; <sup>6</sup>Azienda Sanitaria Universitaria Integrata Di Udine, Udine, Italy; <sup>7</sup>Dipartimento Universitario Clinico Di Scienze Mediche Chirurgiche E Della Salute, Trieste, Italy; <sup>8</sup>Catholic University Of The Sacred Heart, Milano, Italy ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,325,144,340]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Background: ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[62,350,934,406]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CXCR4/CD5 expression on circulating chronic lymphocytic leukemia (CLL) cells discriminate the proliferative fraction (CD5high/CXCR4dim, PF), recently egressed from the lymph node, from the resting fraction (CXCR4high/CD5dim, RF) of older quiescent cells. BTK inhibitors affect the proliferative capacity of CLL cells and cause their redistribution from lymph nodes to peripheral blood (PB). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,416,100,429]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Aims: ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[62,441,892,456]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In long-term Ibrutinib (IB)-treated CLL, to investigate the evolution of PF and correlate with clinical outcome. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,467,122,480]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Methods: ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[62,491,910,534]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PB samples were from CLL patients entering NCT02827617 trial $(n = 31)$ or referred for routine (immuno)genetic analyses $(n = 100)$ . Methods employed include: flow cytometry for PF/RF analyses (Fig.A), reported as median %, DNA and RNA-seq (Illumina) analyzed with R package DESeq2. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,545,115,559]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Results: ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[62,568,927,611]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Longitudinal analysis of the PF under IB from trial CLL (156 sequential samples, median 6/case) showed depletion of the PF, from $17.7\\%$ at pre-treatment to 4.75-0.4-1.5-0.6-1.5% at 0.5-6-12-18-24 months $(p < 0.001)$ with loss of Ki67 expression. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,621,922,690]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By analyzing the PF from IB-treated real-world CLL (296 samples, median 3/case; Fig.A-B), median PF was $12.0\\%$ (range 0.7-50) in pre-IB samples with a drop to $2.7\\%$ at 1 year of treatment and $1.0\\%$ in the second year $(p < 0.001)$ . IB was discontinued within 5 years in 71/100 cases (17 toxicity, 46 progression, 8 other/death). BTK/PLCG2 mutations were detected in 29/100 cases; among these, 24 discontinued IB due to progression/relapse (23/24) or toxicity (1/24). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,700,932,769]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Focusing on samples collected within 12 months before/after IB discontinuation $(n = 86)$ , mutations were represented in progressing cases (22/43, $51\\%$ ) with a PF higher than wild-type (WT) cases discontinuing for toxicity ( $8.9\\%$ vs. $2.4\\%$ ; $p = 0.012$ ) or still on treatment ( $2.8\\%$ , $p = 0.015$ ; Fig.C). Notably, the 21 WT progressing cases also showed higher PF compared to cases discontinuing for toxicity ( $6.9\\%$ vs. $2.4\\%$ ; $p = 0.004$ ) or still on treatment ( $6.9\\%$ vs. $2.8\\%$ ; $p = 0.024$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,779,922,795]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sequential dosages of plasmatic B2M in progressing cases $(n = 5)$ revealed steady levels (median $2.38\\mathrm{ug / mL}$ ) below ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,24,331,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"HemaSphere ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[766,9,942,53]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"EHA EUROPEAN HEMATOLOGY ASSOCIATION ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[53,808,941,855]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Copyright Information: (Online) ISSN: 2572-9241 \n© 2023 the Author(s). Published by Wolters Kluwer Health, Inc. on behalf of the European Hematology Association. This is an open access Abstract Book distributed under the Attribution-NonCommercial-NoDerivs (CC BY-NC-ND) which allows third parties to download the articles and share them with others as long as they credit the author and the Abstract Book, but they cannot change the content in any way or use them commercially. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[53,861,749,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract Book Citations: Authors, Title, HemaSphere, 2023;7(S3):pages. The individual abstract DOIs can be found at https://journals.lww.com/hemisphere/pages/default.aspx. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[53,894,917,916]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Disclaimer: Articles published in the journal HemaSphere exclusively reflect the opinions of the authors. The authors are responsible for all content in their abstracts including accuracy of the facts, statements, citing resources, etc. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[53,928,226,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"HemaSphere | 2023;7(S3) ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[482,926,517,937]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1050 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[769,928,942,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"EHA2023 Hybrid Congress ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[62,115,613,131]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"the $3.5\\mathrm{ug / mL}$ threshold, despite the PF increase (median PF $18\\%$ ; Fig.D). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,138,929,209]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The PF/RF fractions were sorted from 10 cases developing $BTK$ mutations with PF reappearance after prolonged IB treatment (median 50.7 months). Sequencing revealed 27 $BTK$ mutations (mean 1.7/sample, range 1-4), with median VAF higher in the PF (23.0%, 0.3-92.3) than in the RF (5.2%, 0.1-84.8; p=0.003); overall, the PF VAF was 2.8-fold larger than the RF (0.01-38). A similar trend was present for $PLCG2$ mutations (median PF VAF: 2.0%, 0.1-6.2; RF VAF: 1.35%, 0.4-2.8; p=0.014; Fig.E). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,218,924,275]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finally, mRNA-seq was performed on separate PF/RF fractions from samples $(n = 6)$ at pre-IB and progression, the latter enriched in $BTK$ mutations. Differential expression signature of pre-IB PF vs. RF, concordant with published gene sets, drove co-clustering of post-IB PF/RF with their respective counterparts (Fig.F), indicating that PF at progression functionally resembles its pre-treatment counterpart. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,285,205,300]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Summary/Conclusion: ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[62,309,892,393]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Here we report: i) that IB treatment abrogates the PF fraction within 1 year of treatment, expanding previous short-term observations; ii) its reconstitution after long-term treatment may predict clinical progression anticipating B2M increase; iii) the enrichment of $\\text{BTK/PLCG2}$ mutations in the reconstituted PF suggests the presence of active clonal selection dynamics, bona fide taking place within the lymph node microenvironment. Clinically, longitudinal monitoring of the CXCR4/CD5 fractions by flow cytometry may provide a simple tool helping to intercept CLL progression under IB therapy. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,24,331,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"HemaSphere ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[766,13,942,53]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"EHA EUROPEAN HEMATOLOGY ASSOCIATION ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[50,808,941,855]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Copyright Information: (Online) ISSN: 2572-9241 © 2023 the Author(s). Published by Wolters Kluwer Health, Inc. on behalf of the European Hematology Association. This is an open access Abstract Book distributed under the Attribution-NonCommercial-NoDerivs (CC BY-NC-ND) which allows third parties to download the articles and share them with others as long as they credit the author and the Abstract Book, but they cannot change the content in any way or use them commercially. ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[52,862,749,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract Book Citations: Authors, Title, HemaSphere, 2023;7(S3):pages. The individual abstract DOIs can be found at https://journals.lww.com/hemosphere/pages/default.aspx. ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[52,894,917,916]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Disclaimer: Articles published in the journal HemaSphere exclusively reflect the opinions of the authors. 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Published by Wolters Kluwer Health, Inc. on behalf of the European Hematology Association. This is an open access Abstract Book distributed under the Attribution-NonCommercial-NoDerivs (CC BY-NC-ND) which allows third parties to download the articles and share them with others as long as they credit the author and the Abstract Book, but they cannot change the content in any way or use them commercially. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[50,862,749,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract Book Citations: Authors, Title, HemaSphere, 2023;7(S3):pages. The individual abstract DOIs can be found at https://journals.lww.com/hemisphere/pages/default.aspx. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[50,894,917,916]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Disclaimer: Articles published in the journal HemaSphere exclusively reflect the opinions of the authors. The authors are responsible for all content in their abstracts including accuracy of the facts, statements, citing resources, etc. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,24,329,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"HemaSphere ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[768,13,873,52]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"EHA ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[882,29,941,52]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"EUROPEAN HEMATOLOGY ASSOCIATION ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[52,928,225,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"HemaSphere | 2023;7(S3) ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[480,926,517,937]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1052 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[769,928,944,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"EHA2023 Hybrid Congress ","text_format":null,"text_level":null,"type":"footer"}],"string":"[\n {\n \"bbox\": \"[62,115,872,150]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"P594 THE CXCR4-LOW/CD5-HIGH PROLIFERATIVE FRACTION IS ENRICHED IN BTK MUTATIONS AND ANTICIPATES RELAPSE IN IBRUTINIB-TREATED CLL \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,161,616,176]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Topic: 5. Chronic lymphocytic leukemia and related disorders - Biology & Translational Research \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,187,934,236]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Federico Pozzo\\\\*1, Gabriela Forestieri2, Giulia Ianna1, Filippo Vit1, Erika Tissino1, Tamara Bittolo1, Robel Papotti1, Lodovico Terzi Di Bergamo2, Agostino Steffan1, Roberta Laureana3, Agostino Tafuri4, Annalisa Chiarenza5, Francesco Di Raimondo5, Jacopo Olivieri6, Francesco Zaja7, Luca Laurenti8, Maria Ilaria Del Principe3, Riccardo Bomben1, Antonella Zucchetto1, Davide Rossi2, Valter Gattei1 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[58,249,924,315]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"<sup>1</sup>Centro Di Riferimento Oncologico Di Aviano (Cro), Irccs, Aviano, Italy; <sup>2</sup>Eoc Istituto Oncologico Della Svizzera Italiana, Bellinzona, Switzerland; <sup>3</sup>Hematology, Dept Of Biomedicine And Prevention, Universita' Tor Vergata, Roma, Italy; <sup>4</sup>Azienda Ospedaliero - Universitaria Sant'Andrea, Italy; <sup>5</sup>Presidio Ospedaliero Ferrarotto, Catania, Italy; <sup>6</sup>Azienda Sanitaria Universitaria Integrata Di Udine, Udine, Italy; <sup>7</sup>Dipartimento Universitario Clinico Di Scienze Mediche Chirurgiche E Della Salute, Trieste, Italy; <sup>8</sup>Catholic University Of The Sacred Heart, Milano, Italy \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,325,144,340]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Background: \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,350,934,406]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CXCR4/CD5 expression on circulating chronic lymphocytic leukemia (CLL) cells discriminate the proliferative fraction (CD5high/CXCR4dim, PF), recently egressed from the lymph node, from the resting fraction (CXCR4high/CD5dim, RF) of older quiescent cells. BTK inhibitors affect the proliferative capacity of CLL cells and cause their redistribution from lymph nodes to peripheral blood (PB). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,416,100,429]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Aims: \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,441,892,456]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In long-term Ibrutinib (IB)-treated CLL, to investigate the evolution of PF and correlate with clinical outcome. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,467,122,480]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Methods: \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,491,910,534]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PB samples were from CLL patients entering NCT02827617 trial $(n = 31)$ or referred for routine (immuno)genetic analyses $(n = 100)$ . Methods employed include: flow cytometry for PF/RF analyses (Fig.A), reported as median %, DNA and RNA-seq (Illumina) analyzed with R package DESeq2. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,545,115,559]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Results: \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,568,927,611]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Longitudinal analysis of the PF under IB from trial CLL (156 sequential samples, median 6/case) showed depletion of the PF, from $17.7\\\\%$ at pre-treatment to 4.75-0.4-1.5-0.6-1.5% at 0.5-6-12-18-24 months $(p < 0.001)$ with loss of Ki67 expression. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,621,922,690]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By analyzing the PF from IB-treated real-world CLL (296 samples, median 3/case; Fig.A-B), median PF was $12.0\\\\%$ (range 0.7-50) in pre-IB samples with a drop to $2.7\\\\%$ at 1 year of treatment and $1.0\\\\%$ in the second year $(p < 0.001)$ . IB was discontinued within 5 years in 71/100 cases (17 toxicity, 46 progression, 8 other/death). BTK/PLCG2 mutations were detected in 29/100 cases; among these, 24 discontinued IB due to progression/relapse (23/24) or toxicity (1/24). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,700,932,769]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Focusing on samples collected within 12 months before/after IB discontinuation $(n = 86)$ , mutations were represented in progressing cases (22/43, $51\\\\%$ ) with a PF higher than wild-type (WT) cases discontinuing for toxicity ( $8.9\\\\%$ vs. $2.4\\\\%$ ; $p = 0.012$ ) or still on treatment ( $2.8\\\\%$ , $p = 0.015$ ; Fig.C). Notably, the 21 WT progressing cases also showed higher PF compared to cases discontinuing for toxicity ( $6.9\\\\%$ vs. $2.4\\\\%$ ; $p = 0.004$ ) or still on treatment ( $6.9\\\\%$ vs. $2.8\\\\%$ ; $p = 0.024$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,779,922,795]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sequential dosages of plasmatic B2M in progressing cases $(n = 5)$ revealed steady levels (median $2.38\\\\mathrm{ug / mL}$ ) below \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,24,331,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"HemaSphere \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[766,9,942,53]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"EHA EUROPEAN HEMATOLOGY ASSOCIATION \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[53,808,941,855]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Copyright Information: (Online) ISSN: 2572-9241 \\n© 2023 the Author(s). Published by Wolters Kluwer Health, Inc. on behalf of the European Hematology Association. This is an open access Abstract Book distributed under the Attribution-NonCommercial-NoDerivs (CC BY-NC-ND) which allows third parties to download the articles and share them with others as long as they credit the author and the Abstract Book, but they cannot change the content in any way or use them commercially. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[53,861,749,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract Book Citations: Authors, Title, HemaSphere, 2023;7(S3):pages. The individual abstract DOIs can be found at https://journals.lww.com/hemisphere/pages/default.aspx. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[53,894,917,916]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Disclaimer: Articles published in the journal HemaSphere exclusively reflect the opinions of the authors. The authors are responsible for all content in their abstracts including accuracy of the facts, statements, citing resources, etc. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[53,928,226,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"HemaSphere | 2023;7(S3) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[482,926,517,937]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1050 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[769,928,942,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"EHA2023 Hybrid Congress \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[62,115,613,131]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"the $3.5\\\\mathrm{ug / mL}$ threshold, despite the PF increase (median PF $18\\\\%$ ; Fig.D). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,138,929,209]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The PF/RF fractions were sorted from 10 cases developing $BTK$ mutations with PF reappearance after prolonged IB treatment (median 50.7 months). Sequencing revealed 27 $BTK$ mutations (mean 1.7/sample, range 1-4), with median VAF higher in the PF (23.0%, 0.3-92.3) than in the RF (5.2%, 0.1-84.8; p=0.003); overall, the PF VAF was 2.8-fold larger than the RF (0.01-38). A similar trend was present for $PLCG2$ mutations (median PF VAF: 2.0%, 0.1-6.2; RF VAF: 1.35%, 0.4-2.8; p=0.014; Fig.E). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,218,924,275]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally, mRNA-seq was performed on separate PF/RF fractions from samples $(n = 6)$ at pre-IB and progression, the latter enriched in $BTK$ mutations. Differential expression signature of pre-IB PF vs. RF, concordant with published gene sets, drove co-clustering of post-IB PF/RF with their respective counterparts (Fig.F), indicating that PF at progression functionally resembles its pre-treatment counterpart. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,285,205,300]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Summary/Conclusion: \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,309,892,393]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Here we report: i) that IB treatment abrogates the PF fraction within 1 year of treatment, expanding previous short-term observations; ii) its reconstitution after long-term treatment may predict clinical progression anticipating B2M increase; iii) the enrichment of $\\\\text{BTK/PLCG2}$ mutations in the reconstituted PF suggests the presence of active clonal selection dynamics, bona fide taking place within the lymph node microenvironment. Clinically, longitudinal monitoring of the CXCR4/CD5 fractions by flow cytometry may provide a simple tool helping to intercept CLL progression under IB therapy. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,24,331,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"HemaSphere \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[766,13,942,53]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"EHA EUROPEAN HEMATOLOGY ASSOCIATION \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[50,808,941,855]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Copyright Information: (Online) ISSN: 2572-9241 © 2023 the Author(s). Published by Wolters Kluwer Health, Inc. on behalf of the European Hematology Association. This is an open access Abstract Book distributed under the Attribution-NonCommercial-NoDerivs (CC BY-NC-ND) which allows third parties to download the articles and share them with others as long as they credit the author and the Abstract Book, but they cannot change the content in any way or use them commercially. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[52,862,749,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract Book Citations: Authors, Title, HemaSphere, 2023;7(S3):pages. The individual abstract DOIs can be found at https://journals.lww.com/hemosphere/pages/default.aspx. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[52,894,917,916]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Disclaimer: Articles published in the journal HemaSphere exclusively reflect the opinions of the authors. The authors are responsible for all content in their abstracts including accuracy of the facts, statements, citing resources, etc. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[53,928,223,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"HemaSphere | 2023;7(S3) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[482,926,514,937]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1051 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[769,928,942,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"EHA2023 Hybrid Congress \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[94,152,416,220]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"A \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0191/8505de2162d3933106dd084f58fcd74a1aeeb9864c204cd1adf202bb8257f715.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[436,122,685,228]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"B \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0490/747705812bd5ade3108e4eef498efc5471efc62914a342c7c2e3410d13541ce5.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[78,226,356,353]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"C \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0977/fd1ab58459169cb8122ae3a8834b02581b28cc0fe9c3f4a2a93fe35e960ed946.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[379,243,690,350]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0774/c2f209ee45b188a74fc3fcc6b28d0dbb6009d7dc1b9e0e1c6b7252a85e454c2a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[85,372,349,549]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"E \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0157/093cafdb09e9653ccc53734d879b14eeba84a941108c1b64d45f0e8ea92eb919.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[322,362,705,547]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0465/edf2858d56686c2cb9de58a36a26fa9081aae79250ff0ac78913c1007909fcfd.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[50,809,941,855]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Copyright Information: (Online) ISSN: 2572-9241 © 2023 the Author(s). Published by Wolters Kluwer Health, Inc. on behalf of the European Hematology Association. This is an open access Abstract Book distributed under the Attribution-NonCommercial-NoDerivs (CC BY-NC-ND) which allows third parties to download the articles and share them with others as long as they credit the author and the Abstract Book, but they cannot change the content in any way or use them commercially. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[50,862,749,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract Book Citations: Authors, Title, HemaSphere, 2023;7(S3):pages. The individual abstract DOIs can be found at https://journals.lww.com/hemisphere/pages/default.aspx. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[50,894,917,916]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Disclaimer: Articles published in the journal HemaSphere exclusively reflect the opinions of the authors. The authors are responsible for all content in their abstracts including accuracy of the facts, statements, citing resources, etc. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,24,329,58]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"HemaSphere \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[768,13,873,52]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"EHA \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[882,29,941,52]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"EUROPEAN HEMATOLOGY ASSOCIATION \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[52,928,225,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"HemaSphere | 2023;7(S3) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[480,926,517,937]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1052 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[769,928,944,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"EHA2023 Hybrid Congress \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.1097/01.hs9.0000969280.78541.51"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Life Sciences"},"sha256":{"kind":"string","value":"2157901b6dad195a507494b410f4f0c7425c267101893206712e22190ae9ed55"},"title":{"kind":"string","value":"P594: THE CXCR4-LOW/CD5-HIGH PROLIFERATIVE FRACTION IS ENRICHED IN BTK MUTATIONS AND ANTICIPATES RELAPSE IN IBRUTINIB-TREATED CLL"}}},{"rowIdx":53,"cells":{"abstract":{"kind":"string","value":"\n \n We say that a class $\\mathcal{F}$ consisting of analytic functions\n$f(z)=\\sum_{n=0}^{\\infty} a_{n}z^{n}$ in the unit disk $\\mathbb{D}:=\\{z\\in\n\\mathbb{C}: |z|<1\\}$ satisfies a Bohr phenomenon if there exists $r_{f} \\in\n(0,1)$ such that $$\n\n\\sum_{n=1}^{\\infty} |a_{n}z^{n}|\\leq d(f(0),\\partial f(\\mathbb{D})) $$ for\nevery function $f \\in \\mathcal{F}$ and $|z|=r\\leq r_{f}$, where $d$ is the\nEuclidean distance. The largest radius $r_{f}$ is the Bohr radius for the class\n$\\mathcal{F}$. In this paper, we establish the Bohr phenomenon for the classes\nconsisting of Ma-Minda type starlike functions and Ma-Minda type convex\nfunctions as well as for the class of starlike functions with respect to a\nboundary point.\n\n "},"author":{"kind":"string","value":"Vasudevarao Allu|Himadri Halder"},"content_list":{"kind":"list like","value":[{"bbox":"[137,173,815,207]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"BOHR RADIUS FOR CERTAIN CLASSES OF STARLIKE AND CONVEX UNIVALENT FUNCTIONS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[280,228,672,242]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"VASUDEVARAO ALLU AND HIMADRI HALDER ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[174,264,793,307]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ABSTRACT. We say that a class $\\mathcal{F}$ consisting of analytic functions $f(z) = \\sum_{n=0}^{\\infty}a_{n}z^{n}$ in the unit disk $\\mathbb{D} := \\{z\\in \\mathbb{C}:|z| < 1\\}$ satisfies a Bohr phenomenon if there exists $r_f\\in (0,1)$ such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[374,308,579,343]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\sum_ {n = 1} ^ {\\infty} \\left| a _ {n} z ^ {n} \\right| \\leq d (f (0), \\partial f (\\mathbb {D}))\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[174,344,779,416]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for every function $f \\in \\mathcal{F}$ and $|z| = r \\leq r_f$ , where $d$ is the Euclidean distance. The largest radius $r_f$ is the Bohr radius for the class $\\mathcal{F}$ . In this paper, we establish the Bohr phenomenon for the classes consisting of Ma-Minda type starlike functions and Ma-Minda type convex functions as well as for the class of starlike functions with respect to a boundary point. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[297,458,655,472]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. INTRODUCTION AND PRELIMINARIES ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[115,480,836,516]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $\\mathcal{H}$ denote the class of all analytic functions $f$ in the unit disk $\\mathbb{D} := \\{z \\in \\mathbb{C} : |z| < 1\\}$ . Each function $f \\in \\mathcal{H}$ has the following power series representation ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,521,549,562]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nf (z) = \\sum_ {n = 0} ^ {\\infty} a _ {n} z ^ {n}. \\tag {1.1}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,568,836,603]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For $f \\in \\mathcal{H}$ , the majorant series is defined by $M_{f}(r) := \\sum_{n=0}^{\\infty} |a_{n}| r^{n}$ for $|z| = r < 1$ . In 1914, Harald Bohr [16] obtained the following remarkable result. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,609,794,628]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem A. Let $f \\in \\mathcal{H}$ be given by (1.1) and $|f(z)| < 1$ for all $z \\in \\mathbb{D}$ . Then ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,633,581,675]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nM _ {f} (r) := \\sum_ {n = 0} ^ {\\infty} | a _ {n} | r ^ {n} \\leq 1 \\tag {1.2}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,680,838,714]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for all $z \\in \\mathbb{D}$ with $|z| = r \\leq 1/3$ . The constant $1/3$ , called the Bohr radius, cannot be improved. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,720,840,823]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Harald Bohr initially obtained the inequality (1.2) for $r \\leq 1/6$ . Later, Weiner, Riesz and Schur have independently established the inequality (1.2) for $r \\leq 1/3$ and have shown that the constant $1/3$ cannot be improved (see [37, 44, 47]). The inequality (1.2) is popularly known as Bohr's inequality. The Bohr's inequality has been extended to several complex variables by finding the multidimensional Bohr radius (see [5, 6, 7, 14, 15]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[21,267,60,711]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"arXiv:2006.15299v1 [math.CV] 27 Jun 2020 ","text_format":null,"text_level":null,"type":"aside_text"},{"bbox":"[139,833,591,847]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"File: Himadri-Vasu-27-june-2020-1.20-a.m.tex, printed: 2020-6-30, 0.52 ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[139,847,678,861]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1991 Mathematics Subject Classification. Primary 30C45, 30C50, 30C80. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[117,861,836,890]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Key words and phrases. Analytic, univalent, starlike, convex, uniformly starlike, uniformly convex functions; coefficient estimate, growth estimate; majorant; Bohr radius. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[470,923,482,934]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[134,137,774,154]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Bohr inequality (1.2) can be written in the following equivalent form ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,163,690,209]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nd \\left(\\sum_ {n = 0} ^ {\\infty} \\left| a _ {n} z ^ {n} \\right|, \\left| a _ {0} \\right|\\right) = \\sum_ {n = 1} ^ {\\infty} \\left| a _ {n} z ^ {n} \\right| \\leq d (f (0), \\partial f (\\mathbb {D})) \\tag {1.3}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,217,838,334]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for $|z| = r \\leq 1/3$ , where $d$ is the Euclidean distance. We say that $\\mathcal{H}$ satisfies the Bohr phenomenon if every function $f \\in \\mathcal{H}$ satisfies the inequality (1.3). The above equivalent form (1.3) makes the notion of the Bohr phenomenon evident for a class $\\mathcal{F}$ consisting of analytic functions of the form (1.1) which map the unit disk $\\mathbb{D}$ into a given domain $\\Omega \\subseteq \\mathbb{C}$ such that $f(\\mathbb{D}) \\subseteq \\Omega$ . The class $\\mathcal{F}$ is said to satisfy the Bohr phenomenon if there exists $r_{\\Omega} \\in (0,1)$ such that every function $f \\in \\mathcal{F}$ satisfies the inequality ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,344,690,389]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nd \\left(\\sum_ {n = 0} ^ {\\infty} \\left| a _ {n} z ^ {n} \\right|, \\left| a _ {0} \\right|\\right) = \\sum_ {n = 1} ^ {\\infty} \\left| a _ {n} z ^ {n} \\right| \\leq d (f (0), \\partial f (\\mathbb {D})) \\tag {1.4}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,397,838,551]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for all $|z| = r \\leq r_{\\Omega}$ . The largest radius $r_{\\Omega}$ is called the Bohr radius for the class $\\mathcal{F}$ . The question arises as to whether or not the largest radius $r_{\\Omega}$ is changed with respect to different types of domains. The answer is affirmative. For any proper simply connected domain $\\Omega$ , Abu-Muhanna [1] has proved that the sharp radius is $r_{\\Omega} = 3 - 2\\sqrt{2}$ for the class $\\mathcal{F}$ . For a convex domain $\\Omega$ , Aizenberg [7] has shown that $r_{\\Omega}$ coincides with the classical Bohr radius $1/3$ . Bohr phenomenon for harmonic mappings has also been extensively studied by several authors (see [4, 11, 17, 23, 24, 26, 30]). For more information about Bohr phenomenon stated above and further related intriguing aspects, we refer the reader to the articles (see [2, 3, 8, 9, 12, 25]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,564,836,599]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $\\mathcal{A}$ denote the subclass of $\\mathcal{H}$ consisting of functions normalized by $f(0) = f'(0) - 1 = 0$ . Each $f \\in \\mathcal{A}$ has the following form ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,610,566,652]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nf (z) = z + \\sum_ {n = 2} ^ {\\infty} a _ {n} z ^ {n}. \\tag {1.5}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,660,838,878]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $S \\subseteq \\mathcal{A}$ be the family of univalent (i.e. one-to-one) functions. Let $\\mathcal{S}^*(\\alpha)$ and $\\mathcal{C}(\\alpha)$ be the subclasses of $S$ consisting of functions starlike of order $\\alpha$ ( $0 \\leq \\alpha < 1$ ) and convex functions of order $\\alpha$ ( $0 \\leq \\alpha < 1$ ) respectively. A function $f \\in S$ belongs to $\\mathcal{S}^*(\\alpha)$ (respectively $\\mathcal{C}(\\alpha)$ ) if $\\operatorname{Re}(zf'(z)/f(z)) > \\alpha$ for $z \\in \\mathbb{D}$ ( $\\operatorname{Re}(1 + zf''(z)/f'(z)) > \\alpha$ for $z \\in \\mathbb{D}$ respectively). It is known that $f \\in \\mathcal{C}(\\alpha)$ if, and only if, $zf' \\in \\mathcal{S}^*(\\alpha)$ . The classes $\\mathcal{S}^* := \\mathcal{S}^*(0)$ and $\\mathcal{C} := \\mathcal{C}(0)$ are the family of starlike and convex functions in $\\mathbb{D}$ respectively. For more properties of starlike and convex functions, we refer the reader to [18, 21, 46]. For two analytic functions $f$ and $g$ in $\\mathbb{D}$ , we say that $f$ is subordinate to $g$ , written $f \\prec g$ in $\\mathbb{D}$ , if there exists a schwarzian function $\\omega: \\mathbb{D} \\to \\mathbb{D}$ with $\\omega(0) = 0$ such that $f(z) = g(\\omega(z))$ for $z \\in \\mathbb{D}$ . In particular, if $g$ is univalent in $\\mathbb{D}$ , then $f \\prec g$ if, and only if, $f(0) = g(0)$ and $f(\\mathbb{D}) \\subseteq g(\\mathbb{D})$ . In 1992, Ma-Minda [32] introduced the function classes $\\mathcal{S}^*(\\phi)$ and $\\mathcal{C}(\\phi)$ by unifying several earlier results on subordination. For the brevity, we write. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,112,131,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[344,111,608,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Vasudevarao Allu and Himadri Halder ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[114,136,836,171]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 1.1. Let $S^{*}(\\phi)$ and $\\mathcal{C}(\\phi)$ denote the subclasses of $S$ consisting of functions characterized by ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[302,176,653,214]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\frac {z f ^ {\\prime} (z)}{f (z)} \\prec \\phi (z) \\quad \\mathrm {a n d} \\quad 1 + \\frac {z f ^ {\\prime \\prime} (z)}{f ^ {\\prime} (z)} \\prec \\phi (z)\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,219,838,287]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"respectively, where $\\phi : \\mathbb{D} \\to \\mathbb{D}$ is called Ma-Minda function which is analytic and univalent in $\\mathbb{D}$ such that $\\phi(\\mathbb{D})$ has positive real part, symmetric with respect to the real axis, starlike with respect to $\\phi(0) = 1$ and $\\phi'(0) > 0$ . Let the Taylor series expansion of $\\phi(z)$ be of the form ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,294,611,337]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\phi (z) = 1 + \\sum_ {n = 1} ^ {\\infty} B _ {n} z ^ {n} \\quad (B _ {1} > 0) \\tag {1.6}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,342,697,359]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for $z \\in \\mathbb{D}$ . Let $\\mathcal{M}$ denote the class of all Ma-Minda functions in $\\mathbb{D}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,368,838,453]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We call $\\mathcal{S}^*(\\phi)$ and $\\mathcal{C}(\\phi)$ are the Ma-Minda type starlike and Ma-Minda type convex classes associated with $\\phi$ . It is known that $f \\in \\mathcal{C}(\\phi)$ if, and only if, $zf' \\in \\mathcal{S}^*(\\phi)$ . For $\\phi(z) = (1 + z)/(1 - z)$ we have $\\mathcal{C}(\\phi) \\coloneqq \\mathcal{C}$ and $\\mathcal{S}^*(\\phi) \\coloneqq \\mathcal{S}^*$ . Evidently, for every such $\\phi$ described in Definition 1.1, $\\mathcal{S}^*(\\phi)$ and $\\mathcal{C}(\\phi)$ are always subclasses of the classes $\\mathcal{S}^*$ and $\\mathcal{C}(\\phi)$ respectively. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,465,853,583]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is worth noting that for particular choices of $\\phi$ , the classes $S^{*}(\\phi)$ and $\\mathcal{C}(\\phi)$ generate several important subclasses of starlike and convex functions, respectively. For instance, $S^{*}(\\alpha) := S^{*}\\left((1 + (1 - 2\\alpha)) / (1 - z)\\right)$ and $\\mathcal{C}(\\alpha) := \\mathcal{C}\\left((1 + (1 - 2\\alpha)) / (1 - z)\\right)$ , the Janowski starlike class $S^{*}[A,B] := S^{*}((1 + Az) / (1 + Bz))$ and Janowski convex class $\\mathcal{C}[A,B] := \\mathcal{C}((1 + Az) / (1 + Bz))$ , where $-1 \\leq B < A \\leq 1$ . If $\\phi(z) = ((1 + z) / (1 - z))^{\\alpha}$ for $0 < \\alpha \\leq 1$ , then $\\mathcal{C}(\\alpha)$ and $S^{*}(\\alpha)$ are the classes of strongly convex and strongly starlike functions of order $\\alpha$ (see [33]). For ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,589,608,629]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\phi (z) = 1 + \\frac {2}{\\pi^ {2}} \\left(\\log \\frac {1 + \\sqrt {z}}{1 - \\sqrt {z}}\\right) ^ {2}, \\tag {1.7}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,635,838,802]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"the class $\\mathcal{C}(\\phi)$ ( $S^{*}(\\phi)$ respectively) is the family $UCV$ ( $UST$ respectively) of normalized uniformly convex (starlike respectively) functions introduced by Goodman (see [19, 20, 41, 42]). Ma and Minda (see [33, 34]) have studied extensively the class $UCV$ . For $0 \\leq \\alpha < 1$ , Khatter et al. [27] introduced $S_{\\alpha,e}^{*} \\coloneqq S^{*}(\\alpha + (1 - \\alpha)e^{z})$ . For $\\alpha = 0$ , $S_{\\alpha,e}^{*}$ reduces to $S_{e}^{*} \\coloneqq S^{*}(e^{z})$ (see [36]). When $\\phi(z) = 1 + 4z/3 + 2z^{2}/3$ , the unit disk $\\mathbb{D}$ is mapped onto a domain bounded by a cardiod and corresponding Ma-Minda starlike class $S^{*}(\\phi)$ reduces to the class $S_{C}^{*}$ (see [43]). For $\\phi(z) = 1 + z/(1 - \\alpha z^{2})$ , Kargar et al. [22] have introduced the class $\\mathcal{BS}^{*}(\\alpha) \\coloneqq S^{*}(1 + z/(1 - \\alpha z^{2}))$ , which is associated with the Booth lemniscate. In 2016, Kumar and Ravichandran [28] considered the class $S_{R}^{*} \\coloneqq S^{*}(\\phi_{0})$ , where $\\phi_{0}$ is the rational function ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,811,648,846]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\phi_ {0} (z) = 1 + \\frac {z}{k} \\left(\\frac {k + z}{k - z}\\right), \\quad k = \\sqrt {2} + 1. \\tag {1.8}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,853,836,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For $\\phi(z) = (1 + sz)^2$ with $0 < s \\leq 1/\\sqrt{2}$ , the class $\\mathcal{S}^*(\\phi)$ reduces to $\\mathcal{ST}_L(s) := \\mathcal{S}^*((1 + sz)^2)$ (see [35]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[228,111,724,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bohr radius for certain classes of starlike and convex univalent functions ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[823,112,836,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[134,137,689,154]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ma and Minda have defined the following functions $h, k \\in S$ by ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,157,653,195]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\frac {z h ^ {\\prime} (z)}{h (z)} = \\phi (z) \\quad \\text {a n d} \\quad 1 + \\frac {z k ^ {\\prime \\prime} (z)}{k ^ {\\prime} (z)} = \\phi (z). \\tag {1.9}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,199,836,266]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Here $h$ and $k$ play the role of Koebe function for the classes $S^{*}(\\phi)$ and $\\mathcal{C}(\\phi)$ , respectively. Clearly, $h$ and $k$ belong to $S^{*}(\\phi)$ and $\\mathcal{C}(\\phi)$ respectively and $zk'(z) = h(z)$ . The subordination and growth estimate for the class $S^{*}(\\phi)$ have been obtained by Ma and Minda (see [32]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,273,836,309]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 1.10. [32] Let $f \\in S^{*}(\\phi)$ . Then $zf'(z)/f(z) \\prec zh'(z)/h(z)$ and $f(z)/z \\prec h(z)/z$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,315,653,333]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 1.11. [32] Assume $f \\in S^{*}(\\phi)$ and $|z| = r < 1$ . Then ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,339,588,357]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n- h (- r) \\leq | f (z) | \\leq h (r). \\tag {1.12}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,362,680,380]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Equality holds for some $z \\neq 0$ if, and only if, $f$ is a rotation of $h$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,387,836,454]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It has been pointed out in [32] that $-h(-r)$ is increasing in $(0,1)$ and bounded by 1 because each $f\\in S^{*}(\\phi)$ is normalized by $f(0) = f^{\\prime}(0) - 1 = 0$ . Therefore, $\\lim_{r\\to 1} - h(-r)$ exists and denote it by $-h(-1)$ . The following subordination and growth theorem for the class $\\mathcal{C}(\\phi)$ have been established in [32]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,461,836,497]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 1.13. [32] Let $f \\in \\mathcal{C}(\\phi)$ . Then $zf''(z) / f'(z) \\prec zk''(z) / k'(z)$ and $f'(z) \\prec zk'(z)$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,504,642,522]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 1.14. [32] Assume $f \\in \\mathcal{C}(\\phi)$ and $|z| = r < 1$ . Then ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,527,588,545]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n- k (- r) \\leq | f (z) | \\leq k (r). \\tag {1.15}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,549,680,568]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Equality holds for some $z \\neq 0$ if, and only if, $f$ is a rotation of $k$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,574,836,628]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is justified in [32] that $-k(-r)$ is increasing in $(0,1)$ and bounded by 1 because each $f\\in \\mathcal{C}(\\phi)$ is normalized by $f(0) = f'(0) - 1 = 0$ . Therefore, $\\lim_{r\\to 1} - k(-r)$ exists and denote it by $-k(-1)$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,644,836,730]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ma and Minda [32] have introduced the analytic and univalent function $\\phi$ with certain conditions, one of which is $\\phi'(0) > 0$ . Recently, Kumar and Banga [29] have considered a non-Ma-Minda function $\\Phi$ , obtained by altering only one condition, namely $\\Phi'(0) < 0$ , which is merely a rotation. Such a function $\\Phi$ is named (see [29]) a non-Ma-Minda of type- $\\mathbf{A}$ , here $\\mathbf{A}$ signifies the condition $\\Phi'(0) < 0$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,737,836,805]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 1.2. An analytic and univalent function $\\Phi$ defined in the unit disk $\\mathbb{D}$ is said to be a non-Ma-Minda of type $\\mathbf{A}$ if it has positive real part in $\\mathbb{D}$ , $\\Phi(\\mathbb{D})$ is symmetric with respect to the real axis, starlike with respect to $\\Phi(0) = 1$ and $\\Phi'(0) < 0$ . Further, it has a power series expansion of the form: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,809,613,852]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\Phi (z) = 1 + \\sum_ {n = 1} ^ {\\infty} C _ {n} z ^ {n} \\quad \\left(C _ {1} < 0\\right). \\tag {1.16}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,856,836,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The class of all such functions of non-Ma-Minda of type $\\mathbf{A}$ , we shall denote it by $\\mathcal{M}_A$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,112,131,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[344,111,608,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Vasudevarao Allu and Himadri Halder ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[114,136,836,220]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On the similar lines of Definition 1.1, Kumar and Banga has defined the classes $S^{*}(\\Phi)$ and $\\mathcal{C}(\\Phi)$ . It is worth noting that by mere replacing $z$ by $-z$ , i.e. $\\Phi(z) = \\phi(-z)$ , each function in $\\mathcal{M}_A$ reduces to its Ma-Minda counter part. For instance, $\\sqrt{1 + z}$ and $1 - \\log(1 - z)$ belong to the class $\\mathcal{M}$ whereas $\\sqrt{1 - z}$ and $1 - \\log(1 + z)$ belong to $\\mathcal{M}_A$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,237,838,323]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Observe that all the coefficients of $\\phi \\in \\mathcal{M}$ are not necessary to be positive. For instance, some of the coefficients of $\\phi(z) = z + \\sqrt{1 + z^2} = 1 + z + \\frac{1}{2} - z^4 / 8 + \\dots$ and $\\phi(z) = 1 + z - z^3 / 3$ are negative. The class $S^*(z + \\sqrt{1 + z^2})$ has been studied by Raina and Sokol [39]. Wani and Swaminathan [48] have studied the class $S_{Ne}^* := S^*(1 + z - z^3 / 3)$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[134,353,803,372]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In 1990, Silverman and Silvia [45] introduced the following classes $\\mathcal{G}_{\\alpha}^{*}$ and $\\mathcal{G}_{\\alpha}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,380,836,413]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 1.3. Let $\\mathcal{G}_{\\alpha}^{*}$ denote the class of functions $G$ analytic in $\\mathbb{D}$ that satisfy the following conditions: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[139,416,833,484]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"(i) $G$ is normalized by $G(0) = 1$ and $G(1) = \\\\lim_{r\\\\to -1} = 0$ \",\"(ii) $G(\\\\mathbb{D})$ lies in a sector with aperture $2(1 - \\\\alpha)\\\\pi$ and vertex at the origin, and \",\"(iii) $G$ maps $\\\\mathbb{D}$ univalently onto a domain that is starlike with respect to $G(1)$ . In addition, let the constant function 1 belong to $\\\\mathcal{G}_{\\\\alpha}^{*}$ . \"]","page_idx":"4","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[115,491,836,526]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 1.4. For $\\alpha$ , let $\\mathcal{G}_{\\alpha}$ denote the class of functions, $G$ with $G(z) = 1 + \\sum_{n=1}^{\\infty} d_n z^n$ , analytic and non-vanishing in $\\mathbb{D}$ which satisfy ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,530,631,568]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname {R e} \\left(\\frac {z G ^ {\\prime} (z)}{G (z)} + \\frac {(1 - \\alpha) (1 + z)}{1 - z}\\right) > 0 \\tag {1.17}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,572,205,587]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for $z\\in \\mathbb{D}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,597,836,666]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For $\\alpha = 1/2$ , Robertson [40] has studied extensively the classes $\\mathcal{G}_{1/2}^*$ and $\\mathcal{G}_{1/2}$ and has conjectured that they are equal. This conjecture was proved by Lyzzaik in 1984 (see[31]). In more general, for all $\\alpha$ , $0 \\leq \\alpha < 1$ , Silverman and Silvia have shown that $\\mathcal{G}_{\\alpha}^{*} = \\mathcal{G}_{\\alpha}$ by proving closely related property of $\\mathcal{G}_{\\alpha}$ with $\\mathcal{S}^{*}(\\alpha)$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,673,836,706]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 1.18. [45] A function $G$ is in $\\mathcal{G}_{\\alpha}$ if, and only if, there exists a function $s \\in S^{*}(\\alpha)$ such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[364,707,588,739]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nG (z) = (1 - z) ^ {2 (1 - \\alpha)} \\frac {s (z)}{z}.\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,745,836,796]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The regions of variability for the class $\\mathcal{G}_{\\alpha}$ has been studied extensively by Ponnusamy et al. [38]. The following growth theorem for the class $\\mathcal{G}_{\\alpha}$ has been established by Silverman and Silvia [45]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,804,416,821]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 1.19. [45] If $g \\in \\mathcal{G}_{\\alpha}$ then ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,826,653,866]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\left(\\frac {1 - r}{1 + r}\\right) ^ {2 (1 - \\alpha)} \\leq | g (z) | \\leq \\left(\\frac {1 + r}{1 - r}\\right) ^ {2 (1 - \\alpha)} \\tag {1.20}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[112,872,825,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for $|z| = r$ . Equality holds for $g(z) = ((1 - z) / (1 + z))^{2(1 - \\alpha)}$ at $z = r$ and $z = -r$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[228,111,724,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bohr radius for certain classes of starlike and convex univalent functions ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[823,112,836,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[115,137,836,172]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In 2018, Bhowmik and Das [13] proved an interesting result for subordination classes. Let $f$ and $g$ be two analytic functions in $\\mathbb{D}$ such that $g \\prec f$ . Let ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,176,547,218]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\ng (z) = \\sum_ {n = 0} ^ {\\infty} c _ {n} z ^ {n}. \\tag {1.21}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,225,838,260]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 1.22. [13] Let $f$ and $g$ be analytic in $\\mathbb{D}$ with Taylor expansions (1.1) and (1.21) respectively and $g \\prec f$ , then ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,264,574,307]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\sum_ {n = 0} ^ {\\infty} \\left| c _ {n} \\right| r ^ {n} \\leq \\sum_ {n = 0} ^ {\\infty} \\left| a _ {n} \\right| r ^ {n} \\tag {1.23}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,312,267,331]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for $z| = r\\leq 1 / 3$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,338,836,372]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The following coefficients bounds for the class $S^{*}(\\alpha)$ are required to obtain the Bohr radius for the class $\\mathcal{G}_{\\alpha}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,380,636,397]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 1.24. [21] Let $f \\in S^{*}(\\alpha)$ be given by (1.15). Then ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[299,402,653,447]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n| a _ {n} | \\leq \\frac {1}{(n - 1) !} \\prod_ {k = 0} ^ {n} (k - 2 \\alpha) \\quad f o r \\quad n \\geq 2.\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,451,833,471]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The equalities in above estimates are attained for $f(z) = z / (1 - z)^{2(1 - \\alpha)}$ for $z \\in \\mathbb{D}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,477,836,562]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this paper, we establish the Bohr phenomenon for the classes $S^{*}(\\phi)$ and $\\mathcal{C}(\\phi)$ , where all the coefficients of associated Ma-Minda function $\\phi$ are positive such that $\\phi \\in H^2$ , the Hardy class of analytic functions in $\\mathbb{D}$ . As a consequence, we obtain several important corollaries for particular choices of $\\phi$ . We also obtain the Bohr radius for the class $\\mathcal{G}_{\\alpha}$ , starlike functions with respect to the boundary point. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[393,574,559,590]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. MAIN RESULTS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[134,598,811,617]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Using Lemmas 1.10, 1.11 and 1.22, we obtain Bohr radius for the class $S^{*}(\\phi)$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,623,840,659]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2.1. Let $f \\in S^{*}(\\phi)$ be given by (1.5) and $\\phi(z)$ be given by (1.6) with all $B_{n} > 0$ , $n \\geq 1$ such that $\\phi \\in H^{2}$ , the Hardy class of analytic functions in $\\mathbb{D}$ . Then ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,663,621,705]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n| z | + \\sum_ {n = 0} ^ {\\infty} \\left| a _ {n} \\right| | z | ^ {n} \\leq d (f (0), \\partial f (\\mathbb {D})) \\tag {2.2}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,710,700,729]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for $|z| = r \\leq \\min \\{r_f, 1/3\\}$ , where $r_f$ is the smallest positive root of ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,733,552,750]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh (r) + h (- 1) = 0 \\tag {2.3}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,756,436,774]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"in $(0,1)$ and $h(z)$ is defined in (1.9). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,781,836,850]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 2.1. Since $\\Phi \\in \\mathcal{M}_A$ is obtained from $\\phi \\in \\mathcal{M}$ by mere replacing $z$ by $-z$ , the image of $\\mathbb{D}$ under the functions $\\Phi$ and $\\phi$ are identical. Therefore we have $S^{*}(\\Phi) = S^{*}(\\phi)$ and hence the Bohr radius for the class $S^{*}(\\Phi)$ is same as that of $S^{*}(\\phi)$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,856,836,890]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As a consequence of Theorem 2.1, we obtain the following corollaries for particular choices of $\\phi$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,112,131,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[344,111,608,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Vasudevarao Allu and Himadri Halder ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[115,136,840,190]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 2.4. For $\\phi(z) = \\alpha + (1 - \\alpha)e^{z}$ , we have $S_{\\alpha,e}^{*} \\coloneqq S^{*}(\\alpha + (1 - \\alpha)e^{z})$ . Let $f \\in S_{\\alpha,e}^{*}$ be given by (1.5) with $0 \\leq \\alpha < 0.05284$ . Then the inequality (2.2) is satisfied for $|z| = r \\leq r_f$ , where $0 < r_f < 1/3$ . The radius $r_f$ is the best possible. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,198,836,233]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In particular, for $\\alpha = 0$ in Corollary 2.4, we obtain the sharp Bohr radius for the class $S_{e}^{*}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,242,842,293]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 2.5. For $\\phi(z) = 1 + 4z/3 + 2z^2/3$ , $S^*(\\phi)$ reduces to $S_C^*$ . For $f \\in S_C^*$ of the form (1.5), the inequality (2.2) is satisfied for $|z| = r \\leq r_f$ , where $0 < r_f < 1/3$ . The radius $r_f$ is the best possible. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,302,875,356]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 2.6. Let $\\phi$ be the rational function such that $\\phi(z) = 1 + (z / k) \\left( \\frac{(k + z)}{k - z} \\right)$ , where $k = \\sqrt{2} + 1$ . Then $\\mathcal{S}^*(\\phi)$ reduces to the class $\\mathbb{S}_R^*$ . Then the inequality (2.2) is satisfied for $|z| = r \\leq 1/3$ for the class $\\mathbb{S}_R^*$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,364,840,448]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 2.7. For $\\phi(z) = (1 + Az)/(1 + Bz)$ , the class $S^{*}(\\phi)$ reduces to the Janowski starlike class $S^{*}[A, B]$ . Let $f \\in S^{*}[A, B]$ be given by (1.5) with $-1 \\leq B < (1 - 3^{k})/(1 + 3^{k}) < 0$ and $0 \\leq A \\leq 1$ , where $k = B/(B - A)$ . Then the inequality (2.2) is satisfied for $|z| = r \\leq r_{f}$ , where $0 < r_{f} < 1/3$ . The radius $r_{f}$ is the best possible. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,457,840,509]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 2.8. For $\\phi(z) = 1 + z / (1 - \\alpha z^2)$ , the class $S^*(\\phi)$ reduces to $\\mathcal{BS}^*(\\alpha)$ . Let $f \\in \\mathcal{BS}^*(\\alpha)$ be given by (1.5) with $0 \\leq \\alpha < 1$ . Then the inequality (2.2) is satisfied for $|z| = r \\leq r_f$ , where $0 < r_f < 1/3$ . The constant $r_f$ cannot be improved. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,517,838,571]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 2.9. Let $\\phi = (1 + sz)^{2}$ , then $S^{*}(\\phi)$ reduces to the class $\\mathcal{ST}_L(s)$ . Let $f\\in \\mathcal{ST}_L(s)$ be given by (1.5) with $0.444981 < s\\leq 1 / \\sqrt{2}$ . Then the inequality (2.2) satisfies for $|z| = r\\leq r_f$ , where $0 < r_{f} < 1 / 3$ . The radius $r_f$ is the best possible. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[134,579,836,598]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By using Lemmas 1.14 and 1.22, we establish Bhor phenomenon for the class $\\mathcal{C}(\\phi)$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,606,838,641]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2.10. Let $f \\in \\mathcal{C}(\\phi)$ be given by (1.5) and $\\phi(z)$ be given by (1.6) with all $B_n > 0$ , $n \\geq 1$ such that $\\phi \\in H^2$ , the Hardy class of analytic functions in $\\mathbb{D}$ . Then ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,650,621,691]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n| z | + \\sum_ {n = 0} ^ {\\infty} \\left| a _ {n} \\right| | z | ^ {n} \\leq d (f (0), \\partial f (\\mathbb {D})) \\tag {2.11}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,700,700,719]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for $|z| = r \\leq \\min \\{r_f, 1/3\\}$ , where $r_f$ is the smallest positive root of ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,726,552,744]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nk (r) + k (- 1) = 0 \\tag {2.12}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,752,436,771]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"in (0,1) and $k(z)$ is defined in (1.9). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[134,780,556,798]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $G \\in \\mathcal{G}_{\\alpha}$ with the power series representation ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,807,603,847]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nG (z) = 1 + \\sum_ {n = 1} ^ {\\infty} d _ {n} z ^ {n}, \\quad z \\in \\mathbb {D}. \\tag {2.13}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,856,836,890]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By using Lemmas 1.18, 1.19 and 1.24, we obtain the sharp Bhor radius for the class $\\mathcal{G}_{\\alpha}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[228,111,724,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bohr radius for certain classes of starlike and convex univalent functions ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[823,112,836,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[115,137,591,154]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2.14. Let $G \\in \\mathcal{G}_{\\alpha}$ be given by (2.13). Then ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,161,603,203]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\sum_ {n = 1} ^ {\\infty} \\left| d _ {n} \\right| | z | ^ {n} \\leq d (G (0), \\partial G (\\mathbb {D})) \\tag {2.15}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[112,209,838,244]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for $|z| = r \\leq r_f$ , where $r_f = (2^{1/2(1 - \\alpha)} - 1) / (2^{1/2(1 - \\alpha)} + 1)$ . The radius $r_f$ is the best possible. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[322,260,628,275]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3. PROOF OF THE MAIN RESULTS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[115,294,806,312]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof of Theorem 2.1. Let $f \\in S^{*}(\\phi)$ then in view of Lemma 1.11, we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,318,615,336]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n| f (z) | \\geq - h (- r) \\quad \\text {f o r} \\quad | z | < 1. \\tag {3.1}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,342,719,359]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By taking $\\lim \\inf$ as $|z| = r\\to 1$ on both the sides of (3.1), we obtain ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,366,581,391]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\lim _ {| z | \\rightarrow 1} | f (z) | \\geq - h (- 1). \\tag {3.2}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,397,773,416]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Euclidean distance between $f(0)$ and the boundary of $f(\\mathbb{D})$ is given by ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,422,640,448]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nd (f (0), \\partial f (\\mathbb {D})) = \\lim _ {| z | \\rightarrow 1} \\inf | f (z) - f (0) |. \\tag {3.3}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,454,521,472]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since $f(0) = 0$ from (3.2) and (3.3), we obtain ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,479,589,496]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nd (f (0), \\partial f (\\mathbb {D})) \\geq - h (- 1). \\tag {3.4}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,502,457,520]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is known that $\\phi \\in H^2$ if, and only if, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,526,549,568]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\sum_ {n = 1} ^ {\\infty} \\left| B _ {n} \\right| ^ {2} < + \\infty . \\tag {3.5}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,574,289,591]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From (3.5) we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,597,643,650]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\sum_ {n = 1} ^ {\\infty} \\frac {B _ {n}}{n} \\leq \\sqrt {\\sum_ {n = 1} ^ {\\infty} | B _ {n} | ^ {2}} \\sqrt {\\sum_ {n = 1} ^ {\\infty} \\frac {1}{n ^ {2}}} < + \\infty . \\tag {3.6}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,656,149,669]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,676,623,718]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh (z) = z + \\sum_ {n = 2} ^ {\\infty} b _ {n} z ^ {n} \\quad \\text {f o r} \\quad z \\in \\mathbb {D}. \\tag {3.7}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,724,835,742]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In view of (1.9), we obtain $zh'(z) / h(z) = \\phi(z)$ and a simple computation shows that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,747,611,801]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh (z) = z \\exp \\left(\\int_ {0} ^ {\\tilde {z}} \\frac {\\phi (t) - 1}{t} d t\\right), \\tag {3.8}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,806,836,840]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where integration is along the linear segment joing 0 to $z \\in \\mathbb{D}$ . From (1.9) and (3.8), we obtain ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,846,594,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh (z) = z \\exp \\left(\\sum_ {n = 1} ^ {\\infty} B _ {n} \\frac {z ^ {n}}{n}\\right). \\tag {3.9}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[117,112,129,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"8 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[344,111,608,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Vasudevarao Allu and Himadri Halder ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[115,136,400,154]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $H:[0,1]\\to \\mathbb{R}$ be defined by ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[381,163,571,181]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nH (r) = h (r) + h (- 1).\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,188,836,239]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From (3.6), (3.8) and (3.9), it is easy to see that $H$ is continuous in $[0,1]$ and differentiable in $(0,1)$ . Note that $H(0) = h(-1) < 0$ and $H(1) = h(1) + h(-1)$ . Clearly, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[205,245,747,294]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh (1) = \\exp \\left(\\sum_ {n = 1} ^ {\\infty} \\frac {B _ {n}}{n}\\right) \\quad \\mathrm {a n d} \\quad h (- 1) = \\exp \\left(\\sum_ {n = 1} ^ {\\infty} \\frac {(- 1) ^ {n} B _ {n}}{n}\\right).\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,299,324,315]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since $B_{n} > 0$ for $n\\geq 1$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[371,318,581,361]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\sum_ {n = 1} ^ {\\infty} \\frac {B _ {n}}{n} > \\sum_ {n = 1} ^ {\\infty} \\frac {(- 1) ^ {n} B _ {n}}{n}.\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,366,208,382]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Therefore, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[226,389,726,437]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh (1) = \\exp \\left(\\sum_ {n = 1} ^ {\\infty} \\frac {B _ {n}}{n}\\right) > \\exp \\left(\\sum_ {n = 1} ^ {\\infty} \\frac {(- 1) ^ {n} B _ {n}}{n}\\right) = - h (- 1)\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,442,838,494]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"which implies that $H(1) > 0$ . Since $H(0) < 0$ and $H(1) > 0$ , by the intermediate value property, $H$ has one real root in $(0,1)$ . Let $r_f$ be the smallest positive root of $H$ in $(0,1)$ . Thus $H(r_f) = 0$ , which is equivalent to ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,503,549,521]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh \\left(r _ {f}\\right) = - h (- 1). \\tag {3.10}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,529,394,546]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From (3.7) and (3.9), we obtain ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,554,722,600]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh (z) = z + \\sum_ {n = 2} ^ {\\infty} b _ {n} z ^ {n} = z \\exp \\left(\\sum_ {n = 1} ^ {\\infty} B _ {n} \\frac {z ^ {n}}{n}\\right) \\quad \\text {f o r} \\quad z \\in \\mathbb {D}. \\tag {3.11}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,606,208,623]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Therefore, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,631,655,761]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} | z | + \\sum_ {n = 2} ^ {\\infty} \\left| b _ {n} \\right| | z | ^ {n} = | z | \\exp \\left(\\sum_ {n = 1} ^ {\\infty} \\left| B _ {n} \\right| \\frac {| z | ^ {n}}{n}\\right) \\tag {3.12} \\\\ = r \\exp \\left(\\sum_ {n = 1} ^ {\\infty} B _ {n} \\frac {r ^ {n}}{n}\\right) \\\\ = h (r). \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,770,608,788]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In view of Lemma 1.22 and (3.11), for $|z| = r$ , we obtain ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,798,695,840]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nr + \\sum_ {n = 2} ^ {\\infty} \\left| a _ {n} \\right| r ^ {n} \\leq r + \\sum_ {n = 2} ^ {\\infty} \\left| b _ {n} \\right| r ^ {n} = h (r) \\quad \\text {f o r} \\quad r \\leq \\frac {1}{3}. \\tag {3.13}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,847,290,864]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For $r \\leq r_f$ , we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,872,581,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh (r) \\leq h \\left(r _ {f}\\right) = - h (- 1). \\tag {3.14}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[228,111,724,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bohr radius for certain classes of starlike and convex univalent functions ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[823,112,836,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"9 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[115,137,579,154]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Combining (3.4), (3.13) and (3.14), we conclude that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[290,161,658,203]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nr + \\sum_ {n = 2} ^ {\\infty} | a _ {n} | r ^ {n} \\leq - h (- 1) \\leq d (f (0), \\partial f (\\mathbb {D}))\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,209,836,276]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for $r \\leq \\min \\{r_f, 1/3\\}$ . If $r_f$ lies in $(0, 1/3]$ , then $r_f$ is the best possible. Let $0 < r_f \\leq 1/3$ . To show the sharpness of $r_f$ , we choose $f$ to be a suitable rotation $h_\\epsilon$ of the function $h$ i.e. $f = h_\\epsilon$ which is obviously belongs to $S^*(\\phi)$ . Since equality in Lemma 1.11 occurs for a suitable rotation of $h$ , we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,283,596,301]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nd \\left(h _ {\\epsilon} (0), \\partial h _ {\\epsilon} (\\mathbb {D})\\right) = - h _ {\\epsilon} (- 1). \\tag {3.15}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,308,823,326]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A simple computation using (3.10) and (3.15), for $f = h_{\\epsilon}$ and $|z| = r_f$ shows that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[233,333,717,375]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n| z | + \\sum_ {n = 0} ^ {\\infty} | a _ {n} | | z | ^ {n} = h _ {\\epsilon} (r) = - h _ {\\epsilon} (- 1) = d (h _ {\\epsilon} (0), \\partial h _ {\\epsilon} (\\mathbb {D})).\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,381,836,399]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Therefore, the radius $r_f \\in (0, 1/3)$ is the best possible. This completes the proof. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,423,838,475]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof of Corollary 2.4. Given that $\\phi(z) = \\alpha + (1 - \\alpha)e^{z}$ and each coefficient of $\\phi(z)$ is strictly positive for $0 \\leq \\alpha < 1$ . By the similar lines of argument as in the proof of Theorem 2.1, consider $H(r) = h(r) + h(-1)$ . Note that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[205,480,747,543]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh \\left(\\frac {1}{3}\\right) = \\frac {1}{3} \\exp \\left((1 - \\alpha) \\int_ {0} ^ {\\frac {1}{3}} \\left(\\frac {- 1 + e ^ {t}}{t}\\right) d t\\right) \\approx \\frac {1}{3} (1. 4 3 8 0 7) ^ {1 - \\alpha}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,549,154,564]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[184,568,768,624]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh (- 1) = - \\exp \\left((1 - \\alpha) \\int_ {0} ^ {- 1} \\left(\\frac {- 1 + e ^ {t}}{t}\\right) d t\\right) \\approx - (0. 4 5 0 8 5 9 4 6 3) ^ {1 - \\alpha}.\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,628,836,697]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Therefore, using Mathematica we can see that $H(1/3) = h(1/3) + h(-1) > 0$ if, and only if, $0 \\leq \\alpha < 0.05284$ . On the other hand, $H(0) = h(-1) < 0$ . Therefore we conclude that $H$ has a root in $(0,1)$ . Since $r_f$ is the smallest root of $H$ in $(0,1)$ , we have $r_f < 1/3$ . Hence in view of Theorem 2.1, $r_f$ is the best possible. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,720,836,751]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof of Corollary 2.5. Let $\\phi(z) = 1 + 4z/3 + 2z^2/3$ . A simple computation shows that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[363,751,589,788]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh (r) = r \\exp \\left(\\frac {4}{3} r + \\frac {r ^ {2}}{3}\\right).\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,790,655,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note that $h(1/3) \\approx 0.539490$ and $h(-1) \\approx -0.367879441$ . Let ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[379,815,571,834]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nH (r) = h (r) + h (- 1).\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,839,836,890]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Then $H(0) \\approx -0.3678799441 < 0$ and $H(1/3) > 0$ . Therefore, $H$ has a root in $(0,1/3)$ . Let $r_f$ be the smallest root in $(0,1/3)$ . In view of Theorem 2.1, $r_f$ is the best possible. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,112,137,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[344,111,608,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Vasudevarao Allu and Himadri Halder ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[114,137,836,172]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof of Corollary 2.6. For $\\phi(z) = 1 + (z / k) \\left( \\frac{(k + z)}{k - z} \\right)$ , where $k = \\sqrt{2} + 1$ , a simple computation using (3.8) shows that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[376,176,574,217]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh (r) = \\frac {r}{e ^ {r}} \\left(\\frac {k}{k - r}\\right) ^ {2 k}.\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,222,838,274]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note that $h(-1) \\approx -0.5099807$ and $h(1/3) \\approx 0.489391446$ . Hence $H(1/3) = h(1/3) + h(-1) < 0$ and $H(0) < 0$ . Therefore $H$ has no root in $(0,1/3)$ and $r_f > 1/3$ . Hence the inequality (2.2) is satisfied for $r \\leq 1/3$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,296,836,347]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof of Corollary 2.7. Let $\\phi(z) = (1 + Az)/(1 + Bz)$ with $-1 \\leq B < A \\leq 1$ . It is easy to see that the coefficients of $\\phi(z)$ are all positive when $-1 \\leq B \\leq 0$ . A simple computation using (3.8) yields ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[369,352,583,377]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh (r) = r \\left(1 + B r\\right) ^ {\\left(\\frac {A - B}{B}\\right)}.\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,381,524,399]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $H(r) = h(r) + h(-1)$ . Then $H(0) < 0$ and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[198,404,754,448]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nH \\left(\\frac {1}{3}\\right) = h \\left(\\frac {1}{3}\\right) + h (- 1) = \\frac {1}{3} \\left(1 + \\frac {B}{3}\\right) ^ {\\left(\\frac {A - B}{B}\\right)} - (1 - B) ^ {\\left(\\frac {A - B}{B}\\right)}.\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,453,836,521]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A simple computation shows that $H(1/3) > 0$ when $-1 \\leq B \\leq (1 - 3^k) / (1 + 3^k)$ , where $k = (B - A) / B$ . Therefore, $H$ has a root in $(0,1)$ . Let the smallest root of $H$ in $(0,1/3)$ be $r_f$ . In view of Theorem 2.1, we conclude that for $r \\leq r_f$ , the inequality (2.2) is satisfied and the radius $r_f$ is the best possible. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,543,815,562]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof of Corollary 2.8. Here $\\phi(z) = 1 + z / (1 - \\alpha z^2)$ . Using (3.8), we obtain ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[359,567,589,610]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh (r) = r \\left(\\frac {1 + \\sqrt {\\alpha} r}{1 - \\sqrt {\\alpha} r}\\right) ^ {\\frac {1}{2 \\sqrt {\\alpha}}}.\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,615,690,687]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Clearly, $h(-1) = -\\left(\\frac{1 + \\sqrt{\\alpha}}{1 + \\sqrt{\\alpha}}\\right)^{\\frac{1}{2\\sqrt{\\alpha}}}$ and $h\\left(\\frac{1}{3}\\right) = \\left(\\frac{1 + \\frac{\\sqrt{\\alpha}}{3}}{1 + \\frac{\\sqrt{\\alpha}}{3}}\\right)^{\\frac{1}{2\\sqrt{\\alpha}}}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,686,836,737]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $H(r) = h(r) + h(-1)$ . Then $H(0) < 0$ and $H(1/3) = h(1/3) + h(-1) > 0$ for $0 \\leq \\alpha < 1$ . Therefore, $H$ has a root in $(0, 1/3)$ and choose $r_f$ to be the smallest root in $(0, 1/3)$ . By Theorem 2.1, the radius $r_f$ is the best possible. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,761,836,793]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof of Corollary 2.9. Let $\\phi(z) = (1 + sz)^2$ with $0 < s \\leq 1/\\sqrt{2}$ . Using (3.8), we obtain ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[339,792,610,828]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh (r) = r \\exp \\left(s \\left(2 r + \\frac {s r ^ {2}}{2}\\right)\\right).\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,831,563,849]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $H(r) = h(r) + h(-1)$ . Note that $H(0) < 0$ and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[226,853,727,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nH \\left(\\frac {1}{3}\\right) = \\frac {1}{3} \\exp \\left(s \\left(\\frac {s + 1 2}{1 8}\\right)\\right) - \\exp \\left(s \\left(- 2 + \\frac {s}{2}\\right)\\right) > 0,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[226,111,724,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bohr radius for certain classes of starlike and convex univalent functions ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[818,112,835,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"11 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[114,137,836,188]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"if $0.444981 < s \\leq 1 / \\sqrt{2}$ . Therefore, $H$ has a real root in $(0,1/3)$ . Let $r_f$ be the smallest root in $(0,1/3)$ . In view of Theorem 2.1, the radius $r_f$ is the best possible. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,214,836,247]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof of the Theorem 2.10. Let $f \\in \\mathcal{C}(\\phi)$ then from Lemma 1.14, it is evident that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,256,685,281]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nd (f (0), \\partial f (\\mathbb {D})) = \\lim _ {| z | \\rightarrow 1} \\inf | f (z) - f (0) | \\geq - k (- 1). \\tag {3.16}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,289,836,325]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is known that $f \\in \\mathcal{C}(\\phi)$ if, and only if, $zf' \\in S^{*}(\\phi)$ . Therefore, there exists $g \\in S^{*}(\\phi)$ such that $zf'(z) = g(z)$ for $z \\in \\mathbb{D}$ . This relation gives ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,333,554,381]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nf (z) = \\int_ {0} ^ {z} \\frac {g (t)}{t} d t. \\tag {3.17}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,389,836,409]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $f(z) = z + \\sum_{n=2}^{\\infty} a_n z^n$ and $g(z) = z + \\sum_{n=2}^{\\infty} b_n z^n$ . Then from (3.17), we obtain ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,416,600,457]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nz + \\sum_ {n = 2} ^ {\\infty} a _ {n} z ^ {n} = z + \\sum_ {n = 2} ^ {\\infty} \\frac {b _ {n}}{n} z ^ {n}. \\tag {3.18}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,464,788,484]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note that the majorant series for $g$ , we have $M_g(r) \\coloneqq r + \\sum_{n=2}^{\\infty} |b_n| r^n$ . Then ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,492,600,609]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\int_ {0} ^ {r} \\frac {M _ {g} (t)}{t} d t = r + \\sum_ {n = 2} ^ {\\infty} \\left| b _ {n} \\right| \\frac {r ^ {n}}{n} \\tag {3.19} \\\\ = r + \\sum_ {n = 2} ^ {\\infty} | a _ {n} | r ^ {n} \\\\ = M _ {f} (r). \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,616,836,649]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since we have $zk'(z) = h(z)$ , then going by same lines of argument as in (3.19), we obtain ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,655,568,705]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nM _ {k} (r) = \\int_ {0} ^ {r} \\frac {M _ {g} (t)}{t} d t. \\tag {3.20}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,712,400,728]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In view of Lemma 1.22, we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,736,601,767]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nM _ {g} (r) \\leq M _ {h} (r) \\quad \\text {f o r} \\quad r \\leq \\frac {1}{3}. \\tag {3.21}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,773,589,790]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Using (3.19), (3.20) and (3.21), for $r \\leq 1/3$ , we obtain ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,799,638,849]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nM _ {f} (r) \\leq \\int_ {0} ^ {r} \\frac {M _ {h} (t)}{t} d t = M _ {k} (r) = k (r). \\tag {3.22}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,856,836,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By the same lines of argument as in Theorem 2.1, we can show that $H_{1}:[0,1]\\to \\mathbb{R}$ defined by $H_{1}(r) = k(r) + k(-1)$ has a real root in $(0,1)$ . Let $r_f$ be the smallest ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,112,137,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"12 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[344,111,608,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Vasudevarao Allu and Himadri Halder ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[114,136,835,171]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"root of $H_{1}$ in $(0,1)$ and $H_{1}(r_{f}) = 0$ , which follows that $k(r_{f}) = -k(-1)$ . Therefore, for $r \\leq \\min \\{r_f, 1/3\\}$ , we obtain ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[290,176,660,195]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nM _ {f} (r) \\leq k (r) \\leq - k (- 1) \\leq d (f (0), \\partial f (\\mathbb {D})).\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,201,836,252]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $0 < r_f \\leq 1/3$ . To show the sharpness of $r_f$ , we choose $f$ to be a suitable rotation $k_{\\epsilon}$ of the function $k$ i.e. $f = k_{\\epsilon}$ which is obviously belongs to $\\mathcal{C}(\\phi)$ . Since the equality in Lemma 1.14 occurs for a suitable rotation of $k$ , we obtain ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,258,596,277]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nd \\left(k _ {\\epsilon} (0), \\partial k _ {\\epsilon} (\\mathbb {D})\\right) = - k _ {\\epsilon} (- 1). \\tag {3.23}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,283,732,301]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A simple computation using (3.16), for $f = k_{\\epsilon}$ and $|z| = r_f$ , shows that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[235,308,714,351]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n| z | + \\sum_ {n = 0} ^ {\\infty} | a _ {n} | | z | ^ {n} = k _ {\\epsilon} (r) = - k _ {\\epsilon} (- 1) = d (k _ {\\epsilon} (0), \\partial k _ {\\epsilon} (\\mathbb {D})).\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,357,801,375]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Therefore, the radius $r_f \\in (0, 1/3)$ is the best possible. This completes the proof. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,399,836,432]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof of the Theorem 2.14. Let $G \\in \\mathcal{G}_{\\alpha}$ be given by (2.13). Then from Lemma 1.19, it is evident that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,439,665,465]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nd (G (0), \\partial G (\\mathbb {D})) = \\lim _ {| z | \\rightarrow 1} \\inf _ {} | G (z) - G (0) | \\geq 1. \\tag {3.24}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,472,589,488]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In view of Lemma 1.18, we have the following relation ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,494,584,527]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nG (z) = (1 - z) ^ {2 (1 - \\alpha)} \\frac {s (z)}{z}. \\tag {3.25}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,533,500,549]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For $s\\in S^{*}(\\alpha)$ , using Lemma 1.24 we obtain ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,557,569,587]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nM _ {s} (r) \\leq \\frac {r}{(1 - r) ^ {2 (1 - \\alpha)}} \\tag {3.26}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,593,836,628]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for $0 < r < 1$ , where $M_s(r)$ is the associated majorant series of $s$ . It is known that for $f$ and $g$ be two analytic functions in $\\mathbb{D}$ , ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,635,536,653]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nM _ {f g} \\leq M _ {f} M _ {g} \\tag {3.27}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[114,659,836,694]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for $0 < r < 1$ , where $M_{fg}$ , $M_f$ and $M_g$ are the associated majorant series of $fg$ , $f$ and $g$ respectively. In view of (3.25), (3.26) and (3.27), for $|z| = r$ , we obtain ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,700,603,819]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} M _ {G} (r) = 1 + \\sum_ {n = 1} ^ {\\infty} | d _ {n} | | z | ^ {n} \\tag {3.28} \\\\ \\leq (1 + | z |) ^ {2 (1 - \\alpha)} \\frac {M _ {s} (r)}{r} \\\\ \\leq \\frac {(1 + r) ^ {2 (1 - \\alpha)}}{(1 - r) ^ {2 (1 - \\alpha)}}. \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,825,403,840]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Therefore from (3.28), we obtain ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[121,849,613,890]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\sum_ {n = 1} ^ {\\infty} \\left| d _ {n} \\right| | z | ^ {n} \\leq \\frac {(1 + r) ^ {2 (1 - \\alpha)}}{(1 - r) ^ {2 (1 - \\alpha)}} - 1. \\tag {3.29}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[228,111,724,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bohr radius for certain classes of starlike and convex univalent functions ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[818,112,836,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"13 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[115,136,771,155]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The right hand side of (3.29) is less than or equal to $1 \\leq d(G(0), \\partial G(\\mathbb{D}))$ if ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[400,160,552,199]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\frac {(1 + r) ^ {2 (1 - \\alpha)}}{(1 - r) ^ {2 (1 - \\alpha)}} \\leq 2\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,204,835,239]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"i.e. for $r \\leq r_G := (2^{1/2(1 - \\alpha)} - 1) / (2^{1/2(1 - \\alpha)} + 1)$ . To show that $r_G$ is the best possible, we consider the function $G_\\alpha : \\mathbb{D} \\to \\mathbb{C}$ defined by ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[366,245,583,286]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nG _ {\\alpha} (z) = \\left(\\frac {1 - z}{1 + z}\\right) ^ {2 (1 - \\alpha)}.\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,290,470,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From Lemma 1.19, for $G = G_{\\alpha}$ , we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[384,314,566,332]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nd (G (0), \\partial G (\\mathbb {D})) = 1.\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,339,635,356]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For $G = G_{\\alpha}$ and $|z| = r_G$ , a simple computation shows that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[226,362,722,406]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\sum_ {n = 1} ^ {\\infty} \\left| d _ {n} \\right| | z | ^ {n} = \\left(\\frac {1 + r _ {G}}{1 - r _ {G}}\\right) ^ {2 (1 - \\alpha)} - 1 = 1 = d (G (0), \\partial G (\\mathbb {D})).\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,412,771,429]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This shows that the radius $r_f$ is the best possible. This completes the proof. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[816,413,836,426]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0498/a6b4585b741cda4ba06643b87f11bebc88e544ba60e35a9d2e08170f742a235c.jpg","list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[115,444,836,478]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acknowledgement: The first author thank SERB-MATRICS and the second author thank CSIR for their support. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[415,496,539,511]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"REFERENCES ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[117,519,836,889]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"[1] Y. ABU-MUHANNA, Bohr's phenomenon in subordination and bounded harmonic classes, Complex Var. Elliptic Equ. 55 (2010), 1071-1078. \",\"[2] Y. ABU-MUHANNA and R. M. ALI, Bohr's phenomenon for analytic functions into the exterior of a compact convex body, J. Math. Anal. Appl. 379 (2011), 512-517. \",\"[3] Y. ABU MUHANNA and R. M. ALI, Bohr's phenomenon for analytic functions and the hyperbolic metric, Math. Nachr. 286 (2013), 1059-1065. \",\"[4] Y.ABU MUHANNA, R. M. ALI, Z. C. NG and S. F. M HASNI, Bohr radius for subordinating families of analytic functions and bounded harmonic mappings, J. Math. Anal. Appl. 420 (2014), 124-136. \",\"[5] L.AIZENBERG, Multidimensional analogues of Bohr's theorem on power series, Proc. Amer. Math. Soc. 128 (2000), 1147-1155. \",\"[6] L. AIZENBERG, A. AYTUNA and P. DJAKOV, Generalization of theorem on Bohr for bases in spaces of holomorphic functions of several complex variables, J. Math. Anal. Appl. 258 (2001), 429-447. \",\"[7] L.AIZENBERG, Generalization of results about the Bohr radius for power series, Stud. Math. 180 (2007), 161-168. \",\"[8] R. M. ALI, R.W. BARNARD and A.Yu. SOLYNIN, A note on Bohr's phenomenon for power series, J. Math. Anal. Appl. 449 (2017), 154-167. \",\"[9] R. M. ALI, N. K.JAIN and V. RAVICHANDRAN, Bohr radius for classes of analytic functions, Results Math. 74 (2019). \",\"[10] S. A. ALKHALEEFAH, I.R. KAYUMOV and S. PONNUSAMY, On the Bohr inequality with a fixed zero coefficient, Proc. Amer. Math. Soc. 147 (2019), 5263-5274. \",\"[11] VASUDEVARAO ALLU and HIMADRI HALDER, Bhor phenomenon for certain subclasses of Harmonic Mappings, arXiv:2006.11622, 2020 \",\"[12] C. BÉNETEAU, A. DAHLNER and D. KHAVINSON, Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4 (2004), 1-19. \"]","page_idx":"13","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[117,112,137,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"14 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[344,111,608,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Vasudevarao Allu and Himadri Halder ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[117,138,835,878]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"[13] B. BHOWMIK and N. DAS, Bohr phenomenon for subordinating families of certain univalent functions, J. Math. Anal. Appl. 462 (2018), 1087-1098. \",\"[14] H.P. BOAS and D. KHAVINSON, Bohr's power series theorem in several variables, Proc. Amer. Math. Soc 125 (1997), 2975-2979. \",\"[15] H.P. BoAs, Majorant Series, J. Korean Math. Soc. 37 (2000), 321-337. \",\"[16] H. BOHR, A theorem concerning power series, Proc. Lond. Math. Soc. s2-13 (1914), 1-5. \",\"[17] S. EVDORIDIS and S. PONNUSAMY, Improved Bohr's inequality for locally univalent harmonic mappings, Indag. Math. (N.S.) 30 (2019), 201-213. \",\"[18] P. L. DUREN, Univalent Functions (Grundlehren der mathematischen Wisseenschaften 259, New York, Berlin, Heidelberg, Tokyo) Springer-Verlag, 1983. \",\"[19] A. W. GOODMAN, On uniformly convex functions, Annl. Polon. Math., 56 (1991), 87-92. \",\"[20] A. W. GOODMAN, On uniformly starlike functions, J. Math. Anal. Appl., 155 (1991), 364-370. \",\"[21] I. GRAHAM and G. KOHR, Geometric Function Theory in One and Higher Dimensions, Monographs and Textbooks in Pure and Applied Mathematics, 255, Marcel Dekker, Inc., New York, 2003. \",\"[22] R. KARGAR, A. EBADIAN and J. SOKOL, On Both lemniscate and starlike functions, *Anal. Math. Phys.* 9 (2019), 143-154. \",\"[23] I.R. KAYUMOV, S. PONNUSAMY and N. SHAKIROV, Bohr radius for locally unovalent harmonic mappings, Math. Nachr. 291 (2018), 1757-1768. \",\"[24] I.R. KAYUMOV and S. PONNUSAMY, Bohr's inequalities for the analytic functions with lacunary series and harmonic functions, J. Math. Anal. Appl. 465 (2018), 857-871. \",\"[25] I.R. KAYUMOV and S. PONNUSAMY, On a powered Bohr inequality, Ann. Acad. Sci. Fenn. Math. 44 (2019), 301-310. \",\"[26] A. KAYUMOVA, I. R. KAYUMOV and S. PONNUSAMY, Bohr's inequality for harmonic mappings and beyond, Mathematics and Computing, 245-256, Commun. Comput. Inf. Sci., 834, Springer, Singapore, 2018. \",\"[27] K.KHATTER, V. RAVICHANDRAN and S. SIVAPRASAD KUMAR, Starlike functions associated with exponential function and the lemniscate of Bernoulli, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113 (2019), 233-253. \",\"[28] S. KUMAR and V. RAVICHANDRAN, A subclass of starlike functions associated with a rational function, Southeast Asian Bull. Math. 40 (2016), 199-212. \",\"[29] S. SIVAPRASAD KUMAR and S. BANGA, On a special type of Non-Ma-Minda function, arXiv:2006.02111v1, 2020. \",\"[30] Z. LIU and S. PONNUSAMY, Bohr radius for subordination and $k$ -quasiconformal harmonic mappings, Bull. Malays. Math. Sci. Soc. 42 (2019), 2151-2168. \",\"[31] A. LYZZAIL, On a conjecture of M. S. Robertson, Proc. Amer. Math. Soc. 91 (1984), 108-110. \",\"[32] W.C. MA and D. MINDA, A unified treatment of some special classes of univalent functions, in Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157-169, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge. \",\"[33] W.C. Ma and D. MINDA, Uniformly convex functions, Ann. Polon. Math. 57 (1992), 165-175. \",\"[34] W.C. MA and D. MINDA, Uniformly convex functions II, Ann. Polon. Math. 58 (1993), 275-285. \",\"[35] V. S. MAsIH and S. KANAS, Subclasses of Starlike and Convex functions associated with associated with the LimaA $\\\\check{\\\\Lambda}$ gon Domain, Symmetry 12 (2020). \",\"[36] R. MENDIRATTA, S. NAGPAL and V. RAVICHANDRAN, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc. 38 (2015), 365-386. \",\"[37] VERN I. PAULSEN, GELU POPESCU and DINESH SINGH, On Bohr's inequality, Proc. Lond. Math. Soc. s3-85 (2002), 493-512. \",\"[38] S. PONNUSAMY, A. VASUDEVARAO and M. VUORINEN, Region of variability for spirallike functions with respect to a boundary point, Colloq. Math. 116 (2009), 31-46. \"]","page_idx":"14","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[228,111,724,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bohr radius for certain classes of starlike and convex univalent functions ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[818,112,835,122]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"15 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[117,138,836,394]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"[39] R. K.RAINA and J. SOKOL, Some properties related to a certain class of starlike functions, C. R. Math. Acad. Sci. Paris 353 (2015), 973-978. \",\"[40] M. S. Robertson, Univalent functions with respect to a boundary point, J. Math. Anal. Appl. 81 (1981), 327-345. \",\"[41] F. RONNING, On starlike functions associated with parabolic regions, Anna. Univ. Mariae Curie-Sklodowska Sect. A, 45 (1991), 117-122. \",\"[42] F. RONNING, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), 189-196. \",\"[43] K. SHARMA, N. K. JAIN and V. RAVICHANDRAN, Starlike functions associated with cardiod, Afr. Mat. 27 (2016), 923-939. \",\"[44] S. SIDON, Uber einen satz von Hernn Bohr, Math. Zeit. 26 (1927), 731-732. \",\"[45] H. SILVERMAN and E.M. SILVIA, Subclasses of univalent functions starlike with respect to a boundary point, Houston J. Math. 16 (1990), 289-299. \",\"[46] DEREK K. THOMAS, NIKOLA TUNESKI and ALLU VASUDEVARAO, Univalent functions. A primer, De Gruyter Studies in Mathematics, 69. De Gruyter, Berlin, 2018. \",\"[47] M. TOMIC, Sur un theoreme de H. Bohr, Math. Scand. 11 (1962), 103-106. \",\"[48] L. A. WANI and A. SWAMINATHAN, Starlike and convex functions associated with nephroid domain, Bull. Malays. Math. Sci. Soc. doi:10.1007/s40840-020-00935-6. 2020 \"]","page_idx":"15","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[117,407,836,434]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"VASUDEVARAO ALLU, SCHOOL OF BASIC SCIENCE, INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR, BHUBANESWAR-752050, ODISHA, INDIA. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[139,435,421,448]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E-mail address: avrao@iitbbs.ac.in ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,459,889,486]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"HIMADRI HALDER, SCHOOL OF BASIC SCIENCE, INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR, BHUBANESWAR-752050, ODISHA, INDIA. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[139,488,413,500]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E-mail address: hh11@iitbbs.ac.in ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[117,112,137,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"16 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[344,111,608,123]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Vasudevarao Allu and Himadri Halder ","text_format":null,"text_level":null,"type":"header"}],"string":"[\n {\n \"bbox\": \"[137,173,815,207]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"BOHR RADIUS FOR CERTAIN CLASSES OF STARLIKE AND CONVEX UNIVALENT FUNCTIONS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[280,228,672,242]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"VASUDEVARAO ALLU AND HIMADRI HALDER \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[174,264,793,307]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ABSTRACT. We say that a class $\\\\mathcal{F}$ consisting of analytic functions $f(z) = \\\\sum_{n=0}^{\\\\infty}a_{n}z^{n}$ in the unit disk $\\\\mathbb{D} := \\\\{z\\\\in \\\\mathbb{C}:|z| < 1\\\\}$ satisfies a Bohr phenomenon if there exists $r_f\\\\in (0,1)$ such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[374,308,579,343]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\sum_ {n = 1} ^ {\\\\infty} \\\\left| a _ {n} z ^ {n} \\\\right| \\\\leq d (f (0), \\\\partial f (\\\\mathbb {D}))\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[174,344,779,416]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for every function $f \\\\in \\\\mathcal{F}$ and $|z| = r \\\\leq r_f$ , where $d$ is the Euclidean distance. The largest radius $r_f$ is the Bohr radius for the class $\\\\mathcal{F}$ . In this paper, we establish the Bohr phenomenon for the classes consisting of Ma-Minda type starlike functions and Ma-Minda type convex functions as well as for the class of starlike functions with respect to a boundary point. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[297,458,655,472]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. INTRODUCTION AND PRELIMINARIES \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,480,836,516]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $\\\\mathcal{H}$ denote the class of all analytic functions $f$ in the unit disk $\\\\mathbb{D} := \\\\{z \\\\in \\\\mathbb{C} : |z| < 1\\\\}$ . Each function $f \\\\in \\\\mathcal{H}$ has the following power series representation \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,521,549,562]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nf (z) = \\\\sum_ {n = 0} ^ {\\\\infty} a _ {n} z ^ {n}. \\\\tag {1.1}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,568,836,603]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For $f \\\\in \\\\mathcal{H}$ , the majorant series is defined by $M_{f}(r) := \\\\sum_{n=0}^{\\\\infty} |a_{n}| r^{n}$ for $|z| = r < 1$ . In 1914, Harald Bohr [16] obtained the following remarkable result. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,609,794,628]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem A. Let $f \\\\in \\\\mathcal{H}$ be given by (1.1) and $|f(z)| < 1$ for all $z \\\\in \\\\mathbb{D}$ . Then \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,633,581,675]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nM _ {f} (r) := \\\\sum_ {n = 0} ^ {\\\\infty} | a _ {n} | r ^ {n} \\\\leq 1 \\\\tag {1.2}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,680,838,714]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for all $z \\\\in \\\\mathbb{D}$ with $|z| = r \\\\leq 1/3$ . The constant $1/3$ , called the Bohr radius, cannot be improved. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,720,840,823]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Harald Bohr initially obtained the inequality (1.2) for $r \\\\leq 1/6$ . Later, Weiner, Riesz and Schur have independently established the inequality (1.2) for $r \\\\leq 1/3$ and have shown that the constant $1/3$ cannot be improved (see [37, 44, 47]). The inequality (1.2) is popularly known as Bohr's inequality. The Bohr's inequality has been extended to several complex variables by finding the multidimensional Bohr radius (see [5, 6, 7, 14, 15]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[21,267,60,711]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"arXiv:2006.15299v1 [math.CV] 27 Jun 2020 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"aside_text\"\n },\n {\n \"bbox\": \"[139,833,591,847]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"File: Himadri-Vasu-27-june-2020-1.20-a.m.tex, printed: 2020-6-30, 0.52 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[139,847,678,861]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1991 Mathematics Subject Classification. Primary 30C45, 30C50, 30C80. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[117,861,836,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Key words and phrases. Analytic, univalent, starlike, convex, uniformly starlike, uniformly convex functions; coefficient estimate, growth estimate; majorant; Bohr radius. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[470,923,482,934]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[134,137,774,154]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Bohr inequality (1.2) can be written in the following equivalent form \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,163,690,209]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nd \\\\left(\\\\sum_ {n = 0} ^ {\\\\infty} \\\\left| a _ {n} z ^ {n} \\\\right|, \\\\left| a _ {0} \\\\right|\\\\right) = \\\\sum_ {n = 1} ^ {\\\\infty} \\\\left| a _ {n} z ^ {n} \\\\right| \\\\leq d (f (0), \\\\partial f (\\\\mathbb {D})) \\\\tag {1.3}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,217,838,334]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for $|z| = r \\\\leq 1/3$ , where $d$ is the Euclidean distance. We say that $\\\\mathcal{H}$ satisfies the Bohr phenomenon if every function $f \\\\in \\\\mathcal{H}$ satisfies the inequality (1.3). The above equivalent form (1.3) makes the notion of the Bohr phenomenon evident for a class $\\\\mathcal{F}$ consisting of analytic functions of the form (1.1) which map the unit disk $\\\\mathbb{D}$ into a given domain $\\\\Omega \\\\subseteq \\\\mathbb{C}$ such that $f(\\\\mathbb{D}) \\\\subseteq \\\\Omega$ . The class $\\\\mathcal{F}$ is said to satisfy the Bohr phenomenon if there exists $r_{\\\\Omega} \\\\in (0,1)$ such that every function $f \\\\in \\\\mathcal{F}$ satisfies the inequality \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,344,690,389]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nd \\\\left(\\\\sum_ {n = 0} ^ {\\\\infty} \\\\left| a _ {n} z ^ {n} \\\\right|, \\\\left| a _ {0} \\\\right|\\\\right) = \\\\sum_ {n = 1} ^ {\\\\infty} \\\\left| a _ {n} z ^ {n} \\\\right| \\\\leq d (f (0), \\\\partial f (\\\\mathbb {D})) \\\\tag {1.4}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,397,838,551]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for all $|z| = r \\\\leq r_{\\\\Omega}$ . The largest radius $r_{\\\\Omega}$ is called the Bohr radius for the class $\\\\mathcal{F}$ . The question arises as to whether or not the largest radius $r_{\\\\Omega}$ is changed with respect to different types of domains. The answer is affirmative. For any proper simply connected domain $\\\\Omega$ , Abu-Muhanna [1] has proved that the sharp radius is $r_{\\\\Omega} = 3 - 2\\\\sqrt{2}$ for the class $\\\\mathcal{F}$ . For a convex domain $\\\\Omega$ , Aizenberg [7] has shown that $r_{\\\\Omega}$ coincides with the classical Bohr radius $1/3$ . Bohr phenomenon for harmonic mappings has also been extensively studied by several authors (see [4, 11, 17, 23, 24, 26, 30]). For more information about Bohr phenomenon stated above and further related intriguing aspects, we refer the reader to the articles (see [2, 3, 8, 9, 12, 25]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,564,836,599]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $\\\\mathcal{A}$ denote the subclass of $\\\\mathcal{H}$ consisting of functions normalized by $f(0) = f'(0) - 1 = 0$ . Each $f \\\\in \\\\mathcal{A}$ has the following form \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,610,566,652]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nf (z) = z + \\\\sum_ {n = 2} ^ {\\\\infty} a _ {n} z ^ {n}. \\\\tag {1.5}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,660,838,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $S \\\\subseteq \\\\mathcal{A}$ be the family of univalent (i.e. one-to-one) functions. Let $\\\\mathcal{S}^*(\\\\alpha)$ and $\\\\mathcal{C}(\\\\alpha)$ be the subclasses of $S$ consisting of functions starlike of order $\\\\alpha$ ( $0 \\\\leq \\\\alpha < 1$ ) and convex functions of order $\\\\alpha$ ( $0 \\\\leq \\\\alpha < 1$ ) respectively. A function $f \\\\in S$ belongs to $\\\\mathcal{S}^*(\\\\alpha)$ (respectively $\\\\mathcal{C}(\\\\alpha)$ ) if $\\\\operatorname{Re}(zf'(z)/f(z)) > \\\\alpha$ for $z \\\\in \\\\mathbb{D}$ ( $\\\\operatorname{Re}(1 + zf''(z)/f'(z)) > \\\\alpha$ for $z \\\\in \\\\mathbb{D}$ respectively). It is known that $f \\\\in \\\\mathcal{C}(\\\\alpha)$ if, and only if, $zf' \\\\in \\\\mathcal{S}^*(\\\\alpha)$ . The classes $\\\\mathcal{S}^* := \\\\mathcal{S}^*(0)$ and $\\\\mathcal{C} := \\\\mathcal{C}(0)$ are the family of starlike and convex functions in $\\\\mathbb{D}$ respectively. For more properties of starlike and convex functions, we refer the reader to [18, 21, 46]. For two analytic functions $f$ and $g$ in $\\\\mathbb{D}$ , we say that $f$ is subordinate to $g$ , written $f \\\\prec g$ in $\\\\mathbb{D}$ , if there exists a schwarzian function $\\\\omega: \\\\mathbb{D} \\\\to \\\\mathbb{D}$ with $\\\\omega(0) = 0$ such that $f(z) = g(\\\\omega(z))$ for $z \\\\in \\\\mathbb{D}$ . In particular, if $g$ is univalent in $\\\\mathbb{D}$ , then $f \\\\prec g$ if, and only if, $f(0) = g(0)$ and $f(\\\\mathbb{D}) \\\\subseteq g(\\\\mathbb{D})$ . In 1992, Ma-Minda [32] introduced the function classes $\\\\mathcal{S}^*(\\\\phi)$ and $\\\\mathcal{C}(\\\\phi)$ by unifying several earlier results on subordination. For the brevity, we write. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,112,131,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[344,111,608,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Vasudevarao Allu and Himadri Halder \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[114,136,836,171]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 1.1. Let $S^{*}(\\\\phi)$ and $\\\\mathcal{C}(\\\\phi)$ denote the subclasses of $S$ consisting of functions characterized by \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[302,176,653,214]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\frac {z f ^ {\\\\prime} (z)}{f (z)} \\\\prec \\\\phi (z) \\\\quad \\\\mathrm {a n d} \\\\quad 1 + \\\\frac {z f ^ {\\\\prime \\\\prime} (z)}{f ^ {\\\\prime} (z)} \\\\prec \\\\phi (z)\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,219,838,287]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"respectively, where $\\\\phi : \\\\mathbb{D} \\\\to \\\\mathbb{D}$ is called Ma-Minda function which is analytic and univalent in $\\\\mathbb{D}$ such that $\\\\phi(\\\\mathbb{D})$ has positive real part, symmetric with respect to the real axis, starlike with respect to $\\\\phi(0) = 1$ and $\\\\phi'(0) > 0$ . Let the Taylor series expansion of $\\\\phi(z)$ be of the form \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,294,611,337]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\phi (z) = 1 + \\\\sum_ {n = 1} ^ {\\\\infty} B _ {n} z ^ {n} \\\\quad (B _ {1} > 0) \\\\tag {1.6}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,342,697,359]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for $z \\\\in \\\\mathbb{D}$ . Let $\\\\mathcal{M}$ denote the class of all Ma-Minda functions in $\\\\mathbb{D}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,368,838,453]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We call $\\\\mathcal{S}^*(\\\\phi)$ and $\\\\mathcal{C}(\\\\phi)$ are the Ma-Minda type starlike and Ma-Minda type convex classes associated with $\\\\phi$ . It is known that $f \\\\in \\\\mathcal{C}(\\\\phi)$ if, and only if, $zf' \\\\in \\\\mathcal{S}^*(\\\\phi)$ . For $\\\\phi(z) = (1 + z)/(1 - z)$ we have $\\\\mathcal{C}(\\\\phi) \\\\coloneqq \\\\mathcal{C}$ and $\\\\mathcal{S}^*(\\\\phi) \\\\coloneqq \\\\mathcal{S}^*$ . Evidently, for every such $\\\\phi$ described in Definition 1.1, $\\\\mathcal{S}^*(\\\\phi)$ and $\\\\mathcal{C}(\\\\phi)$ are always subclasses of the classes $\\\\mathcal{S}^*$ and $\\\\mathcal{C}(\\\\phi)$ respectively. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,465,853,583]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is worth noting that for particular choices of $\\\\phi$ , the classes $S^{*}(\\\\phi)$ and $\\\\mathcal{C}(\\\\phi)$ generate several important subclasses of starlike and convex functions, respectively. For instance, $S^{*}(\\\\alpha) := S^{*}\\\\left((1 + (1 - 2\\\\alpha)) / (1 - z)\\\\right)$ and $\\\\mathcal{C}(\\\\alpha) := \\\\mathcal{C}\\\\left((1 + (1 - 2\\\\alpha)) / (1 - z)\\\\right)$ , the Janowski starlike class $S^{*}[A,B] := S^{*}((1 + Az) / (1 + Bz))$ and Janowski convex class $\\\\mathcal{C}[A,B] := \\\\mathcal{C}((1 + Az) / (1 + Bz))$ , where $-1 \\\\leq B < A \\\\leq 1$ . If $\\\\phi(z) = ((1 + z) / (1 - z))^{\\\\alpha}$ for $0 < \\\\alpha \\\\leq 1$ , then $\\\\mathcal{C}(\\\\alpha)$ and $S^{*}(\\\\alpha)$ are the classes of strongly convex and strongly starlike functions of order $\\\\alpha$ (see [33]). For \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,589,608,629]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\phi (z) = 1 + \\\\frac {2}{\\\\pi^ {2}} \\\\left(\\\\log \\\\frac {1 + \\\\sqrt {z}}{1 - \\\\sqrt {z}}\\\\right) ^ {2}, \\\\tag {1.7}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,635,838,802]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"the class $\\\\mathcal{C}(\\\\phi)$ ( $S^{*}(\\\\phi)$ respectively) is the family $UCV$ ( $UST$ respectively) of normalized uniformly convex (starlike respectively) functions introduced by Goodman (see [19, 20, 41, 42]). Ma and Minda (see [33, 34]) have studied extensively the class $UCV$ . For $0 \\\\leq \\\\alpha < 1$ , Khatter et al. [27] introduced $S_{\\\\alpha,e}^{*} \\\\coloneqq S^{*}(\\\\alpha + (1 - \\\\alpha)e^{z})$ . For $\\\\alpha = 0$ , $S_{\\\\alpha,e}^{*}$ reduces to $S_{e}^{*} \\\\coloneqq S^{*}(e^{z})$ (see [36]). When $\\\\phi(z) = 1 + 4z/3 + 2z^{2}/3$ , the unit disk $\\\\mathbb{D}$ is mapped onto a domain bounded by a cardiod and corresponding Ma-Minda starlike class $S^{*}(\\\\phi)$ reduces to the class $S_{C}^{*}$ (see [43]). For $\\\\phi(z) = 1 + z/(1 - \\\\alpha z^{2})$ , Kargar et al. [22] have introduced the class $\\\\mathcal{BS}^{*}(\\\\alpha) \\\\coloneqq S^{*}(1 + z/(1 - \\\\alpha z^{2}))$ , which is associated with the Booth lemniscate. In 2016, Kumar and Ravichandran [28] considered the class $S_{R}^{*} \\\\coloneqq S^{*}(\\\\phi_{0})$ , where $\\\\phi_{0}$ is the rational function \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,811,648,846]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\phi_ {0} (z) = 1 + \\\\frac {z}{k} \\\\left(\\\\frac {k + z}{k - z}\\\\right), \\\\quad k = \\\\sqrt {2} + 1. \\\\tag {1.8}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,853,836,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For $\\\\phi(z) = (1 + sz)^2$ with $0 < s \\\\leq 1/\\\\sqrt{2}$ , the class $\\\\mathcal{S}^*(\\\\phi)$ reduces to $\\\\mathcal{ST}_L(s) := \\\\mathcal{S}^*((1 + sz)^2)$ (see [35]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[228,111,724,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bohr radius for certain classes of starlike and convex univalent functions \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[823,112,836,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[134,137,689,154]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ma and Minda have defined the following functions $h, k \\\\in S$ by \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,157,653,195]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\frac {z h ^ {\\\\prime} (z)}{h (z)} = \\\\phi (z) \\\\quad \\\\text {a n d} \\\\quad 1 + \\\\frac {z k ^ {\\\\prime \\\\prime} (z)}{k ^ {\\\\prime} (z)} = \\\\phi (z). \\\\tag {1.9}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,199,836,266]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Here $h$ and $k$ play the role of Koebe function for the classes $S^{*}(\\\\phi)$ and $\\\\mathcal{C}(\\\\phi)$ , respectively. Clearly, $h$ and $k$ belong to $S^{*}(\\\\phi)$ and $\\\\mathcal{C}(\\\\phi)$ respectively and $zk'(z) = h(z)$ . The subordination and growth estimate for the class $S^{*}(\\\\phi)$ have been obtained by Ma and Minda (see [32]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,273,836,309]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 1.10. [32] Let $f \\\\in S^{*}(\\\\phi)$ . Then $zf'(z)/f(z) \\\\prec zh'(z)/h(z)$ and $f(z)/z \\\\prec h(z)/z$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,315,653,333]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 1.11. [32] Assume $f \\\\in S^{*}(\\\\phi)$ and $|z| = r < 1$ . Then \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,339,588,357]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n- h (- r) \\\\leq | f (z) | \\\\leq h (r). \\\\tag {1.12}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,362,680,380]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Equality holds for some $z \\\\neq 0$ if, and only if, $f$ is a rotation of $h$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,387,836,454]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It has been pointed out in [32] that $-h(-r)$ is increasing in $(0,1)$ and bounded by 1 because each $f\\\\in S^{*}(\\\\phi)$ is normalized by $f(0) = f^{\\\\prime}(0) - 1 = 0$ . Therefore, $\\\\lim_{r\\\\to 1} - h(-r)$ exists and denote it by $-h(-1)$ . The following subordination and growth theorem for the class $\\\\mathcal{C}(\\\\phi)$ have been established in [32]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,461,836,497]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 1.13. [32] Let $f \\\\in \\\\mathcal{C}(\\\\phi)$ . Then $zf''(z) / f'(z) \\\\prec zk''(z) / k'(z)$ and $f'(z) \\\\prec zk'(z)$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,504,642,522]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 1.14. [32] Assume $f \\\\in \\\\mathcal{C}(\\\\phi)$ and $|z| = r < 1$ . Then \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,527,588,545]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n- k (- r) \\\\leq | f (z) | \\\\leq k (r). \\\\tag {1.15}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,549,680,568]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Equality holds for some $z \\\\neq 0$ if, and only if, $f$ is a rotation of $k$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,574,836,628]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is justified in [32] that $-k(-r)$ is increasing in $(0,1)$ and bounded by 1 because each $f\\\\in \\\\mathcal{C}(\\\\phi)$ is normalized by $f(0) = f'(0) - 1 = 0$ . Therefore, $\\\\lim_{r\\\\to 1} - k(-r)$ exists and denote it by $-k(-1)$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,644,836,730]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ma and Minda [32] have introduced the analytic and univalent function $\\\\phi$ with certain conditions, one of which is $\\\\phi'(0) > 0$ . Recently, Kumar and Banga [29] have considered a non-Ma-Minda function $\\\\Phi$ , obtained by altering only one condition, namely $\\\\Phi'(0) < 0$ , which is merely a rotation. Such a function $\\\\Phi$ is named (see [29]) a non-Ma-Minda of type- $\\\\mathbf{A}$ , here $\\\\mathbf{A}$ signifies the condition $\\\\Phi'(0) < 0$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,737,836,805]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 1.2. An analytic and univalent function $\\\\Phi$ defined in the unit disk $\\\\mathbb{D}$ is said to be a non-Ma-Minda of type $\\\\mathbf{A}$ if it has positive real part in $\\\\mathbb{D}$ , $\\\\Phi(\\\\mathbb{D})$ is symmetric with respect to the real axis, starlike with respect to $\\\\Phi(0) = 1$ and $\\\\Phi'(0) < 0$ . Further, it has a power series expansion of the form: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,809,613,852]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\Phi (z) = 1 + \\\\sum_ {n = 1} ^ {\\\\infty} C _ {n} z ^ {n} \\\\quad \\\\left(C _ {1} < 0\\\\right). \\\\tag {1.16}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,856,836,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The class of all such functions of non-Ma-Minda of type $\\\\mathbf{A}$ , we shall denote it by $\\\\mathcal{M}_A$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,112,131,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[344,111,608,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Vasudevarao Allu and Himadri Halder \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[114,136,836,220]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On the similar lines of Definition 1.1, Kumar and Banga has defined the classes $S^{*}(\\\\Phi)$ and $\\\\mathcal{C}(\\\\Phi)$ . It is worth noting that by mere replacing $z$ by $-z$ , i.e. $\\\\Phi(z) = \\\\phi(-z)$ , each function in $\\\\mathcal{M}_A$ reduces to its Ma-Minda counter part. For instance, $\\\\sqrt{1 + z}$ and $1 - \\\\log(1 - z)$ belong to the class $\\\\mathcal{M}$ whereas $\\\\sqrt{1 - z}$ and $1 - \\\\log(1 + z)$ belong to $\\\\mathcal{M}_A$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,237,838,323]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Observe that all the coefficients of $\\\\phi \\\\in \\\\mathcal{M}$ are not necessary to be positive. For instance, some of the coefficients of $\\\\phi(z) = z + \\\\sqrt{1 + z^2} = 1 + z + \\\\frac{1}{2} - z^4 / 8 + \\\\dots$ and $\\\\phi(z) = 1 + z - z^3 / 3$ are negative. The class $S^*(z + \\\\sqrt{1 + z^2})$ has been studied by Raina and Sokol [39]. Wani and Swaminathan [48] have studied the class $S_{Ne}^* := S^*(1 + z - z^3 / 3)$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,353,803,372]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In 1990, Silverman and Silvia [45] introduced the following classes $\\\\mathcal{G}_{\\\\alpha}^{*}$ and $\\\\mathcal{G}_{\\\\alpha}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,380,836,413]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 1.3. Let $\\\\mathcal{G}_{\\\\alpha}^{*}$ denote the class of functions $G$ analytic in $\\\\mathbb{D}$ that satisfy the following conditions: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[139,416,833,484]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"(i) $G$ is normalized by $G(0) = 1$ and $G(1) = \\\\\\\\lim_{r\\\\\\\\to -1} = 0$ \\\",\\\"(ii) $G(\\\\\\\\mathbb{D})$ lies in a sector with aperture $2(1 - \\\\\\\\alpha)\\\\\\\\pi$ and vertex at the origin, and \\\",\\\"(iii) $G$ maps $\\\\\\\\mathbb{D}$ univalently onto a domain that is starlike with respect to $G(1)$ . In addition, let the constant function 1 belong to $\\\\\\\\mathcal{G}_{\\\\\\\\alpha}^{*}$ . \\\"]\",\n \"page_idx\": \"4\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[115,491,836,526]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 1.4. For $\\\\alpha$ , let $\\\\mathcal{G}_{\\\\alpha}$ denote the class of functions, $G$ with $G(z) = 1 + \\\\sum_{n=1}^{\\\\infty} d_n z^n$ , analytic and non-vanishing in $\\\\mathbb{D}$ which satisfy \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,530,631,568]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname {R e} \\\\left(\\\\frac {z G ^ {\\\\prime} (z)}{G (z)} + \\\\frac {(1 - \\\\alpha) (1 + z)}{1 - z}\\\\right) > 0 \\\\tag {1.17}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,572,205,587]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for $z\\\\in \\\\mathbb{D}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,597,836,666]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For $\\\\alpha = 1/2$ , Robertson [40] has studied extensively the classes $\\\\mathcal{G}_{1/2}^*$ and $\\\\mathcal{G}_{1/2}$ and has conjectured that they are equal. This conjecture was proved by Lyzzaik in 1984 (see[31]). In more general, for all $\\\\alpha$ , $0 \\\\leq \\\\alpha < 1$ , Silverman and Silvia have shown that $\\\\mathcal{G}_{\\\\alpha}^{*} = \\\\mathcal{G}_{\\\\alpha}$ by proving closely related property of $\\\\mathcal{G}_{\\\\alpha}$ with $\\\\mathcal{S}^{*}(\\\\alpha)$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,673,836,706]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 1.18. [45] A function $G$ is in $\\\\mathcal{G}_{\\\\alpha}$ if, and only if, there exists a function $s \\\\in S^{*}(\\\\alpha)$ such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[364,707,588,739]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nG (z) = (1 - z) ^ {2 (1 - \\\\alpha)} \\\\frac {s (z)}{z}.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,745,836,796]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The regions of variability for the class $\\\\mathcal{G}_{\\\\alpha}$ has been studied extensively by Ponnusamy et al. [38]. The following growth theorem for the class $\\\\mathcal{G}_{\\\\alpha}$ has been established by Silverman and Silvia [45]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,804,416,821]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 1.19. [45] If $g \\\\in \\\\mathcal{G}_{\\\\alpha}$ then \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,826,653,866]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left(\\\\frac {1 - r}{1 + r}\\\\right) ^ {2 (1 - \\\\alpha)} \\\\leq | g (z) | \\\\leq \\\\left(\\\\frac {1 + r}{1 - r}\\\\right) ^ {2 (1 - \\\\alpha)} \\\\tag {1.20}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[112,872,825,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for $|z| = r$ . Equality holds for $g(z) = ((1 - z) / (1 + z))^{2(1 - \\\\alpha)}$ at $z = r$ and $z = -r$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[228,111,724,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bohr radius for certain classes of starlike and convex univalent functions \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[823,112,836,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[115,137,836,172]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In 2018, Bhowmik and Das [13] proved an interesting result for subordination classes. Let $f$ and $g$ be two analytic functions in $\\\\mathbb{D}$ such that $g \\\\prec f$ . Let \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,176,547,218]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\ng (z) = \\\\sum_ {n = 0} ^ {\\\\infty} c _ {n} z ^ {n}. \\\\tag {1.21}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,225,838,260]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 1.22. [13] Let $f$ and $g$ be analytic in $\\\\mathbb{D}$ with Taylor expansions (1.1) and (1.21) respectively and $g \\\\prec f$ , then \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,264,574,307]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\sum_ {n = 0} ^ {\\\\infty} \\\\left| c _ {n} \\\\right| r ^ {n} \\\\leq \\\\sum_ {n = 0} ^ {\\\\infty} \\\\left| a _ {n} \\\\right| r ^ {n} \\\\tag {1.23}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,312,267,331]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for $z| = r\\\\leq 1 / 3$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,338,836,372]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The following coefficients bounds for the class $S^{*}(\\\\alpha)$ are required to obtain the Bohr radius for the class $\\\\mathcal{G}_{\\\\alpha}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,380,636,397]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 1.24. [21] Let $f \\\\in S^{*}(\\\\alpha)$ be given by (1.15). Then \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[299,402,653,447]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n| a _ {n} | \\\\leq \\\\frac {1}{(n - 1) !} \\\\prod_ {k = 0} ^ {n} (k - 2 \\\\alpha) \\\\quad f o r \\\\quad n \\\\geq 2.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,451,833,471]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The equalities in above estimates are attained for $f(z) = z / (1 - z)^{2(1 - \\\\alpha)}$ for $z \\\\in \\\\mathbb{D}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,477,836,562]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this paper, we establish the Bohr phenomenon for the classes $S^{*}(\\\\phi)$ and $\\\\mathcal{C}(\\\\phi)$ , where all the coefficients of associated Ma-Minda function $\\\\phi$ are positive such that $\\\\phi \\\\in H^2$ , the Hardy class of analytic functions in $\\\\mathbb{D}$ . As a consequence, we obtain several important corollaries for particular choices of $\\\\phi$ . We also obtain the Bohr radius for the class $\\\\mathcal{G}_{\\\\alpha}$ , starlike functions with respect to the boundary point. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[393,574,559,590]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. MAIN RESULTS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,598,811,617]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Using Lemmas 1.10, 1.11 and 1.22, we obtain Bohr radius for the class $S^{*}(\\\\phi)$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,623,840,659]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2.1. Let $f \\\\in S^{*}(\\\\phi)$ be given by (1.5) and $\\\\phi(z)$ be given by (1.6) with all $B_{n} > 0$ , $n \\\\geq 1$ such that $\\\\phi \\\\in H^{2}$ , the Hardy class of analytic functions in $\\\\mathbb{D}$ . Then \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,663,621,705]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n| z | + \\\\sum_ {n = 0} ^ {\\\\infty} \\\\left| a _ {n} \\\\right| | z | ^ {n} \\\\leq d (f (0), \\\\partial f (\\\\mathbb {D})) \\\\tag {2.2}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,710,700,729]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for $|z| = r \\\\leq \\\\min \\\\{r_f, 1/3\\\\}$ , where $r_f$ is the smallest positive root of \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,733,552,750]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh (r) + h (- 1) = 0 \\\\tag {2.3}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,756,436,774]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"in $(0,1)$ and $h(z)$ is defined in (1.9). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,781,836,850]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 2.1. Since $\\\\Phi \\\\in \\\\mathcal{M}_A$ is obtained from $\\\\phi \\\\in \\\\mathcal{M}$ by mere replacing $z$ by $-z$ , the image of $\\\\mathbb{D}$ under the functions $\\\\Phi$ and $\\\\phi$ are identical. Therefore we have $S^{*}(\\\\Phi) = S^{*}(\\\\phi)$ and hence the Bohr radius for the class $S^{*}(\\\\Phi)$ is same as that of $S^{*}(\\\\phi)$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,856,836,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As a consequence of Theorem 2.1, we obtain the following corollaries for particular choices of $\\\\phi$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,112,131,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[344,111,608,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Vasudevarao Allu and Himadri Halder \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[115,136,840,190]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 2.4. For $\\\\phi(z) = \\\\alpha + (1 - \\\\alpha)e^{z}$ , we have $S_{\\\\alpha,e}^{*} \\\\coloneqq S^{*}(\\\\alpha + (1 - \\\\alpha)e^{z})$ . Let $f \\\\in S_{\\\\alpha,e}^{*}$ be given by (1.5) with $0 \\\\leq \\\\alpha < 0.05284$ . Then the inequality (2.2) is satisfied for $|z| = r \\\\leq r_f$ , where $0 < r_f < 1/3$ . The radius $r_f$ is the best possible. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,198,836,233]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In particular, for $\\\\alpha = 0$ in Corollary 2.4, we obtain the sharp Bohr radius for the class $S_{e}^{*}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,242,842,293]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 2.5. For $\\\\phi(z) = 1 + 4z/3 + 2z^2/3$ , $S^*(\\\\phi)$ reduces to $S_C^*$ . For $f \\\\in S_C^*$ of the form (1.5), the inequality (2.2) is satisfied for $|z| = r \\\\leq r_f$ , where $0 < r_f < 1/3$ . The radius $r_f$ is the best possible. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,302,875,356]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 2.6. Let $\\\\phi$ be the rational function such that $\\\\phi(z) = 1 + (z / k) \\\\left( \\\\frac{(k + z)}{k - z} \\\\right)$ , where $k = \\\\sqrt{2} + 1$ . Then $\\\\mathcal{S}^*(\\\\phi)$ reduces to the class $\\\\mathbb{S}_R^*$ . Then the inequality (2.2) is satisfied for $|z| = r \\\\leq 1/3$ for the class $\\\\mathbb{S}_R^*$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,364,840,448]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 2.7. For $\\\\phi(z) = (1 + Az)/(1 + Bz)$ , the class $S^{*}(\\\\phi)$ reduces to the Janowski starlike class $S^{*}[A, B]$ . Let $f \\\\in S^{*}[A, B]$ be given by (1.5) with $-1 \\\\leq B < (1 - 3^{k})/(1 + 3^{k}) < 0$ and $0 \\\\leq A \\\\leq 1$ , where $k = B/(B - A)$ . Then the inequality (2.2) is satisfied for $|z| = r \\\\leq r_{f}$ , where $0 < r_{f} < 1/3$ . The radius $r_{f}$ is the best possible. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,457,840,509]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 2.8. For $\\\\phi(z) = 1 + z / (1 - \\\\alpha z^2)$ , the class $S^*(\\\\phi)$ reduces to $\\\\mathcal{BS}^*(\\\\alpha)$ . Let $f \\\\in \\\\mathcal{BS}^*(\\\\alpha)$ be given by (1.5) with $0 \\\\leq \\\\alpha < 1$ . Then the inequality (2.2) is satisfied for $|z| = r \\\\leq r_f$ , where $0 < r_f < 1/3$ . The constant $r_f$ cannot be improved. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,517,838,571]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 2.9. Let $\\\\phi = (1 + sz)^{2}$ , then $S^{*}(\\\\phi)$ reduces to the class $\\\\mathcal{ST}_L(s)$ . Let $f\\\\in \\\\mathcal{ST}_L(s)$ be given by (1.5) with $0.444981 < s\\\\leq 1 / \\\\sqrt{2}$ . Then the inequality (2.2) satisfies for $|z| = r\\\\leq r_f$ , where $0 < r_{f} < 1 / 3$ . The radius $r_f$ is the best possible. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,579,836,598]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By using Lemmas 1.14 and 1.22, we establish Bhor phenomenon for the class $\\\\mathcal{C}(\\\\phi)$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,606,838,641]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2.10. Let $f \\\\in \\\\mathcal{C}(\\\\phi)$ be given by (1.5) and $\\\\phi(z)$ be given by (1.6) with all $B_n > 0$ , $n \\\\geq 1$ such that $\\\\phi \\\\in H^2$ , the Hardy class of analytic functions in $\\\\mathbb{D}$ . Then \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,650,621,691]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n| z | + \\\\sum_ {n = 0} ^ {\\\\infty} \\\\left| a _ {n} \\\\right| | z | ^ {n} \\\\leq d (f (0), \\\\partial f (\\\\mathbb {D})) \\\\tag {2.11}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,700,700,719]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for $|z| = r \\\\leq \\\\min \\\\{r_f, 1/3\\\\}$ , where $r_f$ is the smallest positive root of \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,726,552,744]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nk (r) + k (- 1) = 0 \\\\tag {2.12}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,752,436,771]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"in (0,1) and $k(z)$ is defined in (1.9). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,780,556,798]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $G \\\\in \\\\mathcal{G}_{\\\\alpha}$ with the power series representation \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,807,603,847]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nG (z) = 1 + \\\\sum_ {n = 1} ^ {\\\\infty} d _ {n} z ^ {n}, \\\\quad z \\\\in \\\\mathbb {D}. \\\\tag {2.13}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,856,836,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By using Lemmas 1.18, 1.19 and 1.24, we obtain the sharp Bhor radius for the class $\\\\mathcal{G}_{\\\\alpha}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[228,111,724,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bohr radius for certain classes of starlike and convex univalent functions \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[823,112,836,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[115,137,591,154]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2.14. Let $G \\\\in \\\\mathcal{G}_{\\\\alpha}$ be given by (2.13). Then \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,161,603,203]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\sum_ {n = 1} ^ {\\\\infty} \\\\left| d _ {n} \\\\right| | z | ^ {n} \\\\leq d (G (0), \\\\partial G (\\\\mathbb {D})) \\\\tag {2.15}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[112,209,838,244]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for $|z| = r \\\\leq r_f$ , where $r_f = (2^{1/2(1 - \\\\alpha)} - 1) / (2^{1/2(1 - \\\\alpha)} + 1)$ . The radius $r_f$ is the best possible. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[322,260,628,275]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3. PROOF OF THE MAIN RESULTS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,294,806,312]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof of Theorem 2.1. Let $f \\\\in S^{*}(\\\\phi)$ then in view of Lemma 1.11, we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,318,615,336]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n| f (z) | \\\\geq - h (- r) \\\\quad \\\\text {f o r} \\\\quad | z | < 1. \\\\tag {3.1}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,342,719,359]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By taking $\\\\lim \\\\inf$ as $|z| = r\\\\to 1$ on both the sides of (3.1), we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,366,581,391]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\lim _ {| z | \\\\rightarrow 1} | f (z) | \\\\geq - h (- 1). \\\\tag {3.2}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,397,773,416]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Euclidean distance between $f(0)$ and the boundary of $f(\\\\mathbb{D})$ is given by \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,422,640,448]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nd (f (0), \\\\partial f (\\\\mathbb {D})) = \\\\lim _ {| z | \\\\rightarrow 1} \\\\inf | f (z) - f (0) |. \\\\tag {3.3}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,454,521,472]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since $f(0) = 0$ from (3.2) and (3.3), we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,479,589,496]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nd (f (0), \\\\partial f (\\\\mathbb {D})) \\\\geq - h (- 1). \\\\tag {3.4}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,502,457,520]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is known that $\\\\phi \\\\in H^2$ if, and only if, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,526,549,568]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\sum_ {n = 1} ^ {\\\\infty} \\\\left| B _ {n} \\\\right| ^ {2} < + \\\\infty . \\\\tag {3.5}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,574,289,591]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From (3.5) we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,597,643,650]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\sum_ {n = 1} ^ {\\\\infty} \\\\frac {B _ {n}}{n} \\\\leq \\\\sqrt {\\\\sum_ {n = 1} ^ {\\\\infty} | B _ {n} | ^ {2}} \\\\sqrt {\\\\sum_ {n = 1} ^ {\\\\infty} \\\\frac {1}{n ^ {2}}} < + \\\\infty . \\\\tag {3.6}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,656,149,669]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,676,623,718]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh (z) = z + \\\\sum_ {n = 2} ^ {\\\\infty} b _ {n} z ^ {n} \\\\quad \\\\text {f o r} \\\\quad z \\\\in \\\\mathbb {D}. \\\\tag {3.7}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,724,835,742]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In view of (1.9), we obtain $zh'(z) / h(z) = \\\\phi(z)$ and a simple computation shows that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,747,611,801]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh (z) = z \\\\exp \\\\left(\\\\int_ {0} ^ {\\\\tilde {z}} \\\\frac {\\\\phi (t) - 1}{t} d t\\\\right), \\\\tag {3.8}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,806,836,840]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where integration is along the linear segment joing 0 to $z \\\\in \\\\mathbb{D}$ . From (1.9) and (3.8), we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,846,594,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh (z) = z \\\\exp \\\\left(\\\\sum_ {n = 1} ^ {\\\\infty} B _ {n} \\\\frac {z ^ {n}}{n}\\\\right). \\\\tag {3.9}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[117,112,129,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"8 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[344,111,608,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Vasudevarao Allu and Himadri Halder \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[115,136,400,154]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $H:[0,1]\\\\to \\\\mathbb{R}$ be defined by \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[381,163,571,181]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nH (r) = h (r) + h (- 1).\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,188,836,239]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From (3.6), (3.8) and (3.9), it is easy to see that $H$ is continuous in $[0,1]$ and differentiable in $(0,1)$ . Note that $H(0) = h(-1) < 0$ and $H(1) = h(1) + h(-1)$ . Clearly, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[205,245,747,294]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh (1) = \\\\exp \\\\left(\\\\sum_ {n = 1} ^ {\\\\infty} \\\\frac {B _ {n}}{n}\\\\right) \\\\quad \\\\mathrm {a n d} \\\\quad h (- 1) = \\\\exp \\\\left(\\\\sum_ {n = 1} ^ {\\\\infty} \\\\frac {(- 1) ^ {n} B _ {n}}{n}\\\\right).\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,299,324,315]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since $B_{n} > 0$ for $n\\\\geq 1$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[371,318,581,361]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\sum_ {n = 1} ^ {\\\\infty} \\\\frac {B _ {n}}{n} > \\\\sum_ {n = 1} ^ {\\\\infty} \\\\frac {(- 1) ^ {n} B _ {n}}{n}.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,366,208,382]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Therefore, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[226,389,726,437]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh (1) = \\\\exp \\\\left(\\\\sum_ {n = 1} ^ {\\\\infty} \\\\frac {B _ {n}}{n}\\\\right) > \\\\exp \\\\left(\\\\sum_ {n = 1} ^ {\\\\infty} \\\\frac {(- 1) ^ {n} B _ {n}}{n}\\\\right) = - h (- 1)\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,442,838,494]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"which implies that $H(1) > 0$ . Since $H(0) < 0$ and $H(1) > 0$ , by the intermediate value property, $H$ has one real root in $(0,1)$ . Let $r_f$ be the smallest positive root of $H$ in $(0,1)$ . Thus $H(r_f) = 0$ , which is equivalent to \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,503,549,521]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh \\\\left(r _ {f}\\\\right) = - h (- 1). \\\\tag {3.10}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,529,394,546]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From (3.7) and (3.9), we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,554,722,600]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh (z) = z + \\\\sum_ {n = 2} ^ {\\\\infty} b _ {n} z ^ {n} = z \\\\exp \\\\left(\\\\sum_ {n = 1} ^ {\\\\infty} B _ {n} \\\\frac {z ^ {n}}{n}\\\\right) \\\\quad \\\\text {f o r} \\\\quad z \\\\in \\\\mathbb {D}. \\\\tag {3.11}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,606,208,623]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Therefore, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,631,655,761]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} | z | + \\\\sum_ {n = 2} ^ {\\\\infty} \\\\left| b _ {n} \\\\right| | z | ^ {n} = | z | \\\\exp \\\\left(\\\\sum_ {n = 1} ^ {\\\\infty} \\\\left| B _ {n} \\\\right| \\\\frac {| z | ^ {n}}{n}\\\\right) \\\\tag {3.12} \\\\\\\\ = r \\\\exp \\\\left(\\\\sum_ {n = 1} ^ {\\\\infty} B _ {n} \\\\frac {r ^ {n}}{n}\\\\right) \\\\\\\\ = h (r). \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,770,608,788]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In view of Lemma 1.22 and (3.11), for $|z| = r$ , we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,798,695,840]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nr + \\\\sum_ {n = 2} ^ {\\\\infty} \\\\left| a _ {n} \\\\right| r ^ {n} \\\\leq r + \\\\sum_ {n = 2} ^ {\\\\infty} \\\\left| b _ {n} \\\\right| r ^ {n} = h (r) \\\\quad \\\\text {f o r} \\\\quad r \\\\leq \\\\frac {1}{3}. \\\\tag {3.13}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,847,290,864]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For $r \\\\leq r_f$ , we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,872,581,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh (r) \\\\leq h \\\\left(r _ {f}\\\\right) = - h (- 1). \\\\tag {3.14}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[228,111,724,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bohr radius for certain classes of starlike and convex univalent functions \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[823,112,836,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"9 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[115,137,579,154]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Combining (3.4), (3.13) and (3.14), we conclude that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[290,161,658,203]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nr + \\\\sum_ {n = 2} ^ {\\\\infty} | a _ {n} | r ^ {n} \\\\leq - h (- 1) \\\\leq d (f (0), \\\\partial f (\\\\mathbb {D}))\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,209,836,276]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for $r \\\\leq \\\\min \\\\{r_f, 1/3\\\\}$ . If $r_f$ lies in $(0, 1/3]$ , then $r_f$ is the best possible. Let $0 < r_f \\\\leq 1/3$ . To show the sharpness of $r_f$ , we choose $f$ to be a suitable rotation $h_\\\\epsilon$ of the function $h$ i.e. $f = h_\\\\epsilon$ which is obviously belongs to $S^*(\\\\phi)$ . Since equality in Lemma 1.11 occurs for a suitable rotation of $h$ , we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,283,596,301]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nd \\\\left(h _ {\\\\epsilon} (0), \\\\partial h _ {\\\\epsilon} (\\\\mathbb {D})\\\\right) = - h _ {\\\\epsilon} (- 1). \\\\tag {3.15}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,308,823,326]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A simple computation using (3.10) and (3.15), for $f = h_{\\\\epsilon}$ and $|z| = r_f$ shows that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[233,333,717,375]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n| z | + \\\\sum_ {n = 0} ^ {\\\\infty} | a _ {n} | | z | ^ {n} = h _ {\\\\epsilon} (r) = - h _ {\\\\epsilon} (- 1) = d (h _ {\\\\epsilon} (0), \\\\partial h _ {\\\\epsilon} (\\\\mathbb {D})).\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,381,836,399]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Therefore, the radius $r_f \\\\in (0, 1/3)$ is the best possible. This completes the proof. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,423,838,475]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof of Corollary 2.4. Given that $\\\\phi(z) = \\\\alpha + (1 - \\\\alpha)e^{z}$ and each coefficient of $\\\\phi(z)$ is strictly positive for $0 \\\\leq \\\\alpha < 1$ . By the similar lines of argument as in the proof of Theorem 2.1, consider $H(r) = h(r) + h(-1)$ . Note that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[205,480,747,543]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh \\\\left(\\\\frac {1}{3}\\\\right) = \\\\frac {1}{3} \\\\exp \\\\left((1 - \\\\alpha) \\\\int_ {0} ^ {\\\\frac {1}{3}} \\\\left(\\\\frac {- 1 + e ^ {t}}{t}\\\\right) d t\\\\right) \\\\approx \\\\frac {1}{3} (1. 4 3 8 0 7) ^ {1 - \\\\alpha}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,549,154,564]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[184,568,768,624]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh (- 1) = - \\\\exp \\\\left((1 - \\\\alpha) \\\\int_ {0} ^ {- 1} \\\\left(\\\\frac {- 1 + e ^ {t}}{t}\\\\right) d t\\\\right) \\\\approx - (0. 4 5 0 8 5 9 4 6 3) ^ {1 - \\\\alpha}.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,628,836,697]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Therefore, using Mathematica we can see that $H(1/3) = h(1/3) + h(-1) > 0$ if, and only if, $0 \\\\leq \\\\alpha < 0.05284$ . On the other hand, $H(0) = h(-1) < 0$ . Therefore we conclude that $H$ has a root in $(0,1)$ . Since $r_f$ is the smallest root of $H$ in $(0,1)$ , we have $r_f < 1/3$ . Hence in view of Theorem 2.1, $r_f$ is the best possible. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,720,836,751]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof of Corollary 2.5. Let $\\\\phi(z) = 1 + 4z/3 + 2z^2/3$ . A simple computation shows that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[363,751,589,788]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh (r) = r \\\\exp \\\\left(\\\\frac {4}{3} r + \\\\frac {r ^ {2}}{3}\\\\right).\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,790,655,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note that $h(1/3) \\\\approx 0.539490$ and $h(-1) \\\\approx -0.367879441$ . Let \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[379,815,571,834]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nH (r) = h (r) + h (- 1).\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,839,836,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Then $H(0) \\\\approx -0.3678799441 < 0$ and $H(1/3) > 0$ . Therefore, $H$ has a root in $(0,1/3)$ . Let $r_f$ be the smallest root in $(0,1/3)$ . In view of Theorem 2.1, $r_f$ is the best possible. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,112,137,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[344,111,608,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Vasudevarao Allu and Himadri Halder \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[114,137,836,172]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof of Corollary 2.6. For $\\\\phi(z) = 1 + (z / k) \\\\left( \\\\frac{(k + z)}{k - z} \\\\right)$ , where $k = \\\\sqrt{2} + 1$ , a simple computation using (3.8) shows that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[376,176,574,217]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh (r) = \\\\frac {r}{e ^ {r}} \\\\left(\\\\frac {k}{k - r}\\\\right) ^ {2 k}.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,222,838,274]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note that $h(-1) \\\\approx -0.5099807$ and $h(1/3) \\\\approx 0.489391446$ . Hence $H(1/3) = h(1/3) + h(-1) < 0$ and $H(0) < 0$ . Therefore $H$ has no root in $(0,1/3)$ and $r_f > 1/3$ . Hence the inequality (2.2) is satisfied for $r \\\\leq 1/3$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,296,836,347]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof of Corollary 2.7. Let $\\\\phi(z) = (1 + Az)/(1 + Bz)$ with $-1 \\\\leq B < A \\\\leq 1$ . It is easy to see that the coefficients of $\\\\phi(z)$ are all positive when $-1 \\\\leq B \\\\leq 0$ . A simple computation using (3.8) yields \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[369,352,583,377]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh (r) = r \\\\left(1 + B r\\\\right) ^ {\\\\left(\\\\frac {A - B}{B}\\\\right)}.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,381,524,399]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $H(r) = h(r) + h(-1)$ . Then $H(0) < 0$ and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[198,404,754,448]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nH \\\\left(\\\\frac {1}{3}\\\\right) = h \\\\left(\\\\frac {1}{3}\\\\right) + h (- 1) = \\\\frac {1}{3} \\\\left(1 + \\\\frac {B}{3}\\\\right) ^ {\\\\left(\\\\frac {A - B}{B}\\\\right)} - (1 - B) ^ {\\\\left(\\\\frac {A - B}{B}\\\\right)}.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,453,836,521]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A simple computation shows that $H(1/3) > 0$ when $-1 \\\\leq B \\\\leq (1 - 3^k) / (1 + 3^k)$ , where $k = (B - A) / B$ . Therefore, $H$ has a root in $(0,1)$ . Let the smallest root of $H$ in $(0,1/3)$ be $r_f$ . In view of Theorem 2.1, we conclude that for $r \\\\leq r_f$ , the inequality (2.2) is satisfied and the radius $r_f$ is the best possible. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,543,815,562]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof of Corollary 2.8. Here $\\\\phi(z) = 1 + z / (1 - \\\\alpha z^2)$ . Using (3.8), we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[359,567,589,610]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh (r) = r \\\\left(\\\\frac {1 + \\\\sqrt {\\\\alpha} r}{1 - \\\\sqrt {\\\\alpha} r}\\\\right) ^ {\\\\frac {1}{2 \\\\sqrt {\\\\alpha}}}.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,615,690,687]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Clearly, $h(-1) = -\\\\left(\\\\frac{1 + \\\\sqrt{\\\\alpha}}{1 + \\\\sqrt{\\\\alpha}}\\\\right)^{\\\\frac{1}{2\\\\sqrt{\\\\alpha}}}$ and $h\\\\left(\\\\frac{1}{3}\\\\right) = \\\\left(\\\\frac{1 + \\\\frac{\\\\sqrt{\\\\alpha}}{3}}{1 + \\\\frac{\\\\sqrt{\\\\alpha}}{3}}\\\\right)^{\\\\frac{1}{2\\\\sqrt{\\\\alpha}}}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,686,836,737]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $H(r) = h(r) + h(-1)$ . Then $H(0) < 0$ and $H(1/3) = h(1/3) + h(-1) > 0$ for $0 \\\\leq \\\\alpha < 1$ . Therefore, $H$ has a root in $(0, 1/3)$ and choose $r_f$ to be the smallest root in $(0, 1/3)$ . By Theorem 2.1, the radius $r_f$ is the best possible. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,761,836,793]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof of Corollary 2.9. Let $\\\\phi(z) = (1 + sz)^2$ with $0 < s \\\\leq 1/\\\\sqrt{2}$ . Using (3.8), we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[339,792,610,828]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh (r) = r \\\\exp \\\\left(s \\\\left(2 r + \\\\frac {s r ^ {2}}{2}\\\\right)\\\\right).\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,831,563,849]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $H(r) = h(r) + h(-1)$ . Note that $H(0) < 0$ and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[226,853,727,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nH \\\\left(\\\\frac {1}{3}\\\\right) = \\\\frac {1}{3} \\\\exp \\\\left(s \\\\left(\\\\frac {s + 1 2}{1 8}\\\\right)\\\\right) - \\\\exp \\\\left(s \\\\left(- 2 + \\\\frac {s}{2}\\\\right)\\\\right) > 0,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[226,111,724,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bohr radius for certain classes of starlike and convex univalent functions \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[818,112,835,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[114,137,836,188]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"if $0.444981 < s \\\\leq 1 / \\\\sqrt{2}$ . Therefore, $H$ has a real root in $(0,1/3)$ . Let $r_f$ be the smallest root in $(0,1/3)$ . In view of Theorem 2.1, the radius $r_f$ is the best possible. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,214,836,247]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof of the Theorem 2.10. Let $f \\\\in \\\\mathcal{C}(\\\\phi)$ then from Lemma 1.14, it is evident that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,256,685,281]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nd (f (0), \\\\partial f (\\\\mathbb {D})) = \\\\lim _ {| z | \\\\rightarrow 1} \\\\inf | f (z) - f (0) | \\\\geq - k (- 1). \\\\tag {3.16}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,289,836,325]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is known that $f \\\\in \\\\mathcal{C}(\\\\phi)$ if, and only if, $zf' \\\\in S^{*}(\\\\phi)$ . Therefore, there exists $g \\\\in S^{*}(\\\\phi)$ such that $zf'(z) = g(z)$ for $z \\\\in \\\\mathbb{D}$ . This relation gives \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,333,554,381]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nf (z) = \\\\int_ {0} ^ {z} \\\\frac {g (t)}{t} d t. \\\\tag {3.17}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,389,836,409]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $f(z) = z + \\\\sum_{n=2}^{\\\\infty} a_n z^n$ and $g(z) = z + \\\\sum_{n=2}^{\\\\infty} b_n z^n$ . Then from (3.17), we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,416,600,457]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nz + \\\\sum_ {n = 2} ^ {\\\\infty} a _ {n} z ^ {n} = z + \\\\sum_ {n = 2} ^ {\\\\infty} \\\\frac {b _ {n}}{n} z ^ {n}. \\\\tag {3.18}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,464,788,484]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note that the majorant series for $g$ , we have $M_g(r) \\\\coloneqq r + \\\\sum_{n=2}^{\\\\infty} |b_n| r^n$ . Then \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,492,600,609]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\int_ {0} ^ {r} \\\\frac {M _ {g} (t)}{t} d t = r + \\\\sum_ {n = 2} ^ {\\\\infty} \\\\left| b _ {n} \\\\right| \\\\frac {r ^ {n}}{n} \\\\tag {3.19} \\\\\\\\ = r + \\\\sum_ {n = 2} ^ {\\\\infty} | a _ {n} | r ^ {n} \\\\\\\\ = M _ {f} (r). \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,616,836,649]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since we have $zk'(z) = h(z)$ , then going by same lines of argument as in (3.19), we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,655,568,705]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nM _ {k} (r) = \\\\int_ {0} ^ {r} \\\\frac {M _ {g} (t)}{t} d t. \\\\tag {3.20}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,712,400,728]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In view of Lemma 1.22, we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,736,601,767]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nM _ {g} (r) \\\\leq M _ {h} (r) \\\\quad \\\\text {f o r} \\\\quad r \\\\leq \\\\frac {1}{3}. \\\\tag {3.21}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,773,589,790]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Using (3.19), (3.20) and (3.21), for $r \\\\leq 1/3$ , we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,799,638,849]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nM _ {f} (r) \\\\leq \\\\int_ {0} ^ {r} \\\\frac {M _ {h} (t)}{t} d t = M _ {k} (r) = k (r). \\\\tag {3.22}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,856,836,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By the same lines of argument as in Theorem 2.1, we can show that $H_{1}:[0,1]\\\\to \\\\mathbb{R}$ defined by $H_{1}(r) = k(r) + k(-1)$ has a real root in $(0,1)$ . Let $r_f$ be the smallest \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,112,137,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"12 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[344,111,608,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Vasudevarao Allu and Himadri Halder \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[114,136,835,171]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"root of $H_{1}$ in $(0,1)$ and $H_{1}(r_{f}) = 0$ , which follows that $k(r_{f}) = -k(-1)$ . Therefore, for $r \\\\leq \\\\min \\\\{r_f, 1/3\\\\}$ , we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[290,176,660,195]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nM _ {f} (r) \\\\leq k (r) \\\\leq - k (- 1) \\\\leq d (f (0), \\\\partial f (\\\\mathbb {D})).\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,201,836,252]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $0 < r_f \\\\leq 1/3$ . To show the sharpness of $r_f$ , we choose $f$ to be a suitable rotation $k_{\\\\epsilon}$ of the function $k$ i.e. $f = k_{\\\\epsilon}$ which is obviously belongs to $\\\\mathcal{C}(\\\\phi)$ . Since the equality in Lemma 1.14 occurs for a suitable rotation of $k$ , we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,258,596,277]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nd \\\\left(k _ {\\\\epsilon} (0), \\\\partial k _ {\\\\epsilon} (\\\\mathbb {D})\\\\right) = - k _ {\\\\epsilon} (- 1). \\\\tag {3.23}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,283,732,301]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A simple computation using (3.16), for $f = k_{\\\\epsilon}$ and $|z| = r_f$ , shows that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[235,308,714,351]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n| z | + \\\\sum_ {n = 0} ^ {\\\\infty} | a _ {n} | | z | ^ {n} = k _ {\\\\epsilon} (r) = - k _ {\\\\epsilon} (- 1) = d (k _ {\\\\epsilon} (0), \\\\partial k _ {\\\\epsilon} (\\\\mathbb {D})).\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,357,801,375]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Therefore, the radius $r_f \\\\in (0, 1/3)$ is the best possible. This completes the proof. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,399,836,432]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof of the Theorem 2.14. Let $G \\\\in \\\\mathcal{G}_{\\\\alpha}$ be given by (2.13). Then from Lemma 1.19, it is evident that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,439,665,465]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nd (G (0), \\\\partial G (\\\\mathbb {D})) = \\\\lim _ {| z | \\\\rightarrow 1} \\\\inf _ {} | G (z) - G (0) | \\\\geq 1. \\\\tag {3.24}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,472,589,488]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In view of Lemma 1.18, we have the following relation \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,494,584,527]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nG (z) = (1 - z) ^ {2 (1 - \\\\alpha)} \\\\frac {s (z)}{z}. \\\\tag {3.25}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,533,500,549]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For $s\\\\in S^{*}(\\\\alpha)$ , using Lemma 1.24 we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,557,569,587]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nM _ {s} (r) \\\\leq \\\\frac {r}{(1 - r) ^ {2 (1 - \\\\alpha)}} \\\\tag {3.26}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,593,836,628]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for $0 < r < 1$ , where $M_s(r)$ is the associated majorant series of $s$ . It is known that for $f$ and $g$ be two analytic functions in $\\\\mathbb{D}$ , \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,635,536,653]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nM _ {f g} \\\\leq M _ {f} M _ {g} \\\\tag {3.27}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[114,659,836,694]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for $0 < r < 1$ , where $M_{fg}$ , $M_f$ and $M_g$ are the associated majorant series of $fg$ , $f$ and $g$ respectively. In view of (3.25), (3.26) and (3.27), for $|z| = r$ , we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,700,603,819]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} M _ {G} (r) = 1 + \\\\sum_ {n = 1} ^ {\\\\infty} | d _ {n} | | z | ^ {n} \\\\tag {3.28} \\\\\\\\ \\\\leq (1 + | z |) ^ {2 (1 - \\\\alpha)} \\\\frac {M _ {s} (r)}{r} \\\\\\\\ \\\\leq \\\\frac {(1 + r) ^ {2 (1 - \\\\alpha)}}{(1 - r) ^ {2 (1 - \\\\alpha)}}. \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,825,403,840]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Therefore from (3.28), we obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,849,613,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\sum_ {n = 1} ^ {\\\\infty} \\\\left| d _ {n} \\\\right| | z | ^ {n} \\\\leq \\\\frac {(1 + r) ^ {2 (1 - \\\\alpha)}}{(1 - r) ^ {2 (1 - \\\\alpha)}} - 1. \\\\tag {3.29}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[228,111,724,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bohr radius for certain classes of starlike and convex univalent functions \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[818,112,836,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"13 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[115,136,771,155]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The right hand side of (3.29) is less than or equal to $1 \\\\leq d(G(0), \\\\partial G(\\\\mathbb{D}))$ if \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[400,160,552,199]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\frac {(1 + r) ^ {2 (1 - \\\\alpha)}}{(1 - r) ^ {2 (1 - \\\\alpha)}} \\\\leq 2\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,204,835,239]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"i.e. for $r \\\\leq r_G := (2^{1/2(1 - \\\\alpha)} - 1) / (2^{1/2(1 - \\\\alpha)} + 1)$ . To show that $r_G$ is the best possible, we consider the function $G_\\\\alpha : \\\\mathbb{D} \\\\to \\\\mathbb{C}$ defined by \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[366,245,583,286]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nG _ {\\\\alpha} (z) = \\\\left(\\\\frac {1 - z}{1 + z}\\\\right) ^ {2 (1 - \\\\alpha)}.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,290,470,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From Lemma 1.19, for $G = G_{\\\\alpha}$ , we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[384,314,566,332]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nd (G (0), \\\\partial G (\\\\mathbb {D})) = 1.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,339,635,356]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For $G = G_{\\\\alpha}$ and $|z| = r_G$ , a simple computation shows that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[226,362,722,406]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\sum_ {n = 1} ^ {\\\\infty} \\\\left| d _ {n} \\\\right| | z | ^ {n} = \\\\left(\\\\frac {1 + r _ {G}}{1 - r _ {G}}\\\\right) ^ {2 (1 - \\\\alpha)} - 1 = 1 = d (G (0), \\\\partial G (\\\\mathbb {D})).\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,412,771,429]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This shows that the radius $r_f$ is the best possible. This completes the proof. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[816,413,836,426]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0498/a6b4585b741cda4ba06643b87f11bebc88e544ba60e35a9d2e08170f742a235c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[115,444,836,478]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acknowledgement: The first author thank SERB-MATRICS and the second author thank CSIR for their support. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[415,496,539,511]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"REFERENCES \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,519,836,889]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"[1] Y. ABU-MUHANNA, Bohr's phenomenon in subordination and bounded harmonic classes, Complex Var. Elliptic Equ. 55 (2010), 1071-1078. \\\",\\\"[2] Y. ABU-MUHANNA and R. M. ALI, Bohr's phenomenon for analytic functions into the exterior of a compact convex body, J. Math. Anal. Appl. 379 (2011), 512-517. \\\",\\\"[3] Y. ABU MUHANNA and R. M. ALI, Bohr's phenomenon for analytic functions and the hyperbolic metric, Math. Nachr. 286 (2013), 1059-1065. \\\",\\\"[4] Y.ABU MUHANNA, R. M. ALI, Z. C. NG and S. F. M HASNI, Bohr radius for subordinating families of analytic functions and bounded harmonic mappings, J. Math. Anal. Appl. 420 (2014), 124-136. \\\",\\\"[5] L.AIZENBERG, Multidimensional analogues of Bohr's theorem on power series, Proc. Amer. Math. Soc. 128 (2000), 1147-1155. \\\",\\\"[6] L. AIZENBERG, A. AYTUNA and P. DJAKOV, Generalization of theorem on Bohr for bases in spaces of holomorphic functions of several complex variables, J. Math. Anal. Appl. 258 (2001), 429-447. \\\",\\\"[7] L.AIZENBERG, Generalization of results about the Bohr radius for power series, Stud. Math. 180 (2007), 161-168. \\\",\\\"[8] R. M. ALI, R.W. BARNARD and A.Yu. SOLYNIN, A note on Bohr's phenomenon for power series, J. Math. Anal. Appl. 449 (2017), 154-167. \\\",\\\"[9] R. M. ALI, N. K.JAIN and V. RAVICHANDRAN, Bohr radius for classes of analytic functions, Results Math. 74 (2019). \\\",\\\"[10] S. A. ALKHALEEFAH, I.R. KAYUMOV and S. PONNUSAMY, On the Bohr inequality with a fixed zero coefficient, Proc. Amer. Math. Soc. 147 (2019), 5263-5274. \\\",\\\"[11] VASUDEVARAO ALLU and HIMADRI HALDER, Bhor phenomenon for certain subclasses of Harmonic Mappings, arXiv:2006.11622, 2020 \\\",\\\"[12] C. BÉNETEAU, A. DAHLNER and D. KHAVINSON, Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4 (2004), 1-19. \\\"]\",\n \"page_idx\": \"13\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[117,112,137,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"14 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[344,111,608,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Vasudevarao Allu and Himadri Halder \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[117,138,835,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"[13] B. BHOWMIK and N. DAS, Bohr phenomenon for subordinating families of certain univalent functions, J. Math. Anal. Appl. 462 (2018), 1087-1098. \\\",\\\"[14] H.P. BOAS and D. KHAVINSON, Bohr's power series theorem in several variables, Proc. Amer. Math. Soc 125 (1997), 2975-2979. \\\",\\\"[15] H.P. BoAs, Majorant Series, J. Korean Math. Soc. 37 (2000), 321-337. \\\",\\\"[16] H. BOHR, A theorem concerning power series, Proc. Lond. Math. Soc. s2-13 (1914), 1-5. \\\",\\\"[17] S. EVDORIDIS and S. PONNUSAMY, Improved Bohr's inequality for locally univalent harmonic mappings, Indag. Math. (N.S.) 30 (2019), 201-213. \\\",\\\"[18] P. L. DUREN, Univalent Functions (Grundlehren der mathematischen Wisseenschaften 259, New York, Berlin, Heidelberg, Tokyo) Springer-Verlag, 1983. \\\",\\\"[19] A. W. GOODMAN, On uniformly convex functions, Annl. Polon. Math., 56 (1991), 87-92. \\\",\\\"[20] A. W. GOODMAN, On uniformly starlike functions, J. Math. Anal. Appl., 155 (1991), 364-370. \\\",\\\"[21] I. GRAHAM and G. KOHR, Geometric Function Theory in One and Higher Dimensions, Monographs and Textbooks in Pure and Applied Mathematics, 255, Marcel Dekker, Inc., New York, 2003. \\\",\\\"[22] R. KARGAR, A. EBADIAN and J. SOKOL, On Both lemniscate and starlike functions, *Anal. Math. Phys.* 9 (2019), 143-154. \\\",\\\"[23] I.R. KAYUMOV, S. PONNUSAMY and N. SHAKIROV, Bohr radius for locally unovalent harmonic mappings, Math. Nachr. 291 (2018), 1757-1768. \\\",\\\"[24] I.R. KAYUMOV and S. PONNUSAMY, Bohr's inequalities for the analytic functions with lacunary series and harmonic functions, J. Math. Anal. Appl. 465 (2018), 857-871. \\\",\\\"[25] I.R. KAYUMOV and S. PONNUSAMY, On a powered Bohr inequality, Ann. Acad. Sci. Fenn. Math. 44 (2019), 301-310. \\\",\\\"[26] A. KAYUMOVA, I. R. KAYUMOV and S. PONNUSAMY, Bohr's inequality for harmonic mappings and beyond, Mathematics and Computing, 245-256, Commun. Comput. Inf. Sci., 834, Springer, Singapore, 2018. \\\",\\\"[27] K.KHATTER, V. RAVICHANDRAN and S. SIVAPRASAD KUMAR, Starlike functions associated with exponential function and the lemniscate of Bernoulli, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113 (2019), 233-253. \\\",\\\"[28] S. KUMAR and V. RAVICHANDRAN, A subclass of starlike functions associated with a rational function, Southeast Asian Bull. Math. 40 (2016), 199-212. \\\",\\\"[29] S. SIVAPRASAD KUMAR and S. BANGA, On a special type of Non-Ma-Minda function, arXiv:2006.02111v1, 2020. \\\",\\\"[30] Z. LIU and S. PONNUSAMY, Bohr radius for subordination and $k$ -quasiconformal harmonic mappings, Bull. Malays. Math. Sci. Soc. 42 (2019), 2151-2168. \\\",\\\"[31] A. LYZZAIL, On a conjecture of M. S. Robertson, Proc. Amer. Math. Soc. 91 (1984), 108-110. \\\",\\\"[32] W.C. MA and D. MINDA, A unified treatment of some special classes of univalent functions, in Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157-169, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge. \\\",\\\"[33] W.C. Ma and D. MINDA, Uniformly convex functions, Ann. Polon. Math. 57 (1992), 165-175. \\\",\\\"[34] W.C. MA and D. MINDA, Uniformly convex functions II, Ann. Polon. Math. 58 (1993), 275-285. \\\",\\\"[35] V. S. MAsIH and S. KANAS, Subclasses of Starlike and Convex functions associated with associated with the LimaA $\\\\\\\\check{\\\\\\\\Lambda}$ gon Domain, Symmetry 12 (2020). \\\",\\\"[36] R. MENDIRATTA, S. NAGPAL and V. RAVICHANDRAN, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc. 38 (2015), 365-386. \\\",\\\"[37] VERN I. PAULSEN, GELU POPESCU and DINESH SINGH, On Bohr's inequality, Proc. Lond. Math. Soc. s3-85 (2002), 493-512. \\\",\\\"[38] S. PONNUSAMY, A. VASUDEVARAO and M. VUORINEN, Region of variability for spirallike functions with respect to a boundary point, Colloq. Math. 116 (2009), 31-46. \\\"]\",\n \"page_idx\": \"14\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[228,111,724,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bohr radius for certain classes of starlike and convex univalent functions \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[818,112,835,122]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"15 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[117,138,836,394]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"[39] R. K.RAINA and J. SOKOL, Some properties related to a certain class of starlike functions, C. R. Math. Acad. Sci. Paris 353 (2015), 973-978. \\\",\\\"[40] M. S. Robertson, Univalent functions with respect to a boundary point, J. Math. Anal. Appl. 81 (1981), 327-345. \\\",\\\"[41] F. RONNING, On starlike functions associated with parabolic regions, Anna. Univ. Mariae Curie-Sklodowska Sect. A, 45 (1991), 117-122. \\\",\\\"[42] F. RONNING, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), 189-196. \\\",\\\"[43] K. SHARMA, N. K. JAIN and V. RAVICHANDRAN, Starlike functions associated with cardiod, Afr. Mat. 27 (2016), 923-939. \\\",\\\"[44] S. SIDON, Uber einen satz von Hernn Bohr, Math. Zeit. 26 (1927), 731-732. \\\",\\\"[45] H. SILVERMAN and E.M. SILVIA, Subclasses of univalent functions starlike with respect to a boundary point, Houston J. Math. 16 (1990), 289-299. \\\",\\\"[46] DEREK K. THOMAS, NIKOLA TUNESKI and ALLU VASUDEVARAO, Univalent functions. A primer, De Gruyter Studies in Mathematics, 69. De Gruyter, Berlin, 2018. \\\",\\\"[47] M. TOMIC, Sur un theoreme de H. Bohr, Math. Scand. 11 (1962), 103-106. \\\",\\\"[48] L. A. WANI and A. SWAMINATHAN, Starlike and convex functions associated with nephroid domain, Bull. Malays. Math. Sci. Soc. doi:10.1007/s40840-020-00935-6. 2020 \\\"]\",\n \"page_idx\": \"15\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[117,407,836,434]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"VASUDEVARAO ALLU, SCHOOL OF BASIC SCIENCE, INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR, BHUBANESWAR-752050, ODISHA, INDIA. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[139,435,421,448]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E-mail address: avrao@iitbbs.ac.in \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,459,889,486]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"HIMADRI HALDER, SCHOOL OF BASIC SCIENCE, INDIAN INSTITUTE OF TECHNOLOGY BHUBANESWAR, BHUBANESWAR-752050, ODISHA, INDIA. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[139,488,413,500]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E-mail address: hh11@iitbbs.ac.in \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[117,112,137,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"16 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[344,111,608,123]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Vasudevarao Allu and Himadri Halder \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.48550/arxiv.2006.15299"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Mathematics and Computational Science"},"sha256":{"kind":"string","value":"888c00add6b1726c5d300792c267556b72bdfe44651e075f7bf56c07b306c487"},"title":{"kind":"string","value":"Bohr radius for certain classes of starlike and convex univalent\\n functions"}}},{"rowIdx":54,"cells":{"abstract":{"kind":"string","value":""},"author":{"kind":"string","value":"Hallysson Oliveira|Stiven S. Dias|Marcelo G. S. Bruno"},"content_list":{"kind":"list like","value":[{"bbox":"[102,83,493,231]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"GNSS-Denied Joint Cooperative Terrain Navigation and Target Tracking Using Factor Graph Geometric Average Fusion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[102,292,364,308]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"HALLYSSON OLIVEIRA, Member, IEEE ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[102,309,467,321]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Instituto Tecnológico de Aeronáutica, São José dos Campos, Brazil ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[102,328,310,345]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"STIVEN S. DIAS, Member, IEEE ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[102,345,341,357]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Embraer S.A., São José dos Campos, Brazil ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[102,358,464,369]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Eindhoven University of Technology, Eindhoven, The Netherlands ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[102,377,401,393]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"MARCELO G. S. BRUNO, Senior Member, IEEE ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[102,393,467,406]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Instituto Tecnológico de Aeronáutica, São José dos Campos, Brazil ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[101,450,500,638]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We propose a fully distributed methodology based on factor graphs for joint cooperative localization and distributed noncooperative target tracking in a 3-D scenario where multiple surveillance aircraft fly in formation without access to global navigation satellite system (GNSS) measurements or communication with anchor nodes. Our approach is based on the adapt-then-combine (ATC) diffusion scheme, which is integrated into the factor graph by the introduction of special combine factors to perform geometric average fusion of the target beliefs over a partially connected network. The updated target belief held by each aircraft following the combine step is also fed back to improve the aircraft's own self-localization, assimilating the target measurements. Simulation results show that the proposed distributed algorithm performed close to the posterior Cramér-Rao lower bound of the optimal centralized solution and that the agents approached a consensus about the target state estimate. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[101,666,500,705]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Manuscript received 5 May 2023; revised 31 August 2023 and 30 October 2023; accepted 4 November 2023. Date of publication 17 November 2023; date of current version 9 February 2024. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[102,712,321,724]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DOI. No. 10.1109/TAES.2023.3333286 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[102,732,406,744]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Refereeing of this contribution was handled by P. Braca. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[101,752,498,790]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The work of Marcelo G. S. Bruno was supported by São Paulo Research Foundation (FAPESP) under Grant 2020/09838-0 (BIOS - Brazilian Institute of Data Science). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[101,797,500,886]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Authors' addresses: Hallysson Oliveira and Marcelo G. S. Bruno are with Instituto Tecnológico de Aeronáutica, São José dos Campos 12228-900, Brazil, E-mail: (hallysson@ieee.org; bruno@ita.br); Stiven S. Dias is with Embraer S.A., São José dos Campos 12247-016, Brazil, and also with the Eindhoven University of Technology, 5600 Eindhoven, The Netherlands, E-mail: (stiven.dias@embraer.com.br). (Corresponding author: Hallysson Oliveira.) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[102,906,241,919]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"0018-9251 © 2023 IEEE ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,44,665,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"I. INTRODUCTION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,65,911,383]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In a previous work [1], we introduced an algorithm for anchorless, GNSS-free, 2-D cooperative self-localization of multiple aircraft using terrain-aided navigation, inertial sensor measurements, and peer-to-peer relative distance measurements. In this article, we extend that previous algorithm to a more challenging scenario where the surveillance aircraft not only estimate their own position, but also jointly track a noncollaborative target in a distributed fashion. To account for more realistic evasive maneuvers that may be taken by the tracked target, including possible changes in altitude, we perform joint cooperative terrain-aided navigation and distributed target tracking (J-CTAN-DTT) in a 3-D surveillance space, as opposed to the previously considered 2-D scenario. As in [1], we use a Bayesian approach that employs factor graphs [2], [3], [4]. The main original contribution of the article is, however, to handle the distributed target tracking task using an adapt-then Combine (ATC) diffusion [5], [6], [7], which is completely integrated into the self-localization factor graph framework, and implemented in a fully distributed fashion by the introduction in the graph of artificial factors to perform the Combine step. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,383,910,594]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We maintain, as in [1], a scenario where none of the surveillance aircraft have available global navigation satellite system (GNSS) data, relying only on inertial unit measurements and terrain-aided navigation (TAN) techniques [8], [9], [10], [11], [12] in addition to peer-to-peer relative distance measurements. To account for the new 3-D scenario and the additional integrated joint distributed target tracking task, we employ an additional set of sensor measurements compared to [1] at each aircraft node, namely, barometric altitude, ground clearance (measured by a radar altimeter and used in conjunction with a 2-D terrain elevation map), and target measurements (slant range, azimuth angle, and elevation angle) collected at each node by an airborne 3-D radar. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,626,816,641]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. Related Work and Article Contributions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,648,911,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Advances in communication technologies have enabled the use of wireless sensor networks in many different engineering applications [13], [14], [15]. Multiagent networks are of special interest for tactical coordination over military data links. In addition to self-localizing, agents—typically military units such as aircraft that are partially connected by a communication network with no data fusion center—must also cooperate [16] to simultaneously track a noncooperative target and agree with each other regarding the tracked target state. The joint cooperative localization and distributed target tracking problem is particularly challenging because tracking performance depends greatly on the ability of each agent that is observing the target to maintain an accurate estimate of its own location, which is unknown a priori in the considered scenario. On top of that, GNSS jamming and spoofing techniques are becoming ubiquitous in modern battlefield, so GNSS-free localization techniques are crucial in contemporary military applications. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[102,939,736,952]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.60, NO.1 FEBRUARY 2024 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[883,940,906,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"991 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[84,44,482,150]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Previous references in the literature, which use factor graphs combined with belief propagation, usually address cooperative location and distributed target tracking separately as fully decoupled problems. While [16], [17], [18], [19], and [20] only address cooperative localization based on factor graphs, [21], [22], and [23], for example, only address target tracking based on factor graphs. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,151,482,435]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The previous works [24], [25], and [26] handled the joint localization and distributed target tracking problem within a factor graph framework using a consensus approach that has a high communication cost, and might not be adequate for real-time systems. In contrast to the aforementioned references, our proposed diffusion scheme performs joint cooperative self-localization and distributed noncooperative target tracking with a much lower internode communication cost than the consensus algorithms in [24], [25], and [26], while maintaining at the same time a performance that is close to the posterior Cramér-Rao lower bound (PCRLB) for both the localization and tracking errors of the optimal nonlinear filter that uses a central data fusion hub to process all measurements in the network, in batch, at each time instant. Furthermore, we empirically show that the different agents approach, over time, a consensus about the target state without the need, as in previous literature, of multiple iterative data exchange between nodes, between consecutive sensor observations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,436,482,737]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Specifically, our algorithm seeks to approximate the adapt step of a classic distributed ATC tracker by replacing the likelihood of the local target observations at each agent node given the target and the agent positions by its expected value with respect to the latest belief (i.e., posterior probability distribution) that the agent has about its own state. Next, the aforementioned special factors combine the updated target state beliefs following the adapt step at each node using a geometric average (GA) fusion criterion [27], which can be shown to minimize an average Kullback-Leibler (KL) divergence metric [28], [29]. For Gaussian beliefs, the employed combine step rule reduces to the well-known method of covariance intersection in sensor fusion [30]. Furthermore, after the distributed combine process, each aircraft uses its updated target belief to improve its own self-localization. As in [1] and [33], we implement the sum-product algorithm (SPA) operations using a scheme that combines Gaussian mixture models (GMMs) and sequential Monte Carlo methods, which significantly reduces the amount of data exchanged over the network. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,739,482,919]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The rest of this article is organized as follows. We present, in section II, the problem statement and detail the state and observation models. In section III, we derive the joint localization and tracking algorithm using a distributed message passing approach over factor graphs. A communication-efficient implementation of the said algorithm using GMMs and sequential Monte Carlo methods is then presented in section IV. Performance results in a simulated 3-D scenario with real terrain elevation maps are presented and discussed in section V, assuming different cooperative and noncooperative operation modes and no available GNSS data. Finally, section VI concludes this article. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[496,44,692,56]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"II. PROBLEM STATEMENT ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[495,65,893,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the cooperative terrain-aided navigation and distributed target tracking scenario considered in this article, we have multiple cooperative aircraft and a single noncooperative target flying in a 3-D environment where GNSS data are unavailable, either due to hostile actions such as spoofing and jamming, or natural causes such as low-latitude ionospheric interference that degrades GNSS signals. The aircraft must, therefore, cooperate within local neighborhoods over a wireless communication network to estimate both their own position and the position of the noncooperative target. As in [1], the topology of the communication network is modeled as a graph $\\mathcal{G}$ consisting of a set of vertices $\\mathcal{V} \\triangleq \\{1, \\dots, A\\}$ , representing the location of each cooperative agent and a set of undirected edges $\\mathcal{E} = \\{(a, b) \\in \\mathcal{V} \\times \\mathcal{V}\\}$ that represent the communication links between agent nodes. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[493,306,893,428]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Each aircraft measures the distance to the terrain beneath them, also known as ground clearance, using an onboard radar altimeter. A barometric sensor is employed in turn to measure the aircraft's altitude with respect to sea level. In addition, each cooperative aircraft has an onboard 3-D radar to measure the slant range as well as the azimuth and elevation angles to the noncooperative target's 3-D location based on the received echoed radar pulses from the target. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[493,429,893,625]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"REMARK Although we consider a single target scenario, it should be stressed that the solution presented in this article could be extended in a straightforward way to a scenario with multiple targets provided that, as considered in [24], perfect a priori target data association is available and there is no clutter, missed detections, or false alarms. In a scenario where the data association problem also has to be solved jointly with the cooperative localization and target tracking tasks, a different algorithm would have to be derived, but that is not the focus of this present article. A comprehensive review of state-of-the-art approaches for multitarget tracking using belief propagation methods may be found in [22], [31], and [32]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[495,635,702,650]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. Distributed Signal Model ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[493,656,893,792]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Aircraft dynamics: Let $\\mathbf{x}_{n,a} \\triangleq [x_{n,a} y_{n,a} z_{n,a}]^T$ be the unknown 3-D position of the aircraft $a$ at the time stamp $t_n = nT_s$ , $n \\in \\mathbb{Z}_+$ , where $T_s$ is the sampling period. The deterministic and known input vector $\\mathbf{u}_{n,a} \\triangleq [\\Delta x_{n,a} \\Delta y_{n,a} \\Delta z_{n,a}]^T$ represents, on the other hand, the displacement in the 3-D space from instant $n$ to $n + 1$ , which is measured by an inertial navigation system (INS) processor. As in [1], [9], and [10], the state vector evolves over time according to the model ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[599,801,890,816]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf {x} _ {n + 1, a} = \\mathbf {x} _ {n, a} + \\mathbf {u} _ {n, a} + \\mathbf {w} _ {n, a} \\tag {1}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[493,823,893,869]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\{\\mathbf{w}_{n,a}\\}$ are Gaussian random vectors that are mutually independent and have identical distribution $\\mathbf{w}_{n,a} \\sim \\mathcal{N}(\\mathbf{0}, \\mathbf{Q}_{n,a})$ , for all $a \\in V$ and all $n \\in \\mathbb{Z}_+$ . Furthermore, as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[495,893,892,919]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"<sup>1</sup>For simplicity, we use the same notation in this article for deterministic and random vectors, with the proper interpretation implied in context. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[88,939,112,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"992 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[256,939,890,952]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.60, NO.1 FEBRUARY 2024 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[99,44,498,73]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"argued in [9], the state noise vector $\\mathbf{w}_{n,a}$ in (1) also includes the INS measurement error. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[99,75,500,195]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Target dynamics: Since the target is noncooperative, each aircraft also needs to estimate the target position and velocity. We then specify the target kinematics using the constant velocity model [34]. More precisely, let $\\mathbf{x}_{n,o} \\triangleq [x_{n,o} \\dot{x}_{n,o} y_{n,o} \\dot{y}_{n,o} z_{n,o} \\dot{z}_{n,o}]^T$ denote, at instant $n$ , the unknown position and velocity of the noncooperative target $o$ in 3-D Cartesian coordinates. The kinematic dynamic model for the target is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[215,205,498,220]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf {x} _ {n + 1, o} = \\mathbf {F} _ {n, o} \\mathbf {x} _ {n, o} + \\mathbf {w} _ {n, o} \\tag {2}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[99,229,498,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\mathbf{F}_{n,o} = \\mathrm{diag}([\\tilde{\\mathbf{F}}_{n,o}\\tilde{\\mathbf{F}}_{n,o}\\tilde{\\mathbf{F}}_{n,o}])$ and the noise $\\{\\mathbf{w}_{n,o}\\}$ $n\\geq 0$ is a sequence of independent and identically distributed (i.i.d.) zero-mean Gaussian random variables with covariance $\\mathbf{Q}_{n,o} = \\sigma_{\\mathrm{acc},o}^2 *\\mathrm{diag}([\\tilde{\\mathbf{Q}}_{n,o}\\tilde{\\mathbf{Q}}_{n,o}\\tilde{\\mathbf{Q}}_{n,o}])$ in which $\\sigma_{\\mathrm{acc},o}^{2}$ is the variance of the target acceleration noise, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[101,306,413,339]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\tilde {\\mathbf {F}} _ {n, o} = \\left[ \\begin{array}{c c} 1 & T _ {s} \\\\ 0 & 1 \\end{array} \\right] \\text {a n d} \\tilde {\\mathbf {Q}} _ {n, o} = \\left[ \\begin{array}{c c} T _ {s} ^ {3} / 3 & T _ {s} ^ {2} / 2 \\\\ T _ {s} ^ {2} / 2 & T _ {s} \\end{array} \\right].\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[99,340,498,400]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Barometric altitude observation: For the sake of simplicity, the barometric altitude $y_{n,a \\leftarrow ba}^2$ measured at time $n$ at node $a$ is assumed to be an unbiased observation of the aircraft altitude $z_{n,a}$ such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[200,425,498,441]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\ny _ {n, a \\leftarrow b a} = b a \\left(\\mathbf {x} _ {n, a}\\right) + v _ {n, a \\leftarrow b a} \\tag {3}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[99,449,498,479]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $ba(\\mathbf{x}_{n,a}) = z_{n,a}$ and $v_{n,a\\leftarrow ba}$ is a Gaussian noise with zero mean and variance $\\sigma_{ba}^2$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[99,479,500,554]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ground clearance observation: An onboard radar altimeter measures the ground clearance, i.e., the distance from the aircraft to the terrain directly below it. So, the ground clearance is modeled as the real aircraft altitude $z_{n,a}$ minus the elevation of the terrain below the aircraft ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[187,566,498,580]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\ny _ {n, a \\leftarrow g c} = g c \\left(\\mathbf {x} _ {n, a}; \\mathcal {M}\\right) + v _ {n, a \\leftarrow g c} \\tag {4}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[99,589,498,695]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $gc(\\mathbf{x}_{n,a};\\mathcal{M}) = z_{n,a} - ge(\\mathbf{x}_{n,a};\\mathcal{M})$ and $ge(\\mathbf{x};\\mathcal{M})$ is a function that outputs the terrain elevation, given the position and a known 2-D digital elevation map $\\mathcal{M}$ , and the noise $\\{v_{n,a\\leftarrow gc}\\}$ , $n\\geq 0$ , is an i.i.d. sequence with Gaussian distribution $v_{n,a\\leftarrow gc}\\sim \\mathcal{N}(0,\\sigma_{gc}^2)$ . The discrete terrain elevation map is interpolated at each continuous position location as described in [1, Appendix A]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[99,696,500,739]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Aircraft distance observations: The 3-D peer-to-peer distance, a.k.a. slant range, between agents $a$ and $b$ measured from node $a$ at time $n$ is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[189,751,498,767]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\ny _ {n, a \\leftarrow b} = r g \\left(\\mathbf {x} _ {n, a}, \\mathbf {x} _ {n, b}\\right) + v _ {n, a \\leftarrow b} \\tag {5}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[99,775,498,853]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\{v_{n,a\\leftarrow b}\\}$ , $n\\geq 0$ , is a sequence of mutually independent Gaussian random variables with identical distributions $v_{n,a\\leftarrow b}\\sim \\mathcal{N}\\left(\\mathbf{0},\\sigma_{ad}^{2}\\right)$ $\\forall n\\in \\mathbb{Z}_{+}$ , $a\\in \\mathcal{V}$ ; $rg(\\mathbf{x},\\mathbf{x}^{\\prime}) = \\|\\mathbf{x} - \\mathbf{x}^{\\prime}\\|$ , and $\\| \\mathbf{x}\\| ^2\\triangleq \\mathbf{x}^T\\mathbf{x}$ $\\forall \\mathbf{x}\\in \\mathbb{R}^3$ . To avoid clock synchronization issues [18], [35], [37], the measurements ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,44,906,73]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"in (5) are obtained using round-trip time of arrival (RTOA) technique. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,75,910,239]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $\\bar{\\mathbf{N}}(n, a)$ be the open neighborhood of node $a$ at instant $n$ defined as the set of indexes of all nodes in the communication network that are connected to node $a$ at that same instant. For each node $a$ , we collect the state vectors $\\{\\mathbf{x}_{n,b}\\}$ for all nodes $b \\in \\bar{\\mathbf{N}}(n, a)$ , in an extended vector denoted $X_{n,a}$ . In a similar fashion, the internode distance measurements $\\{y_{n,a \\leftarrow b}, \\forall b \\in \\bar{\\mathbf{N}}(n, a)\\}$ , are stacked into the extended vector $Y_{n,a}$ at node $a$ . It follows from (5) that the observations in $Y_{n,a}$ are independent conditioned on any pair of states of the type $(\\mathbf{x}_{n,a}, \\mathbf{x}_{n,b}), b \\in \\bar{\\mathbf{N}}(n, a)$ , at instant $n$ . Thus, we can write ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[565,253,908,287]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\np \\left(\\mathbf {Y} _ {n, a} \\mid \\mathbf {X} _ {n, a}\\right) = \\prod_ {b \\in \\bar {\\mathbf {N}} (n, a)} p \\left(y _ {n, a \\leftarrow b} \\mid \\mathbf {x} _ {n, a}, \\mathbf {x} _ {n, b}\\right). \\tag {6}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,301,910,375]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Target observation: Each aircraft has a radar sensor capable of making 3-D observations of the target location. The target observation, measured by the cooperative aircraft $a$ , to the noncooperative target $o$ at instant $n$ is modeled then as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[601,392,908,407]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf {y} _ {n, a \\leftarrow o} = \\mathbf {t o} (\\mathbf {x} _ {n, a}, \\mathbf {x} _ {n, o}) + \\mathbf {v} _ {n, a \\leftarrow o} \\tag {7}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,421,910,580]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\mathbf{y}_{n,a\\leftarrow o} = \\left[y_{n,a\\leftarrow o}^{(rg)}y_{n,a\\leftarrow o}^{(az)}y_{n,a\\leftarrow o}^{(el)}\\right]^T,\\mathbf{t o}(\\mathbf{x},\\mathbf{x}^{\\prime}) =$ $\\left[rg(\\mathbf{x},\\mathbf{x}^{\\prime})az(\\mathbf{x},\\mathbf{x}^{\\prime})el(\\mathbf{x},\\mathbf{x}^{\\prime})\\right]^{T},az(\\mathbf{x},\\mathbf{x}^{\\prime}) = \\tan^{-1}\\left(\\frac{y^{\\prime} - y}{x^{\\prime} - x}\\right)$ is the azimuth angle and $el(\\mathbf{x},\\mathbf{x}^{\\prime}) = \\tan^{-1}\\left(\\frac{z^{\\prime} - z}{\\sqrt{(x^{\\prime} - x)^{2} + (y^{\\prime} - y)^{2}}}\\right)$ the elevation angle. We further assume that the covariance matrix $\\mathbf{R}_{to}$ has a time-invariant diagonal structure such that $\\mathbf{R}_{to} = diag\\left([ \\sigma_{ro}^{2}\\sigma_{az}^{2}\\sigma_{el}^{2}] \\right]$ . Lastly, note that all cooperative aircraft $a\\in \\mathcal{V}$ make observations of the same single target state $\\mathbf{x}_{n,o}$ independently. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,597,795,611]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"III. JOINT CTAN-DTT FACTOR GRAPH ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,618,908,691]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section, we propose a fully distributed algorithm that enables joint cooperative estimation of the aircraft states, referred to in this article as cooperative terrain-aided navigation, or CTAN, and distributed target tracking, abbreviated DTT. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,693,910,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The DTT routine is based on an adapt-then Combine (ATC) diffusion technique with a modified adapt step where the likelihood of the target measurements for a known aircraft state is approximated by its expected value with respect to (w.r.t.) the latest available belief of the aircraft state after aircraft motion prediction and assimilation of the local barometric altitude and ground clearance sensor measurements. The combine step is then implemented via geometric average (GA) fusion in local network neighborhoods. We show how the proposed ATC scheme can be represented exactly by a factor graph model in which we introduce special factor nodes that enable a distributed implementation of the GA fusion step and the computation of the aforementioned expected values of the likelihood of the target observations at each aircraft node. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[99,880,500,919]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2To shorten notation, we denote an observation vector of the physical property (or of the entity) $t$ at node $a$ at time $n$ as $\\mathbf{y}_{n,a\\leftarrow t}$ , highlighting that node $a$ observed $t$ . ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[101,938,750,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"OLIVEIRA ET AL.: GNSS-DENIED JOINT COOPERATIVE TERRAIN NAVIGATION AND TARGET TRACKING ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[883,940,906,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"993 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[84,44,483,330]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The CTAN routine, without feedback from the DTT subgraph—the connection between CTAN and DTT routines/subgraphs will be discussed later—computes in turn the marginal posterior p.d.f. of the aircraft state at each node $a$ via message passing over a factor subgraph that represents the factorization of the join p.d.f. of the aircraft states and barometric altitude, ground clearance, and internode distance measurements. The computed marginal is further refined by adding an extra message in the graph that represents the expected value of the likelihood of the target observations w.r.t. the latest available belief about the target state outputted by the DTT subgraph. By introducing a special factor node that enables the computation of the aforementioned expected value, we are able to connect the CTAN and DTT subgraphs and arrive at a full joint CTAN-DTT (J-CTAN-DTT) factor graph in which, by following a noniterative message passing schedule according to the usual SPA rules, we can implement the J-CTAN-DTT algorithm described in this section exactly. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,330,483,454]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To enable a fully distributed implementation, each node $a$ in the network tracks a local copy $\\mathbf{x}_{n,o}^{(a)}$ of the target state vector $\\mathbf{x}_{n,o}$ . As an artifact of our distributed implementation, we also denote the target state $\\mathbf{x}_{n,o}^{(a)}$ before the Combine step as $\\tilde{\\mathbf{x}}_{n,o}^{(a)}$ , and the aircraft state $\\mathbf{x}_{n,a}$ before the assimilation of the internode distance measurements as $\\tilde{\\mathbf{x}}_{n,a}$ . We emphasize, however, that both notations, with and without the diacritic mark, represent the same physical state vector. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,468,227,481]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. Factor Notation ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[84,489,483,563]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To obtain a factor graph representation of the operations in this section, we consider the variable nodes $\\mathbf{x}_{n - 1,a}$ ; $\\tilde{\\mathbf{x}}_{n,a}$ ; $\\mathbf{x}_{n,a}$ ; $\\mathbf{x}_{n - 1,o}^{(a)}$ ; $\\tilde{\\mathbf{x}}_{n,o}^{(a)}$ and $\\mathbf{x}_{n,o}^{(a)}$ at each network node $a\\in \\{1,2,\\dots,A\\}$ , and define the following local factor nodes at the same node. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[86,565,256,579]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1) Aircraft motion factor ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[202,589,482,628]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} f _ {n, a} \\triangleq f _ {n, a} (\\tilde {\\mathbf {x}} _ {n, a}, \\mathbf {x} _ {n - 1, a}) \\\\ \\triangleq p \\left(\\tilde {\\mathbf {x}} _ {n, a} \\mid \\mathbf {x} _ {n - 1, a}\\right). \\tag {8} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[86,638,271,651]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2) Ground clearance factor ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[202,662,482,700]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} g c _ {n, a} \\triangleq g c _ {n, a} (\\tilde {\\mathbf {x}} _ {n, a}) \\\\ \\triangleq p \\left(y _ {n, a \\leftarrow g c} | \\tilde {\\mathbf {x}} _ {n, a}\\right). \\tag {9} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[86,710,282,724]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3) Barometric altitude factor ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[200,734,482,772]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} b a _ {n, a} \\triangleq b a _ {n, a} (\\tilde {\\mathbf {x}} _ {n, a}) \\\\ \\triangleq p \\left(y _ {n, a \\leftarrow b a} \\mid \\tilde {\\mathbf {x}} _ {n, a}\\right). \\tag {10} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[84,782,462,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4) Internode distance factor between node $a$ and node $b$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[179,806,482,843]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} a d _ {n, a \\leftarrow b} \\triangleq a d _ {n, a \\leftarrow b} (\\mathbf {x} _ {n, a}, \\tilde {\\mathbf {x}} _ {n, b}) \\\\ = p \\left(y _ {n, a \\leftarrow b} | \\mathbf {x} _ {n, a}, \\tilde {\\mathbf {x}} _ {n, b}\\right). \\tag {11} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[86,853,276,868]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5) Target observation factor ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[184,876,482,917]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} t o _ {n, o} ^ {(a)} \\triangleq t o _ {n, o} ^ {(a)} \\left(\\tilde {\\mathbf {x}} _ {n, a}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right) \\\\ \\triangleq p \\left(\\mathbf {y} _ {n, a \\leftarrow o} | \\tilde {\\mathbf {x}} _ {n, a}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right). \\tag {12} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[496,44,656,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6) Target motion factor ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[614,65,892,107]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} f _ {n, o} ^ {(a)} \\triangleq f _ {n, o} ^ {(a)} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}, \\mathbf {x} _ {n - 1, o} ^ {(a)}\\right) \\\\ \\triangleq p \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)} \\mid \\mathbf {x} _ {n - 1, o} ^ {(a)}\\right). \\tag {13} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[496,116,781,132]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B. Decoupled Cooperative Localization ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[495,138,893,213]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $\\gamma_{n,a}(\\mathbf{x}_{n,a})$ denote the aircraft state belief at each node $a$ at instant $n$ . To update the belief $\\gamma_{n,a}$ sequentially in time, the aircraft network may perform cooperative localization independently of the distributed tracking task using the factor graph-based CTAN algorithm in [1]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[495,214,893,335]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Specifically, in a decoupled cooperative localization scenario, let $\\mathbf{X}_n$ denote the collection of all aircraft states $\\{\\mathbf{x}_{k,a}\\} ,\\mathbf{Z}_n$ denote the collection of all local ground clearance and barometric altitude measurements $\\{y_{k,a\\leftarrow gc};y_{k,a\\leftarrow ba}\\}$ and let $\\mathbf{Y}_n$ represent the set of all internode measurements $\\{y_{k,a\\leftarrow b}\\}$ for $a\\in \\{1,\\dots ,A\\}$ $b\\in \\{1,\\dots ,A\\}$ $a\\neq b$ and $k\\in$ $\\{0,\\ldots ,n\\}$ . Given the conditional independence assumptions in the model, the joint p.d.f. can be written as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,342,892,443]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} p \\left(\\mathbf {X} _ {n}, \\mathbf {Y} _ {n}, \\mathbf {Z} _ {n}\\right) = \\prod_ {k = 1} ^ {n} \\prod_ {a = 1} ^ {A} p \\left(\\mathbf {x} _ {k, a} | \\mathbf {x} _ {k - 1, a}\\right) \\\\ \\times p \\left(y _ {k, a \\leftarrow g c} | \\mathbf {x} _ {k, a}\\right) p \\left(y _ {k, a \\leftarrow b a} | \\mathbf {x} _ {k, a}\\right) \\\\ \\times \\prod_ {b \\in \\tilde {\\mathbf {N}} (k, a)} p \\left(y _ {k, a \\leftarrow b} | \\mathbf {x} _ {k, a}, \\mathbf {x} _ {k, b}\\right). \\tag {14} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[495,449,893,510]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As explained before, for convenience and to enable the integration between the CTAN and DTT subgraphs, we introduce an auxiliary variable $\\tilde{\\mathbf{x}}_{n,a}$ which is actually identical to $\\mathbf{x}_{n,a}$ . We then rewrite the factorization in (14) as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[506,518,892,732]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} p \\left(\\mathbf {X} _ {n}, \\tilde {\\mathbf {X}} _ {n}, \\mathbf {Y} _ {n}, \\mathbf {Z} _ {n}\\right) = \\prod_ {k = 1} ^ {n} \\prod_ {a = 1} ^ {A} p \\left(\\tilde {\\mathbf {x}} _ {k, a} | \\mathbf {x} _ {k - 1, a}\\right) \\delta \\left(\\mathbf {x} _ {k, a} - \\tilde {\\mathbf {x}} _ {k, a}\\right) \\\\ \\times p \\left(y _ {k, a \\leftarrow g c} | \\tilde {\\mathbf {x}} _ {k, a}\\right) p \\left(y _ {k, a \\leftarrow b a} | \\tilde {\\mathbf {x}} _ {k, a}\\right) \\\\ \\times \\prod_ {b \\in \\tilde {\\mathbf {N}} (k, a)} p \\left(y _ {k, a \\leftarrow b} | \\mathbf {x} _ {k, a}, \\tilde {\\mathbf {x}} _ {k, b}\\right) \\\\ \\equiv \\prod_ {k = 1} ^ {n} \\prod_ {a = 1} ^ {A} f _ {k, a} (\\tilde {\\mathbf {x}} _ {k, a}, \\mathbf {x} _ {k - 1, a}) \\delta (\\mathbf {x} _ {k, a} \\\\ \\left. - \\tilde {\\mathbf {x}} _ {k, a}\\right) \\times g c _ {k, a} \\left(\\tilde {\\mathbf {x}} _ {k, a}\\right) b a _ {k, a} \\left(\\tilde {\\mathbf {x}} _ {k, a}\\right) \\\\ \\times \\prod_ {b \\in \\tilde {\\mathbf {N}} (k, a)} a d _ {k, a \\leftarrow b} \\left(\\mathbf {x} _ {k, a}, \\tilde {\\mathbf {x}} _ {k, b}\\right) \\tag {15} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[495,739,892,768]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $f_{n,a}$ , $gc_{n,a}$ , $ba_{n,a}$ , and $ad_{n,a \\leftarrow b}$ are the factors previously introduced in (8), (9), (10), and (11), respectively. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[495,768,893,875]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The factorization in (15) is represented by the factor graph in Fig. 1, shown here for two neighboring network nodes $a$ and $b$ . The messages computed at nodes $a$ and $b$ are highlighted in blue and red, respectively, with messages that are passed between nodes shown in dashed lines and intranode messages in solid lines. The factor represented by \" $=$ \" symbol corresponds to the Dirac delta operator in (15). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[495,875,892,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By applying the standard SPA rules, the noniterative message passing schedule at node $a$ corresponding to the factor graph in Fig. 1 runs as described in the sequel. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[88,940,111,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"994 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[256,939,888,952]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.60, NO.1 FEBRUARY 2024 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[120,46,478,251]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 1. CTAN factor subgraph without DTT feedback. \"]","image_footnote":"[]","img_path":"images/0215/8398557ae5c91a563fffc0e7cb08acd8c0f8471328776a68172c3c53a5a88584.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[101,294,498,325]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"First, we compute the aircraft motion prediction message $\\alpha_{n,a}$ such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,330,498,375]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\alpha_ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) \\propto \\int f _ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}, \\mathbf {x} _ {n - 1, a}\\right) \\gamma_ {n - 1, a} \\left(\\mathbf {x} _ {n - 1, a}\\right) d \\mathbf {x} _ {n - 1, a}. \\tag {16}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[101,376,498,407]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Next, we define the barometric altitude and ground clearance messages, respectively, $\\kappa_{n,a}$ and $\\beta_{n,a}$ , such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[200,412,496,450]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\kappa_ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) \\triangleq p \\left(y _ {n, a - b a} | \\tilde {\\mathbf {x}} _ {n, a}\\right) \\\\ = b a _ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) \\tag {17} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[200,454,498,493]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\beta_ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) \\triangleq p \\left(y _ {n, a \\leftarrow g c} | \\tilde {\\mathbf {x}} _ {n, a}\\right) \\\\ = g c _ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right). \\tag {18} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[101,500,498,529]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The assimilation of the intranode distance measurements is done by computing the messages ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[137,537,498,554]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\iota_ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) \\propto \\alpha_ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) \\kappa_ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) \\beta_ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) \\tag {19}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[135,556,498,587]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\epsilon_ {n, a} \\left(\\mathbf {x} _ {n, a}\\right) \\propto \\int \\delta \\left(\\mathbf {x} _ {n, a} - \\tilde {\\mathbf {x}} _ {n, a}\\right) \\iota_ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) d \\tilde {\\mathbf {x}} _ {n, a} \\tag {20}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[119,589,496,636]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\xi_ {n, a \\leftarrow b} (\\mathbf {x} _ {n, a}) \\propto \\int a d _ {n, a \\leftarrow b} (\\mathbf {x} _ {n, a}, \\tilde {\\mathbf {x}} _ {n, b}) \\lambda_ {n, a \\leftarrow b} (\\tilde {\\mathbf {x}} _ {n, b}) d \\tilde {\\mathbf {x}} _ {n, b} \\tag {21}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[101,643,498,672]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\lambda_{n,a\\leftarrow b}(\\tilde{\\mathbf{x}}_{n,b})$ is identical to the message $\\iota_{n,b}(\\tilde{\\mathbf{x}}_{n,b})$ and the functions $\\epsilon_{n,a}$ and $\\iota_{n,a}$ are also identical. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[125,672,491,689]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The final update belief of the state $\\mathbf{x}_{n,a}$ is the message ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[156,695,498,729]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\gamma_ {n, a} (\\mathbf {x} _ {n, a}) \\propto \\epsilon_ {n, a} (\\mathbf {x} _ {n, a}) \\prod_ {b \\in \\bar {\\mathbf {N}} (n, a)} \\xi_ {n, a \\leftarrow b} (\\mathbf {x} _ {n, a}). \\tag {22}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[101,736,498,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note that messages (16)-(18) correspond to the self-localization (self-loc) subgraph highlighted in Fig. 1, whereas messages (19)-(22) belong to the cooperative localization (coop-loc) subgraph. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[101,808,473,824]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Adapt-Then-Combine Distributed Target Tracking ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[101,829,500,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ATC diffusion is a well-known method for distributed target tracking in the literature [5], [6], [7]. Unlike consensus methods, diffusion techniques do not force all network nodes to have the same target state estimate at each time instant. Instead, each node computes its own posterior p.d.f. of the target state, which is, in principle, different ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,44,908,194]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"from the posterior computed by the other nodes. Ideally, over time however, as $n \\to \\infty$ , the local posteriors should converge to a consensus posterior [6] so that the variance of the local state estimates is reduced. The practical advantage of diffusion methods over consensus techniques is that they do not require multiple iterative communication between network nodes in the time interval between consecutive sensor measurements and, therefore, are more suitable for use in real-time scenarios due to their reduced internode communication cost. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,196,908,268]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the sequel, we propose an approach to integrate ATC diffusion with factor graph cooperative localization. We detail the proposed adapt and combine steps separately and then show how to represent them exactly by a factor graph model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,271,908,359]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1) Adapt Step: For known aircraft positions, the algorithm would start with a local adapt step at each node $a$ consisting of the computation of the predicted target state p.d.f. using the target motion transition p.d.f. followed by the correction of that predicted p.d.f, using the likelihood function of the target observations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,361,910,465]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In our signal model, however, since the aircraft states are unknown, we propose to replace the ideal likelihood of the target observations given the true (deterministic) aircraft state by the expected value of the same likelihood given the unknown (random) aircraft state w.r.t. to the latest available local posterior p.d.f. $\\tilde{p}_{n,a}$ of the aircraft $a$ at node $a$ at instant $n$ , i.e., ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[578,472,908,503]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\int p \\left(\\mathbf {y} _ {n, a \\leftarrow o} | \\tilde {\\mathbf {x}} _ {n, a}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right) \\tilde {p} _ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) d \\tilde {\\mathbf {x}} _ {n, a}. \\tag {23}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,507,910,583]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We obtain $\\tilde{p}_{n,a}(\\tilde{\\mathbf{x}}_{n,a})$ from the previous posterior $p_{n - 1,a}$ $(\\mathbf{x}_{n - 1,a})$ using the dynamic model for the agent $a$ and assimilating the ground clearance and barometric altitude observations, respectively, $y_{n,a\\leftarrow gc}$ and $y_{n,a\\leftarrow ba}$ , which are available at node $a$ at instant $n$ . Specifically, we make ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[514,590,908,657]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\tilde {p} _ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) \\propto p \\left(y _ {n, a \\leftarrow g c} \\mid \\tilde {\\mathbf {x}} _ {n, a}\\right) p \\left(y _ {n, a \\leftarrow b a} \\mid \\tilde {\\mathbf {x}} _ {n, a}\\right) \\\\ \\times \\left[ \\int p \\left(\\tilde {\\mathbf {x}} _ {n, a} \\mid \\mathbf {x} _ {n - 1, a}\\right) p _ {n - 1, a} \\left(\\mathbf {x} _ {n - 1, a}\\right) d \\mathbf {x} _ {n - 1, a} \\right]. \\tag {24} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,664,872,679]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The complete adapt step expression is written then as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[519,684,908,768]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\tilde {p} _ {n, o} ^ {(a)} (\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}) \\propto \\left[ \\int p (\\mathbf {y} _ {n, a \\leftarrow o} | \\tilde {\\mathbf {x}} _ {n, a}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}) \\tilde {p} _ {n, a} (\\tilde {\\mathbf {x}} _ {n, a}) d \\tilde {\\mathbf {x}} _ {n, a} \\right] \\\\ \\times \\left[ \\int p \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)} \\mid \\mathbf {x} _ {n - 1, o} ^ {(a)}\\right) p _ {n - 1, a} \\left(\\mathbf {x} _ {n - 1, o} ^ {(a)}\\right) d \\mathbf {x} _ {n - 1, o} ^ {(a)} \\right] \\tag {25} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,773,754,789]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"with $\\tilde{p}_{n,a}(\\tilde{\\mathbf{x}}_{n,a})$ computed as in (24). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,789,908,869]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2) Distributed Combine Step: Let $\\mathbf{N}(n,a)$ denote the closed neighborhood of the node $a$ , defined as in [1] as $\\tilde{\\mathbf{N}}(n,a) \\cup \\{a\\}$ , at instant $n$ . Given now the target state p.d.f.'s $\\tilde{p}_{n,o}^{(b)}(\\tilde{\\mathbf{x}}_{n,o}^{(b)})$ at each node $b \\in \\mathbf{N}(n,a)$ , we want to obtain an artificial transition p.d.f. $c_{n,o}^{(a,b)}(\\mathbf{x}_{n,o}^{(a)}, \\tilde{\\mathbf{x}}_{n,o}^{(b)})$ at node $a$ such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[532,875,906,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\int c _ {n, o} ^ {(a, b)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}\\right) \\tilde {p} _ {n, o} ^ {(b)} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}\\right) d \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)} \\propto \\left[ \\tilde {p} _ {n, o} ^ {(b)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) \\right] ^ {r _ {a, b}} \\tag {26}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[102,938,750,952]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"OLIVEIRA ET AL.: GNSS-DENIED JOINT COOPERATIVE TERRAIN NAVIGATION AND TARGET TRACKING ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[885,940,906,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"995 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[84,44,482,89]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $r_{a,b}$ is a set of network combination coefficients satisfying $\\sum_{b\\in \\mathbf{N}(n,a)}r_{a,b} = 1$ , and $r_{a,b} > 0\\forall a$ and $\\forall b\\in$ $\\mathbf{N}(n,a)$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,89,482,133]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It follows then that the desire merged p.d.f. is computed at node $a$ using geometric average fusion [27], [28], [29] as: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[86,137,482,214]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} p _ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) \\propto \\prod_ {b \\in \\mathbf {N} (n, a)} \\left[ \\int c _ {n, o} ^ {(a, b)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}\\right) \\tilde {p} _ {n, o} ^ {(b)} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}\\right) d \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)} \\right] \\\\ \\propto \\prod_ {b \\in \\mathbf {N} (n, a)} \\left[ \\tilde {p} _ {n, o} ^ {(b)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) \\right] ^ {r _ {a, b}}. \\tag {27} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[84,220,483,281]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"After some algebraic manipulations, it is possible to show [28], [29] that the fusion rule in (27) yields the merged p.d.f. $p$ that minimizes the weighted average Kullback-Leibler divergence ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[200,287,367,321]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\sum_ {b \\in \\mathbf {N} (n, a)} r _ {a, b} D _ {K L} \\left(p | | \\tilde {p} _ {n, o} ^ {(b)}\\right)\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[84,328,482,381]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $D_{KL}(f||g)\\triangleq \\mathbb{E}_f\\left[\\log \\left(\\frac{f}{g}\\right)\\right]$ and $\\mathbb{E}_f[\\cdot ]$ denotes the expectation of its argument, which is computed using the p.d.f. $f$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,381,483,434]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In particular, if $\\tilde{p}_{n,o}^{(b)}\\left(\\tilde{\\mathbf{x}}_{n,o}^{(b)}\\right)$ has a Gaussian parametric representation of the form $\\mathcal{N}\\left(\\tilde{\\mathbf{x}}_{n,o}^{(b)}|\\tilde{\\mathbf{m}}_{n,o}^{(b)},\\tilde{\\mathbf{\\Sigma}}_{n,o}^{(b)}\\right)$ , it suffices to make ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[151,440,482,460]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nc _ {n, o} ^ {(a, b)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}\\right) \\triangleq \\mathcal {N} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)} | \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}, \\tilde {\\mathbf {Q}} _ {n, o} ^ {(a, b)}\\right) \\tag {28}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[86,465,151,479]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[197,484,482,516]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\tilde {\\mathbf {Q}} _ {n, o} ^ {(a, b)} + \\tilde {\\boldsymbol {\\Sigma}} _ {n, o} ^ {(b)} = \\frac {1}{r _ {a , b}} \\tilde {\\boldsymbol {\\Sigma}} _ {n, o} ^ {(b)} \\tag {29}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[86,521,194,537]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"or, equivalently ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[197,542,482,578]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\tilde {\\mathbf {Q}} _ {n, o} ^ {(a, b)} = \\left[ \\frac {1}{r _ {a , b}} - 1 \\right] \\tilde {\\boldsymbol {\\Sigma}} _ {n, o} ^ {(b)}. \\tag {30}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[86,582,168,595]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In that case ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[109,599,482,752]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\int c _ {n, o} ^ {(a, b)} (\\mathbf {\\tilde {x}} _ {n, o} ^ {(a)}, \\mathbf {\\tilde {x}} _ {n, o} ^ {(b)}) \\tilde {p} _ {n, o} ^ {(b)} \\left(\\mathbf {\\tilde {x}} _ {n, o} ^ {(b)}\\right) d \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)} \\\\ = \\int \\mathcal {N} (\\mathbf {x} _ {n, o} ^ {(a)} | \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}, \\tilde {\\mathbf {Q}} _ {n, o} ^ {(a, b)}) \\mathcal {N} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(b)} | \\tilde {\\mathbf {m}} _ {n, o} ^ {(b)}, \\tilde {\\mathbf {z}} _ {n, o} ^ {(b)}\\right) d \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)} \\\\ = \\mathcal {N} \\left(\\mathbf {x} _ {n, o} ^ {(a)} | \\tilde {\\mathbf {m}} _ {n, o} ^ {(b)}, \\tilde {\\mathbf {Q}} _ {n, o} ^ {(a, b)} + \\tilde {\\boldsymbol {\\Sigma}} _ {n, o} ^ {(b)}\\right) \\\\ = \\mathcal {N} \\left(\\mathbf {x} _ {n, o} ^ {(a)} | \\tilde {\\mathbf {m}} _ {n, o} ^ {(b)}, \\frac {1}{r _ {a , b}} \\tilde {\\boldsymbol {\\Sigma}} _ {n, o} ^ {(b)}\\right) \\\\ \\propto \\left[ \\tilde {p} _ {n, o} ^ {(b)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) \\right] ^ {r _ {a, b}}. \\tag {31} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[84,756,483,803]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3) DTT Factor Graph Representation: To obtain a factor graph representation of the ATC distributed target tracking algorithm, we start with the incoming messages ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[173,808,482,827]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\gamma_ {n - 1, a} \\left(\\mathbf {x} _ {n - 1, a}\\right) \\triangleq p _ {n - 1, a} \\left(\\mathbf {x} _ {n - 1, a}\\right) \\tag {32}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[173,829,482,854]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\gamma_ {n - 1, o} ^ {(a)} \\left(\\mathbf {x} _ {n - 1, o} ^ {(a)}\\right) \\triangleq p _ {n - 1, o} ^ {(a)} \\left(\\mathbf {x} _ {n - 1, o} ^ {(a)}\\right) \\tag {33}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[84,859,482,888]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"at instant $n - 1$ , where $\\gamma_{n - 1,a}$ is the same message previously used in the CTAN routine in (16). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,890,482,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Next, using the messages $\\alpha_{n,a},\\kappa_{n,a}$ , and $\\beta_{n,a}$ also previously defined in the CTAN subgraph, respectively, in (16), ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[495,44,893,89]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(17), and (18), it follows from (24) and the definition of the factor $f_{n,a}$ in (8) that the message $\\zeta_{n,a}(\\tilde{\\mathbf{x}}_{n,a}) \\triangleq \\tilde{p}_{n,a}(\\tilde{\\mathbf{x}}_{n,a})$ is computed as: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[553,98,892,114]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\zeta_ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) \\propto \\alpha_ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) \\kappa_ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) \\beta_ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right). \\tag {34}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[495,121,893,167]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On the other hand, defining the message $\\nu_{n,o}^{(a)}\\left(\\tilde{\\mathbf{x}}_{n,o}^{(a)}\\right) \\triangleq \\tilde{p}_{n,o}^{(a)}\\left(\\tilde{\\mathbf{x}}_{n,o}^{(a)}\\right)$ , we can rewrite (25) using (12), (13), and (33) as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,172,892,265]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\nu_ {n, o} ^ {(a)} (\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}) \\propto \\left[ \\int f _ {n, o} ^ {(a)} (\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}, \\mathbf {x} _ {n - 1, o} ^ {(a)}) \\gamma_ {n - 1, o} ^ {(a)} (\\mathbf {x} _ {n - 1, o} ^ {(a)}) d \\mathbf {x} _ {n - 1, o} ^ {(a)} \\right] \\\\ \\times \\left[ \\int t o _ {n, o} ^ {(a)} \\left(\\tilde {\\mathbf {x}} _ {n, a}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right) \\zeta_ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) d \\tilde {\\mathbf {x}} _ {n, a} \\right] \\\\ \\triangleq \\alpha_ {n, o} ^ {(a)} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right) \\phi_ {n, o} ^ {(a)} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right) \\tag {35} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[496,271,544,284]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[501,289,892,335]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\alpha_ {n, o} ^ {(a)} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right) \\triangleq \\int f _ {n, o} ^ {(a)} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}, \\mathbf {x} _ {n - 1, o} ^ {(a)}\\right) \\gamma_ {n - 1, o} ^ {(a)} \\left(\\mathbf {x} _ {n - 1, o} ^ {(a)}\\right) d \\mathbf {x} _ {n - 1, o} ^ {(a)} \\tag {36}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[503,339,892,371]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\phi_ {n, o} ^ {(a)} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right) \\triangleq \\int t o _ {n, o} ^ {(a)} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}, \\tilde {\\mathbf {x}} _ {n, a}\\right) \\zeta_ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) d \\tilde {\\mathbf {x}} _ {n, a}. \\tag {37}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[495,376,893,406]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the sequel, we introduce the local and remote combine factors, respectively, as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[616,412,772,431]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nc l _ {n, o} ^ {(a)} \\triangleq c l _ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right)\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[651,434,892,452]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\triangleq c _ {n, o} ^ {(a, a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right) \\tag {38}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[594,455,794,474]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nc d _ {n, o} ^ {(a \\leftarrow b)} \\triangleq c d _ {n, o} ^ {(a \\leftarrow b)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}\\right)\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[651,476,890,494]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\triangleq c _ {n, o} ^ {(a, b)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}\\right). \\tag {39}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[496,501,879,517]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finally, using (38) and (39) and introducing the message ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[614,522,892,541]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\gamma_ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) \\triangleq p _ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) \\tag {40}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[496,547,622,561]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"we rewrite (27) as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[500,568,892,694]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\gamma_ {n, o} ^ {(a)} \\left(\\mathbf {\\tilde {x}} _ {n, o} ^ {(a)}\\right) \\triangleq \\prod_ {b \\in \\tilde {\\mathbf {N}} (n, a)} \\int c d _ {n, o} ^ {(a \\leftarrow b)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}\\right) \\lambda_ {n, o} ^ {(a \\leftarrow b)} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}\\right) d \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)} \\\\ \\times \\int c l _ {n, o} ^ {(a)} \\left(\\mathbf {\\tilde {x}} _ {n, o} ^ {(a)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right) v _ {n, o} ^ {(a)} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right) d \\tilde {\\mathbf {x}} _ {n, o} ^ {(a)} \\\\ \\triangleq \\left[ \\prod_ {b \\in \\overline {{\\mathbf {N}}} (n, a)} \\theta_ {n, o} ^ {(a \\leftarrow b)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) \\right] \\psi_ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) \\tag {41} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[495,700,893,731]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\bar{\\mathbf{N}}(n, a)$ denotes again the open neighborhood of the node $a$ at instant $n$ (i.e., excluding node $a$ itself) and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,734,892,756]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\lambda_ {n, o} ^ {(a \\leftarrow b)} (\\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}) \\triangleq v _ {n, o} ^ {(b)} (\\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}) \\tag {42}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[509,757,890,804]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\theta_ {n, o} ^ {(a \\leftarrow b)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) \\triangleq \\int c d _ {n, o} ^ {(a \\leftarrow b)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}\\right) \\lambda_ {n, o} ^ {(a \\leftarrow b)} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}\\right) d \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)} \\tag {43}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[524,808,892,840]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\psi_ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) \\triangleq \\int c l _ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right) v _ {n, o} ^ {(a)} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right) d \\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}. \\tag {44}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[495,844,893,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Equations (32)-(44) can be represented by a DTT factor subgraph as illustrated in Fig. 2 with the output messages computed using the usual SPA rules. Specifically, messages (35)-(37) implement the SPA over the local target tracking (local- tt) subgraph. Messages (41)-(44) in turn implement ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[88,939,111,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"996 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[256,939,888,952]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.60, NO.1 FEBRUARY 2024 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[122,46,480,421]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 2. DTT factor subgraph. \"]","image_footnote":"[]","img_path":"images/0945/7f94358a1a62c654d71a0c35e724bb955ffaf017ffe8fa16e7fa44ce8e6ad2d2.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[101,453,498,484]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"the SPA over the distributed target tracking (dist- tt) subgraph. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[102,494,500,526]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"D. Cooperative Localization With Distributed Target Tracking Feedback ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[101,531,500,638]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $\\mathbf{O}_n$ represent the set of all target measurements $\\{\\mathbf{y}_{k,a\\leftarrow o}\\}$ for $a\\in \\{1,\\dots ,A\\}$ , and $k\\in \\{0,\\ldots ,n\\}$ . The belief of the unknown aircraft state at node $a$ and instant $n$ may be improved if we also assimilate the target observations $\\{\\mathbf{y}_{n,a\\leftarrow o}\\}$ to infer the state $\\mathbf{x}_{n,a}$ . In fact, if the target state were perfectly known, say $\\bar{\\mathbf{x}}_{n,o}$ , we could marginalize instead the joint p.d.f. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,643,498,768]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} p \\left(\\mathbf {X} _ {n}, \\mathbf {Y} _ {n}, \\mathbf {Z} _ {n}, \\mathbf {O} _ {n}\\right) = \\prod_ {k = 1} ^ {n} \\prod_ {a = 1} ^ {A} p \\left(\\mathbf {x} _ {k, a} \\mid \\mathbf {x} _ {k - 1, a}\\right) \\\\ \\times p \\left(y _ {k, a \\leftarrow g c} \\mid \\mathbf {x} _ {k, a}\\right) p \\left(y _ {k, a \\leftarrow b a} \\mid \\mathbf {x} _ {k, a}\\right) \\\\ \\times \\left[ \\prod_ {b \\in \\tilde {\\mathbf {N}} (k, a)} p \\left(y _ {k, a - b} \\mid \\mathbf {x} _ {k, a}, \\mathbf {x} _ {k, b}\\right) \\right] \\\\ \\times p \\left(\\mathbf {y} _ {k, a \\leftarrow o} \\mid \\mathbf {x} _ {k, a}, \\bar {\\mathbf {x}} _ {k, o}\\right) \\tag {45} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[101,773,500,851]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where, for the purpose of this discussion, $\\bar{\\mathbf{x}}_{k,o}$ in (45) is interpreted as being deterministic and known, as the position of an anchor node, for example. The factorization in (45) introduces an additional factor in Fig. 1 and modifies the expression for $\\gamma_{n,a}(\\mathbf{x}_{n,a})$ in (22) to ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,854,498,917]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\gamma_ {n, a} (\\mathbf {x} _ {n, a}) \\propto \\epsilon_ {n, a} (\\mathbf {x} _ {n, a}) \\left[ \\prod_ {b \\in \\tilde {\\mathbf {N}} (n, a)} \\xi_ {n, a \\leftarrow b} (\\mathbf {x} _ {n, a}) \\right] \\\\ \\times p \\left(\\mathbf {y} _ {n, a \\leftarrow o} \\mid \\mathbf {x} _ {n, a}, \\bar {\\mathbf {x}} _ {n, o}\\right). \\tag {46} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,44,908,164]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"However, as we do not know the actual $\\bar{\\mathbf{x}}_{k,o}$ , we propose again an approximation where we replace the last term on the right-hand side of (46) with its expected value with respect to the most recent belief that node $a$ has about the state of $\\mathbf{x}_{n,o}$ at instant $n$ . Specifically, we propose to compute that expectation using the belief $\\gamma_{n,o}^{(a)}\\left(\\mathbf{x}_{n,o}^{(a)}\\right)$ in (41) obtained at the end of the ATC distributed target tracking procedure in section III-C. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,164,908,210]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For convenience, we introduce a message $\\rho_{n,o}^{(a)}\\left(\\mathbf{x}_{n,o}^{(a)}\\right)$ which is identical to $\\gamma_{n,o}^{(a)}\\left(\\mathbf{x}_{n,o}^{(a)}\\right)$ . We then define the so-called feedback factor ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[609,222,908,263]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} f b _ {n, o} ^ {(a)} \\triangleq f b _ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, a}, \\mathbf {x} _ {n, o} ^ {(a)}\\right) \\\\ \\triangleq p \\left(\\mathbf {y} _ {n, a \\leftarrow o} \\mid \\mathbf {x} _ {n, a}, \\mathbf {x} _ {n, o} ^ {(a)}\\right). \\tag {47} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,272,908,301]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The modified updated belief of $\\mathbf{x}_{n,a}$ at node $a$ at instant $n$ is now ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[529,310,908,367]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\gamma_ {n, a} \\left(\\mathbf {x} _ {n, a}\\right) \\propto \\left[ \\prod_ {b \\in \\tilde {\\mathbf {N}} (n, a)} \\xi_ {n, a \\leftarrow b} \\left(\\mathbf {x} _ {n, a}\\right) \\right] \\epsilon_ {n, a} \\left(\\mathbf {x} _ {n, a}\\right) \\tau_ {n, a} \\left(\\mathbf {x} _ {n, a}\\right) \\tag {48}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,378,643,393]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where the message ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[547,402,906,455]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\tau_ {n, a} (\\mathbf {x} _ {n, a}) \\triangleq \\int f b _ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, a}, \\mathbf {x} _ {n, o} ^ {(a)}\\right) \\rho_ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) d \\mathbf {x} _ {n, o} ^ {(a)} \\\\ = \\mathrm {E} _ {\\rho_ {n, o} ^ {(a)}} \\left[ p \\left(\\mathbf {y} _ {n, a \\leftarrow o} | \\mathbf {x} _ {n, a}, \\mathbf {x} _ {n, o} ^ {(a)}\\right) \\right]. \\tag {49} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,467,910,590]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By adding the factor $fb_{n,o}^{(a)}$ and the previous factor $to_{n,o}^{(a)}$ in (12), and defining the messages $\\tau_{n,a}$ in (49) and $\\phi_{n,o}^{(a)}$ in (37), it is possible to further connect the dist-rtt and coop-loc subgraphs into a single joint CTAN-DTT (J-CTAN-DTT) factor graph shown in Fig. 3, with the message passing equations obtained from the application of the SPA to this graph being identical to the message equations shown in sections III-B, III-C, and III-D. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,606,823,619]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IV.MESSAGE PASSING IMPLEMENTATION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[511,627,908,674]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section, we propose a GMM/SMC implementation for the message passing schedule in the factor graph in Fig. 3 that represents the J-CTAN-DTT algorithm. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,688,782,703]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. Prediction and Local Observations ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[511,710,908,786]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In our proposed implementation, we assume that the incoming messages $\\gamma_{n - 1,a}$ and $\\gamma_{n - 1,o}^{(a)}$ , which represent, respectively, the final beliefs of the aircraft $a$ and the target state representation on aircraft $a$ at instant $n - 1$ , are modeled by parametric GMMs as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[524,796,908,853]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\check {\\gamma} _ {n - 1, a} \\left(\\mathbf {x} _ {n - 1, a}\\right) = \\sum_ {l = 1} ^ {L} \\eta_ {n - 1, a} ^ {(l)} \\mathcal {N} \\left(\\mathbf {x} _ {n - 1, a} \\mid \\mathbf {m} _ {n - 1, a} ^ {(l)}, \\boldsymbol {\\Sigma} _ {n - 1, a} ^ {(l)}\\right) \\tag {50}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[521,858,908,916]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\check {\\gamma} _ {n - 1, o} ^ {(a)} \\left(\\mathbf {x} _ {n - 1, o} ^ {(a)}\\right) = \\sum_ {l = 1} ^ {L} \\eta_ {n - 1, o} ^ {(a, l)} \\mathcal {N} \\left(\\mathbf {x} _ {n - 1, o} ^ {(a)} | \\mathbf {m} _ {n - 1, o} ^ {(a, l)}, \\boldsymbol {\\Sigma} _ {n - 1, o} ^ {(a, l)}\\right) \\tag {51}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[102,938,750,952]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"OLIVEIRA ET AL.: GNSS-DENIED JOINT COOPERATIVE TERRAIN NAVIGATION AND TARGET TRACKING ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[883,940,906,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"997 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[106,45,467,417]","code_body":null,"code_caption":null,"image_caption":"[\"Fig.3.J-CTAN-DTT full factor graph. \"]","image_footnote":"[]","img_path":"images/0790/a5faa338d7cc43a3c7b48186f2dff8c79fa4498508c0f0fccfae0f51cd0387af.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[84,463,482,508]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $L$ denotes the number of modes in the GMM model. On the other hand, from the aircraft kinematic model in (1), it follows that: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[94,520,482,560]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} f _ {n, a} (\\tilde {\\mathbf {x}} _ {n, a}, \\mathbf {x} _ {n - 1, a}) = p (\\tilde {\\mathbf {x}} _ {n, a} | \\mathbf {x} _ {n - 1, a}) \\\\ = \\mathcal {N} \\left(\\tilde {\\mathbf {x}} _ {n, a} \\mid \\mathbf {x} _ {n - 1, a} + \\mathbf {u} _ {n - 1, a}, \\mathbf {Q} _ {n - 1, a}\\right). \\tag {52} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[84,571,483,602]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Therefore, by substituting (52) and (50) into (16), we approximate the message $\\alpha_{n,a}(\\tilde{\\mathbf{x}}_{n,a})$ as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,611,483,758]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\check {\\alpha} _ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) = \\int \\left[ \\mathcal {N} \\left(\\tilde {\\mathbf {x}} _ {n, a} \\mid \\mathbf {x} _ {n - 1, a} + \\mathbf {u} _ {n - 1, a}, \\mathbf {Q} _ {a}\\right) \\right. \\\\ \\left. \\times \\sum_ {l = 1} ^ {L} \\eta_ {n - 1, a} ^ {(l)} \\mathcal {N} \\left(\\mathbf {x} _ {n - 1, a} \\mid \\mathbf {m} _ {n - 1, a} ^ {(l)}, \\boldsymbol {\\Sigma} _ {n - 1, a} ^ {(l)}\\right) \\right] d \\mathbf {x} _ {n - 1, a} \\\\ = \\sum_ {l = 1} ^ {L} \\eta_ {n | n - 1, a} ^ {(l)} \\mathcal {N} \\left(\\tilde {\\mathbf {x}} _ {n, a} \\mid \\mathbf {m} _ {n | n - 1, a} ^ {(l)}, \\boldsymbol {\\Sigma} _ {n | n - 1, a} ^ {(l)}\\right) \\tag {53} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[86,768,482,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\eta_{n|n-1,a}^{(l)} = \\eta_{n-1,a}^{(l)}$ , $\\mathbf{m}_{n|n-1,a}^{(l)} = \\mathbf{m}_{n-1,a}^{(l)} + \\mathbf{u}_{n-1,a}$ , and $\\boldsymbol{\\Sigma}_{n|n-1,a}^{(l)} = \\boldsymbol{\\Sigma}_{n-1,a}^{(l)} + \\mathbf{Q}_a$ $\\forall l \\in \\{1, \\dots, L\\}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[86,806,482,835]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Similarly, from the target dynamic model in (2), we write ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[94,843,483,902]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} f _ {n, o} ^ {(a)} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}, \\mathbf {x} _ {n - 1, o} ^ {(a)}\\right) = p \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)} | \\mathbf {x} _ {n - 1, o} ^ {(a)}\\right) \\\\ = \\mathcal {N} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)} \\mid \\mathbf {F} _ {n - 1, o} ^ {(a)} \\mathbf {x} _ {n - 1, o} ^ {(a)}, \\mathbf {Q} _ {n - 1, o} ^ {(a)}\\right). \\tag {54} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[495,44,893,75]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Thus, by substituting (54) and (51) into (36), the target prediction message $\\alpha_{n,o}^{(a)}(\\mathbf{x}_{n,o}^{(a)})$ is approximated as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[496,82,892,220]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\check {\\alpha} _ {n, o} ^ {(a)} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right) = \\int \\left[ \\mathcal {N} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)} | \\mathbf {F} _ {n - 1, o} ^ {(a)} \\mathbf {x} _ {n - 1, o} ^ {(a)}, \\mathbf {Q} _ {n - 1, o} ^ {(a)}\\right) \\right. \\\\ \\left. \\times \\sum_ {l = 1} ^ {L} \\eta_ {n - 1, o} ^ {(a, l)} \\mathcal {N} \\left(\\mathbf {x} _ {n - 1, o} ^ {(a)} \\mid \\mathbf {m} _ {n - 1, o} ^ {(a, l)}, \\boldsymbol {\\Sigma} _ {n - 1, o} ^ {(a, l)}\\right) \\right] d \\mathbf {x} _ {n - 1, o} ^ {(a)} \\\\ = \\sum_ {l = 1} ^ {L} \\eta_ {n | n - 1, o} ^ {(a, l)} \\mathcal {N} \\left(\\tilde {\\mathbf {x}} _ {n, o} ^ {(a)} \\mid \\mathbf {m} _ {n | n - 1, a} ^ {(a, l)}, \\boldsymbol {\\Sigma} _ {n | n - 1, a} ^ {(a, l)}\\right) \\tag {55} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[495,227,893,286]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"in which $\\eta_{n|n-1,o}^{(a,l)} = \\eta_{n-1,o}^{(a,l)}$ , $\\mathbf{m}_{n|n-1,o}^{(a,l)} = \\mathbf{F}_{n-1,o}^{(a)}\\mathbf{m}_{n-1,o}^{(a,l)}$ and $\\boldsymbol{\\Sigma}_{n|n-1,o}^{(a,l)} = \\mathbf{F}_{n-1,o}^{(a)}\\boldsymbol{\\Sigma}_{n-1,o}^{(a,l)}\\left[\\mathbf{F}_{n-1,o}^{(a)}\\right]^T + \\mathbf{Q}_{n-1,o}^{(a)} \\quad \\forall l \\in \\{1, \\ldots, L\\}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[495,286,893,431]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the sequel, at node $a$ , we sample $\\tilde{\\mathbf{x}}_{n,a}^{(i)} \\sim \\check{\\alpha}_{n,a}\\left(\\tilde{\\mathbf{x}}_{n,a}\\right)$ and compute the associated importance weights $\\tilde{w}_{n,a}^{(i)} \\propto \\beta_{n,a}\\left(\\tilde{\\mathbf{x}}_{n,a}^{(i)}\\right) \\kappa_{n,a}\\left(\\tilde{\\mathbf{x}}_{n,a}^{(i)}\\right)$ to create a Monte Carlo (MC) representation $\\left\\{\\left(\\tilde{w}_{n,a}^{(i)}, \\tilde{\\mathbf{x}}_{n,a}^{(i)}\\right)\\right\\}$ , $i \\in \\{1, \\dots, N_a\\}$ , of the aircraft belief $\\zeta_{n,a}\\left(\\tilde{\\mathbf{x}}_{n,a}\\right)$ in (34) following the assimilation of the ground clearance and barometric altitude observations, respectively. As usual in sequential Monte Carlo methods, the importance weights undergo normalization in such a way that $\\sum_{i=1}^{N_a} \\tilde{w}_{n,a}^{(i)} = 1$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[495,431,893,537]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Next, for the target, we build another MC representation $\\{(\\tilde{w}_{n,o}^{(a,k)},\\tilde{\\mathbf{x}}_{n,o}^{(a,k)})\\}$ $k\\in \\{1,\\dots ,N_o\\}$ by sampling $\\tilde{\\mathbf{x}}_{n,o}^{(a,k)}\\sim$ $\\check{\\alpha}_{n,o}^{(a)}\\left(\\tilde{\\mathbf{x}}_{n,o}^{(a)}\\right)$ and setting the weights $\\tilde{w}_{n,o}^{(a,k)}\\propto \\phi_{n,o}^{(a)}\\bigl (\\tilde{\\mathbf{x}}_{n,o}^{(a,k)}\\bigr),$ where the message $\\phi_{n,o}^{(a)}\\left(\\tilde{\\mathbf{x}}_{n,o}^{(a)}\\right)$ introduced in (37), assimilates the target observation, and we normalize the weights again such that $\\sum_{i = 1}^{N_o}\\tilde{w}_{n,o}^{(a,k)} = 1$ . Since the target observation factor ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[514,545,892,590]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} t o _ {n, o} ^ {(a)} \\left(\\tilde {\\mathbf {x}} _ {n, a}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right) = p \\left(\\mathbf {y} _ {n, a \\leftarrow o} | \\tilde {\\mathbf {x}} _ {n, a}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right) \\\\ = \\mathcal {N} \\left(\\mathbf {y} _ {n, a \\leftarrow o} | \\mathbf {t o} \\left(\\tilde {\\mathbf {x}} _ {n, a}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right), \\mathbf {R} _ {t o}\\right) \\tag {56} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[495,595,893,645]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"also depends on $\\tilde{\\mathbf{x}}_{n,a}$ and we already have an MC representation $\\{(\\tilde{w}_{n,a}^{(i)},\\tilde{\\mathbf{x}}_{n,a}^{(i)})\\}$ , $i\\in \\{1,\\dots ,N_a\\}$ , of the posterior $\\zeta_{n,a}(\\tilde{\\mathbf{x}}_{n,a})$ , it results that, for each particle $\\tilde{\\mathbf{x}}_{n,o}^{(a,k)}$ , we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,650,892,853]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\phi_ {n, o} ^ {(a)} (\\tilde {\\mathbf {x}} _ {n, o} ^ {(a, k)}) \\propto \\int t o _ {n, o} ^ {(a)} \\left(\\tilde {\\mathbf {x}} _ {n, a}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a, k)}\\right) \\zeta_ {n, a} \\left(\\tilde {\\mathbf {x}} _ {n, a}\\right) d \\tilde {\\mathbf {x}} _ {n, a} \\\\ \\approx \\sum_ {i = 1} ^ {N _ {a}} \\left[ \\mathcal {N} \\left(\\mathbf {y} _ {n, a \\leftarrow o} | \\mathbf {t o} \\left(\\tilde {\\mathbf {x}} _ {n, a} ^ {(i)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a, k)}\\right), \\mathbf {R} _ {t o}\\right) \\tilde {w} _ {n, a} ^ {(i)} \\right] \\\\ = \\sum_ {i = 1} ^ {N _ {a}} \\tilde {w} _ {n, a} ^ {(i)} \\left[ \\mathcal {N} \\left(y _ {n, a \\leftarrow o} ^ {(r g)} \\mid r g \\left(\\tilde {\\mathbf {x}} _ {n, a} ^ {(i)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a, k)}\\right), \\sigma_ {r o} ^ {2}\\right) \\right. \\\\ \\times \\mathcal {N} \\left(y _ {n, a \\leftarrow o} ^ {(a z)} | a z \\left(\\tilde {\\mathbf {x}} _ {n, a} ^ {(i)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a, k)}\\right), \\sigma_ {a z} ^ {2}\\right) \\\\ \\left. \\times \\mathcal {N} \\left(y _ {n, a \\leftarrow o} ^ {(e l)} \\mid e l \\left(\\tilde {\\mathbf {x}} _ {n, a} ^ {(i)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a, k)}\\right), \\sigma_ {e l} ^ {2}\\right) \\right]. \\tag {57} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[495,859,893,892]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the sequel, we create the GMM approximations $\\check{\\lambda}_{n,a}\\left(\\tilde{\\mathbf{x}}_{n,a}\\right)$ and $\\check{\\lambda}_{n,o}^{(a)}\\left(\\tilde{\\mathbf{x}}_{n,o}^{(a)}\\right)$ from the MC representations ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[88,940,109,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"998 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[256,939,888,952]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.60, NO.1 FEBRUARY 2024 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[102,44,501,77]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\left\\{\\left(\\tilde{w}_{n,a}^{(i)}, \\tilde{\\mathbf{x}}_{n,a}^{(i)}\\right)\\right\\}$ of the aircraft belief and $\\left\\{\\left(\\tilde{w}_{n,o}^{(a,k)}, \\tilde{\\mathbf{x}}_{n,o}^{(a,k)}\\right)\\right\\}$ of the target belief, respectively. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[102,88,307,103]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B. Cooperative Localization ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[101,109,500,256]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The cooperative localization is analogous to the procedure described in [1]. Each aircraft $a$ sends the parametric representation $\\check{\\lambda}_{n,a}(\\tilde{\\mathbf{x}}_{n,a})$ to the nodes in their neighborhood, instead of sending $N_{a}$ particles, to decrease the communication costs. After receiving $\\{\\check{\\lambda}_{n,b}(\\tilde{\\mathbf{x}}_{n,b})\\}$ from each node $b\\in \\mathbf{N}(n,a),N_b$ particles are resampled at node $a$ from each of the received GMMs. That procedure results in a new set of weighted particles $\\left\\{\\left(\\tilde{w}_{n,b}^{(j)},\\tilde{\\mathbf{x}}_{n,b}^{(j)}\\right)\\right\\}$ for each $b\\in \\mathbf{N}(n,a)$ where $\\tilde{\\mathbf{x}}_{n,b}^{(j)}\\sim \\check{\\lambda}_{n,b}(\\tilde{\\mathbf{x}}_{n,b})$ and $\\tilde{w}_{n,b}^{(j)} = 1 / N_b\\forall j\\in \\{1,\\dots ,N_b\\}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[125,256,472,270]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From the internode distance model in (5), we write ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[109,277,498,316]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} a d _ {n, a \\leftarrow b} (\\mathbf {x} _ {n, a}, \\tilde {\\mathbf {x}} _ {n, b}) = p (y _ {n, a \\leftarrow b} | \\mathbf {x} _ {n, a}, \\tilde {\\mathbf {x}} _ {n, b}) \\\\ = \\mathcal {N} \\left(y _ {n, a \\leftarrow b} \\mid r g \\left(\\mathbf {x} _ {n, a}, \\tilde {\\mathbf {x}} _ {n, b}\\right), \\sigma_ {a d} ^ {2}\\right). \\tag {58} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[101,321,498,357]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Then, the message $\\xi_{n,a\\leftarrow b}\\left(\\mathbf{x}_{n,a}^{(i)}\\right)$ , defined in (21) is computed, for each particle $\\mathbf{x}_{n,a}^{(i)}\\triangleq \\tilde{\\mathbf{x}}_{n,a}^{(i)}$ , as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[109,362,498,518]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\xi_ {n, a \\leftarrow b} (\\mathbf {x} _ {n, a} ^ {(i)}) \\propto \\int a d _ {n, a \\leftarrow b} \\left(\\mathbf {x} _ {n, a} ^ {(i)}, \\tilde {\\mathbf {x}} _ {n, b}\\right) \\lambda_ {n, a \\leftarrow b} \\left(\\tilde {\\mathbf {x}} _ {n, b}\\right) d \\tilde {\\mathbf {x}} _ {n, b} \\\\ = \\int \\left[ \\mathcal {N} \\left(y _ {n, a \\leftarrow b} | r g \\left(\\mathbf {x} _ {n, a} ^ {(i)}, \\tilde {\\mathbf {x}} _ {n, b}\\right), \\sigma_ {a d} ^ {2}\\right) \\right. \\\\ \\left. \\times \\breve {\\lambda} _ {n, b} (\\tilde {\\mathbf {x}} _ {n, b}) \\right] d \\tilde {\\mathbf {x}} _ {n, b} \\\\ \\approx \\sum_ {j = 1} ^ {N _ {b}} \\mathcal {N} \\left(y _ {n, a \\leftarrow b} \\mid r g \\left(\\mathbf {x} _ {n, a} ^ {(i)}, \\tilde {\\mathbf {x}} _ {n, b} ^ {(j)}\\right), \\sigma_ {a d} ^ {2}\\right) \\tilde {w} _ {n, b} ^ {(j)}. \\tag {59} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[101,527,500,626]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Next, we make $w_{n,a}^{(i)} \\propto \\tilde{w}_{n,a}^{(i)} \\prod_{b \\in \\bar{\\mathbf{N}}(n,a)} \\xi_{n,a \\leftarrow b}\\left(\\mathbf{x}_{n,a}^{(i)}\\right)$ with proportionality constant such that $\\sum_{i=1}^{N_a} w_{n,a}^{(i)} = 1$ . The new weighted particle set $\\left\\{\\left(w_{n,a}^{(i)}, \\mathbf{x}_{n,a}^{(i)}\\right)\\right\\}, i \\in \\{1, \\dots, N_a\\}$ , is now the updated MC representation of the belief of the position of the aircraft $a$ following the assimilation of the distance measurements between the neighboring aircraft. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[102,636,321,652]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Distributed Target Tracking ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[101,657,498,781]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Each aircraft $a$ also broadcasts the parametric representations $\\check{\\lambda}_{n,o}^{(a)}\\left(\\tilde{\\mathbf{x}}_{n,o}^{(a)}\\right)$ . After receiving $\\{\\check{\\lambda}_{n,o}^{(b)}\\left(\\tilde{\\mathbf{x}}_{n,o}^{(b)}\\right)\\}$ from all $b \\in \\bar{\\mathbf{N}}(n,a)$ , we set the network coefficients $r_{a,b} \\triangleq 1/|\\mathbf{N}(n,a)|$ , where $|\\mathbf{N}(n,a)|$ denotes the cardinality of the closed neighborhood of $a$ , $\\mathbf{N}(n,a)$ , which we recall is given by $\\mathbf{N}(n,a) \\triangleq \\bar{\\mathbf{N}}(n,a) \\cup \\{a\\}$ . In this case, $|\\mathbf{N}(n,a)|$ is simply equal to the number of received messages plus one and, then, the factors $cl_{n,o}^{(a)}$ and $cd_{n,o}^{(a \\leftarrow b)}$ can be computed analytically. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[101,782,500,876]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Assuming that the input message $\\nu_{n,o}^{(a)}\\left(\\tilde{\\mathbf{x}}_{n,o}^{(a)}\\right)$ is represented by a Gaussian p.d.f. $\\mathcal{N}\\left(\\tilde{\\mathbf{x}}_{n,o}^{(a)}|\\tilde{\\mathbf{m}}_{n,o}^{(a)},\\tilde{\\mathbf{\\Sigma}}_{n,o}^{(a)}\\right)$ and, similarly, $\\lambda_{n,o}^{(a\\leftarrow b)}\\left(\\tilde{\\mathbf{x}}_{n,o}^{(b)}\\right)\\approx \\check{\\lambda}_{n,o}^{(b)}\\left(\\tilde{\\mathbf{x}}_{n,o}^{(b)}\\right) = \\mathcal{N}\\left(\\tilde{\\mathbf{x}}_{n,o}^{(b)}|\\tilde{\\mathbf{m}}_{n,o}^{(b)},\\tilde{\\mathbf{\\Sigma}}_{n,o}^{(b)}\\right)$ , it follows from (28) and (29) setting $r_{a,b} = 1 / |\\mathbf{N}(n,a)|\\forall b\\in$ $\\mathbf{N}(n,a)$ , that: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[138,882,500,907]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nc l _ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}\\right) = \\mathcal {N} \\left(\\mathbf {x} _ {n, o} ^ {(a)} | \\tilde {\\mathbf {x}} _ {n, o} ^ {(a)}, | \\bar {\\mathbf {N}} (n, a) | \\tilde {\\mathbf {\\Sigma}} _ {n, o} ^ {(a)}\\right) \\tag {60}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[527,45,908,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nc d _ {n, o} ^ {(a \\leftarrow b)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}, \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}\\right) = \\mathcal {N} \\left(\\mathbf {x} _ {n, o} ^ {(a)} | \\tilde {\\mathbf {x}} _ {n, o} ^ {(b)}, | \\bar {\\mathbf {N}} (n, a) | \\tilde {\\mathbf {\\Sigma}} _ {n, o} ^ {(b)}\\right) \\tag {61}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,78,906,108]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $|\\tilde{\\mathbf{N}}(n, a)| = |\\mathbf{N}(n, a)| - 1$ is the cardinality of the open neighborhood of node $a$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,109,906,138]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Using (43) and (44), the local and remote combine messages are then parameterized as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[566,146,908,212]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\psi_ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) = \\mathcal {N} \\left(\\mathbf {x} _ {n, o} ^ {(a)} | \\tilde {\\mathbf {m}} _ {n, o} ^ {(a)}, | \\mathbf {N} (n, a) | \\tilde {\\boldsymbol {\\Sigma}} _ {n, o} ^ {(a)}\\right) \\\\ \\propto \\left[ \\tilde {p} _ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) \\right] ^ {\\frac {1}{| \\mathbf {N} (n , a) |}} \\tag {62} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,217,542,231]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[558,236,908,294]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\theta_ {n, o} ^ {(a \\leftarrow b)} (\\mathbf {x} _ {n, o} ^ {(a)}) = \\mathcal {N} \\left(\\mathbf {x} _ {n, o} ^ {(a)} | \\tilde {\\mathbf {m}} _ {n, o} ^ {(b)}, | \\mathbf {N} (n, a) | \\tilde {\\boldsymbol {\\Sigma}} _ {n, o} ^ {(b)}\\right) \\\\ \\propto \\left[ \\tilde {p} _ {n, o} ^ {(b)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) \\right] ^ {\\frac {1}{| \\mathbf {N} (n , a) |}}. \\tag {63} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,301,676,318]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"D. Feedback Message ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[511,323,910,422]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The combined estimate of the target state $\\mathbf{x}_{n,o}^{(a)}$ is used to improve the self-localization estimate of the state $\\mathbf{x}_{n,a}$ of the aircraft $a$ via the factor $fb_{n,o}^{(a)}$ in (47). The input message $\\rho_{n,o}^{(a)}\\left(\\mathbf{x}_{n,o}^{(a)}\\right)$ , which is a copy of $\\gamma_{n,o}^{(a)}\\left(\\mathbf{x}_{n,o}^{(a)}\\right)$ in (41) can now be calculated as the product of the messages $\\psi_{n,o}^{(a)}\\left(\\mathbf{x}_{n,o}^{(a)}\\right)$ and $\\theta_{n,o}^{(a\\leftarrow b)}(\\mathbf{x}_{n,o}^{(a)})$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[534,429,908,575]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\rho_ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) \\propto \\psi_ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) \\prod_ {b \\in \\tilde {\\mathbf {N}} (n, a)} \\theta_ {n, o} ^ {(a \\leftarrow b)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) \\\\ = \\mathcal {N} \\left(\\mathbf {x} _ {n, o} ^ {(a)} | \\tilde {\\mathbf {m}} _ {n, o} ^ {(a)}, | \\mathbf {N} (n, a) | \\tilde {\\boldsymbol {\\Sigma}} _ {n, o} ^ {(a)}\\right) \\\\ \\times \\prod_ {b \\in \\tilde {\\mathbf {N}} (n, a)} \\mathcal {N} \\left(\\mathbf {x} _ {n, o} ^ {(a)} | \\tilde {\\mathbf {m}} _ {n, o} ^ {(b)}, | \\mathbf {N} (n, a) | \\tilde {\\boldsymbol {\\Sigma}} _ {n, o} ^ {(b)}\\right) \\\\ = \\mathcal {N} \\left(\\mathbf {x} _ {n, o} ^ {(a)} \\mid \\mathbf {m} _ {n, o} ^ {(a)}, \\boldsymbol {\\Sigma} _ {n, o} ^ {(a)}\\right) \\tag {64} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,583,558,595]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[566,601,908,638]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\left[ \\boldsymbol {\\Sigma} _ {n, o} ^ {(a)} \\right] ^ {- 1} = \\frac {1}{| \\mathbf {N} (n , a) |} \\sum_ {u \\in \\mathbf {N} (n, a)} \\left[ \\tilde {\\boldsymbol {\\Sigma}} _ {n, o} ^ {(u)} \\right] ^ {- 1} \\tag {65}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[534,648,908,686]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\left[ \\boldsymbol {\\Sigma} _ {n, o} ^ {(a)} \\right] ^ {- 1} \\mathbf {m} _ {n, o} ^ {(a)} = \\frac {1}{| \\mathbf {N} (n , a) |} \\sum_ {u \\in \\mathbf {N} (n, a)} \\left[ \\tilde {\\boldsymbol {\\Sigma}} _ {n, o} ^ {(u)} \\right] ^ {- 1} \\tilde {\\mathbf {m}} _ {n, o} ^ {(u)}. \\tag {66}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,696,906,727]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The output message $\\tau_{n,a}(\\mathbf{x}_{n,a})$ of the factor $fb_{n,o}^{(a)}$ is computed using (47) and (49) such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[549,733,908,820]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\tau_ {n, a} (\\mathbf {x} _ {n, a}) \\propto \\int f b _ {n, o} ^ {(a)} (\\mathbf {x} _ {n, a}, \\mathbf {x} _ {n, o} ^ {(a)}) \\rho_ {n, o} ^ {(a)} (\\mathbf {x} _ {n, o} ^ {(a)}) d \\mathbf {x} _ {n, o} ^ {(a)} \\\\ = \\int \\left[ \\mathcal {N} \\left(\\mathbf {y} _ {n, a \\leftarrow o} | \\mathbf {t o} \\left(\\mathbf {x} _ {n, a}, \\mathbf {x} _ {n, o} ^ {(a)}\\right), \\mathbf {R} _ {t o}\\right) \\right. \\\\ \\left. \\times \\mathcal {N} \\left(\\mathbf {x} _ {n, o} ^ {(a)} \\mid \\mathbf {m} _ {n, o} ^ {(a)}, \\boldsymbol {\\Sigma} _ {n, o} ^ {(a)}\\right) \\right] d \\mathbf {x} _ {n, o} ^ {(a)}. \\tag {67} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,827,910,907]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"However, since the target observation $\\mathbf{t o}\\left(\\mathbf{x}_{n,a},\\mathbf{x}_{n,o}^{(a)}\\right)$ defined in (7) is a nonlinear function, we employ the previously introduced approach in section IV-B to solve for the right-hand side of (67). Thus, to process the factor $fb_{n,o}^{(a)}$ ,we create an MC representation $\\left\\{\\left(w_{n,o}^{(a,k)},\\mathbf{x}_{n,o}^{(a,k)}\\right)\\right\\} ,k\\in \\{1,\\dots ,N_o\\}$ , of ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[102,938,750,952]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"OLIVEIRA ET AL.: GNSS-DENIED JOINT COOPERATIVE TERRAIN NAVIGATION AND TARGET TRACKING ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[885,940,908,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"999 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[84,45,486,159]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"the belief of the target state $\\mathbf{x}_{n,o}^{(a)}$ , by sampling $\\mathbf{x}_{n,o}^{(a,k)} \\sim \\rho_{n,o}^{(a)}(\\mathbf{x}_{n,o}^{(a)})$ and making $w_{n,o}^{(a,k)} = 1 / N_o$ . Since we already have an MC representation $\\{(w_{n,a}^{(i)},\\mathbf{x}_{n,a}^{(i)})\\}$ , $i \\in \\{1,\\dots,N_a\\}$ of the belief of the aircraft state estimate $\\mathbf{x}_{n,a}$ , just after the cooperative localization message assimilation described in section IV-B, we can approximate the output message $\\tau_{n,a}(\\mathbf{x}_{n,a})$ , by computing $\\tau_{n,a}(\\mathbf{x}_{n,a}^{(i)})$ for each particle $\\mathbf{x}_{n,a}^{(i)}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[96,164,482,386]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\tau_ {n, a} (\\mathbf {x} _ {n, a} ^ {(i)}) \\propto \\int f b _ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, a} ^ {(i)}, \\mathbf {x} _ {n, o} ^ {(a)}\\right) \\rho_ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) d \\mathbf {x} _ {n, o} ^ {(a)} \\\\ = \\int \\left[ \\mathcal {N} \\left(\\mathbf {y} _ {n, a \\leftarrow o} | \\mathbf {t o} \\left(\\mathbf {x} _ {n, a} ^ {(i)}, \\mathbf {x} _ {n, o} ^ {(a)}\\right), \\mathbf {R} _ {t o}\\right) \\right. \\\\ \\times \\mathcal {N} \\left(\\mathbf {x} _ {n, o} ^ {(a)} \\mid \\mathbf {m} _ {n, o} ^ {(a)}, \\boldsymbol {\\Sigma} _ {n, o} ^ {(a)}\\right) ] d \\mathbf {x} _ {n, o} ^ {(a)} \\\\ \\approx \\sum_ {k = 1} ^ {N _ {o}} \\left[ w _ {n, o} ^ {(a, k)} \\mathcal {N} \\left(\\mathbf {y} _ {n, a \\leftarrow o} | \\mathbf {t o} \\left(\\mathbf {x} _ {n, a} ^ {(i)}, \\mathbf {x} _ {n, o} ^ {(a, k)}\\right), \\mathbf {R} _ {t o}\\right) \\right] \\\\ = \\sum_ {k = 1} ^ {N _ {o}} \\left[ w _ {n, o} ^ {(a, k)} \\mathcal {N} \\left(y _ {n, a \\leftarrow o} ^ {(r g)} \\right\\rvert r g \\left(\\mathbf {x} _ {n, a} ^ {(i)}, \\mathbf {x} _ {n, o} ^ {(a, k)}\\right), \\sigma_ {r o} ^ {2}\\right) \\\\ \\times \\mathcal {N} \\left(y _ {n, a \\leftarrow o} ^ {(a z)} | a z \\left(\\mathbf {x} _ {n, a} ^ {(i)}, \\mathbf {x} _ {n, o} ^ {(a, k)}\\right), \\sigma_ {a z} ^ {2}\\right) \\\\ \\left. \\times \\mathcal {N} \\left(y _ {n, a \\leftarrow o} ^ {(e l)} \\mid e l \\left(\\mathbf {x} _ {n, a} ^ {(i)}, \\mathbf {x} _ {n, o} ^ {(a, k)}\\right), \\sigma_ {e l} ^ {2}\\right) \\right]. \\tag {68} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[86,395,220,411]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Final Messages ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[84,415,483,463]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The final message $\\gamma_{n,o}^{(a)}\\left(\\mathbf{x}_{n,o}^{(a)}\\right)$ , which represents the belief of the target state at node $a$ after the combine step, is exactly the same as the message $\\rho_{n,o}^{(a)}\\left(\\mathbf{x}_{n,o}^{(a)}\\right)$ defined in (64), i.e., ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[199,470,482,491]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\gamma_ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right) = \\rho_ {n, o} ^ {(a)} \\left(\\mathbf {x} _ {n, o} ^ {(a)}\\right). \\tag {69}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[84,496,483,640]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To compute the final message $\\gamma_{n,a}(\\mathbf{x}_{n,a})$ , which represents the final belief of the state of the aircraft $a$ at instant $n$ , we first update the weights $w_{n,a}^{(i)}$ of the MC representation $\\left\\{\\left(w_{n,a}^{(i)},\\mathbf{x}_{n,a}^{(i)}\\right)\\right\\}$ , assimilating the feedback message $\\tau_{n,a}\\left(\\mathbf{x}_{n,a}^{(i)}\\right)$ by making $w_{n,a}^{(i)}\\propto w_{n,a}^{(i)}\\tau_{n,a}\\left(\\mathbf{x}_{n,a}^{(i)}\\right)$ and normalizing the weights such that they add up to one. Before the time step is incremented, we fit then a GMM representation $\\check{\\gamma}_{n,a}\\left(\\mathbf{x}_{n,a}\\right)$ to the message $\\gamma_{n,a}\\left(\\mathbf{x}_{n,a}\\right)$ using the updated MC representation $\\left\\{\\left(w_{n,a}^{(i)},\\mathbf{x}_{n,a}^{(i)}\\right)\\right\\}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[86,648,279,664]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"V. SIMULATION RESULTS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[84,670,483,867]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To test the proposed algorithm, we used the simulation setup described in [1], consisting of 12 cooperative aircraft flying over a terrain in the state of São Paulo, Brazil, for which we have a real digital elevation map provided by INPE. Unlike in [1], however, we now simulate 3-D trajectories for the surveillance aircraft, including variations in altitude, and also add to the scenario a noncooperative target. The terrain map, initial target and aircraft positions and the communication network topology are shown in Fig. 4. We used 200 independent MC runs in our performance studies. In Fig. 5, we show all 200 simulated trajectories of each aircraft and the target. The target starts flying from the left side of the scenario, as shown in gray, while the ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[496,46,895,224]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0804/d545cbb80c94cf2337d431a1ff081344129a101a9d30c1c3579a8991c7add180.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[496,271,893,428]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 4. Terrain map, initial target position (red), aircraft initial positions (blue), and the communication network topology (orange). \",\"Fig. 5. All 200 trajectories of the target (gray) and of each aircraft. \"]","image_footnote":"[]","img_path":"images/0210/f19329ef494ae8ad8fda3dedba51028aec7a27146fa9978bd55987f0fb5efe2b.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[496,492,893,594]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0794/4e47b77b98553fb729fe232f25c2ee23a9a1df60bd5271a265642b781b18a1db.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":"<table><tr><td></td><td>CL</td><td>DTT</td><td>Feedback</td></tr><tr><td>Factors</td><td>adn,a←b</td><td>cd(n,o)(a←b)</td><td>fb(n,o)(a)</td></tr><tr><td>Messages</td><td>λn,aandξn,a←b</td><td>λ(n,o)(a) and θ(n,o)(a←b)</td><td>ρ(n,o)(a) and τn,a</td></tr><tr><td>NC</td><td>False</td><td>False</td><td>False</td></tr><tr><td>CTAN</td><td>True</td><td>False</td><td>False</td></tr><tr><td>J-CTAN-DTT</td><td>True</td><td>True</td><td>True</td></tr></table>","table_caption":"[\"TABLEI Factors and Messages Enabling for Each Factor Graph Configuration \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[493,618,893,708]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"aircraft are flying in formation, starting from the bottom left side of the scenario, monitoring the area. The target and aircraft speeds are set as 50 and $100\\mathrm{m / s}$ , respectively. The total simulation window lasts $600\\mathrm{s}$ and $T_{s} = 1\\mathrm{s}$ . The surveillance aircraft maneuvers were simulated using a 3-D coordinated turn model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[493,709,893,844]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The proposed algorithm was evaluated using three different factor graph configurations in the \"6-TAN 6-INS\" scenario, where none of the 12 aircraft have GNSS measurements, but six aircraft perform terrain navigation while the six remaining aircraft navigate with INS data only. Each of the proposed configurations or operation modes correspond to a subset of the full factor graph shown in Fig. 3, in which we add or remove the corresponding factors and messages according to Table I. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[493,845,893,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We employed $N_{a} = 3000$ particles to represent the state vector of each surveillance aircraft and $N_{o} = 1000$ particles for the local representation of the target state at each node. In the cooperative localization loop, the GMMs representing the aircraft beliefs that are received from neighboring ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,892,482,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"<sup>3</sup>Source INPE: http://www.dsr.inpe.br/topodata/cesso.php [March 1, 2023]. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[88,939,119,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1000 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[256,939,890,952]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.60, NO.1 FEBRUARY 2024 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[102,73,496,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0442/0b4be56ce99f68984b0237f60aef868974a7172d5857ea0c664aaf29863192c9.jpg","list_items":null,"page_idx":"10","sub_type":null,"table_body":"<table><tr><td>Time step</td><td>σacc factor</td><td>Na factor</td><td>σacc,o factor</td><td>No factor</td></tr><tr><td>0</td><td>2.00</td><td>2.00</td><td>2.00</td><td>2.00</td></tr><tr><td>30</td><td>1.50</td><td>1.50</td><td>1.50</td><td>1.50</td></tr><tr><td>60</td><td>1.25</td><td>1.25</td><td>1.25</td><td>1.25</td></tr><tr><td>90</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr></table>","table_caption":"[\"TABLE II Covariance and Particle Number Schedule \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[99,191,496,234]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"nodes were resampled with $N_{b} = 3000$ particles. We use GMMs with one single mode for both aircraft and target representations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[99,236,496,356]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We also computed the PCRLB by performing 1000 independent MC runs for the centralized version of the joint cooperative terrain-aided navigation and distributed target tracking (J-CTAN-DTT) problem, see Appendix A. In the aircraft dynamic model (1), we assumed covariance matrix $\\mathbf{Q}_a = \\sigma_{\\mathrm{acc},a}^2 * \\mathbf{I}_3$ with $\\sigma_{\\mathrm{acc},a} = 1\\mathrm{m}\\forall a$ . For the target's constant velocity model in (2), we set $\\sigma_{\\mathrm{acc},o} = 0.1\\mathrm{m}$ in the process noise $\\mathbf{w}_{n,o}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[101,357,496,402]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We used $\\sigma_{ba} = 2\\mathrm{m}$ , $\\sigma_{gc} = 2\\mathrm{m}$ and $\\sigma_{ad} = 1\\mathrm{m}$ for all aircraft, and $\\sigma_{ro} = 10\\mathrm{m}$ , $\\sigma_{az} = \\pi /360\\mathrm{rad}$ $(0.5^{\\circ})$ , $\\sigma_{el} = \\pi /360\\mathrm{rad}$ $(0.5^{\\circ})$ for the target. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[101,402,496,462]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For filtering, we used $\\sigma_{\\mathrm{acc}} = 1.5\\mathrm{m / s}^2$ for the NC configuration and $\\sigma_{\\mathrm{acc}} = 6\\mathrm{m / s}^2$ for both the CTAN and J-CTAN-DTT configurations. For the target, we used $\\sigma_{\\mathrm{acc},o} = 0.75\\mathrm{m / s}^2$ in all configurations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[101,462,496,575]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We sampled the estimate of the initial state of each aircraft $\\hat{\\mathbf{x}}_{0,a}$ using a Gaussian distribution with mean $\\mathbf{x}_{0,a}$ and covariance $\\mathbf{P}_{0,a} \\triangleq \\sigma_{\\mathrm{pos}}^2 * \\mathbf{I}$ , while the estimate of the initial target representation of each aircraft $\\hat{\\mathbf{x}}_{0,o}^{(a)}$ was sampled from a Gaussian distribution with mean $\\mathbf{x}_{0,o}^{(a)}$ and covariance $\\mathbf{P}_{0,o} \\triangleq \\mathrm{diag}([\\sigma_{\\mathrm{pos}}^2 \\sigma_{\\mathrm{vel}}^2 \\sigma_{\\mathrm{pos}}^2 \\sigma_{\\mathrm{vel}}^2 \\sigma_{\\mathrm{pos}}^2 \\sigma_{\\mathrm{vel}}^2])$ , with $\\sigma_{\\mathrm{pos}} = 50\\mathrm{m}$ and $\\sigma_{\\mathrm{vel}} = 5\\mathrm{m/s}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[99,577,496,712]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We implemented a scheduler for the process noise covariances and number of particles to improve the filter robustness, specially in the initial time steps, when the uncertainty of the state variables/vectors are bigger. We apply a scale factor to the parameters that define the covariance matrices and the amount of particles according to a simple schedule such that, as time progresses, the scaling factor is reduced and then removed. Table II shows the schedule and the scale factors. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[99,713,496,803]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The performance of each scenario configuration was assessed using the root-mean-square error (RMSE) of the position estimates for both cooperative localization $e_n$ and target tracking $e_{n,o}$ , averaged as a function of time, average over all network nodes and all MC simulations, as explained in detail in [1]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[101,804,496,834]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moreover, the performance metrics $e_n$ and $e_{n,o}$ are minorized by the centralized PCRLB as follows: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[222,848,496,881]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\bar {e} _ {n} \\geq \\sqrt {\\frac {1}{A} \\sum_ {a = 1} ^ {A} \\operatorname {t r} \\left(\\mathbf {P} _ {n , a}\\right)} \\tag {70}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[212,885,496,917]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\bar {e} _ {n, o} \\geq \\sqrt {\\frac {1}{A} \\sum_ {a = 1} ^ {A} \\operatorname {t r} \\left(\\mathbf {P} _ {n , o}\\right)} \\tag {71}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,45,911,248]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0331/66bd048eb90c4b8832066d27e98ca469b44b11db41ac261bfc4d1c426efb24dc.jpg","list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[522,287,897,478]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 6. RMSE for cooperative localization over time for different configurations. \",\"Fig. 7. RMSE for distributed target tracking over time for different configurations. \"]","image_footnote":"[]","img_path":"images/0994/433286ebafae803667cd94e007b3fcd77da35faac35d4cde1bd6b06a79e61f77.jpg","list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[511,536,908,613]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\operatorname{tr}(\\mathbf{R})$ represents the trace of the matrix $\\mathbf{R}$ , $\\mathbf{P}_{n,a}$ , and $\\mathbf{P}_{n,o}$ , associated, respectively, with the aircraft $a$ position and the target state (excluding velocities), are $3 \\times 3$ diagonal blocks of $[\\mathbf{J}_n]^{-1}$ , where $\\mathbf{J}_n$ is the Fisher information matrix (see Appendix A). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,627,777,641]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. Numerical Results and Discussion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,648,908,691]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The cooperative localization experiments results are presented in Fig. 6 and compared to the PCRLB, in gray dashed line - as in (70). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,694,908,768]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the noncooperative (NC) configuration, since only half of the aircraft perform TAN, the results diverge. The results of CTAN and J-CTAN-DTT are in turn very similar to each other, with J-CTAN-DTT showing only a small improvement, mainly in the beginning of the simulation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,768,908,813]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The results of the target tracking experiments are shown in Fig. 7 and compared to the associated PCRLB - gray dashed line - as in (71). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,814,908,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In Fig. 7, the results of the CTAN configuration are better than those of NC, even without the distributed target tracking messages, because the target position estimate is based on the aircraft position estimate. Without cooperative localization, the six aircraft that do not perform TAN (i.e., are flying only with INS) cannot estimate their own position accurately, and therefore do not perform good target ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[102,938,750,952]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"OLIVEIRA ET AL.: GNSS-DENIED JOINT COOPERATIVE TERRAIN NAVIGATION AND TARGET TRACKING ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[879,940,906,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1001 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[124,45,339,176]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0949/c4ea4addb2a10743480874767ee51db6f7b0cd2ce448b26e35f153a302f9e650.jpg","list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[380,45,596,176]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0120/96253634d0e70604dc87781ee9647e7099b8dd5f4baac35467aaf5e80724dc79.jpg","list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[640,45,856,176]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 8. Target tracking dispersion results. The black curve shows the corresponding scenario configuration RMSE result, while the gray curves show the RMSE results of each aircraft. J-CTAN-DTT has a significantly lower dispersion than NC and CTAN. \"]","image_footnote":"[]","img_path":"images/0279/6783b76e6882a1e12999409539dc4fb79e47fe9f478dc299cd89652d0ee5f6da.jpg","list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[117,267,454,309]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0158/d4a2d67d896460a81df71aeea9ca6671b77642e71bd3df4c40a2aa55ff167bf5.jpg","list_items":null,"page_idx":"11","sub_type":null,"table_body":"<table><tr><td></td><td>NC</td><td>CTAN</td><td>J-CTAN-DTT</td></tr><tr><td>Time/node/iter [s]</td><td>0.10567</td><td>0.46486</td><td>0.56919</td></tr></table>","table_caption":"[\"TABLE III Average Execution Time Per Node Per Iteration \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[84,352,483,458]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"tracking. In addition, for the other six aircraft that perform TAN, cooperative localization improves the aircraft position estimate, which in turn, allows for a better target observation assimilation. In the J-CTAN-DTT configuration, where the cooperative localization, distributed target tracking, and feedback messages are enabled, the results are close to the PCRLB, shown in gray. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,458,483,579]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Besides the accuracy improvement in target tracking, the proposed algorithm also decreased the dispersion of the target state estimate for each aircraft, suggesting that the aircraft approach a weak consensus on the target state estimate. To highlight this precision improvement, we show in Fig. 8, the target tracking results for all configurations, comparing the average network tracking RMSE (in black) with target tracking RMSE $e_{n,q}^{(a)}$ for each aircraft $a$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,579,483,640]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In Table III, we show the average execution time, per iteration, per node, for the proposed algorithm, in a $\\mathrm{C + + }$ multithread implementation, running in a Intel i9-12900H computer. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[86,672,323,688]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B. Communication Cost Analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[84,694,482,755]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1) Comm Cost $J$ -CTAN-DTT: The proposed algorithm J-CTAN-DTT described in section III has some characteristics that are desirable in real-time distributed algorithms. For example: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[109,784,482,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1) It does not need random number generation synchronization between the different network nodes since each of them stores a different set of particles to represent its own state and the state of the target. \",\"2) It does not require iterative message passing between nodes at the same time stamp $n$ (i.e., between two consecutive sets of sensor measurements). \",\"3) The number of network nodes can be unknown and vary over time. \"]","page_idx":"11","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[495,238,893,300]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In terms of internode communication overhead, at each instant $n$ , each aircraft $a$ needs to broadcast to the neighborhood only the parameters that define the local state variables $\\tilde{\\mathbf{x}}_{n,a}$ and $\\tilde{\\mathbf{x}}_{n,o}^{(a)}$ . Let ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[519,306,890,470]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1) $D_{a}$ denote the dimension of the aircraft state vector; \",\"2) $D_{o}$ denote the dimension of the target state vector; \",\"3) $D_{\\\\mathrm{cov},a}$ denote the number of real-valued free elements of the (symmetric) aircraft state covariance matrix, i.e., $(D_a^2 + D_a) / 2$ ; \",\"4) $D_{\\\\mathrm{cov},o}$ denote the number of real-valued free elements of the (symmetric) target state covariance matrix, i.e., $(D_o^2 +D_o) / 2$ , and \",\"5) $|O|$ denote the number of objects, which, as noted before, could be greater than one if there are no data association issues. \"]","page_idx":"11","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[495,479,893,556]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Using the aforementioned notation and, since the state p.d.f's are represented by Gaussian functions, it follows that the total number of real values that the aircraft $a$ broadcasts at instant $n$ over a given internode communication link is $(D_{a} + D_{\\mathrm{cov},a}) + [(D_{o} + D_{\\mathrm{cov},o})*|O|]$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[495,556,893,614]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Considering the scenario described in this article where $D_{a} = 3$ , $D_{o} = 6$ , and $|O| = 1$ , the total number of real values the aircraft $a$ broadcasts over each internode communication link is 36. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[495,616,893,675]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2) Comm Cost Average Consensus: As a comparison, using the average consensus approach described in [24], the total number of real values the agent $a$ broadcasts at instant $n$ is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[575,683,890,700]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nN ^ {c} + N ^ {N B P} = P J ((C + I) | O | + L) \\tag {72}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[496,708,772,723]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where using the same notation as in [24] ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[519,731,890,866]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1) $\\\\mathrm{P}$ is number of message passing iterations; \",\"2) C is number of consensus iterations; \",\"3) J is the number of particles used to represent either target or agent states; \",\"4) I is diameter of the communication graph; \",\"5) L is the dimension of the agent states (only position); \",\"6) $|O|$ is the number of objects; \",\"7) $N^c \\\\triangleq P(C + I)J|O|$ ; and \",\"8) $N^{NBP} \\\\triangleq PJL$ . \"]","page_idx":"11","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[495,875,893,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Considering the same scenario described in this article, with the aircraft being the agents and the object being the target, we have $P = 1$ , $I = 4$ , $L = D_{a} = 3$ , $|O| = 1$ . In ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[88,940,117,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1002 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[256,939,890,952]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.60, NO.1 FEBRUARY 2024 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[99,44,500,194]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"addition, for the purposes of this discussion, we assume the same number of consensus iterations $C = 6$ and the same number of particles $J = 1000$ used in the dynamic scenario described in [24]. Thus, using the formula in (72), the total number of real values that the agent $a$ would have to broadcast over each network link in the same consensus approach as in [24] is 13 000. Note that even though in our scenario we use 3000 particles for the aircraft representation, we are assuming $J = 1000$ for both aircraft and target states, which decreases the communication cost. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[101,195,500,301]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Even if we considered only 1 consensus iteration ( $C = 1$ ) and also considered a fixed network configuration with known network diameter size, in which case we could set $I$ to zero in (72), see [24], the total number of real values that the agent $a$ would have to broadcast over each link using the average consensus approach would be 4000 in the same scenario as in our article. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[99,301,500,650]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The high communication cost presented in [24] is driven by the need to have $J$ average consensus protocols—one for each particle—between consecutive sets of sensor observations, where each average consensus protocol requires, in turn, $C$ iterations. Moreover, the $J$ particles have to be identical in all nodes, thus requiring random number generators that are synchronized between the different nodes. The internode communication cost in [24] can be significantly reduced if an alternative scheme called likelihood consensus proposed in [38] is used. Specifically, likelihood consensus eliminates the linear growth of the internode communication cost with the number of particles $J$ and also eliminates the need for random number generator synchronization across nodes. Furthermore, likelihood consensus also eliminates the need to do $I$ additional max consensus iterations to determine the number of nodes in the network. However, even in a likelihood consensus implementation, the internode communication cost still grows linearly with the number of consensus iterations $C$ and, if that parameter is high, for example, $C = 15$ as in the second scenario considered in [24], the communication requirements between nodes are still about an order of magnitude higher than in the diffusion approach proposed in this article. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[101,672,246,686]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"VI. CONCLUSION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[99,693,500,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We proposed in this work an approach based on ATC diffusion [6], [7], which is seamlessly integrated into the factor graph framework and implemented in a fully distributed fashion using the SPA by introducing novel special factors to perform the distributed combine step. The proposed solution approximates the adapt step of a classic distributed ATC tracker by replacing the likelihood of the local target observations at each agent node given the target and the agent positions by its expected value with respect to the latest belief (i.e., posterior probability distribution) that the agent has about its own state. Next, the aforementioned special factors combine the updated target state beliefs following the adapt step at each node using an optimal minimum average KL divergence criterion, which, for Gaussian beliefs, reduces to the well-known method of ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,44,911,194]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"covariance intersection in sensor fusion. With the proposed factor-graph-based combine step, we improved the overall target tracking accuracy and precision, in such a way that all aircraft in the network have a better and almost identical target state estimate, agreeing with each other on their target position estimates, using a low-cost communication scheme. In addition, after the combine step, each agent uses their updated target belief as if it were an anchor, to improve their self-localization, thus assimilating the target measurements. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,195,911,393]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We extended the scenario from our previous work [1] and tested three different configurations. In all cases, the error performance was compared to the PCRLB for the error of the optimal data fusion filter that processes all available measurements on the network at each time step on a central hub. The proposed distributed algorithm was shown to perform close to the centralized PCRLB, for both self-localization and target tracking errors, in a simulated GNSS-denied scenario and a partially connected network with a low-cost communication scheme. Even aircraft that cannot locate themselves alone, using the proposed algorithm, they can locate themselves, track the target, and agree with each other on their estimates of the target's position. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,404,607,417]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"APPENDIX A ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[511,426,691,439]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. Performance Bounds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,446,911,601]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1) Centralized Signal Model: As in [1], for the centralized filtering approach, we stack together all cooperative aircraft and noncooperative target state vectors in an extended state vector and do the same for the observations, now including also target radar measurements and barometric altitude measurements. We then make the factor graph corresponding to the distributed signal model into a tree-like graph by removing all loops between variable nodes. So, the extended state vector $\\mathbf{x}_n^c$ and the extended observation vector $\\mathbf{y}_n^c$ are defined, respectively, as<sup>4</sup> ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[591,606,908,628]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf {x} _ {n} ^ {c} = \\left[ \\begin{array}{l l l l} \\mathbf {x} _ {n, 1} ^ {T} & \\mathbf {x} _ {n, 2} ^ {T} & \\dots & \\mathbf {x} _ {n, 1 2} ^ {T} & \\mathbf {x} _ {n, o} ^ {T} \\end{array} \\right] ^ {T} \\tag {73}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[591,628,908,751]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf {y} _ {n} ^ {c} = \\left[ \\begin{array}{l} {\\left[ \\begin{array}{l l l} y _ {n, 1} \\leftarrow b a & \\dots & y _ {n, 1 2} \\leftarrow b a \\\\ \\left[ \\begin{array}{l l l} y _ {n, 1} \\leftarrow g c & \\dots & y _ {n, 1 2} \\leftarrow g c \\end{array} \\right] ^ {T} \\end{array} \\right]} \\\\ {\\left[ \\begin{array}{l l l} y _ {n, 1} \\leftarrow 2 & \\dots & y _ {n, 1 2} \\leftarrow 1 1 \\end{array} \\right] ^ {T}} \\\\ {\\left[ \\begin{array}{l l l} y _ {n, 1} \\leftarrow r o & \\dots & y _ {n, 1 2} \\leftarrow r o \\end{array} \\right] ^ {T}} \\\\ {\\left[ \\begin{array}{l l l} y _ {n, 1} \\leftarrow a z & \\dots & y _ {n, 1 2} \\leftarrow a z \\end{array} \\right] ^ {T}} \\\\ {\\left[ \\begin{array}{l l l} y _ {n, 1} \\leftarrow e l & \\dots & y _ {n, 1 2} \\leftarrow e l \\end{array} \\right] ^ {T}} \\end{array} \\right]. \\tag {74}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,757,911,789]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The kinematics of the sequence of extended state vectors $\\{\\mathbf{x}_n^c\\}$ , $n \\in \\mathbb{Z}_+$ , tracked by the centralized filter is modeled as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[625,795,908,814]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf {x} _ {n + 1} ^ {c} = \\mathbf {F} _ {n} ^ {c} \\mathbf {x} _ {n} ^ {c} + \\mathbf {u} _ {n} ^ {c} + \\mathbf {w} _ {n} ^ {c} \\tag {75}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[511,821,911,856]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"in which $\\mathbf{u}^c = \\left[\\mathbf{u}_1^T\\ldots \\mathbf{u}_{12}^T\\mathbf{0}_{6\\times 1}^T\\right]^T$ is the extended input for all aircraft, concatenated with $\\mathbf{0}_{6\\times 1}$ , since we do not have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,880,911,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note that, for the centralized factor graph, it is not necessary to consider multiple copies $\\mathbf{x}_{n,o}^{(a)}$ of the single target state vector $\\mathbf{x}_{n,o}$ at each node $a$ as in the factor graph for the distributed filter. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[102,938,750,952]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"OLIVEIRA ET AL.: GNSS-DENIED JOINT COOPERATIVE TERRAIN NAVIGATION AND TARGET TRACKING ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[879,940,908,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1003 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[84,44,482,119]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"access to the target's INS sensor, $\\mathbf{w}_n^c\\sim \\mathcal{N}(\\mathbf{0},\\mathbf{Q}_n^c)$ is the random vector corresponding the dynamic model noise, $\\mathbf{F}_n^c =$ diag([I36x36 $\\mathbf{F}_{n,o}]$ ) and $\\mathbf{Q}_n^c = \\mathrm{diag}([\\mathbf{Q}_{n,1}\\dots \\mathbf{Q}_{n,12}\\mathbf{Q}_{n,o}])$ with $\\mathbf{Q}_{n,a} = \\sigma_{\\mathrm{acc},a}^{2}*\\mathbf{I}_{3\\times 3}$ $\\forall a\\in \\{1,\\ldots ,A\\}$ , and both $\\mathbf{F}_{n,o}$ and $\\mathbf{Q}_{n,o}$ defined as introduced in (2). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[84,119,482,148]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The extended observation vector assimilated by the centralized filter at each instant $n$ is modeled in turn as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[220,157,480,176]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf {y} _ {n} ^ {c} = \\mathbf {h} ^ {c} \\left(\\mathbf {x} _ {n} ^ {c}\\right) + \\mathbf {v} _ {n} ^ {c} \\tag {76}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[84,183,482,248]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\mathbf{v}_n^c\\sim \\mathcal{N}(\\mathbf{0},\\mathbf{R}_n^c),\\quad \\mathbf{R}_n^c = \\mathrm{diag}([\\mathbf{R}_{ba}\\mathbf{R}_{gc}\\mathbf{R}_{ad}\\mathbf{R}_{to}])$ $\\mathbf{R}_{ba} = \\sigma_{ba}^{2}*\\mathbf{I}_{12\\times 12},\\mathbf{R}_{gc} = \\sigma_{gc}^{2}*\\mathbf{I}_{12\\times 12},\\mathbf{R}_{ad} = \\sigma_{ad}^{2}*\\mathbf{I}_{46\\times 46}$ and $\\mathbf{R}_{to} = \\mathrm{diag}([\\sigma_{ro}^{2}*\\mathbf{I}_{12\\times 12}\\sigma_{az}^{2}*\\mathbf{I}_{12\\times 12}\\sigma_{el}^{2}*\\mathbf{I}_{12\\times 12}])$ Now, let ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[132,253,482,272]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf {b a} \\left(\\mathbf {x} _ {n} ^ {c}\\right) = \\left[ b a \\left(\\mathbf {x} _ {n, 1}\\right) \\dots b a \\left(\\mathbf {x} _ {n, 1 2}\\right) \\right] \\tag {77}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[104,275,482,292]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf {g c} \\left(\\mathbf {x} _ {n} ^ {c}; \\mathcal {M}\\right) = \\left[ g c \\left(\\mathbf {x} _ {n, 1}; \\mathcal {M}\\right) \\dots g c \\left(\\mathbf {x} _ {n, 1 2}; \\mathcal {M}\\right) \\right] \\tag {78}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[133,295,480,314]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\left. \\mathbf {a d} \\left(\\mathbf {x} _ {n} ^ {c}\\right) = \\left[ r g \\left(\\mathbf {x} _ {n, 1}, \\mathbf {x} _ {n, 2}\\right) \\dots r g \\left(\\mathbf {x} _ {n, 1 2}, \\mathbf {x} _ {n, 1 1}\\right) \\right] \\right. \\tag {79}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[135,316,480,391]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf {t o} \\left(\\mathbf {x} _ {n} ^ {c}\\right) = \\left[ \\begin{array}{l} {\\left[ r g \\left(\\mathbf {x} _ {n, o}, \\mathbf {x} _ {n, 1}\\right) \\dots r g \\left(\\mathbf {x} _ {n, o}, \\mathbf {x} _ {n, 1 2}\\right) \\right] ^ {T}} \\\\ {\\left[ a z \\left(\\mathbf {x} _ {n, o}, \\mathbf {x} _ {n, 1}\\right) \\dots a z \\left(\\mathbf {x} _ {n, o}, \\mathbf {x} _ {n, 1 2}\\right) \\right] ^ {T}} \\\\ {\\left[ e l \\left(\\mathbf {x} _ {n, o}, \\mathbf {x} _ {n, 1}\\right) \\dots e l \\left(\\mathbf {x} _ {n, o}, \\mathbf {x} _ {n, 1 2}\\right) \\right] ^ {T}} \\end{array} \\right] \\tag {80}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[84,400,482,477]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"collect, respectively, the functions modeling all barometric altitude, ground clearances, aircraft pairwise distances, and target observations available over the network at a given instant $n$ . Lastly, we concatenate these functions into an extended observation function $\\mathbf{h}^c(\\mathbf{x}_n^c)$ such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[106,484,480,517]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf {h} ^ {c} \\left(\\mathbf {x} _ {n} ^ {c}\\right) = \\left[ \\mathbf {b a} \\left(\\mathbf {x} _ {n} ^ {c}\\right) ^ {T} \\mathbf {g c} \\left(\\mathbf {x} _ {n} ^ {c}; \\mathcal {M}\\right) ^ {T} \\mathbf {a d} \\left(\\mathbf {x} _ {n} ^ {c}\\right) ^ {T} \\mathbf {t o} \\left(\\mathbf {x} _ {n} ^ {c}\\right) ^ {T} \\right] ^ {T}. \\tag {81}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[84,518,482,563]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2) PCRLB: The posterior Cramér-Rao lower bound of the state estimation error for the centralized filtering problem has the form ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[192,571,480,590]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathrm {E} \\left\\{\\left| \\left| \\mathbf {x} _ {n} ^ {c} - \\hat {\\mathbf {x}} _ {n} ^ {c} \\right| \\right| ^ {2} \\right\\} \\geq \\operatorname {t r} \\left[ \\mathbf {J} _ {n} ^ {- 1} \\right] \\tag {82}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[84,597,482,628]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"in which, as derived in [39], the Fisher matrix $\\mathbf{J}_{n + 1}$ at instant $n + 1$ is recursively computed from $\\mathbf{J}_n$ as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[158,636,482,656]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf {J} _ {n + 1} = \\mathbf {D} _ {n} ^ {2 2} - \\mathbf {D} _ {n} ^ {2 1} \\left(\\mathbf {J} _ {n} + \\mathbf {D} _ {n} ^ {1 1}\\right) ^ {- 1} \\mathbf {D} _ {n} ^ {1 2}. \\tag {83}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[84,662,482,709]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The coefficients $\\mathbf{D}_n^{11},\\mathbf{D}_n^{12},\\mathbf{D}_n^{21}$ , and $\\mathbf{D}_n^{22}$ in (83), induced by the dynamic and observation models, defined in model (75) and (76), respectively, are computed such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[127,717,482,738]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf {D} _ {n} ^ {1 1} = \\left[ \\mathbf {F} _ {n} ^ {c} \\right] ^ {T} \\left[ \\mathbf {Q} _ {n} ^ {c} \\right] ^ {- 1} \\left[ \\mathbf {F} _ {n} ^ {c} \\right] \\tag {84}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[127,739,482,760]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf {D} _ {n} ^ {1 2} = - \\left[ \\mathbf {F} _ {n} ^ {c} \\right] ^ {T} \\left[ \\mathbf {Q} _ {n} ^ {c} \\right] ^ {- 1} \\tag {85}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[127,763,480,784]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf {D} _ {n} ^ {2 1} = \\left[ \\mathbf {D} _ {n} ^ {1 2} \\right] ^ {T} = - \\left[ \\mathbf {Q} _ {n} ^ {c} \\right] ^ {- 1} \\mathbf {F} _ {n} ^ {c} \\tag {86}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[127,786,480,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf {D} _ {n} ^ {2 2} = \\left[ \\mathbf {Q} _ {n} ^ {c} \\right] ^ {- 1} + \\mathrm {E} \\left\\{\\left[ \\mathbf {H} _ {n + 1} ^ {c} \\right] ^ {T} \\left[ \\mathbf {R} _ {n} ^ {c} \\right] ^ {- 1} \\mathbf {H} _ {n + 1} ^ {c} \\right\\} \\tag {87}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[86,815,477,833]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\mathbf{H}_{n + 1}^{c} = \\nabla_{\\mathbf{x}_{n + 1}^{c}}\\left[\\mathbf{h}^{c}\\left(\\mathbf{x}_{n + 1}^{c}\\right)\\right]$ . Then, we compute $\\mathbf{H}_n^c$ as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[119,840,480,883]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} \\mathbf {H} _ {n} ^ {c} = \\nabla_ {\\mathbf {x} _ {n} ^ {c}} \\mathbf {h} ^ {c} \\left(\\mathbf {x} _ {n} ^ {c}\\right) \\\\ = \\left[ \\left[ \\mathbf {H} _ {n, b a} ^ {c} \\right] ^ {T} \\left[ \\mathbf {H} _ {n, g c} ^ {c} \\right] ^ {T} \\left[ \\mathbf {H} _ {n, a d} ^ {c} \\right] ^ {T} \\left[ \\mathbf {H} _ {n, t o} ^ {c} \\right] ^ {T} \\right] ^ {T} \\tag {88} \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[86,890,482,922]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\mathbf{H}_{n,ba}^{c} = \\nabla_{\\mathbf{x}_{n}^{c}}\\mathbf{b}\\mathbf{a}(\\mathbf{x}_{n}^{c}),\\mathbf{H}_{n,gc}^{c} = \\nabla_{\\mathbf{x}_{n}^{c}}\\mathbf{g}\\mathbf{c}(\\mathbf{x}_{n}^{c};\\mathcal{M}),\\mathbf{H}_{n,ad}^{c} =$ $\\nabla_{\\mathbf{x}_n^c}\\mathbf{ad}(\\mathbf{x}_n^c)$ , and $\\mathbf{H}_{n,to}^{c} = \\nabla_{\\mathbf{x}_n^c}\\mathbf{t}\\mathbf{o}(\\mathbf{x}_n^c)$ are the Jacobians related ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[496,44,893,119]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"to the barometric altitude, ground clearance, aircraft distance, and target observations, respectively. Note that the Jacobian matrix $\\mathbf{H}_{n,to}^{c}$ includes block diagonal Jacobian submatrices for all available target observations—slant-range, azimuth, and elevation—at instant $n$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[498,155,584,167]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"REFERENCES ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[498,176,893,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"[1] H. Oliveira, S. S. Dias, and M. G. S. Bruno, \\\"Cooperative terrain navigation using hybrid GMM/SMC message passing on factor graphs,\\\" IEEE Trans. Aerosp. Electron. Syst., vol. 56, no. 5, pp. 3958-3970, Oct. 2020. \",\"[2] B. J. Frey, F. R. Kschischang, H.-A. Loeliger, and N. Wiberg, \\\"Factor graphs and algorithms,\\\" in Proc. Annu. Allerton Conf. Commun. Control Comput., 1997, pp. 666-680. \",\"[3] H.-A. Loeliger, “An introduction to factor graphs,” IEEE Signal Process. Mag., vol. 21, no. 1, pp. 28–41, Jan. 2004. \",\"[4] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 498–519, Feb. 2001. \",\"[5] F. S. Cattivelli and A. H. Sayed, \\\"Diffusion strategies for distributed Kalman filtering and smoothing,\\\" IEEE Trans. Autom. Control, vol. 55, no. 9, pp. 2069-2084, Sep. 2010. \",\"[6] K. Dedecius and P. M. Djuric, \\\"Sequential estimation and diffusion of information over networks: A Bayesian approach with exponential family of distributions,\\\" IEEE Trans. Signal Process., vol. 65, no. 7, pp. 1795-1809, Apr. 2017. \",\"[7] M. G. S. Bruno and S. S. Dias, “A Bayesian interpretation of distributed diffusion filtering algorithms [Lecture Notes],” IEEE Signal Process. Mag., vol. 35, no. 3, pp. 118–123, May 2018. \",\"[8] N. Bergman, “A Bayesian approach to terrain-aided navigation,” IFAC Proc. Volumes, vol. 30, no. 11, pp. 1457–1462, Jul. 1997. \",\"[9] N. Bergman, L. Ljung, and F. Gustafsson, \\\"Terrain navigation using Bayesian statistics,\\\" IEEE Control Syst., vol. 19, no. 3, pp. 33-40, Jun. 1999. \",\"[10] B. Copp and K. Subbarao, \\\"Nonlinear adaptive filtering in terrain-referenced navigation,\\\" IEEE Trans. Aerosp. Electron. Syst., vol. 51, no. 4, pp. 3461-3469, Oct. 2015. \",\"[11] P.J. Nordlund and F. Gustafsson, \\\"Marginalized particle filter for accurate and reliable terrain-aided navigation,\\\" IEEE Trans. Aerosp. Electron. Syst., vol. 45, no. 4, pp. 1385-1399, Oct. 2009. \",\"[12] J. Dunífák, M. Soták, M. Vesely, O. Straka, and W. Hawkins, \\\"Design of Rao-Blackwellized point-mass filter with application in terrain aided navigation,\\\" IEEE Trans. Aerosp. Electron. Syst., vol. 55, no. 1, pp. 251-272, Feb. 2019. \",\"[13] N. Correal and N. Patwari, \\\"Wireless sensor networks: Challenges and opportunities,\\\" in Proc. MPRG/Virginia Tech Wireless Symp., 2001, pp. 1-9. \",\"[14] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, \\\"Wireless sensor networks: A survey,\\\" Comput. Netw., vol. 38, no. 4, pp. 393-422, 2002. \",\"[15] M. Perkins, N. Correal, and B. O'Dea, \\\"Emergent wireless sensor network limitations: A plea for advancement in core technologies,\\\" in Proc. SENSORS, 2002, pp. 1505-1509. \",\"[16] M. Z. Win, F. Meyer, Z. Liu, W. Dai, S. Bartoletti, and A. Conti, \\\"Efficient multisensor localization for the Internet of Things: Exploring a new class of scalable localization algorithms,\\\" IEEE Signal Process. Mag., vol. 35, no. 5, pp. 153-167, Sep. 2018. \",\"[17] U. J. Ferner, H. Wymeersch, and M. Z. Win, \\\"Cooperative anchorless localization for large dynamic networks,\\\" in Proc. IEEE Int. Conf. Ultra-Wideband, 2008, pp. 181-185. \",\"[18] H. Wymeersch, J. Lien, and M. Z. Win, \\\"Cooperative localization in wireless networks,\\\" Proc. IEEE, vol. 97, no. 2, pp. 427-450, Feb. 2009. \",\"[19] F. Meyer, Z. Liu, and M. Z. Win, \\\"Network localization and navigation using measurements with uncertain origin,\\\" in Proc. 21st Int. Conf. Inf. Fusion, 2018, pp. 1-7. \"]","page_idx":"13","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[88,940,117,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1004 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[256,939,888,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.60, NO.1 FEBRUARY 2024 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[104,46,498,902]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"[20] G. C. Fernandes, S. S. Dias, M. R. Maximo, and M. G. S. Bruno, \\\"Cooperative localization for multiple soccer agents using factor graphs and sequential Monte Carlo,\\\" IEEE Access, vol. 8, pp. 213168-213184, 2020. \",\"[21] F. Meyer, P. Braca, P. Willett, and F. Hlawatsch, \\\"A scalable algorithm for tracking an unknown number of targets using multiple sensors,\\\" IEEE Trans. Signal Process., vol. 65, no. 13, pp. 3478-3493, Jul. 2017. \",\"[22] F. Meyer et al., \\\"Message passing algorithms for scalable multitarget tracking,\\\" Proc. IEEE, vol. 106, no. 2, pp. 221-259, Feb. 2018. \",\"[23] G. Soldi et al., \\\"Heterogeneous information fusion for multitarget tracking using the sum-product algorithm,\\\" in Proc. IEEE Int. Conf. Acoust. Speech Signal Process., 2019, pp. 5471-5475. \",\"[24] F. Meyer, O. Hlinka, H. Wymeersch, E. Riegler, and F. Hlawatsch, \\\"Distributed localization and tracking of mobile networks including noncooperative objects,\\\" IEEE Trans. Signal Inf. Process. Netw., vol. 2, no. 1, pp. 57-71, Mar. 2016. \",\"[25] J. Zhang, J. Cui, Z. Wang, Y. Ding, and Y. Xia, \\\"Distributed joint cooperative self-localization and target tracking algorithm for mobile networks,\\\" Sensors, vol. 19, no. 18, 2019, Art. no. 3829. \",\"[26] P. Sharma, A. A. Saucan, D. J. Bucci, and P. K. Varshney, \\\"Decentralized Gaussian filters for cooperative self-localization and multi-target tracking,\\\" IEEE Trans. Signal Process., vol. 67, no. 22, pp. 5896-5911, Nov. 2019. \",\"[27] G. Battistelli, L. Chisci, C. Fantacci, A. Farina, and B. Vo, \\\"Average Kullback-Leibler divergence for random finite sets,\\\" in Proc. 18th Int. Conf. Inf. Fusion, 2015, pp. 1359-1366. \",\"[28] G. Koliander, Y. El-Laham, P. Djuric, and F. Hlawatsch, “Fusion of probability density functions,” Proc. IEEE, vol. 110, no. 4, pp. 404–453, Apr. 2022. \",\"[29] M. Kayaalp, Y. Inan, E. Telatar, and A. Sayed, \\\"On the arithmetic and geometric fusion of beliefs for distributed inference,\\\" IEEE Trans. Autom. Contr., early access, Nov. 6, 2023, doi: 10.1109/TAC.2023.3330405. \",\"[30] S. J. Julier and J. K. Uhlmann, “A non-divergent estimation algorithm in the presence of unknown correlations,” in Proc. Amer. Control Conf., 1997, pp. 2369–2373. \",\"[31] T. Kropfreiter, F. Meyer, S. Coraluppi, C. Carthel, R. Mendrzik, and P. Willett, \\\"Track coalescence and repulsion: MHT, JPDA, and BP,\\\" in Proc. IEEE 24th Int. Conf. Inf. Fusion, Sun City, South Africa, 2021, pp. 1-8. \",\"[32] T. Kropfreiter, F. Meyer, D. Crouse, S. Coraluppi, F. Hlawatsch, and P. Willett, \\\"To coalesce or to repel? An analysis of MHT, JPDA, and belief propagation multitarget tracking methods,\\\" Aug. 2023, arXiv:2308.06326. \",\"[33] S. S. Dias and M. G. S. Bruno, \\\"Distributed bernoulli filters for joint detection and tracking in sensor networks,\\\" IEEE Trans. Signal Inf. Process. over Netw., vol. 2, no. 3, pp. 260-275, Sep. 2016. \",\"[34] S. Blackman and R. Popoli, Design and Analysis of Modern Tracking Systems. Boston, MA, USA: Artech House, 1999. \",\"[35] N. Patwari, J. N. Ash, S. Kyperountas, A. O. Hero, R. L. Moses, and N. S. Correal, \\\"Locating the nodes: Cooperative localization in wireless sensor networks,\\\" IEEE Signal Process. Mag., vol. 22, no. 4, pp. 54-69, Jul. 2005. \",\"[36] H. Wymeersch, U. J. Ferner, and M. Z. Win, \\\"Cooperative Bayesian self-tracking for wireless networks,\\\" IEEE Commun. Lett., vol. 12, no. 7, pp. 505-507, Jul. 2008. \",\"[37] S. Zhang, E. Staudinger, S. Sand, R. Raulefs, and A. Dammann, \\\"Anchor-free localization using round-trip delay measurements for martian swarm exploration,\\\" in Proc. IEEE/ION Position Location Navigation Symp., 2014, pp. 1130-1139. \",\"[38] O. Hlinka, O. Sluckak, F. Hlawatsch, P. M. Djuric, and M. Rupp, \\\"Likelihood consensus and its application to distributed particle filtering,\\\" IEEE Trans. Signal Process., vol. 60, no. 8, pp. 4334-4349, Aug. 2012. \",\"[39] P. Tichavsky, C. H. Muravchik, and A. Nehorai, \\\"Posterior Cramér-Rao bounds for discrete-time nonlinear filtering,\\\" IEEE Trans. Signal Process., vol. 46, no. 5, pp. 1386-1396, May 1998. \"]","page_idx":"14","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[514,47,635,162]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0506/9fdfaa5c626cddee685ec3068655de35200f669ddefb53e0ac64b2f3960c5f7d.jpg","list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[648,46,908,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hallysson Oliveira (Member, IEEE) received the bachelor's degree in computer engineering and the master's degree in computer science from Universidade Federal do Espirito Santo, Vitoria, Brazil, in 2002 and 2005, respectively, and the master's degree in aeronautical engineering and the Ph.D. degree in electrical and computer engineering from Instituto Tecnologico de Aeronáutica (ITA), São José dos Campos, Brazil, in 2007 and 2022, respectively. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[663,160,908,171]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From 2006 to 2021, he worked for Embraer ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,172,908,239]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"S.A. as a Research Software Engineer working in R&T projects related to different technologies and areas of application, such as image processing, pattern recognition, machine learning, sensor fusion, and autonomous navigation. Since 2021 he has been with ADASI working with software simulation and integration and currently leading the embedded software team. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[514,292,633,410]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0188/efdc6c15403c6898a71a2606bbb0333d91ff63c3094e1d5a79b8360f7c605c0f.jpg","list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[648,292,908,395]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Stiven S. Dias (Member, IEEE) received the bachelor's degree in computer engineering from Universidade Federal do Espirito Santo, Vitoria, Brazil, in 2006, the master's degree in aeronautical engineering, and the Ph.D. degree in electrical and computer engineering from Instituto Tecnologico de Aeronáutica (ITA), São José dos Campos, Brazil, in 2008 and 2014, respectively. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[648,395,908,417]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"He has been with Eindhoven University of Technology (TUE), Eindhoven, Netherlands, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,417,908,532]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"since 2021 as a Machine Learning Lecturer, where he has supervised several graduation projects mainly focused on anomaly detection and prediction applications. From 2008 to 2021, he worked for the Embraer Group, São Paulo, Brazil, as System Engineer Specialist designing coastal surveillance applications and as a Research Engineer developing advanced data fusion systems for embedded command and control (C2) systems. In 2023, he has changed his role at TUe to a Research Fellow and has rejoined Embraer S.A. as a Data Fusion Research Engineer to lead R&T projects. His main research interests include distributed estimation in sensor networks, Bayesian inference, and machine learning. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[514,585,633,700]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0011/a38bc944a8b427de2cc37234724fd45dc59c1bb7656f33af182a1f229d68a5d7.jpg","list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[648,584,908,664]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Marcelo G. S. Bruno (Senior Member, IEEE) received the bachelor's and master's degrees in electrical engineering from the University of São Paulo, São Paulo, Brazil, in 1992 and 1993 respectively, and the Ph.D. degree in electrical and computer engineering from Carnegie Mellon University, Pittsburgh, PA, USA, in 1998. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[648,664,908,710]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"He is currently affiliated with the Electrical Engineering Division, Instituto Tecnológico de Aeronáutica (ITA), São José dos Campos, Brazil, where he is an Associate Professor. His ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,710,908,868]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"research interests include statistical signal processing, particularly Markov random fields (Mrfs), hidden Markov models (HMMs), particle filters/sequential Monte Carlo methods (SMC), Markov Chain Monte Carlo (MCMC), statistics on manifolds, random finite sets (RFS), Bernoulli and multi-Bernoulli filters, nonparametric belief propagation, and their applications in target detection/tracking, image processing, machine learning, mobile robotics, and telecommunications. His current research is focused on distributed detection, estimation and filtering in sensor networks combining Bayesian methods with iterative consensus, diffusion and random information dissemination strategies. Particular areas under investigation include distributed filtering and estimation in Matrix Lie groups; joint cooperative localization and noncooperative detection and tracking in sensor networks; and manifold-constrained message passing on factor graphs. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,868,908,902]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Dr. Bruno served as an Associate Editor of IEEE Signal Processing Letters, IEEE TRANSACTIONS ON SIGNAL PROCESSING, and IEEE TRANSACTIONS ON SIGNAL AND INFORMATION PROCESSING OVER NETWORKS. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[102,939,750,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"OLIVEIRA ET AL.: GNSS-DENIED JOINT COOPERATIVE TERRAIN NAVIGATION AND TARGET TRACKING ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[879,940,908,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1005 ","text_format":null,"text_level":null,"type":"page_number"}],"string":"[\n {\n \"bbox\": \"[102,83,493,231]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"GNSS-Denied Joint Cooperative Terrain Navigation and Target Tracking Using Factor Graph Geometric Average Fusion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,292,364,308]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"HALLYSSON OLIVEIRA, Member, IEEE \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,309,467,321]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Instituto Tecnológico de Aeronáutica, São José dos Campos, Brazil \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,328,310,345]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"STIVEN S. DIAS, Member, IEEE \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,345,341,357]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Embraer S.A., São José dos Campos, Brazil \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,358,464,369]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Eindhoven University of Technology, Eindhoven, The Netherlands \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,377,401,393]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"MARCELO G. S. BRUNO, Senior Member, IEEE \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,393,467,406]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Instituto Tecnológico de Aeronáutica, São José dos Campos, Brazil \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,450,500,638]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We propose a fully distributed methodology based on factor graphs for joint cooperative localization and distributed noncooperative target tracking in a 3-D scenario where multiple surveillance aircraft fly in formation without access to global navigation satellite system (GNSS) measurements or communication with anchor nodes. Our approach is based on the adapt-then-combine (ATC) diffusion scheme, which is integrated into the factor graph by the introduction of special combine factors to perform geometric average fusion of the target beliefs over a partially connected network. The updated target belief held by each aircraft following the combine step is also fed back to improve the aircraft's own self-localization, assimilating the target measurements. Simulation results show that the proposed distributed algorithm performed close to the posterior Cramér-Rao lower bound of the optimal centralized solution and that the agents approached a consensus about the target state estimate. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,666,500,705]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Manuscript received 5 May 2023; revised 31 August 2023 and 30 October 2023; accepted 4 November 2023. Date of publication 17 November 2023; date of current version 9 February 2024. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,712,321,724]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DOI. No. 10.1109/TAES.2023.3333286 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,732,406,744]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Refereeing of this contribution was handled by P. Braca. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,752,498,790]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The work of Marcelo G. S. Bruno was supported by São Paulo Research Foundation (FAPESP) under Grant 2020/09838-0 (BIOS - Brazilian Institute of Data Science). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,797,500,886]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Authors' addresses: Hallysson Oliveira and Marcelo G. S. Bruno are with Instituto Tecnológico de Aeronáutica, São José dos Campos 12228-900, Brazil, E-mail: (hallysson@ieee.org; bruno@ita.br); Stiven S. Dias is with Embraer S.A., São José dos Campos 12247-016, Brazil, and also with the Eindhoven University of Technology, 5600 Eindhoven, The Netherlands, E-mail: (stiven.dias@embraer.com.br). (Corresponding author: Hallysson Oliveira.) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,906,241,919]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"0018-9251 © 2023 IEEE \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,44,665,58]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"I. INTRODUCTION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,65,911,383]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In a previous work [1], we introduced an algorithm for anchorless, GNSS-free, 2-D cooperative self-localization of multiple aircraft using terrain-aided navigation, inertial sensor measurements, and peer-to-peer relative distance measurements. In this article, we extend that previous algorithm to a more challenging scenario where the surveillance aircraft not only estimate their own position, but also jointly track a noncollaborative target in a distributed fashion. To account for more realistic evasive maneuvers that may be taken by the tracked target, including possible changes in altitude, we perform joint cooperative terrain-aided navigation and distributed target tracking (J-CTAN-DTT) in a 3-D surveillance space, as opposed to the previously considered 2-D scenario. As in [1], we use a Bayesian approach that employs factor graphs [2], [3], [4]. The main original contribution of the article is, however, to handle the distributed target tracking task using an adapt-then Combine (ATC) diffusion [5], [6], [7], which is completely integrated into the self-localization factor graph framework, and implemented in a fully distributed fashion by the introduction in the graph of artificial factors to perform the Combine step. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,383,910,594]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We maintain, as in [1], a scenario where none of the surveillance aircraft have available global navigation satellite system (GNSS) data, relying only on inertial unit measurements and terrain-aided navigation (TAN) techniques [8], [9], [10], [11], [12] in addition to peer-to-peer relative distance measurements. To account for the new 3-D scenario and the additional integrated joint distributed target tracking task, we employ an additional set of sensor measurements compared to [1] at each aircraft node, namely, barometric altitude, ground clearance (measured by a radar altimeter and used in conjunction with a 2-D terrain elevation map), and target measurements (slant range, azimuth angle, and elevation angle) collected at each node by an airborne 3-D radar. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,626,816,641]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. Related Work and Article Contributions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,648,911,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Advances in communication technologies have enabled the use of wireless sensor networks in many different engineering applications [13], [14], [15]. Multiagent networks are of special interest for tactical coordination over military data links. In addition to self-localizing, agents—typically military units such as aircraft that are partially connected by a communication network with no data fusion center—must also cooperate [16] to simultaneously track a noncooperative target and agree with each other regarding the tracked target state. The joint cooperative localization and distributed target tracking problem is particularly challenging because tracking performance depends greatly on the ability of each agent that is observing the target to maintain an accurate estimate of its own location, which is unknown a priori in the considered scenario. On top of that, GNSS jamming and spoofing techniques are becoming ubiquitous in modern battlefield, so GNSS-free localization techniques are crucial in contemporary military applications. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,939,736,952]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.60, NO.1 FEBRUARY 2024 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[883,940,906,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"991 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[84,44,482,150]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Previous references in the literature, which use factor graphs combined with belief propagation, usually address cooperative location and distributed target tracking separately as fully decoupled problems. While [16], [17], [18], [19], and [20] only address cooperative localization based on factor graphs, [21], [22], and [23], for example, only address target tracking based on factor graphs. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,151,482,435]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The previous works [24], [25], and [26] handled the joint localization and distributed target tracking problem within a factor graph framework using a consensus approach that has a high communication cost, and might not be adequate for real-time systems. In contrast to the aforementioned references, our proposed diffusion scheme performs joint cooperative self-localization and distributed noncooperative target tracking with a much lower internode communication cost than the consensus algorithms in [24], [25], and [26], while maintaining at the same time a performance that is close to the posterior Cramér-Rao lower bound (PCRLB) for both the localization and tracking errors of the optimal nonlinear filter that uses a central data fusion hub to process all measurements in the network, in batch, at each time instant. Furthermore, we empirically show that the different agents approach, over time, a consensus about the target state without the need, as in previous literature, of multiple iterative data exchange between nodes, between consecutive sensor observations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,436,482,737]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Specifically, our algorithm seeks to approximate the adapt step of a classic distributed ATC tracker by replacing the likelihood of the local target observations at each agent node given the target and the agent positions by its expected value with respect to the latest belief (i.e., posterior probability distribution) that the agent has about its own state. Next, the aforementioned special factors combine the updated target state beliefs following the adapt step at each node using a geometric average (GA) fusion criterion [27], which can be shown to minimize an average Kullback-Leibler (KL) divergence metric [28], [29]. For Gaussian beliefs, the employed combine step rule reduces to the well-known method of covariance intersection in sensor fusion [30]. Furthermore, after the distributed combine process, each aircraft uses its updated target belief to improve its own self-localization. As in [1] and [33], we implement the sum-product algorithm (SPA) operations using a scheme that combines Gaussian mixture models (GMMs) and sequential Monte Carlo methods, which significantly reduces the amount of data exchanged over the network. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,739,482,919]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The rest of this article is organized as follows. We present, in section II, the problem statement and detail the state and observation models. In section III, we derive the joint localization and tracking algorithm using a distributed message passing approach over factor graphs. A communication-efficient implementation of the said algorithm using GMMs and sequential Monte Carlo methods is then presented in section IV. Performance results in a simulated 3-D scenario with real terrain elevation maps are presented and discussed in section V, assuming different cooperative and noncooperative operation modes and no available GNSS data. Finally, section VI concludes this article. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[496,44,692,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"II. PROBLEM STATEMENT \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[495,65,893,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the cooperative terrain-aided navigation and distributed target tracking scenario considered in this article, we have multiple cooperative aircraft and a single noncooperative target flying in a 3-D environment where GNSS data are unavailable, either due to hostile actions such as spoofing and jamming, or natural causes such as low-latitude ionospheric interference that degrades GNSS signals. The aircraft must, therefore, cooperate within local neighborhoods over a wireless communication network to estimate both their own position and the position of the noncooperative target. As in [1], the topology of the communication network is modeled as a graph $\\\\mathcal{G}$ consisting of a set of vertices $\\\\mathcal{V} \\\\triangleq \\\\{1, \\\\dots, A\\\\}$ , representing the location of each cooperative agent and a set of undirected edges $\\\\mathcal{E} = \\\\{(a, b) \\\\in \\\\mathcal{V} \\\\times \\\\mathcal{V}\\\\}$ that represent the communication links between agent nodes. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[493,306,893,428]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Each aircraft measures the distance to the terrain beneath them, also known as ground clearance, using an onboard radar altimeter. A barometric sensor is employed in turn to measure the aircraft's altitude with respect to sea level. In addition, each cooperative aircraft has an onboard 3-D radar to measure the slant range as well as the azimuth and elevation angles to the noncooperative target's 3-D location based on the received echoed radar pulses from the target. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[493,429,893,625]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"REMARK Although we consider a single target scenario, it should be stressed that the solution presented in this article could be extended in a straightforward way to a scenario with multiple targets provided that, as considered in [24], perfect a priori target data association is available and there is no clutter, missed detections, or false alarms. In a scenario where the data association problem also has to be solved jointly with the cooperative localization and target tracking tasks, a different algorithm would have to be derived, but that is not the focus of this present article. A comprehensive review of state-of-the-art approaches for multitarget tracking using belief propagation methods may be found in [22], [31], and [32]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[495,635,702,650]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. Distributed Signal Model \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[493,656,893,792]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Aircraft dynamics: Let $\\\\mathbf{x}_{n,a} \\\\triangleq [x_{n,a} y_{n,a} z_{n,a}]^T$ be the unknown 3-D position of the aircraft $a$ at the time stamp $t_n = nT_s$ , $n \\\\in \\\\mathbb{Z}_+$ , where $T_s$ is the sampling period. The deterministic and known input vector $\\\\mathbf{u}_{n,a} \\\\triangleq [\\\\Delta x_{n,a} \\\\Delta y_{n,a} \\\\Delta z_{n,a}]^T$ represents, on the other hand, the displacement in the 3-D space from instant $n$ to $n + 1$ , which is measured by an inertial navigation system (INS) processor. As in [1], [9], and [10], the state vector evolves over time according to the model \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[599,801,890,816]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf {x} _ {n + 1, a} = \\\\mathbf {x} _ {n, a} + \\\\mathbf {u} _ {n, a} + \\\\mathbf {w} _ {n, a} \\\\tag {1}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[493,823,893,869]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\{\\\\mathbf{w}_{n,a}\\\\}$ are Gaussian random vectors that are mutually independent and have identical distribution $\\\\mathbf{w}_{n,a} \\\\sim \\\\mathcal{N}(\\\\mathbf{0}, \\\\mathbf{Q}_{n,a})$ , for all $a \\\\in V$ and all $n \\\\in \\\\mathbb{Z}_+$ . Furthermore, as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[495,893,892,919]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"<sup>1</sup>For simplicity, we use the same notation in this article for deterministic and random vectors, with the proper interpretation implied in context. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[88,939,112,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"992 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[256,939,890,952]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.60, NO.1 FEBRUARY 2024 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[99,44,498,73]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"argued in [9], the state noise vector $\\\\mathbf{w}_{n,a}$ in (1) also includes the INS measurement error. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[99,75,500,195]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Target dynamics: Since the target is noncooperative, each aircraft also needs to estimate the target position and velocity. We then specify the target kinematics using the constant velocity model [34]. More precisely, let $\\\\mathbf{x}_{n,o} \\\\triangleq [x_{n,o} \\\\dot{x}_{n,o} y_{n,o} \\\\dot{y}_{n,o} z_{n,o} \\\\dot{z}_{n,o}]^T$ denote, at instant $n$ , the unknown position and velocity of the noncooperative target $o$ in 3-D Cartesian coordinates. The kinematic dynamic model for the target is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[215,205,498,220]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf {x} _ {n + 1, o} = \\\\mathbf {F} _ {n, o} \\\\mathbf {x} _ {n, o} + \\\\mathbf {w} _ {n, o} \\\\tag {2}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[99,229,498,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\mathbf{F}_{n,o} = \\\\mathrm{diag}([\\\\tilde{\\\\mathbf{F}}_{n,o}\\\\tilde{\\\\mathbf{F}}_{n,o}\\\\tilde{\\\\mathbf{F}}_{n,o}])$ and the noise $\\\\{\\\\mathbf{w}_{n,o}\\\\}$ $n\\\\geq 0$ is a sequence of independent and identically distributed (i.i.d.) zero-mean Gaussian random variables with covariance $\\\\mathbf{Q}_{n,o} = \\\\sigma_{\\\\mathrm{acc},o}^2 *\\\\mathrm{diag}([\\\\tilde{\\\\mathbf{Q}}_{n,o}\\\\tilde{\\\\mathbf{Q}}_{n,o}\\\\tilde{\\\\mathbf{Q}}_{n,o}])$ in which $\\\\sigma_{\\\\mathrm{acc},o}^{2}$ is the variance of the target acceleration noise, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,306,413,339]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\tilde {\\\\mathbf {F}} _ {n, o} = \\\\left[ \\\\begin{array}{c c} 1 & T _ {s} \\\\\\\\ 0 & 1 \\\\end{array} \\\\right] \\\\text {a n d} \\\\tilde {\\\\mathbf {Q}} _ {n, o} = \\\\left[ \\\\begin{array}{c c} T _ {s} ^ {3} / 3 & T _ {s} ^ {2} / 2 \\\\\\\\ T _ {s} ^ {2} / 2 & T _ {s} \\\\end{array} \\\\right].\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[99,340,498,400]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Barometric altitude observation: For the sake of simplicity, the barometric altitude $y_{n,a \\\\leftarrow ba}^2$ measured at time $n$ at node $a$ is assumed to be an unbiased observation of the aircraft altitude $z_{n,a}$ such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[200,425,498,441]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\ny _ {n, a \\\\leftarrow b a} = b a \\\\left(\\\\mathbf {x} _ {n, a}\\\\right) + v _ {n, a \\\\leftarrow b a} \\\\tag {3}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[99,449,498,479]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $ba(\\\\mathbf{x}_{n,a}) = z_{n,a}$ and $v_{n,a\\\\leftarrow ba}$ is a Gaussian noise with zero mean and variance $\\\\sigma_{ba}^2$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[99,479,500,554]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ground clearance observation: An onboard radar altimeter measures the ground clearance, i.e., the distance from the aircraft to the terrain directly below it. So, the ground clearance is modeled as the real aircraft altitude $z_{n,a}$ minus the elevation of the terrain below the aircraft \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[187,566,498,580]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\ny _ {n, a \\\\leftarrow g c} = g c \\\\left(\\\\mathbf {x} _ {n, a}; \\\\mathcal {M}\\\\right) + v _ {n, a \\\\leftarrow g c} \\\\tag {4}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[99,589,498,695]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $gc(\\\\mathbf{x}_{n,a};\\\\mathcal{M}) = z_{n,a} - ge(\\\\mathbf{x}_{n,a};\\\\mathcal{M})$ and $ge(\\\\mathbf{x};\\\\mathcal{M})$ is a function that outputs the terrain elevation, given the position and a known 2-D digital elevation map $\\\\mathcal{M}$ , and the noise $\\\\{v_{n,a\\\\leftarrow gc}\\\\}$ , $n\\\\geq 0$ , is an i.i.d. sequence with Gaussian distribution $v_{n,a\\\\leftarrow gc}\\\\sim \\\\mathcal{N}(0,\\\\sigma_{gc}^2)$ . The discrete terrain elevation map is interpolated at each continuous position location as described in [1, Appendix A]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[99,696,500,739]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Aircraft distance observations: The 3-D peer-to-peer distance, a.k.a. slant range, between agents $a$ and $b$ measured from node $a$ at time $n$ is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[189,751,498,767]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\ny _ {n, a \\\\leftarrow b} = r g \\\\left(\\\\mathbf {x} _ {n, a}, \\\\mathbf {x} _ {n, b}\\\\right) + v _ {n, a \\\\leftarrow b} \\\\tag {5}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[99,775,498,853]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\{v_{n,a\\\\leftarrow b}\\\\}$ , $n\\\\geq 0$ , is a sequence of mutually independent Gaussian random variables with identical distributions $v_{n,a\\\\leftarrow b}\\\\sim \\\\mathcal{N}\\\\left(\\\\mathbf{0},\\\\sigma_{ad}^{2}\\\\right)$ $\\\\forall n\\\\in \\\\mathbb{Z}_{+}$ , $a\\\\in \\\\mathcal{V}$ ; $rg(\\\\mathbf{x},\\\\mathbf{x}^{\\\\prime}) = \\\\|\\\\mathbf{x} - \\\\mathbf{x}^{\\\\prime}\\\\|$ , and $\\\\| \\\\mathbf{x}\\\\| ^2\\\\triangleq \\\\mathbf{x}^T\\\\mathbf{x}$ $\\\\forall \\\\mathbf{x}\\\\in \\\\mathbb{R}^3$ . To avoid clock synchronization issues [18], [35], [37], the measurements \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,44,906,73]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"in (5) are obtained using round-trip time of arrival (RTOA) technique. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,75,910,239]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $\\\\bar{\\\\mathbf{N}}(n, a)$ be the open neighborhood of node $a$ at instant $n$ defined as the set of indexes of all nodes in the communication network that are connected to node $a$ at that same instant. For each node $a$ , we collect the state vectors $\\\\{\\\\mathbf{x}_{n,b}\\\\}$ for all nodes $b \\\\in \\\\bar{\\\\mathbf{N}}(n, a)$ , in an extended vector denoted $X_{n,a}$ . In a similar fashion, the internode distance measurements $\\\\{y_{n,a \\\\leftarrow b}, \\\\forall b \\\\in \\\\bar{\\\\mathbf{N}}(n, a)\\\\}$ , are stacked into the extended vector $Y_{n,a}$ at node $a$ . It follows from (5) that the observations in $Y_{n,a}$ are independent conditioned on any pair of states of the type $(\\\\mathbf{x}_{n,a}, \\\\mathbf{x}_{n,b}), b \\\\in \\\\bar{\\\\mathbf{N}}(n, a)$ , at instant $n$ . Thus, we can write \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[565,253,908,287]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\np \\\\left(\\\\mathbf {Y} _ {n, a} \\\\mid \\\\mathbf {X} _ {n, a}\\\\right) = \\\\prod_ {b \\\\in \\\\bar {\\\\mathbf {N}} (n, a)} p \\\\left(y _ {n, a \\\\leftarrow b} \\\\mid \\\\mathbf {x} _ {n, a}, \\\\mathbf {x} _ {n, b}\\\\right). \\\\tag {6}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,301,910,375]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Target observation: Each aircraft has a radar sensor capable of making 3-D observations of the target location. The target observation, measured by the cooperative aircraft $a$ , to the noncooperative target $o$ at instant $n$ is modeled then as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[601,392,908,407]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf {y} _ {n, a \\\\leftarrow o} = \\\\mathbf {t o} (\\\\mathbf {x} _ {n, a}, \\\\mathbf {x} _ {n, o}) + \\\\mathbf {v} _ {n, a \\\\leftarrow o} \\\\tag {7}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,421,910,580]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\mathbf{y}_{n,a\\\\leftarrow o} = \\\\left[y_{n,a\\\\leftarrow o}^{(rg)}y_{n,a\\\\leftarrow o}^{(az)}y_{n,a\\\\leftarrow o}^{(el)}\\\\right]^T,\\\\mathbf{t o}(\\\\mathbf{x},\\\\mathbf{x}^{\\\\prime}) =$ $\\\\left[rg(\\\\mathbf{x},\\\\mathbf{x}^{\\\\prime})az(\\\\mathbf{x},\\\\mathbf{x}^{\\\\prime})el(\\\\mathbf{x},\\\\mathbf{x}^{\\\\prime})\\\\right]^{T},az(\\\\mathbf{x},\\\\mathbf{x}^{\\\\prime}) = \\\\tan^{-1}\\\\left(\\\\frac{y^{\\\\prime} - y}{x^{\\\\prime} - x}\\\\right)$ is the azimuth angle and $el(\\\\mathbf{x},\\\\mathbf{x}^{\\\\prime}) = \\\\tan^{-1}\\\\left(\\\\frac{z^{\\\\prime} - z}{\\\\sqrt{(x^{\\\\prime} - x)^{2} + (y^{\\\\prime} - y)^{2}}}\\\\right)$ the elevation angle. We further assume that the covariance matrix $\\\\mathbf{R}_{to}$ has a time-invariant diagonal structure such that $\\\\mathbf{R}_{to} = diag\\\\left([ \\\\sigma_{ro}^{2}\\\\sigma_{az}^{2}\\\\sigma_{el}^{2}] \\\\right]$ . Lastly, note that all cooperative aircraft $a\\\\in \\\\mathcal{V}$ make observations of the same single target state $\\\\mathbf{x}_{n,o}$ independently. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,597,795,611]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"III. JOINT CTAN-DTT FACTOR GRAPH \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,618,908,691]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section, we propose a fully distributed algorithm that enables joint cooperative estimation of the aircraft states, referred to in this article as cooperative terrain-aided navigation, or CTAN, and distributed target tracking, abbreviated DTT. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,693,910,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The DTT routine is based on an adapt-then Combine (ATC) diffusion technique with a modified adapt step where the likelihood of the target measurements for a known aircraft state is approximated by its expected value with respect to (w.r.t.) the latest available belief of the aircraft state after aircraft motion prediction and assimilation of the local barometric altitude and ground clearance sensor measurements. The combine step is then implemented via geometric average (GA) fusion in local network neighborhoods. We show how the proposed ATC scheme can be represented exactly by a factor graph model in which we introduce special factor nodes that enable a distributed implementation of the GA fusion step and the computation of the aforementioned expected values of the likelihood of the target observations at each aircraft node. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[99,880,500,919]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2To shorten notation, we denote an observation vector of the physical property (or of the entity) $t$ at node $a$ at time $n$ as $\\\\mathbf{y}_{n,a\\\\leftarrow t}$ , highlighting that node $a$ observed $t$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[101,938,750,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"OLIVEIRA ET AL.: GNSS-DENIED JOINT COOPERATIVE TERRAIN NAVIGATION AND TARGET TRACKING \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[883,940,906,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"993 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[84,44,483,330]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The CTAN routine, without feedback from the DTT subgraph—the connection between CTAN and DTT routines/subgraphs will be discussed later—computes in turn the marginal posterior p.d.f. of the aircraft state at each node $a$ via message passing over a factor subgraph that represents the factorization of the join p.d.f. of the aircraft states and barometric altitude, ground clearance, and internode distance measurements. The computed marginal is further refined by adding an extra message in the graph that represents the expected value of the likelihood of the target observations w.r.t. the latest available belief about the target state outputted by the DTT subgraph. By introducing a special factor node that enables the computation of the aforementioned expected value, we are able to connect the CTAN and DTT subgraphs and arrive at a full joint CTAN-DTT (J-CTAN-DTT) factor graph in which, by following a noniterative message passing schedule according to the usual SPA rules, we can implement the J-CTAN-DTT algorithm described in this section exactly. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,330,483,454]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To enable a fully distributed implementation, each node $a$ in the network tracks a local copy $\\\\mathbf{x}_{n,o}^{(a)}$ of the target state vector $\\\\mathbf{x}_{n,o}$ . As an artifact of our distributed implementation, we also denote the target state $\\\\mathbf{x}_{n,o}^{(a)}$ before the Combine step as $\\\\tilde{\\\\mathbf{x}}_{n,o}^{(a)}$ , and the aircraft state $\\\\mathbf{x}_{n,a}$ before the assimilation of the internode distance measurements as $\\\\tilde{\\\\mathbf{x}}_{n,a}$ . We emphasize, however, that both notations, with and without the diacritic mark, represent the same physical state vector. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,468,227,481]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. Factor Notation \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,489,483,563]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To obtain a factor graph representation of the operations in this section, we consider the variable nodes $\\\\mathbf{x}_{n - 1,a}$ ; $\\\\tilde{\\\\mathbf{x}}_{n,a}$ ; $\\\\mathbf{x}_{n,a}$ ; $\\\\mathbf{x}_{n - 1,o}^{(a)}$ ; $\\\\tilde{\\\\mathbf{x}}_{n,o}^{(a)}$ and $\\\\mathbf{x}_{n,o}^{(a)}$ at each network node $a\\\\in \\\\{1,2,\\\\dots,A\\\\}$ , and define the following local factor nodes at the same node. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[86,565,256,579]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1) Aircraft motion factor \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[202,589,482,628]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} f _ {n, a} \\\\triangleq f _ {n, a} (\\\\tilde {\\\\mathbf {x}} _ {n, a}, \\\\mathbf {x} _ {n - 1, a}) \\\\\\\\ \\\\triangleq p \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a} \\\\mid \\\\mathbf {x} _ {n - 1, a}\\\\right). \\\\tag {8} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[86,638,271,651]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2) Ground clearance factor \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[202,662,482,700]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} g c _ {n, a} \\\\triangleq g c _ {n, a} (\\\\tilde {\\\\mathbf {x}} _ {n, a}) \\\\\\\\ \\\\triangleq p \\\\left(y _ {n, a \\\\leftarrow g c} | \\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right). \\\\tag {9} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[86,710,282,724]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3) Barometric altitude factor \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[200,734,482,772]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} b a _ {n, a} \\\\triangleq b a _ {n, a} (\\\\tilde {\\\\mathbf {x}} _ {n, a}) \\\\\\\\ \\\\triangleq p \\\\left(y _ {n, a \\\\leftarrow b a} \\\\mid \\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right). \\\\tag {10} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[84,782,462,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4) Internode distance factor between node $a$ and node $b$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[179,806,482,843]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} a d _ {n, a \\\\leftarrow b} \\\\triangleq a d _ {n, a \\\\leftarrow b} (\\\\mathbf {x} _ {n, a}, \\\\tilde {\\\\mathbf {x}} _ {n, b}) \\\\\\\\ = p \\\\left(y _ {n, a \\\\leftarrow b} | \\\\mathbf {x} _ {n, a}, \\\\tilde {\\\\mathbf {x}} _ {n, b}\\\\right). \\\\tag {11} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[86,853,276,868]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5) Target observation factor \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[184,876,482,917]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} t o _ {n, o} ^ {(a)} \\\\triangleq t o _ {n, o} ^ {(a)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right) \\\\\\\\ \\\\triangleq p \\\\left(\\\\mathbf {y} _ {n, a \\\\leftarrow o} | \\\\tilde {\\\\mathbf {x}} _ {n, a}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right). \\\\tag {12} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[496,44,656,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6) Target motion factor \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[614,65,892,107]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} f _ {n, o} ^ {(a)} \\\\triangleq f _ {n, o} ^ {(a)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}, \\\\mathbf {x} _ {n - 1, o} ^ {(a)}\\\\right) \\\\\\\\ \\\\triangleq p \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)} \\\\mid \\\\mathbf {x} _ {n - 1, o} ^ {(a)}\\\\right). \\\\tag {13} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[496,116,781,132]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B. Decoupled Cooperative Localization \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[495,138,893,213]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $\\\\gamma_{n,a}(\\\\mathbf{x}_{n,a})$ denote the aircraft state belief at each node $a$ at instant $n$ . To update the belief $\\\\gamma_{n,a}$ sequentially in time, the aircraft network may perform cooperative localization independently of the distributed tracking task using the factor graph-based CTAN algorithm in [1]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[495,214,893,335]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Specifically, in a decoupled cooperative localization scenario, let $\\\\mathbf{X}_n$ denote the collection of all aircraft states $\\\\{\\\\mathbf{x}_{k,a}\\\\} ,\\\\mathbf{Z}_n$ denote the collection of all local ground clearance and barometric altitude measurements $\\\\{y_{k,a\\\\leftarrow gc};y_{k,a\\\\leftarrow ba}\\\\}$ and let $\\\\mathbf{Y}_n$ represent the set of all internode measurements $\\\\{y_{k,a\\\\leftarrow b}\\\\}$ for $a\\\\in \\\\{1,\\\\dots ,A\\\\}$ $b\\\\in \\\\{1,\\\\dots ,A\\\\}$ $a\\\\neq b$ and $k\\\\in$ $\\\\{0,\\\\ldots ,n\\\\}$ . Given the conditional independence assumptions in the model, the joint p.d.f. can be written as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,342,892,443]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} p \\\\left(\\\\mathbf {X} _ {n}, \\\\mathbf {Y} _ {n}, \\\\mathbf {Z} _ {n}\\\\right) = \\\\prod_ {k = 1} ^ {n} \\\\prod_ {a = 1} ^ {A} p \\\\left(\\\\mathbf {x} _ {k, a} | \\\\mathbf {x} _ {k - 1, a}\\\\right) \\\\\\\\ \\\\times p \\\\left(y _ {k, a \\\\leftarrow g c} | \\\\mathbf {x} _ {k, a}\\\\right) p \\\\left(y _ {k, a \\\\leftarrow b a} | \\\\mathbf {x} _ {k, a}\\\\right) \\\\\\\\ \\\\times \\\\prod_ {b \\\\in \\\\tilde {\\\\mathbf {N}} (k, a)} p \\\\left(y _ {k, a \\\\leftarrow b} | \\\\mathbf {x} _ {k, a}, \\\\mathbf {x} _ {k, b}\\\\right). \\\\tag {14} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[495,449,893,510]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As explained before, for convenience and to enable the integration between the CTAN and DTT subgraphs, we introduce an auxiliary variable $\\\\tilde{\\\\mathbf{x}}_{n,a}$ which is actually identical to $\\\\mathbf{x}_{n,a}$ . We then rewrite the factorization in (14) as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[506,518,892,732]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} p \\\\left(\\\\mathbf {X} _ {n}, \\\\tilde {\\\\mathbf {X}} _ {n}, \\\\mathbf {Y} _ {n}, \\\\mathbf {Z} _ {n}\\\\right) = \\\\prod_ {k = 1} ^ {n} \\\\prod_ {a = 1} ^ {A} p \\\\left(\\\\tilde {\\\\mathbf {x}} _ {k, a} | \\\\mathbf {x} _ {k - 1, a}\\\\right) \\\\delta \\\\left(\\\\mathbf {x} _ {k, a} - \\\\tilde {\\\\mathbf {x}} _ {k, a}\\\\right) \\\\\\\\ \\\\times p \\\\left(y _ {k, a \\\\leftarrow g c} | \\\\tilde {\\\\mathbf {x}} _ {k, a}\\\\right) p \\\\left(y _ {k, a \\\\leftarrow b a} | \\\\tilde {\\\\mathbf {x}} _ {k, a}\\\\right) \\\\\\\\ \\\\times \\\\prod_ {b \\\\in \\\\tilde {\\\\mathbf {N}} (k, a)} p \\\\left(y _ {k, a \\\\leftarrow b} | \\\\mathbf {x} _ {k, a}, \\\\tilde {\\\\mathbf {x}} _ {k, b}\\\\right) \\\\\\\\ \\\\equiv \\\\prod_ {k = 1} ^ {n} \\\\prod_ {a = 1} ^ {A} f _ {k, a} (\\\\tilde {\\\\mathbf {x}} _ {k, a}, \\\\mathbf {x} _ {k - 1, a}) \\\\delta (\\\\mathbf {x} _ {k, a} \\\\\\\\ \\\\left. - \\\\tilde {\\\\mathbf {x}} _ {k, a}\\\\right) \\\\times g c _ {k, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {k, a}\\\\right) b a _ {k, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {k, a}\\\\right) \\\\\\\\ \\\\times \\\\prod_ {b \\\\in \\\\tilde {\\\\mathbf {N}} (k, a)} a d _ {k, a \\\\leftarrow b} \\\\left(\\\\mathbf {x} _ {k, a}, \\\\tilde {\\\\mathbf {x}} _ {k, b}\\\\right) \\\\tag {15} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[495,739,892,768]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $f_{n,a}$ , $gc_{n,a}$ , $ba_{n,a}$ , and $ad_{n,a \\\\leftarrow b}$ are the factors previously introduced in (8), (9), (10), and (11), respectively. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[495,768,893,875]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The factorization in (15) is represented by the factor graph in Fig. 1, shown here for two neighboring network nodes $a$ and $b$ . The messages computed at nodes $a$ and $b$ are highlighted in blue and red, respectively, with messages that are passed between nodes shown in dashed lines and intranode messages in solid lines. The factor represented by \\\" $=$ \\\" symbol corresponds to the Dirac delta operator in (15). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[495,875,892,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By applying the standard SPA rules, the noniterative message passing schedule at node $a$ corresponding to the factor graph in Fig. 1 runs as described in the sequel. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[88,940,111,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"994 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[256,939,888,952]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.60, NO.1 FEBRUARY 2024 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[120,46,478,251]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 1. CTAN factor subgraph without DTT feedback. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0215/8398557ae5c91a563fffc0e7cb08acd8c0f8471328776a68172c3c53a5a88584.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[101,294,498,325]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"First, we compute the aircraft motion prediction message $\\\\alpha_{n,a}$ such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,330,498,375]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\alpha_ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\propto \\\\int f _ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}, \\\\mathbf {x} _ {n - 1, a}\\\\right) \\\\gamma_ {n - 1, a} \\\\left(\\\\mathbf {x} _ {n - 1, a}\\\\right) d \\\\mathbf {x} _ {n - 1, a}. \\\\tag {16}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[101,376,498,407]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Next, we define the barometric altitude and ground clearance messages, respectively, $\\\\kappa_{n,a}$ and $\\\\beta_{n,a}$ , such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[200,412,496,450]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\kappa_ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\triangleq p \\\\left(y _ {n, a - b a} | \\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\\\\\ = b a _ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\tag {17} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[200,454,498,493]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\beta_ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\triangleq p \\\\left(y _ {n, a \\\\leftarrow g c} | \\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\\\\\ = g c _ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right). \\\\tag {18} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[101,500,498,529]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The assimilation of the intranode distance measurements is done by computing the messages \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[137,537,498,554]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\iota_ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\propto \\\\alpha_ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\kappa_ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\beta_ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\tag {19}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[135,556,498,587]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\epsilon_ {n, a} \\\\left(\\\\mathbf {x} _ {n, a}\\\\right) \\\\propto \\\\int \\\\delta \\\\left(\\\\mathbf {x} _ {n, a} - \\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\iota_ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) d \\\\tilde {\\\\mathbf {x}} _ {n, a} \\\\tag {20}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[119,589,496,636]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\xi_ {n, a \\\\leftarrow b} (\\\\mathbf {x} _ {n, a}) \\\\propto \\\\int a d _ {n, a \\\\leftarrow b} (\\\\mathbf {x} _ {n, a}, \\\\tilde {\\\\mathbf {x}} _ {n, b}) \\\\lambda_ {n, a \\\\leftarrow b} (\\\\tilde {\\\\mathbf {x}} _ {n, b}) d \\\\tilde {\\\\mathbf {x}} _ {n, b} \\\\tag {21}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[101,643,498,672]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\lambda_{n,a\\\\leftarrow b}(\\\\tilde{\\\\mathbf{x}}_{n,b})$ is identical to the message $\\\\iota_{n,b}(\\\\tilde{\\\\mathbf{x}}_{n,b})$ and the functions $\\\\epsilon_{n,a}$ and $\\\\iota_{n,a}$ are also identical. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[125,672,491,689]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The final update belief of the state $\\\\mathbf{x}_{n,a}$ is the message \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[156,695,498,729]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\gamma_ {n, a} (\\\\mathbf {x} _ {n, a}) \\\\propto \\\\epsilon_ {n, a} (\\\\mathbf {x} _ {n, a}) \\\\prod_ {b \\\\in \\\\bar {\\\\mathbf {N}} (n, a)} \\\\xi_ {n, a \\\\leftarrow b} (\\\\mathbf {x} _ {n, a}). \\\\tag {22}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[101,736,498,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note that messages (16)-(18) correspond to the self-localization (self-loc) subgraph highlighted in Fig. 1, whereas messages (19)-(22) belong to the cooperative localization (coop-loc) subgraph. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,808,473,824]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Adapt-Then-Combine Distributed Target Tracking \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,829,500,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ATC diffusion is a well-known method for distributed target tracking in the literature [5], [6], [7]. Unlike consensus methods, diffusion techniques do not force all network nodes to have the same target state estimate at each time instant. Instead, each node computes its own posterior p.d.f. of the target state, which is, in principle, different \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,44,908,194]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"from the posterior computed by the other nodes. Ideally, over time however, as $n \\\\to \\\\infty$ , the local posteriors should converge to a consensus posterior [6] so that the variance of the local state estimates is reduced. The practical advantage of diffusion methods over consensus techniques is that they do not require multiple iterative communication between network nodes in the time interval between consecutive sensor measurements and, therefore, are more suitable for use in real-time scenarios due to their reduced internode communication cost. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,196,908,268]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the sequel, we propose an approach to integrate ATC diffusion with factor graph cooperative localization. We detail the proposed adapt and combine steps separately and then show how to represent them exactly by a factor graph model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,271,908,359]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1) Adapt Step: For known aircraft positions, the algorithm would start with a local adapt step at each node $a$ consisting of the computation of the predicted target state p.d.f. using the target motion transition p.d.f. followed by the correction of that predicted p.d.f, using the likelihood function of the target observations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,361,910,465]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In our signal model, however, since the aircraft states are unknown, we propose to replace the ideal likelihood of the target observations given the true (deterministic) aircraft state by the expected value of the same likelihood given the unknown (random) aircraft state w.r.t. to the latest available local posterior p.d.f. $\\\\tilde{p}_{n,a}$ of the aircraft $a$ at node $a$ at instant $n$ , i.e., \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[578,472,908,503]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\int p \\\\left(\\\\mathbf {y} _ {n, a \\\\leftarrow o} | \\\\tilde {\\\\mathbf {x}} _ {n, a}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right) \\\\tilde {p} _ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) d \\\\tilde {\\\\mathbf {x}} _ {n, a}. \\\\tag {23}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,507,910,583]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We obtain $\\\\tilde{p}_{n,a}(\\\\tilde{\\\\mathbf{x}}_{n,a})$ from the previous posterior $p_{n - 1,a}$ $(\\\\mathbf{x}_{n - 1,a})$ using the dynamic model for the agent $a$ and assimilating the ground clearance and barometric altitude observations, respectively, $y_{n,a\\\\leftarrow gc}$ and $y_{n,a\\\\leftarrow ba}$ , which are available at node $a$ at instant $n$ . Specifically, we make \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,590,908,657]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\tilde {p} _ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\propto p \\\\left(y _ {n, a \\\\leftarrow g c} \\\\mid \\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) p \\\\left(y _ {n, a \\\\leftarrow b a} \\\\mid \\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\\\\\ \\\\times \\\\left[ \\\\int p \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a} \\\\mid \\\\mathbf {x} _ {n - 1, a}\\\\right) p _ {n - 1, a} \\\\left(\\\\mathbf {x} _ {n - 1, a}\\\\right) d \\\\mathbf {x} _ {n - 1, a} \\\\right]. \\\\tag {24} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,664,872,679]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The complete adapt step expression is written then as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[519,684,908,768]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\tilde {p} _ {n, o} ^ {(a)} (\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}) \\\\propto \\\\left[ \\\\int p (\\\\mathbf {y} _ {n, a \\\\leftarrow o} | \\\\tilde {\\\\mathbf {x}} _ {n, a}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}) \\\\tilde {p} _ {n, a} (\\\\tilde {\\\\mathbf {x}} _ {n, a}) d \\\\tilde {\\\\mathbf {x}} _ {n, a} \\\\right] \\\\\\\\ \\\\times \\\\left[ \\\\int p \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)} \\\\mid \\\\mathbf {x} _ {n - 1, o} ^ {(a)}\\\\right) p _ {n - 1, a} \\\\left(\\\\mathbf {x} _ {n - 1, o} ^ {(a)}\\\\right) d \\\\mathbf {x} _ {n - 1, o} ^ {(a)} \\\\right] \\\\tag {25} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,773,754,789]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"with $\\\\tilde{p}_{n,a}(\\\\tilde{\\\\mathbf{x}}_{n,a})$ computed as in (24). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,789,908,869]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2) Distributed Combine Step: Let $\\\\mathbf{N}(n,a)$ denote the closed neighborhood of the node $a$ , defined as in [1] as $\\\\tilde{\\\\mathbf{N}}(n,a) \\\\cup \\\\{a\\\\}$ , at instant $n$ . Given now the target state p.d.f.'s $\\\\tilde{p}_{n,o}^{(b)}(\\\\tilde{\\\\mathbf{x}}_{n,o}^{(b)})$ at each node $b \\\\in \\\\mathbf{N}(n,a)$ , we want to obtain an artificial transition p.d.f. $c_{n,o}^{(a,b)}(\\\\mathbf{x}_{n,o}^{(a)}, \\\\tilde{\\\\mathbf{x}}_{n,o}^{(b)})$ at node $a$ such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[532,875,906,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\int c _ {n, o} ^ {(a, b)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}\\\\right) \\\\tilde {p} _ {n, o} ^ {(b)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}\\\\right) d \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)} \\\\propto \\\\left[ \\\\tilde {p} _ {n, o} ^ {(b)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\right] ^ {r _ {a, b}} \\\\tag {26}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[102,938,750,952]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"OLIVEIRA ET AL.: GNSS-DENIED JOINT COOPERATIVE TERRAIN NAVIGATION AND TARGET TRACKING \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[885,940,906,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"995 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[84,44,482,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $r_{a,b}$ is a set of network combination coefficients satisfying $\\\\sum_{b\\\\in \\\\mathbf{N}(n,a)}r_{a,b} = 1$ , and $r_{a,b} > 0\\\\forall a$ and $\\\\forall b\\\\in$ $\\\\mathbf{N}(n,a)$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,89,482,133]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It follows then that the desire merged p.d.f. is computed at node $a$ using geometric average fusion [27], [28], [29] as: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[86,137,482,214]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} p _ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\propto \\\\prod_ {b \\\\in \\\\mathbf {N} (n, a)} \\\\left[ \\\\int c _ {n, o} ^ {(a, b)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}\\\\right) \\\\tilde {p} _ {n, o} ^ {(b)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}\\\\right) d \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)} \\\\right] \\\\\\\\ \\\\propto \\\\prod_ {b \\\\in \\\\mathbf {N} (n, a)} \\\\left[ \\\\tilde {p} _ {n, o} ^ {(b)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\right] ^ {r _ {a, b}}. \\\\tag {27} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[84,220,483,281]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"After some algebraic manipulations, it is possible to show [28], [29] that the fusion rule in (27) yields the merged p.d.f. $p$ that minimizes the weighted average Kullback-Leibler divergence \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[200,287,367,321]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\sum_ {b \\\\in \\\\mathbf {N} (n, a)} r _ {a, b} D _ {K L} \\\\left(p | | \\\\tilde {p} _ {n, o} ^ {(b)}\\\\right)\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[84,328,482,381]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $D_{KL}(f||g)\\\\triangleq \\\\mathbb{E}_f\\\\left[\\\\log \\\\left(\\\\frac{f}{g}\\\\right)\\\\right]$ and $\\\\mathbb{E}_f[\\\\cdot ]$ denotes the expectation of its argument, which is computed using the p.d.f. $f$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,381,483,434]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In particular, if $\\\\tilde{p}_{n,o}^{(b)}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,o}^{(b)}\\\\right)$ has a Gaussian parametric representation of the form $\\\\mathcal{N}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,o}^{(b)}|\\\\tilde{\\\\mathbf{m}}_{n,o}^{(b)},\\\\tilde{\\\\mathbf{\\\\Sigma}}_{n,o}^{(b)}\\\\right)$ , it suffices to make \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[151,440,482,460]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nc _ {n, o} ^ {(a, b)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}\\\\right) \\\\triangleq \\\\mathcal {N} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)} | \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}, \\\\tilde {\\\\mathbf {Q}} _ {n, o} ^ {(a, b)}\\\\right) \\\\tag {28}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[86,465,151,479]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[197,484,482,516]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\tilde {\\\\mathbf {Q}} _ {n, o} ^ {(a, b)} + \\\\tilde {\\\\boldsymbol {\\\\Sigma}} _ {n, o} ^ {(b)} = \\\\frac {1}{r _ {a , b}} \\\\tilde {\\\\boldsymbol {\\\\Sigma}} _ {n, o} ^ {(b)} \\\\tag {29}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[86,521,194,537]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"or, equivalently \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[197,542,482,578]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\tilde {\\\\mathbf {Q}} _ {n, o} ^ {(a, b)} = \\\\left[ \\\\frac {1}{r _ {a , b}} - 1 \\\\right] \\\\tilde {\\\\boldsymbol {\\\\Sigma}} _ {n, o} ^ {(b)}. \\\\tag {30}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[86,582,168,595]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In that case \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[109,599,482,752]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\int c _ {n, o} ^ {(a, b)} (\\\\mathbf {\\\\tilde {x}} _ {n, o} ^ {(a)}, \\\\mathbf {\\\\tilde {x}} _ {n, o} ^ {(b)}) \\\\tilde {p} _ {n, o} ^ {(b)} \\\\left(\\\\mathbf {\\\\tilde {x}} _ {n, o} ^ {(b)}\\\\right) d \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)} \\\\\\\\ = \\\\int \\\\mathcal {N} (\\\\mathbf {x} _ {n, o} ^ {(a)} | \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}, \\\\tilde {\\\\mathbf {Q}} _ {n, o} ^ {(a, b)}) \\\\mathcal {N} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)} | \\\\tilde {\\\\mathbf {m}} _ {n, o} ^ {(b)}, \\\\tilde {\\\\mathbf {z}} _ {n, o} ^ {(b)}\\\\right) d \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)} \\\\\\\\ = \\\\mathcal {N} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)} | \\\\tilde {\\\\mathbf {m}} _ {n, o} ^ {(b)}, \\\\tilde {\\\\mathbf {Q}} _ {n, o} ^ {(a, b)} + \\\\tilde {\\\\boldsymbol {\\\\Sigma}} _ {n, o} ^ {(b)}\\\\right) \\\\\\\\ = \\\\mathcal {N} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)} | \\\\tilde {\\\\mathbf {m}} _ {n, o} ^ {(b)}, \\\\frac {1}{r _ {a , b}} \\\\tilde {\\\\boldsymbol {\\\\Sigma}} _ {n, o} ^ {(b)}\\\\right) \\\\\\\\ \\\\propto \\\\left[ \\\\tilde {p} _ {n, o} ^ {(b)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\right] ^ {r _ {a, b}}. \\\\tag {31} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[84,756,483,803]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3) DTT Factor Graph Representation: To obtain a factor graph representation of the ATC distributed target tracking algorithm, we start with the incoming messages \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[173,808,482,827]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\gamma_ {n - 1, a} \\\\left(\\\\mathbf {x} _ {n - 1, a}\\\\right) \\\\triangleq p _ {n - 1, a} \\\\left(\\\\mathbf {x} _ {n - 1, a}\\\\right) \\\\tag {32}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[173,829,482,854]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\gamma_ {n - 1, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n - 1, o} ^ {(a)}\\\\right) \\\\triangleq p _ {n - 1, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n - 1, o} ^ {(a)}\\\\right) \\\\tag {33}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[84,859,482,888]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"at instant $n - 1$ , where $\\\\gamma_{n - 1,a}$ is the same message previously used in the CTAN routine in (16). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,890,482,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Next, using the messages $\\\\alpha_{n,a},\\\\kappa_{n,a}$ , and $\\\\beta_{n,a}$ also previously defined in the CTAN subgraph, respectively, in (16), \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[495,44,893,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(17), and (18), it follows from (24) and the definition of the factor $f_{n,a}$ in (8) that the message $\\\\zeta_{n,a}(\\\\tilde{\\\\mathbf{x}}_{n,a}) \\\\triangleq \\\\tilde{p}_{n,a}(\\\\tilde{\\\\mathbf{x}}_{n,a})$ is computed as: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[553,98,892,114]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\zeta_ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\propto \\\\alpha_ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\kappa_ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\beta_ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right). \\\\tag {34}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[495,121,893,167]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On the other hand, defining the message $\\\\nu_{n,o}^{(a)}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,o}^{(a)}\\\\right) \\\\triangleq \\\\tilde{p}_{n,o}^{(a)}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,o}^{(a)}\\\\right)$ , we can rewrite (25) using (12), (13), and (33) as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,172,892,265]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\nu_ {n, o} ^ {(a)} (\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}) \\\\propto \\\\left[ \\\\int f _ {n, o} ^ {(a)} (\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}, \\\\mathbf {x} _ {n - 1, o} ^ {(a)}) \\\\gamma_ {n - 1, o} ^ {(a)} (\\\\mathbf {x} _ {n - 1, o} ^ {(a)}) d \\\\mathbf {x} _ {n - 1, o} ^ {(a)} \\\\right] \\\\\\\\ \\\\times \\\\left[ \\\\int t o _ {n, o} ^ {(a)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right) \\\\zeta_ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) d \\\\tilde {\\\\mathbf {x}} _ {n, a} \\\\right] \\\\\\\\ \\\\triangleq \\\\alpha_ {n, o} ^ {(a)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right) \\\\phi_ {n, o} ^ {(a)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right) \\\\tag {35} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[496,271,544,284]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[501,289,892,335]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\alpha_ {n, o} ^ {(a)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right) \\\\triangleq \\\\int f _ {n, o} ^ {(a)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}, \\\\mathbf {x} _ {n - 1, o} ^ {(a)}\\\\right) \\\\gamma_ {n - 1, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n - 1, o} ^ {(a)}\\\\right) d \\\\mathbf {x} _ {n - 1, o} ^ {(a)} \\\\tag {36}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[503,339,892,371]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\phi_ {n, o} ^ {(a)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right) \\\\triangleq \\\\int t o _ {n, o} ^ {(a)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}, \\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) \\\\zeta_ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) d \\\\tilde {\\\\mathbf {x}} _ {n, a}. \\\\tag {37}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[495,376,893,406]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the sequel, we introduce the local and remote combine factors, respectively, as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[616,412,772,431]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nc l _ {n, o} ^ {(a)} \\\\triangleq c l _ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right)\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[651,434,892,452]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\triangleq c _ {n, o} ^ {(a, a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right) \\\\tag {38}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[594,455,794,474]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nc d _ {n, o} ^ {(a \\\\leftarrow b)} \\\\triangleq c d _ {n, o} ^ {(a \\\\leftarrow b)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}\\\\right)\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[651,476,890,494]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\triangleq c _ {n, o} ^ {(a, b)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}\\\\right). \\\\tag {39}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[496,501,879,517]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally, using (38) and (39) and introducing the message \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[614,522,892,541]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\gamma_ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\triangleq p _ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\tag {40}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[496,547,622,561]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"we rewrite (27) as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[500,568,892,694]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\gamma_ {n, o} ^ {(a)} \\\\left(\\\\mathbf {\\\\tilde {x}} _ {n, o} ^ {(a)}\\\\right) \\\\triangleq \\\\prod_ {b \\\\in \\\\tilde {\\\\mathbf {N}} (n, a)} \\\\int c d _ {n, o} ^ {(a \\\\leftarrow b)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}\\\\right) \\\\lambda_ {n, o} ^ {(a \\\\leftarrow b)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}\\\\right) d \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)} \\\\\\\\ \\\\times \\\\int c l _ {n, o} ^ {(a)} \\\\left(\\\\mathbf {\\\\tilde {x}} _ {n, o} ^ {(a)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right) v _ {n, o} ^ {(a)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right) d \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)} \\\\\\\\ \\\\triangleq \\\\left[ \\\\prod_ {b \\\\in \\\\overline {{\\\\mathbf {N}}} (n, a)} \\\\theta_ {n, o} ^ {(a \\\\leftarrow b)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\right] \\\\psi_ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\tag {41} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[495,700,893,731]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\bar{\\\\mathbf{N}}(n, a)$ denotes again the open neighborhood of the node $a$ at instant $n$ (i.e., excluding node $a$ itself) and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,734,892,756]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\lambda_ {n, o} ^ {(a \\\\leftarrow b)} (\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}) \\\\triangleq v _ {n, o} ^ {(b)} (\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}) \\\\tag {42}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[509,757,890,804]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\theta_ {n, o} ^ {(a \\\\leftarrow b)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\triangleq \\\\int c d _ {n, o} ^ {(a \\\\leftarrow b)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}\\\\right) \\\\lambda_ {n, o} ^ {(a \\\\leftarrow b)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}\\\\right) d \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)} \\\\tag {43}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[524,808,892,840]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\psi_ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\triangleq \\\\int c l _ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right) v _ {n, o} ^ {(a)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right) d \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}. \\\\tag {44}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[495,844,893,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Equations (32)-(44) can be represented by a DTT factor subgraph as illustrated in Fig. 2 with the output messages computed using the usual SPA rules. Specifically, messages (35)-(37) implement the SPA over the local target tracking (local- tt) subgraph. Messages (41)-(44) in turn implement \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[88,939,111,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"996 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[256,939,888,952]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.60, NO.1 FEBRUARY 2024 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[122,46,480,421]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 2. DTT factor subgraph. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0945/7f94358a1a62c654d71a0c35e724bb955ffaf017ffe8fa16e7fa44ce8e6ad2d2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[101,453,498,484]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"the SPA over the distributed target tracking (dist- tt) subgraph. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,494,500,526]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"D. Cooperative Localization With Distributed Target Tracking Feedback \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,531,500,638]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $\\\\mathbf{O}_n$ represent the set of all target measurements $\\\\{\\\\mathbf{y}_{k,a\\\\leftarrow o}\\\\}$ for $a\\\\in \\\\{1,\\\\dots ,A\\\\}$ , and $k\\\\in \\\\{0,\\\\ldots ,n\\\\}$ . The belief of the unknown aircraft state at node $a$ and instant $n$ may be improved if we also assimilate the target observations $\\\\{\\\\mathbf{y}_{n,a\\\\leftarrow o}\\\\}$ to infer the state $\\\\mathbf{x}_{n,a}$ . In fact, if the target state were perfectly known, say $\\\\bar{\\\\mathbf{x}}_{n,o}$ , we could marginalize instead the joint p.d.f. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,643,498,768]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} p \\\\left(\\\\mathbf {X} _ {n}, \\\\mathbf {Y} _ {n}, \\\\mathbf {Z} _ {n}, \\\\mathbf {O} _ {n}\\\\right) = \\\\prod_ {k = 1} ^ {n} \\\\prod_ {a = 1} ^ {A} p \\\\left(\\\\mathbf {x} _ {k, a} \\\\mid \\\\mathbf {x} _ {k - 1, a}\\\\right) \\\\\\\\ \\\\times p \\\\left(y _ {k, a \\\\leftarrow g c} \\\\mid \\\\mathbf {x} _ {k, a}\\\\right) p \\\\left(y _ {k, a \\\\leftarrow b a} \\\\mid \\\\mathbf {x} _ {k, a}\\\\right) \\\\\\\\ \\\\times \\\\left[ \\\\prod_ {b \\\\in \\\\tilde {\\\\mathbf {N}} (k, a)} p \\\\left(y _ {k, a - b} \\\\mid \\\\mathbf {x} _ {k, a}, \\\\mathbf {x} _ {k, b}\\\\right) \\\\right] \\\\\\\\ \\\\times p \\\\left(\\\\mathbf {y} _ {k, a \\\\leftarrow o} \\\\mid \\\\mathbf {x} _ {k, a}, \\\\bar {\\\\mathbf {x}} _ {k, o}\\\\right) \\\\tag {45} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[101,773,500,851]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where, for the purpose of this discussion, $\\\\bar{\\\\mathbf{x}}_{k,o}$ in (45) is interpreted as being deterministic and known, as the position of an anchor node, for example. The factorization in (45) introduces an additional factor in Fig. 1 and modifies the expression for $\\\\gamma_{n,a}(\\\\mathbf{x}_{n,a})$ in (22) to \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,854,498,917]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\gamma_ {n, a} (\\\\mathbf {x} _ {n, a}) \\\\propto \\\\epsilon_ {n, a} (\\\\mathbf {x} _ {n, a}) \\\\left[ \\\\prod_ {b \\\\in \\\\tilde {\\\\mathbf {N}} (n, a)} \\\\xi_ {n, a \\\\leftarrow b} (\\\\mathbf {x} _ {n, a}) \\\\right] \\\\\\\\ \\\\times p \\\\left(\\\\mathbf {y} _ {n, a \\\\leftarrow o} \\\\mid \\\\mathbf {x} _ {n, a}, \\\\bar {\\\\mathbf {x}} _ {n, o}\\\\right). \\\\tag {46} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,44,908,164]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"However, as we do not know the actual $\\\\bar{\\\\mathbf{x}}_{k,o}$ , we propose again an approximation where we replace the last term on the right-hand side of (46) with its expected value with respect to the most recent belief that node $a$ has about the state of $\\\\mathbf{x}_{n,o}$ at instant $n$ . Specifically, we propose to compute that expectation using the belief $\\\\gamma_{n,o}^{(a)}\\\\left(\\\\mathbf{x}_{n,o}^{(a)}\\\\right)$ in (41) obtained at the end of the ATC distributed target tracking procedure in section III-C. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,164,908,210]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For convenience, we introduce a message $\\\\rho_{n,o}^{(a)}\\\\left(\\\\mathbf{x}_{n,o}^{(a)}\\\\right)$ which is identical to $\\\\gamma_{n,o}^{(a)}\\\\left(\\\\mathbf{x}_{n,o}^{(a)}\\\\right)$ . We then define the so-called feedback factor \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[609,222,908,263]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} f b _ {n, o} ^ {(a)} \\\\triangleq f b _ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, a}, \\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\\\\\ \\\\triangleq p \\\\left(\\\\mathbf {y} _ {n, a \\\\leftarrow o} \\\\mid \\\\mathbf {x} _ {n, a}, \\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right). \\\\tag {47} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,272,908,301]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The modified updated belief of $\\\\mathbf{x}_{n,a}$ at node $a$ at instant $n$ is now \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[529,310,908,367]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\gamma_ {n, a} \\\\left(\\\\mathbf {x} _ {n, a}\\\\right) \\\\propto \\\\left[ \\\\prod_ {b \\\\in \\\\tilde {\\\\mathbf {N}} (n, a)} \\\\xi_ {n, a \\\\leftarrow b} \\\\left(\\\\mathbf {x} _ {n, a}\\\\right) \\\\right] \\\\epsilon_ {n, a} \\\\left(\\\\mathbf {x} _ {n, a}\\\\right) \\\\tau_ {n, a} \\\\left(\\\\mathbf {x} _ {n, a}\\\\right) \\\\tag {48}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,378,643,393]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where the message \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[547,402,906,455]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\tau_ {n, a} (\\\\mathbf {x} _ {n, a}) \\\\triangleq \\\\int f b _ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, a}, \\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\rho_ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) d \\\\mathbf {x} _ {n, o} ^ {(a)} \\\\\\\\ = \\\\mathrm {E} _ {\\\\rho_ {n, o} ^ {(a)}} \\\\left[ p \\\\left(\\\\mathbf {y} _ {n, a \\\\leftarrow o} | \\\\mathbf {x} _ {n, a}, \\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\right]. \\\\tag {49} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,467,910,590]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By adding the factor $fb_{n,o}^{(a)}$ and the previous factor $to_{n,o}^{(a)}$ in (12), and defining the messages $\\\\tau_{n,a}$ in (49) and $\\\\phi_{n,o}^{(a)}$ in (37), it is possible to further connect the dist-rtt and coop-loc subgraphs into a single joint CTAN-DTT (J-CTAN-DTT) factor graph shown in Fig. 3, with the message passing equations obtained from the application of the SPA to this graph being identical to the message equations shown in sections III-B, III-C, and III-D. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,606,823,619]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IV.MESSAGE PASSING IMPLEMENTATION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,627,908,674]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section, we propose a GMM/SMC implementation for the message passing schedule in the factor graph in Fig. 3 that represents the J-CTAN-DTT algorithm. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,688,782,703]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. Prediction and Local Observations \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,710,908,786]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In our proposed implementation, we assume that the incoming messages $\\\\gamma_{n - 1,a}$ and $\\\\gamma_{n - 1,o}^{(a)}$ , which represent, respectively, the final beliefs of the aircraft $a$ and the target state representation on aircraft $a$ at instant $n - 1$ , are modeled by parametric GMMs as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[524,796,908,853]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\check {\\\\gamma} _ {n - 1, a} \\\\left(\\\\mathbf {x} _ {n - 1, a}\\\\right) = \\\\sum_ {l = 1} ^ {L} \\\\eta_ {n - 1, a} ^ {(l)} \\\\mathcal {N} \\\\left(\\\\mathbf {x} _ {n - 1, a} \\\\mid \\\\mathbf {m} _ {n - 1, a} ^ {(l)}, \\\\boldsymbol {\\\\Sigma} _ {n - 1, a} ^ {(l)}\\\\right) \\\\tag {50}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[521,858,908,916]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\check {\\\\gamma} _ {n - 1, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n - 1, o} ^ {(a)}\\\\right) = \\\\sum_ {l = 1} ^ {L} \\\\eta_ {n - 1, o} ^ {(a, l)} \\\\mathcal {N} \\\\left(\\\\mathbf {x} _ {n - 1, o} ^ {(a)} | \\\\mathbf {m} _ {n - 1, o} ^ {(a, l)}, \\\\boldsymbol {\\\\Sigma} _ {n - 1, o} ^ {(a, l)}\\\\right) \\\\tag {51}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[102,938,750,952]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"OLIVEIRA ET AL.: GNSS-DENIED JOINT COOPERATIVE TERRAIN NAVIGATION AND TARGET TRACKING \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[883,940,906,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"997 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[106,45,467,417]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig.3.J-CTAN-DTT full factor graph. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0790/a5faa338d7cc43a3c7b48186f2dff8c79fa4498508c0f0fccfae0f51cd0387af.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[84,463,482,508]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $L$ denotes the number of modes in the GMM model. On the other hand, from the aircraft kinematic model in (1), it follows that: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[94,520,482,560]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} f _ {n, a} (\\\\tilde {\\\\mathbf {x}} _ {n, a}, \\\\mathbf {x} _ {n - 1, a}) = p (\\\\tilde {\\\\mathbf {x}} _ {n, a} | \\\\mathbf {x} _ {n - 1, a}) \\\\\\\\ = \\\\mathcal {N} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a} \\\\mid \\\\mathbf {x} _ {n - 1, a} + \\\\mathbf {u} _ {n - 1, a}, \\\\mathbf {Q} _ {n - 1, a}\\\\right). \\\\tag {52} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[84,571,483,602]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Therefore, by substituting (52) and (50) into (16), we approximate the message $\\\\alpha_{n,a}(\\\\tilde{\\\\mathbf{x}}_{n,a})$ as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,611,483,758]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\check {\\\\alpha} _ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) = \\\\int \\\\left[ \\\\mathcal {N} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a} \\\\mid \\\\mathbf {x} _ {n - 1, a} + \\\\mathbf {u} _ {n - 1, a}, \\\\mathbf {Q} _ {a}\\\\right) \\\\right. \\\\\\\\ \\\\left. \\\\times \\\\sum_ {l = 1} ^ {L} \\\\eta_ {n - 1, a} ^ {(l)} \\\\mathcal {N} \\\\left(\\\\mathbf {x} _ {n - 1, a} \\\\mid \\\\mathbf {m} _ {n - 1, a} ^ {(l)}, \\\\boldsymbol {\\\\Sigma} _ {n - 1, a} ^ {(l)}\\\\right) \\\\right] d \\\\mathbf {x} _ {n - 1, a} \\\\\\\\ = \\\\sum_ {l = 1} ^ {L} \\\\eta_ {n | n - 1, a} ^ {(l)} \\\\mathcal {N} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a} \\\\mid \\\\mathbf {m} _ {n | n - 1, a} ^ {(l)}, \\\\boldsymbol {\\\\Sigma} _ {n | n - 1, a} ^ {(l)}\\\\right) \\\\tag {53} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[86,768,482,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\eta_{n|n-1,a}^{(l)} = \\\\eta_{n-1,a}^{(l)}$ , $\\\\mathbf{m}_{n|n-1,a}^{(l)} = \\\\mathbf{m}_{n-1,a}^{(l)} + \\\\mathbf{u}_{n-1,a}$ , and $\\\\boldsymbol{\\\\Sigma}_{n|n-1,a}^{(l)} = \\\\boldsymbol{\\\\Sigma}_{n-1,a}^{(l)} + \\\\mathbf{Q}_a$ $\\\\forall l \\\\in \\\\{1, \\\\dots, L\\\\}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[86,806,482,835]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Similarly, from the target dynamic model in (2), we write \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[94,843,483,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} f _ {n, o} ^ {(a)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}, \\\\mathbf {x} _ {n - 1, o} ^ {(a)}\\\\right) = p \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)} | \\\\mathbf {x} _ {n - 1, o} ^ {(a)}\\\\right) \\\\\\\\ = \\\\mathcal {N} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)} \\\\mid \\\\mathbf {F} _ {n - 1, o} ^ {(a)} \\\\mathbf {x} _ {n - 1, o} ^ {(a)}, \\\\mathbf {Q} _ {n - 1, o} ^ {(a)}\\\\right). \\\\tag {54} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[495,44,893,75]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Thus, by substituting (54) and (51) into (36), the target prediction message $\\\\alpha_{n,o}^{(a)}(\\\\mathbf{x}_{n,o}^{(a)})$ is approximated as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[496,82,892,220]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\check {\\\\alpha} _ {n, o} ^ {(a)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right) = \\\\int \\\\left[ \\\\mathcal {N} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)} | \\\\mathbf {F} _ {n - 1, o} ^ {(a)} \\\\mathbf {x} _ {n - 1, o} ^ {(a)}, \\\\mathbf {Q} _ {n - 1, o} ^ {(a)}\\\\right) \\\\right. \\\\\\\\ \\\\left. \\\\times \\\\sum_ {l = 1} ^ {L} \\\\eta_ {n - 1, o} ^ {(a, l)} \\\\mathcal {N} \\\\left(\\\\mathbf {x} _ {n - 1, o} ^ {(a)} \\\\mid \\\\mathbf {m} _ {n - 1, o} ^ {(a, l)}, \\\\boldsymbol {\\\\Sigma} _ {n - 1, o} ^ {(a, l)}\\\\right) \\\\right] d \\\\mathbf {x} _ {n - 1, o} ^ {(a)} \\\\\\\\ = \\\\sum_ {l = 1} ^ {L} \\\\eta_ {n | n - 1, o} ^ {(a, l)} \\\\mathcal {N} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)} \\\\mid \\\\mathbf {m} _ {n | n - 1, a} ^ {(a, l)}, \\\\boldsymbol {\\\\Sigma} _ {n | n - 1, a} ^ {(a, l)}\\\\right) \\\\tag {55} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[495,227,893,286]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"in which $\\\\eta_{n|n-1,o}^{(a,l)} = \\\\eta_{n-1,o}^{(a,l)}$ , $\\\\mathbf{m}_{n|n-1,o}^{(a,l)} = \\\\mathbf{F}_{n-1,o}^{(a)}\\\\mathbf{m}_{n-1,o}^{(a,l)}$ and $\\\\boldsymbol{\\\\Sigma}_{n|n-1,o}^{(a,l)} = \\\\mathbf{F}_{n-1,o}^{(a)}\\\\boldsymbol{\\\\Sigma}_{n-1,o}^{(a,l)}\\\\left[\\\\mathbf{F}_{n-1,o}^{(a)}\\\\right]^T + \\\\mathbf{Q}_{n-1,o}^{(a)} \\\\quad \\\\forall l \\\\in \\\\{1, \\\\ldots, L\\\\}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[495,286,893,431]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the sequel, at node $a$ , we sample $\\\\tilde{\\\\mathbf{x}}_{n,a}^{(i)} \\\\sim \\\\check{\\\\alpha}_{n,a}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,a}\\\\right)$ and compute the associated importance weights $\\\\tilde{w}_{n,a}^{(i)} \\\\propto \\\\beta_{n,a}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,a}^{(i)}\\\\right) \\\\kappa_{n,a}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,a}^{(i)}\\\\right)$ to create a Monte Carlo (MC) representation $\\\\left\\\\{\\\\left(\\\\tilde{w}_{n,a}^{(i)}, \\\\tilde{\\\\mathbf{x}}_{n,a}^{(i)}\\\\right)\\\\right\\\\}$ , $i \\\\in \\\\{1, \\\\dots, N_a\\\\}$ , of the aircraft belief $\\\\zeta_{n,a}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,a}\\\\right)$ in (34) following the assimilation of the ground clearance and barometric altitude observations, respectively. As usual in sequential Monte Carlo methods, the importance weights undergo normalization in such a way that $\\\\sum_{i=1}^{N_a} \\\\tilde{w}_{n,a}^{(i)} = 1$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[495,431,893,537]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Next, for the target, we build another MC representation $\\\\{(\\\\tilde{w}_{n,o}^{(a,k)},\\\\tilde{\\\\mathbf{x}}_{n,o}^{(a,k)})\\\\}$ $k\\\\in \\\\{1,\\\\dots ,N_o\\\\}$ by sampling $\\\\tilde{\\\\mathbf{x}}_{n,o}^{(a,k)}\\\\sim$ $\\\\check{\\\\alpha}_{n,o}^{(a)}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,o}^{(a)}\\\\right)$ and setting the weights $\\\\tilde{w}_{n,o}^{(a,k)}\\\\propto \\\\phi_{n,o}^{(a)}\\\\bigl (\\\\tilde{\\\\mathbf{x}}_{n,o}^{(a,k)}\\\\bigr),$ where the message $\\\\phi_{n,o}^{(a)}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,o}^{(a)}\\\\right)$ introduced in (37), assimilates the target observation, and we normalize the weights again such that $\\\\sum_{i = 1}^{N_o}\\\\tilde{w}_{n,o}^{(a,k)} = 1$ . Since the target observation factor \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,545,892,590]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} t o _ {n, o} ^ {(a)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right) = p \\\\left(\\\\mathbf {y} _ {n, a \\\\leftarrow o} | \\\\tilde {\\\\mathbf {x}} _ {n, a}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right) \\\\\\\\ = \\\\mathcal {N} \\\\left(\\\\mathbf {y} _ {n, a \\\\leftarrow o} | \\\\mathbf {t o} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right), \\\\mathbf {R} _ {t o}\\\\right) \\\\tag {56} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[495,595,893,645]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"also depends on $\\\\tilde{\\\\mathbf{x}}_{n,a}$ and we already have an MC representation $\\\\{(\\\\tilde{w}_{n,a}^{(i)},\\\\tilde{\\\\mathbf{x}}_{n,a}^{(i)})\\\\}$ , $i\\\\in \\\\{1,\\\\dots ,N_a\\\\}$ , of the posterior $\\\\zeta_{n,a}(\\\\tilde{\\\\mathbf{x}}_{n,a})$ , it results that, for each particle $\\\\tilde{\\\\mathbf{x}}_{n,o}^{(a,k)}$ , we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,650,892,853]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\phi_ {n, o} ^ {(a)} (\\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a, k)}) \\\\propto \\\\int t o _ {n, o} ^ {(a)} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a, k)}\\\\right) \\\\zeta_ {n, a} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a}\\\\right) d \\\\tilde {\\\\mathbf {x}} _ {n, a} \\\\\\\\ \\\\approx \\\\sum_ {i = 1} ^ {N _ {a}} \\\\left[ \\\\mathcal {N} \\\\left(\\\\mathbf {y} _ {n, a \\\\leftarrow o} | \\\\mathbf {t o} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a} ^ {(i)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a, k)}\\\\right), \\\\mathbf {R} _ {t o}\\\\right) \\\\tilde {w} _ {n, a} ^ {(i)} \\\\right] \\\\\\\\ = \\\\sum_ {i = 1} ^ {N _ {a}} \\\\tilde {w} _ {n, a} ^ {(i)} \\\\left[ \\\\mathcal {N} \\\\left(y _ {n, a \\\\leftarrow o} ^ {(r g)} \\\\mid r g \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a} ^ {(i)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a, k)}\\\\right), \\\\sigma_ {r o} ^ {2}\\\\right) \\\\right. \\\\\\\\ \\\\times \\\\mathcal {N} \\\\left(y _ {n, a \\\\leftarrow o} ^ {(a z)} | a z \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a} ^ {(i)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a, k)}\\\\right), \\\\sigma_ {a z} ^ {2}\\\\right) \\\\\\\\ \\\\left. \\\\times \\\\mathcal {N} \\\\left(y _ {n, a \\\\leftarrow o} ^ {(e l)} \\\\mid e l \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, a} ^ {(i)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a, k)}\\\\right), \\\\sigma_ {e l} ^ {2}\\\\right) \\\\right]. \\\\tag {57} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[495,859,893,892]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the sequel, we create the GMM approximations $\\\\check{\\\\lambda}_{n,a}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,a}\\\\right)$ and $\\\\check{\\\\lambda}_{n,o}^{(a)}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,o}^{(a)}\\\\right)$ from the MC representations \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[88,940,109,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"998 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[256,939,888,952]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.60, NO.1 FEBRUARY 2024 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[102,44,501,77]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\left\\\\{\\\\left(\\\\tilde{w}_{n,a}^{(i)}, \\\\tilde{\\\\mathbf{x}}_{n,a}^{(i)}\\\\right)\\\\right\\\\}$ of the aircraft belief and $\\\\left\\\\{\\\\left(\\\\tilde{w}_{n,o}^{(a,k)}, \\\\tilde{\\\\mathbf{x}}_{n,o}^{(a,k)}\\\\right)\\\\right\\\\}$ of the target belief, respectively. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,88,307,103]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B. Cooperative Localization \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,109,500,256]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The cooperative localization is analogous to the procedure described in [1]. Each aircraft $a$ sends the parametric representation $\\\\check{\\\\lambda}_{n,a}(\\\\tilde{\\\\mathbf{x}}_{n,a})$ to the nodes in their neighborhood, instead of sending $N_{a}$ particles, to decrease the communication costs. After receiving $\\\\{\\\\check{\\\\lambda}_{n,b}(\\\\tilde{\\\\mathbf{x}}_{n,b})\\\\}$ from each node $b\\\\in \\\\mathbf{N}(n,a),N_b$ particles are resampled at node $a$ from each of the received GMMs. That procedure results in a new set of weighted particles $\\\\left\\\\{\\\\left(\\\\tilde{w}_{n,b}^{(j)},\\\\tilde{\\\\mathbf{x}}_{n,b}^{(j)}\\\\right)\\\\right\\\\}$ for each $b\\\\in \\\\mathbf{N}(n,a)$ where $\\\\tilde{\\\\mathbf{x}}_{n,b}^{(j)}\\\\sim \\\\check{\\\\lambda}_{n,b}(\\\\tilde{\\\\mathbf{x}}_{n,b})$ and $\\\\tilde{w}_{n,b}^{(j)} = 1 / N_b\\\\forall j\\\\in \\\\{1,\\\\dots ,N_b\\\\}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[125,256,472,270]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From the internode distance model in (5), we write \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[109,277,498,316]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} a d _ {n, a \\\\leftarrow b} (\\\\mathbf {x} _ {n, a}, \\\\tilde {\\\\mathbf {x}} _ {n, b}) = p (y _ {n, a \\\\leftarrow b} | \\\\mathbf {x} _ {n, a}, \\\\tilde {\\\\mathbf {x}} _ {n, b}) \\\\\\\\ = \\\\mathcal {N} \\\\left(y _ {n, a \\\\leftarrow b} \\\\mid r g \\\\left(\\\\mathbf {x} _ {n, a}, \\\\tilde {\\\\mathbf {x}} _ {n, b}\\\\right), \\\\sigma_ {a d} ^ {2}\\\\right). \\\\tag {58} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[101,321,498,357]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Then, the message $\\\\xi_{n,a\\\\leftarrow b}\\\\left(\\\\mathbf{x}_{n,a}^{(i)}\\\\right)$ , defined in (21) is computed, for each particle $\\\\mathbf{x}_{n,a}^{(i)}\\\\triangleq \\\\tilde{\\\\mathbf{x}}_{n,a}^{(i)}$ , as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[109,362,498,518]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\xi_ {n, a \\\\leftarrow b} (\\\\mathbf {x} _ {n, a} ^ {(i)}) \\\\propto \\\\int a d _ {n, a \\\\leftarrow b} \\\\left(\\\\mathbf {x} _ {n, a} ^ {(i)}, \\\\tilde {\\\\mathbf {x}} _ {n, b}\\\\right) \\\\lambda_ {n, a \\\\leftarrow b} \\\\left(\\\\tilde {\\\\mathbf {x}} _ {n, b}\\\\right) d \\\\tilde {\\\\mathbf {x}} _ {n, b} \\\\\\\\ = \\\\int \\\\left[ \\\\mathcal {N} \\\\left(y _ {n, a \\\\leftarrow b} | r g \\\\left(\\\\mathbf {x} _ {n, a} ^ {(i)}, \\\\tilde {\\\\mathbf {x}} _ {n, b}\\\\right), \\\\sigma_ {a d} ^ {2}\\\\right) \\\\right. \\\\\\\\ \\\\left. \\\\times \\\\breve {\\\\lambda} _ {n, b} (\\\\tilde {\\\\mathbf {x}} _ {n, b}) \\\\right] d \\\\tilde {\\\\mathbf {x}} _ {n, b} \\\\\\\\ \\\\approx \\\\sum_ {j = 1} ^ {N _ {b}} \\\\mathcal {N} \\\\left(y _ {n, a \\\\leftarrow b} \\\\mid r g \\\\left(\\\\mathbf {x} _ {n, a} ^ {(i)}, \\\\tilde {\\\\mathbf {x}} _ {n, b} ^ {(j)}\\\\right), \\\\sigma_ {a d} ^ {2}\\\\right) \\\\tilde {w} _ {n, b} ^ {(j)}. \\\\tag {59} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[101,527,500,626]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Next, we make $w_{n,a}^{(i)} \\\\propto \\\\tilde{w}_{n,a}^{(i)} \\\\prod_{b \\\\in \\\\bar{\\\\mathbf{N}}(n,a)} \\\\xi_{n,a \\\\leftarrow b}\\\\left(\\\\mathbf{x}_{n,a}^{(i)}\\\\right)$ with proportionality constant such that $\\\\sum_{i=1}^{N_a} w_{n,a}^{(i)} = 1$ . The new weighted particle set $\\\\left\\\\{\\\\left(w_{n,a}^{(i)}, \\\\mathbf{x}_{n,a}^{(i)}\\\\right)\\\\right\\\\}, i \\\\in \\\\{1, \\\\dots, N_a\\\\}$ , is now the updated MC representation of the belief of the position of the aircraft $a$ following the assimilation of the distance measurements between the neighboring aircraft. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,636,321,652]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Distributed Target Tracking \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,657,498,781]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Each aircraft $a$ also broadcasts the parametric representations $\\\\check{\\\\lambda}_{n,o}^{(a)}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,o}^{(a)}\\\\right)$ . After receiving $\\\\{\\\\check{\\\\lambda}_{n,o}^{(b)}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,o}^{(b)}\\\\right)\\\\}$ from all $b \\\\in \\\\bar{\\\\mathbf{N}}(n,a)$ , we set the network coefficients $r_{a,b} \\\\triangleq 1/|\\\\mathbf{N}(n,a)|$ , where $|\\\\mathbf{N}(n,a)|$ denotes the cardinality of the closed neighborhood of $a$ , $\\\\mathbf{N}(n,a)$ , which we recall is given by $\\\\mathbf{N}(n,a) \\\\triangleq \\\\bar{\\\\mathbf{N}}(n,a) \\\\cup \\\\{a\\\\}$ . In this case, $|\\\\mathbf{N}(n,a)|$ is simply equal to the number of received messages plus one and, then, the factors $cl_{n,o}^{(a)}$ and $cd_{n,o}^{(a \\\\leftarrow b)}$ can be computed analytically. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,782,500,876]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Assuming that the input message $\\\\nu_{n,o}^{(a)}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,o}^{(a)}\\\\right)$ is represented by a Gaussian p.d.f. $\\\\mathcal{N}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,o}^{(a)}|\\\\tilde{\\\\mathbf{m}}_{n,o}^{(a)},\\\\tilde{\\\\mathbf{\\\\Sigma}}_{n,o}^{(a)}\\\\right)$ and, similarly, $\\\\lambda_{n,o}^{(a\\\\leftarrow b)}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,o}^{(b)}\\\\right)\\\\approx \\\\check{\\\\lambda}_{n,o}^{(b)}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,o}^{(b)}\\\\right) = \\\\mathcal{N}\\\\left(\\\\tilde{\\\\mathbf{x}}_{n,o}^{(b)}|\\\\tilde{\\\\mathbf{m}}_{n,o}^{(b)},\\\\tilde{\\\\mathbf{\\\\Sigma}}_{n,o}^{(b)}\\\\right)$ , it follows from (28) and (29) setting $r_{a,b} = 1 / |\\\\mathbf{N}(n,a)|\\\\forall b\\\\in$ $\\\\mathbf{N}(n,a)$ , that: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[138,882,500,907]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nc l _ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}\\\\right) = \\\\mathcal {N} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)} | \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(a)}, | \\\\bar {\\\\mathbf {N}} (n, a) | \\\\tilde {\\\\mathbf {\\\\Sigma}} _ {n, o} ^ {(a)}\\\\right) \\\\tag {60}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[527,45,908,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nc d _ {n, o} ^ {(a \\\\leftarrow b)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}, \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}\\\\right) = \\\\mathcal {N} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)} | \\\\tilde {\\\\mathbf {x}} _ {n, o} ^ {(b)}, | \\\\bar {\\\\mathbf {N}} (n, a) | \\\\tilde {\\\\mathbf {\\\\Sigma}} _ {n, o} ^ {(b)}\\\\right) \\\\tag {61}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,78,906,108]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $|\\\\tilde{\\\\mathbf{N}}(n, a)| = |\\\\mathbf{N}(n, a)| - 1$ is the cardinality of the open neighborhood of node $a$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,109,906,138]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Using (43) and (44), the local and remote combine messages are then parameterized as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[566,146,908,212]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\psi_ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) = \\\\mathcal {N} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)} | \\\\tilde {\\\\mathbf {m}} _ {n, o} ^ {(a)}, | \\\\mathbf {N} (n, a) | \\\\tilde {\\\\boldsymbol {\\\\Sigma}} _ {n, o} ^ {(a)}\\\\right) \\\\\\\\ \\\\propto \\\\left[ \\\\tilde {p} _ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\right] ^ {\\\\frac {1}{| \\\\mathbf {N} (n , a) |}} \\\\tag {62} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,217,542,231]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[558,236,908,294]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\theta_ {n, o} ^ {(a \\\\leftarrow b)} (\\\\mathbf {x} _ {n, o} ^ {(a)}) = \\\\mathcal {N} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)} | \\\\tilde {\\\\mathbf {m}} _ {n, o} ^ {(b)}, | \\\\mathbf {N} (n, a) | \\\\tilde {\\\\boldsymbol {\\\\Sigma}} _ {n, o} ^ {(b)}\\\\right) \\\\\\\\ \\\\propto \\\\left[ \\\\tilde {p} _ {n, o} ^ {(b)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\right] ^ {\\\\frac {1}{| \\\\mathbf {N} (n , a) |}}. \\\\tag {63} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,301,676,318]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"D. Feedback Message \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,323,910,422]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The combined estimate of the target state $\\\\mathbf{x}_{n,o}^{(a)}$ is used to improve the self-localization estimate of the state $\\\\mathbf{x}_{n,a}$ of the aircraft $a$ via the factor $fb_{n,o}^{(a)}$ in (47). The input message $\\\\rho_{n,o}^{(a)}\\\\left(\\\\mathbf{x}_{n,o}^{(a)}\\\\right)$ , which is a copy of $\\\\gamma_{n,o}^{(a)}\\\\left(\\\\mathbf{x}_{n,o}^{(a)}\\\\right)$ in (41) can now be calculated as the product of the messages $\\\\psi_{n,o}^{(a)}\\\\left(\\\\mathbf{x}_{n,o}^{(a)}\\\\right)$ and $\\\\theta_{n,o}^{(a\\\\leftarrow b)}(\\\\mathbf{x}_{n,o}^{(a)})$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[534,429,908,575]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\rho_ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\propto \\\\psi_ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\prod_ {b \\\\in \\\\tilde {\\\\mathbf {N}} (n, a)} \\\\theta_ {n, o} ^ {(a \\\\leftarrow b)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\\\\\ = \\\\mathcal {N} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)} | \\\\tilde {\\\\mathbf {m}} _ {n, o} ^ {(a)}, | \\\\mathbf {N} (n, a) | \\\\tilde {\\\\boldsymbol {\\\\Sigma}} _ {n, o} ^ {(a)}\\\\right) \\\\\\\\ \\\\times \\\\prod_ {b \\\\in \\\\tilde {\\\\mathbf {N}} (n, a)} \\\\mathcal {N} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)} | \\\\tilde {\\\\mathbf {m}} _ {n, o} ^ {(b)}, | \\\\mathbf {N} (n, a) | \\\\tilde {\\\\boldsymbol {\\\\Sigma}} _ {n, o} ^ {(b)}\\\\right) \\\\\\\\ = \\\\mathcal {N} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)} \\\\mid \\\\mathbf {m} _ {n, o} ^ {(a)}, \\\\boldsymbol {\\\\Sigma} _ {n, o} ^ {(a)}\\\\right) \\\\tag {64} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,583,558,595]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[566,601,908,638]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left[ \\\\boldsymbol {\\\\Sigma} _ {n, o} ^ {(a)} \\\\right] ^ {- 1} = \\\\frac {1}{| \\\\mathbf {N} (n , a) |} \\\\sum_ {u \\\\in \\\\mathbf {N} (n, a)} \\\\left[ \\\\tilde {\\\\boldsymbol {\\\\Sigma}} _ {n, o} ^ {(u)} \\\\right] ^ {- 1} \\\\tag {65}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[534,648,908,686]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left[ \\\\boldsymbol {\\\\Sigma} _ {n, o} ^ {(a)} \\\\right] ^ {- 1} \\\\mathbf {m} _ {n, o} ^ {(a)} = \\\\frac {1}{| \\\\mathbf {N} (n , a) |} \\\\sum_ {u \\\\in \\\\mathbf {N} (n, a)} \\\\left[ \\\\tilde {\\\\boldsymbol {\\\\Sigma}} _ {n, o} ^ {(u)} \\\\right] ^ {- 1} \\\\tilde {\\\\mathbf {m}} _ {n, o} ^ {(u)}. \\\\tag {66}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,696,906,727]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The output message $\\\\tau_{n,a}(\\\\mathbf{x}_{n,a})$ of the factor $fb_{n,o}^{(a)}$ is computed using (47) and (49) such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[549,733,908,820]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\tau_ {n, a} (\\\\mathbf {x} _ {n, a}) \\\\propto \\\\int f b _ {n, o} ^ {(a)} (\\\\mathbf {x} _ {n, a}, \\\\mathbf {x} _ {n, o} ^ {(a)}) \\\\rho_ {n, o} ^ {(a)} (\\\\mathbf {x} _ {n, o} ^ {(a)}) d \\\\mathbf {x} _ {n, o} ^ {(a)} \\\\\\\\ = \\\\int \\\\left[ \\\\mathcal {N} \\\\left(\\\\mathbf {y} _ {n, a \\\\leftarrow o} | \\\\mathbf {t o} \\\\left(\\\\mathbf {x} _ {n, a}, \\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right), \\\\mathbf {R} _ {t o}\\\\right) \\\\right. \\\\\\\\ \\\\left. \\\\times \\\\mathcal {N} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)} \\\\mid \\\\mathbf {m} _ {n, o} ^ {(a)}, \\\\boldsymbol {\\\\Sigma} _ {n, o} ^ {(a)}\\\\right) \\\\right] d \\\\mathbf {x} _ {n, o} ^ {(a)}. \\\\tag {67} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,827,910,907]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"However, since the target observation $\\\\mathbf{t o}\\\\left(\\\\mathbf{x}_{n,a},\\\\mathbf{x}_{n,o}^{(a)}\\\\right)$ defined in (7) is a nonlinear function, we employ the previously introduced approach in section IV-B to solve for the right-hand side of (67). Thus, to process the factor $fb_{n,o}^{(a)}$ ,we create an MC representation $\\\\left\\\\{\\\\left(w_{n,o}^{(a,k)},\\\\mathbf{x}_{n,o}^{(a,k)}\\\\right)\\\\right\\\\} ,k\\\\in \\\\{1,\\\\dots ,N_o\\\\}$ , of \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,938,750,952]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"OLIVEIRA ET AL.: GNSS-DENIED JOINT COOPERATIVE TERRAIN NAVIGATION AND TARGET TRACKING \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[885,940,908,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"999 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[84,45,486,159]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"the belief of the target state $\\\\mathbf{x}_{n,o}^{(a)}$ , by sampling $\\\\mathbf{x}_{n,o}^{(a,k)} \\\\sim \\\\rho_{n,o}^{(a)}(\\\\mathbf{x}_{n,o}^{(a)})$ and making $w_{n,o}^{(a,k)} = 1 / N_o$ . Since we already have an MC representation $\\\\{(w_{n,a}^{(i)},\\\\mathbf{x}_{n,a}^{(i)})\\\\}$ , $i \\\\in \\\\{1,\\\\dots,N_a\\\\}$ of the belief of the aircraft state estimate $\\\\mathbf{x}_{n,a}$ , just after the cooperative localization message assimilation described in section IV-B, we can approximate the output message $\\\\tau_{n,a}(\\\\mathbf{x}_{n,a})$ , by computing $\\\\tau_{n,a}(\\\\mathbf{x}_{n,a}^{(i)})$ for each particle $\\\\mathbf{x}_{n,a}^{(i)}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[96,164,482,386]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\tau_ {n, a} (\\\\mathbf {x} _ {n, a} ^ {(i)}) \\\\propto \\\\int f b _ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, a} ^ {(i)}, \\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) \\\\rho_ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) d \\\\mathbf {x} _ {n, o} ^ {(a)} \\\\\\\\ = \\\\int \\\\left[ \\\\mathcal {N} \\\\left(\\\\mathbf {y} _ {n, a \\\\leftarrow o} | \\\\mathbf {t o} \\\\left(\\\\mathbf {x} _ {n, a} ^ {(i)}, \\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right), \\\\mathbf {R} _ {t o}\\\\right) \\\\right. \\\\\\\\ \\\\times \\\\mathcal {N} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)} \\\\mid \\\\mathbf {m} _ {n, o} ^ {(a)}, \\\\boldsymbol {\\\\Sigma} _ {n, o} ^ {(a)}\\\\right) ] d \\\\mathbf {x} _ {n, o} ^ {(a)} \\\\\\\\ \\\\approx \\\\sum_ {k = 1} ^ {N _ {o}} \\\\left[ w _ {n, o} ^ {(a, k)} \\\\mathcal {N} \\\\left(\\\\mathbf {y} _ {n, a \\\\leftarrow o} | \\\\mathbf {t o} \\\\left(\\\\mathbf {x} _ {n, a} ^ {(i)}, \\\\mathbf {x} _ {n, o} ^ {(a, k)}\\\\right), \\\\mathbf {R} _ {t o}\\\\right) \\\\right] \\\\\\\\ = \\\\sum_ {k = 1} ^ {N _ {o}} \\\\left[ w _ {n, o} ^ {(a, k)} \\\\mathcal {N} \\\\left(y _ {n, a \\\\leftarrow o} ^ {(r g)} \\\\right\\\\rvert r g \\\\left(\\\\mathbf {x} _ {n, a} ^ {(i)}, \\\\mathbf {x} _ {n, o} ^ {(a, k)}\\\\right), \\\\sigma_ {r o} ^ {2}\\\\right) \\\\\\\\ \\\\times \\\\mathcal {N} \\\\left(y _ {n, a \\\\leftarrow o} ^ {(a z)} | a z \\\\left(\\\\mathbf {x} _ {n, a} ^ {(i)}, \\\\mathbf {x} _ {n, o} ^ {(a, k)}\\\\right), \\\\sigma_ {a z} ^ {2}\\\\right) \\\\\\\\ \\\\left. \\\\times \\\\mathcal {N} \\\\left(y _ {n, a \\\\leftarrow o} ^ {(e l)} \\\\mid e l \\\\left(\\\\mathbf {x} _ {n, a} ^ {(i)}, \\\\mathbf {x} _ {n, o} ^ {(a, k)}\\\\right), \\\\sigma_ {e l} ^ {2}\\\\right) \\\\right]. \\\\tag {68} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[86,395,220,411]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Final Messages \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,415,483,463]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The final message $\\\\gamma_{n,o}^{(a)}\\\\left(\\\\mathbf{x}_{n,o}^{(a)}\\\\right)$ , which represents the belief of the target state at node $a$ after the combine step, is exactly the same as the message $\\\\rho_{n,o}^{(a)}\\\\left(\\\\mathbf{x}_{n,o}^{(a)}\\\\right)$ defined in (64), i.e., \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[199,470,482,491]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\gamma_ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right) = \\\\rho_ {n, o} ^ {(a)} \\\\left(\\\\mathbf {x} _ {n, o} ^ {(a)}\\\\right). \\\\tag {69}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[84,496,483,640]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To compute the final message $\\\\gamma_{n,a}(\\\\mathbf{x}_{n,a})$ , which represents the final belief of the state of the aircraft $a$ at instant $n$ , we first update the weights $w_{n,a}^{(i)}$ of the MC representation $\\\\left\\\\{\\\\left(w_{n,a}^{(i)},\\\\mathbf{x}_{n,a}^{(i)}\\\\right)\\\\right\\\\}$ , assimilating the feedback message $\\\\tau_{n,a}\\\\left(\\\\mathbf{x}_{n,a}^{(i)}\\\\right)$ by making $w_{n,a}^{(i)}\\\\propto w_{n,a}^{(i)}\\\\tau_{n,a}\\\\left(\\\\mathbf{x}_{n,a}^{(i)}\\\\right)$ and normalizing the weights such that they add up to one. Before the time step is incremented, we fit then a GMM representation $\\\\check{\\\\gamma}_{n,a}\\\\left(\\\\mathbf{x}_{n,a}\\\\right)$ to the message $\\\\gamma_{n,a}\\\\left(\\\\mathbf{x}_{n,a}\\\\right)$ using the updated MC representation $\\\\left\\\\{\\\\left(w_{n,a}^{(i)},\\\\mathbf{x}_{n,a}^{(i)}\\\\right)\\\\right\\\\}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[86,648,279,664]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"V. SIMULATION RESULTS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,670,483,867]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To test the proposed algorithm, we used the simulation setup described in [1], consisting of 12 cooperative aircraft flying over a terrain in the state of São Paulo, Brazil, for which we have a real digital elevation map provided by INPE. Unlike in [1], however, we now simulate 3-D trajectories for the surveillance aircraft, including variations in altitude, and also add to the scenario a noncooperative target. The terrain map, initial target and aircraft positions and the communication network topology are shown in Fig. 4. We used 200 independent MC runs in our performance studies. In Fig. 5, we show all 200 simulated trajectories of each aircraft and the target. The target starts flying from the left side of the scenario, as shown in gray, while the \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[496,46,895,224]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0804/d545cbb80c94cf2337d431a1ff081344129a101a9d30c1c3579a8991c7add180.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[496,271,893,428]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 4. Terrain map, initial target position (red), aircraft initial positions (blue), and the communication network topology (orange). \\\",\\\"Fig. 5. All 200 trajectories of the target (gray) and of each aircraft. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0210/f19329ef494ae8ad8fda3dedba51028aec7a27146fa9978bd55987f0fb5efe2b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[496,492,893,594]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0794/4e47b77b98553fb729fe232f25c2ee23a9a1df60bd5271a265642b781b18a1db.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td></td><td>CL</td><td>DTT</td><td>Feedback</td></tr><tr><td>Factors</td><td>adn,a←b</td><td>cd(n,o)(a←b)</td><td>fb(n,o)(a)</td></tr><tr><td>Messages</td><td>λn,aandξn,a←b</td><td>λ(n,o)(a) and θ(n,o)(a←b)</td><td>ρ(n,o)(a) and τn,a</td></tr><tr><td>NC</td><td>False</td><td>False</td><td>False</td></tr><tr><td>CTAN</td><td>True</td><td>False</td><td>False</td></tr><tr><td>J-CTAN-DTT</td><td>True</td><td>True</td><td>True</td></tr></table>\",\n \"table_caption\": \"[\\\"TABLEI Factors and Messages Enabling for Each Factor Graph Configuration \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[493,618,893,708]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"aircraft are flying in formation, starting from the bottom left side of the scenario, monitoring the area. The target and aircraft speeds are set as 50 and $100\\\\mathrm{m / s}$ , respectively. The total simulation window lasts $600\\\\mathrm{s}$ and $T_{s} = 1\\\\mathrm{s}$ . The surveillance aircraft maneuvers were simulated using a 3-D coordinated turn model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[493,709,893,844]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The proposed algorithm was evaluated using three different factor graph configurations in the \\\"6-TAN 6-INS\\\" scenario, where none of the 12 aircraft have GNSS measurements, but six aircraft perform terrain navigation while the six remaining aircraft navigate with INS data only. Each of the proposed configurations or operation modes correspond to a subset of the full factor graph shown in Fig. 3, in which we add or remove the corresponding factors and messages according to Table I. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[493,845,893,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We employed $N_{a} = 3000$ particles to represent the state vector of each surveillance aircraft and $N_{o} = 1000$ particles for the local representation of the target state at each node. In the cooperative localization loop, the GMMs representing the aircraft beliefs that are received from neighboring \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,892,482,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"<sup>3</sup>Source INPE: http://www.dsr.inpe.br/topodata/cesso.php [March 1, 2023]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[88,939,119,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1000 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[256,939,890,952]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.60, NO.1 FEBRUARY 2024 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[102,73,496,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0442/0b4be56ce99f68984b0237f60aef868974a7172d5857ea0c664aaf29863192c9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Time step</td><td>σacc factor</td><td>Na factor</td><td>σacc,o factor</td><td>No factor</td></tr><tr><td>0</td><td>2.00</td><td>2.00</td><td>2.00</td><td>2.00</td></tr><tr><td>30</td><td>1.50</td><td>1.50</td><td>1.50</td><td>1.50</td></tr><tr><td>60</td><td>1.25</td><td>1.25</td><td>1.25</td><td>1.25</td></tr><tr><td>90</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr></table>\",\n \"table_caption\": \"[\\\"TABLE II Covariance and Particle Number Schedule \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[99,191,496,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"nodes were resampled with $N_{b} = 3000$ particles. We use GMMs with one single mode for both aircraft and target representations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[99,236,496,356]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We also computed the PCRLB by performing 1000 independent MC runs for the centralized version of the joint cooperative terrain-aided navigation and distributed target tracking (J-CTAN-DTT) problem, see Appendix A. In the aircraft dynamic model (1), we assumed covariance matrix $\\\\mathbf{Q}_a = \\\\sigma_{\\\\mathrm{acc},a}^2 * \\\\mathbf{I}_3$ with $\\\\sigma_{\\\\mathrm{acc},a} = 1\\\\mathrm{m}\\\\forall a$ . For the target's constant velocity model in (2), we set $\\\\sigma_{\\\\mathrm{acc},o} = 0.1\\\\mathrm{m}$ in the process noise $\\\\mathbf{w}_{n,o}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,357,496,402]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We used $\\\\sigma_{ba} = 2\\\\mathrm{m}$ , $\\\\sigma_{gc} = 2\\\\mathrm{m}$ and $\\\\sigma_{ad} = 1\\\\mathrm{m}$ for all aircraft, and $\\\\sigma_{ro} = 10\\\\mathrm{m}$ , $\\\\sigma_{az} = \\\\pi /360\\\\mathrm{rad}$ $(0.5^{\\\\circ})$ , $\\\\sigma_{el} = \\\\pi /360\\\\mathrm{rad}$ $(0.5^{\\\\circ})$ for the target. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,402,496,462]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For filtering, we used $\\\\sigma_{\\\\mathrm{acc}} = 1.5\\\\mathrm{m / s}^2$ for the NC configuration and $\\\\sigma_{\\\\mathrm{acc}} = 6\\\\mathrm{m / s}^2$ for both the CTAN and J-CTAN-DTT configurations. For the target, we used $\\\\sigma_{\\\\mathrm{acc},o} = 0.75\\\\mathrm{m / s}^2$ in all configurations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,462,496,575]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We sampled the estimate of the initial state of each aircraft $\\\\hat{\\\\mathbf{x}}_{0,a}$ using a Gaussian distribution with mean $\\\\mathbf{x}_{0,a}$ and covariance $\\\\mathbf{P}_{0,a} \\\\triangleq \\\\sigma_{\\\\mathrm{pos}}^2 * \\\\mathbf{I}$ , while the estimate of the initial target representation of each aircraft $\\\\hat{\\\\mathbf{x}}_{0,o}^{(a)}$ was sampled from a Gaussian distribution with mean $\\\\mathbf{x}_{0,o}^{(a)}$ and covariance $\\\\mathbf{P}_{0,o} \\\\triangleq \\\\mathrm{diag}([\\\\sigma_{\\\\mathrm{pos}}^2 \\\\sigma_{\\\\mathrm{vel}}^2 \\\\sigma_{\\\\mathrm{pos}}^2 \\\\sigma_{\\\\mathrm{vel}}^2 \\\\sigma_{\\\\mathrm{pos}}^2 \\\\sigma_{\\\\mathrm{vel}}^2])$ , with $\\\\sigma_{\\\\mathrm{pos}} = 50\\\\mathrm{m}$ and $\\\\sigma_{\\\\mathrm{vel}} = 5\\\\mathrm{m/s}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[99,577,496,712]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We implemented a scheduler for the process noise covariances and number of particles to improve the filter robustness, specially in the initial time steps, when the uncertainty of the state variables/vectors are bigger. We apply a scale factor to the parameters that define the covariance matrices and the amount of particles according to a simple schedule such that, as time progresses, the scaling factor is reduced and then removed. Table II shows the schedule and the scale factors. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[99,713,496,803]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The performance of each scenario configuration was assessed using the root-mean-square error (RMSE) of the position estimates for both cooperative localization $e_n$ and target tracking $e_{n,o}$ , averaged as a function of time, average over all network nodes and all MC simulations, as explained in detail in [1]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,804,496,834]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moreover, the performance metrics $e_n$ and $e_{n,o}$ are minorized by the centralized PCRLB as follows: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[222,848,496,881]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\bar {e} _ {n} \\\\geq \\\\sqrt {\\\\frac {1}{A} \\\\sum_ {a = 1} ^ {A} \\\\operatorname {t r} \\\\left(\\\\mathbf {P} _ {n , a}\\\\right)} \\\\tag {70}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[212,885,496,917]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\bar {e} _ {n, o} \\\\geq \\\\sqrt {\\\\frac {1}{A} \\\\sum_ {a = 1} ^ {A} \\\\operatorname {t r} \\\\left(\\\\mathbf {P} _ {n , o}\\\\right)} \\\\tag {71}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,45,911,248]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0331/66bd048eb90c4b8832066d27e98ca469b44b11db41ac261bfc4d1c426efb24dc.jpg\",\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[522,287,897,478]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 6. RMSE for cooperative localization over time for different configurations. \\\",\\\"Fig. 7. RMSE for distributed target tracking over time for different configurations. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0994/433286ebafae803667cd94e007b3fcd77da35faac35d4cde1bd6b06a79e61f77.jpg\",\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[511,536,908,613]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\operatorname{tr}(\\\\mathbf{R})$ represents the trace of the matrix $\\\\mathbf{R}$ , $\\\\mathbf{P}_{n,a}$ , and $\\\\mathbf{P}_{n,o}$ , associated, respectively, with the aircraft $a$ position and the target state (excluding velocities), are $3 \\\\times 3$ diagonal blocks of $[\\\\mathbf{J}_n]^{-1}$ , where $\\\\mathbf{J}_n$ is the Fisher information matrix (see Appendix A). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,627,777,641]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. Numerical Results and Discussion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,648,908,691]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The cooperative localization experiments results are presented in Fig. 6 and compared to the PCRLB, in gray dashed line - as in (70). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,694,908,768]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the noncooperative (NC) configuration, since only half of the aircraft perform TAN, the results diverge. The results of CTAN and J-CTAN-DTT are in turn very similar to each other, with J-CTAN-DTT showing only a small improvement, mainly in the beginning of the simulation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,768,908,813]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The results of the target tracking experiments are shown in Fig. 7 and compared to the associated PCRLB - gray dashed line - as in (71). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,814,908,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In Fig. 7, the results of the CTAN configuration are better than those of NC, even without the distributed target tracking messages, because the target position estimate is based on the aircraft position estimate. Without cooperative localization, the six aircraft that do not perform TAN (i.e., are flying only with INS) cannot estimate their own position accurately, and therefore do not perform good target \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,938,750,952]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"OLIVEIRA ET AL.: GNSS-DENIED JOINT COOPERATIVE TERRAIN NAVIGATION AND TARGET TRACKING \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[879,940,906,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1001 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[124,45,339,176]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0949/c4ea4addb2a10743480874767ee51db6f7b0cd2ce448b26e35f153a302f9e650.jpg\",\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[380,45,596,176]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0120/96253634d0e70604dc87781ee9647e7099b8dd5f4baac35467aaf5e80724dc79.jpg\",\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[640,45,856,176]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 8. Target tracking dispersion results. The black curve shows the corresponding scenario configuration RMSE result, while the gray curves show the RMSE results of each aircraft. J-CTAN-DTT has a significantly lower dispersion than NC and CTAN. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0279/6783b76e6882a1e12999409539dc4fb79e47fe9f478dc299cd89652d0ee5f6da.jpg\",\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[117,267,454,309]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0158/d4a2d67d896460a81df71aeea9ca6671b77642e71bd3df4c40a2aa55ff167bf5.jpg\",\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td></td><td>NC</td><td>CTAN</td><td>J-CTAN-DTT</td></tr><tr><td>Time/node/iter [s]</td><td>0.10567</td><td>0.46486</td><td>0.56919</td></tr></table>\",\n \"table_caption\": \"[\\\"TABLE III Average Execution Time Per Node Per Iteration \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[84,352,483,458]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"tracking. In addition, for the other six aircraft that perform TAN, cooperative localization improves the aircraft position estimate, which in turn, allows for a better target observation assimilation. In the J-CTAN-DTT configuration, where the cooperative localization, distributed target tracking, and feedback messages are enabled, the results are close to the PCRLB, shown in gray. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,458,483,579]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Besides the accuracy improvement in target tracking, the proposed algorithm also decreased the dispersion of the target state estimate for each aircraft, suggesting that the aircraft approach a weak consensus on the target state estimate. To highlight this precision improvement, we show in Fig. 8, the target tracking results for all configurations, comparing the average network tracking RMSE (in black) with target tracking RMSE $e_{n,q}^{(a)}$ for each aircraft $a$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,579,483,640]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In Table III, we show the average execution time, per iteration, per node, for the proposed algorithm, in a $\\\\mathrm{C + + }$ multithread implementation, running in a Intel i9-12900H computer. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[86,672,323,688]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B. Communication Cost Analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,694,482,755]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1) Comm Cost $J$ -CTAN-DTT: The proposed algorithm J-CTAN-DTT described in section III has some characteristics that are desirable in real-time distributed algorithms. For example: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[109,784,482,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1) It does not need random number generation synchronization between the different network nodes since each of them stores a different set of particles to represent its own state and the state of the target. \\\",\\\"2) It does not require iterative message passing between nodes at the same time stamp $n$ (i.e., between two consecutive sets of sensor measurements). \\\",\\\"3) The number of network nodes can be unknown and vary over time. \\\"]\",\n \"page_idx\": \"11\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[495,238,893,300]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In terms of internode communication overhead, at each instant $n$ , each aircraft $a$ needs to broadcast to the neighborhood only the parameters that define the local state variables $\\\\tilde{\\\\mathbf{x}}_{n,a}$ and $\\\\tilde{\\\\mathbf{x}}_{n,o}^{(a)}$ . Let \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[519,306,890,470]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1) $D_{a}$ denote the dimension of the aircraft state vector; \\\",\\\"2) $D_{o}$ denote the dimension of the target state vector; \\\",\\\"3) $D_{\\\\\\\\mathrm{cov},a}$ denote the number of real-valued free elements of the (symmetric) aircraft state covariance matrix, i.e., $(D_a^2 + D_a) / 2$ ; \\\",\\\"4) $D_{\\\\\\\\mathrm{cov},o}$ denote the number of real-valued free elements of the (symmetric) target state covariance matrix, i.e., $(D_o^2 +D_o) / 2$ , and \\\",\\\"5) $|O|$ denote the number of objects, which, as noted before, could be greater than one if there are no data association issues. \\\"]\",\n \"page_idx\": \"11\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[495,479,893,556]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Using the aforementioned notation and, since the state p.d.f's are represented by Gaussian functions, it follows that the total number of real values that the aircraft $a$ broadcasts at instant $n$ over a given internode communication link is $(D_{a} + D_{\\\\mathrm{cov},a}) + [(D_{o} + D_{\\\\mathrm{cov},o})*|O|]$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[495,556,893,614]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Considering the scenario described in this article where $D_{a} = 3$ , $D_{o} = 6$ , and $|O| = 1$ , the total number of real values the aircraft $a$ broadcasts over each internode communication link is 36. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[495,616,893,675]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2) Comm Cost Average Consensus: As a comparison, using the average consensus approach described in [24], the total number of real values the agent $a$ broadcasts at instant $n$ is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[575,683,890,700]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nN ^ {c} + N ^ {N B P} = P J ((C + I) | O | + L) \\\\tag {72}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[496,708,772,723]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where using the same notation as in [24] \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[519,731,890,866]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1) $\\\\\\\\mathrm{P}$ is number of message passing iterations; \\\",\\\"2) C is number of consensus iterations; \\\",\\\"3) J is the number of particles used to represent either target or agent states; \\\",\\\"4) I is diameter of the communication graph; \\\",\\\"5) L is the dimension of the agent states (only position); \\\",\\\"6) $|O|$ is the number of objects; \\\",\\\"7) $N^c \\\\\\\\triangleq P(C + I)J|O|$ ; and \\\",\\\"8) $N^{NBP} \\\\\\\\triangleq PJL$ . \\\"]\",\n \"page_idx\": \"11\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[495,875,893,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Considering the same scenario described in this article, with the aircraft being the agents and the object being the target, we have $P = 1$ , $I = 4$ , $L = D_{a} = 3$ , $|O| = 1$ . In \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[88,940,117,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1002 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[256,939,890,952]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.60, NO.1 FEBRUARY 2024 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[99,44,500,194]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"addition, for the purposes of this discussion, we assume the same number of consensus iterations $C = 6$ and the same number of particles $J = 1000$ used in the dynamic scenario described in [24]. Thus, using the formula in (72), the total number of real values that the agent $a$ would have to broadcast over each network link in the same consensus approach as in [24] is 13 000. Note that even though in our scenario we use 3000 particles for the aircraft representation, we are assuming $J = 1000$ for both aircraft and target states, which decreases the communication cost. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,195,500,301]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Even if we considered only 1 consensus iteration ( $C = 1$ ) and also considered a fixed network configuration with known network diameter size, in which case we could set $I$ to zero in (72), see [24], the total number of real values that the agent $a$ would have to broadcast over each link using the average consensus approach would be 4000 in the same scenario as in our article. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[99,301,500,650]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The high communication cost presented in [24] is driven by the need to have $J$ average consensus protocols—one for each particle—between consecutive sets of sensor observations, where each average consensus protocol requires, in turn, $C$ iterations. Moreover, the $J$ particles have to be identical in all nodes, thus requiring random number generators that are synchronized between the different nodes. The internode communication cost in [24] can be significantly reduced if an alternative scheme called likelihood consensus proposed in [38] is used. Specifically, likelihood consensus eliminates the linear growth of the internode communication cost with the number of particles $J$ and also eliminates the need for random number generator synchronization across nodes. Furthermore, likelihood consensus also eliminates the need to do $I$ additional max consensus iterations to determine the number of nodes in the network. However, even in a likelihood consensus implementation, the internode communication cost still grows linearly with the number of consensus iterations $C$ and, if that parameter is high, for example, $C = 15$ as in the second scenario considered in [24], the communication requirements between nodes are still about an order of magnitude higher than in the diffusion approach proposed in this article. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[101,672,246,686]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"VI. CONCLUSION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[99,693,500,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We proposed in this work an approach based on ATC diffusion [6], [7], which is seamlessly integrated into the factor graph framework and implemented in a fully distributed fashion using the SPA by introducing novel special factors to perform the distributed combine step. The proposed solution approximates the adapt step of a classic distributed ATC tracker by replacing the likelihood of the local target observations at each agent node given the target and the agent positions by its expected value with respect to the latest belief (i.e., posterior probability distribution) that the agent has about its own state. Next, the aforementioned special factors combine the updated target state beliefs following the adapt step at each node using an optimal minimum average KL divergence criterion, which, for Gaussian beliefs, reduces to the well-known method of \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,44,911,194]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"covariance intersection in sensor fusion. With the proposed factor-graph-based combine step, we improved the overall target tracking accuracy and precision, in such a way that all aircraft in the network have a better and almost identical target state estimate, agreeing with each other on their target position estimates, using a low-cost communication scheme. In addition, after the combine step, each agent uses their updated target belief as if it were an anchor, to improve their self-localization, thus assimilating the target measurements. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,195,911,393]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We extended the scenario from our previous work [1] and tested three different configurations. In all cases, the error performance was compared to the PCRLB for the error of the optimal data fusion filter that processes all available measurements on the network at each time step on a central hub. The proposed distributed algorithm was shown to perform close to the centralized PCRLB, for both self-localization and target tracking errors, in a simulated GNSS-denied scenario and a partially connected network with a low-cost communication scheme. Even aircraft that cannot locate themselves alone, using the proposed algorithm, they can locate themselves, track the target, and agree with each other on their estimates of the target's position. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,404,607,417]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"APPENDIX A \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,426,691,439]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. Performance Bounds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,446,911,601]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1) Centralized Signal Model: As in [1], for the centralized filtering approach, we stack together all cooperative aircraft and noncooperative target state vectors in an extended state vector and do the same for the observations, now including also target radar measurements and barometric altitude measurements. We then make the factor graph corresponding to the distributed signal model into a tree-like graph by removing all loops between variable nodes. So, the extended state vector $\\\\mathbf{x}_n^c$ and the extended observation vector $\\\\mathbf{y}_n^c$ are defined, respectively, as<sup>4</sup> \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[591,606,908,628]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf {x} _ {n} ^ {c} = \\\\left[ \\\\begin{array}{l l l l} \\\\mathbf {x} _ {n, 1} ^ {T} & \\\\mathbf {x} _ {n, 2} ^ {T} & \\\\dots & \\\\mathbf {x} _ {n, 1 2} ^ {T} & \\\\mathbf {x} _ {n, o} ^ {T} \\\\end{array} \\\\right] ^ {T} \\\\tag {73}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[591,628,908,751]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf {y} _ {n} ^ {c} = \\\\left[ \\\\begin{array}{l} {\\\\left[ \\\\begin{array}{l l l} y _ {n, 1} \\\\leftarrow b a & \\\\dots & y _ {n, 1 2} \\\\leftarrow b a \\\\\\\\ \\\\left[ \\\\begin{array}{l l l} y _ {n, 1} \\\\leftarrow g c & \\\\dots & y _ {n, 1 2} \\\\leftarrow g c \\\\end{array} \\\\right] ^ {T} \\\\end{array} \\\\right]} \\\\\\\\ {\\\\left[ \\\\begin{array}{l l l} y _ {n, 1} \\\\leftarrow 2 & \\\\dots & y _ {n, 1 2} \\\\leftarrow 1 1 \\\\end{array} \\\\right] ^ {T}} \\\\\\\\ {\\\\left[ \\\\begin{array}{l l l} y _ {n, 1} \\\\leftarrow r o & \\\\dots & y _ {n, 1 2} \\\\leftarrow r o \\\\end{array} \\\\right] ^ {T}} \\\\\\\\ {\\\\left[ \\\\begin{array}{l l l} y _ {n, 1} \\\\leftarrow a z & \\\\dots & y _ {n, 1 2} \\\\leftarrow a z \\\\end{array} \\\\right] ^ {T}} \\\\\\\\ {\\\\left[ \\\\begin{array}{l l l} y _ {n, 1} \\\\leftarrow e l & \\\\dots & y _ {n, 1 2} \\\\leftarrow e l \\\\end{array} \\\\right] ^ {T}} \\\\end{array} \\\\right]. \\\\tag {74}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,757,911,789]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The kinematics of the sequence of extended state vectors $\\\\{\\\\mathbf{x}_n^c\\\\}$ , $n \\\\in \\\\mathbb{Z}_+$ , tracked by the centralized filter is modeled as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[625,795,908,814]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf {x} _ {n + 1} ^ {c} = \\\\mathbf {F} _ {n} ^ {c} \\\\mathbf {x} _ {n} ^ {c} + \\\\mathbf {u} _ {n} ^ {c} + \\\\mathbf {w} _ {n} ^ {c} \\\\tag {75}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[511,821,911,856]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"in which $\\\\mathbf{u}^c = \\\\left[\\\\mathbf{u}_1^T\\\\ldots \\\\mathbf{u}_{12}^T\\\\mathbf{0}_{6\\\\times 1}^T\\\\right]^T$ is the extended input for all aircraft, concatenated with $\\\\mathbf{0}_{6\\\\times 1}$ , since we do not have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,880,911,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note that, for the centralized factor graph, it is not necessary to consider multiple copies $\\\\mathbf{x}_{n,o}^{(a)}$ of the single target state vector $\\\\mathbf{x}_{n,o}$ at each node $a$ as in the factor graph for the distributed filter. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[102,938,750,952]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"OLIVEIRA ET AL.: GNSS-DENIED JOINT COOPERATIVE TERRAIN NAVIGATION AND TARGET TRACKING \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[879,940,908,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1003 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[84,44,482,119]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"access to the target's INS sensor, $\\\\mathbf{w}_n^c\\\\sim \\\\mathcal{N}(\\\\mathbf{0},\\\\mathbf{Q}_n^c)$ is the random vector corresponding the dynamic model noise, $\\\\mathbf{F}_n^c =$ diag([I36x36 $\\\\mathbf{F}_{n,o}]$ ) and $\\\\mathbf{Q}_n^c = \\\\mathrm{diag}([\\\\mathbf{Q}_{n,1}\\\\dots \\\\mathbf{Q}_{n,12}\\\\mathbf{Q}_{n,o}])$ with $\\\\mathbf{Q}_{n,a} = \\\\sigma_{\\\\mathrm{acc},a}^{2}*\\\\mathbf{I}_{3\\\\times 3}$ $\\\\forall a\\\\in \\\\{1,\\\\ldots ,A\\\\}$ , and both $\\\\mathbf{F}_{n,o}$ and $\\\\mathbf{Q}_{n,o}$ defined as introduced in (2). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[84,119,482,148]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The extended observation vector assimilated by the centralized filter at each instant $n$ is modeled in turn as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[220,157,480,176]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf {y} _ {n} ^ {c} = \\\\mathbf {h} ^ {c} \\\\left(\\\\mathbf {x} _ {n} ^ {c}\\\\right) + \\\\mathbf {v} _ {n} ^ {c} \\\\tag {76}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[84,183,482,248]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\mathbf{v}_n^c\\\\sim \\\\mathcal{N}(\\\\mathbf{0},\\\\mathbf{R}_n^c),\\\\quad \\\\mathbf{R}_n^c = \\\\mathrm{diag}([\\\\mathbf{R}_{ba}\\\\mathbf{R}_{gc}\\\\mathbf{R}_{ad}\\\\mathbf{R}_{to}])$ $\\\\mathbf{R}_{ba} = \\\\sigma_{ba}^{2}*\\\\mathbf{I}_{12\\\\times 12},\\\\mathbf{R}_{gc} = \\\\sigma_{gc}^{2}*\\\\mathbf{I}_{12\\\\times 12},\\\\mathbf{R}_{ad} = \\\\sigma_{ad}^{2}*\\\\mathbf{I}_{46\\\\times 46}$ and $\\\\mathbf{R}_{to} = \\\\mathrm{diag}([\\\\sigma_{ro}^{2}*\\\\mathbf{I}_{12\\\\times 12}\\\\sigma_{az}^{2}*\\\\mathbf{I}_{12\\\\times 12}\\\\sigma_{el}^{2}*\\\\mathbf{I}_{12\\\\times 12}])$ Now, let \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[132,253,482,272]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf {b a} \\\\left(\\\\mathbf {x} _ {n} ^ {c}\\\\right) = \\\\left[ b a \\\\left(\\\\mathbf {x} _ {n, 1}\\\\right) \\\\dots b a \\\\left(\\\\mathbf {x} _ {n, 1 2}\\\\right) \\\\right] \\\\tag {77}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[104,275,482,292]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf {g c} \\\\left(\\\\mathbf {x} _ {n} ^ {c}; \\\\mathcal {M}\\\\right) = \\\\left[ g c \\\\left(\\\\mathbf {x} _ {n, 1}; \\\\mathcal {M}\\\\right) \\\\dots g c \\\\left(\\\\mathbf {x} _ {n, 1 2}; \\\\mathcal {M}\\\\right) \\\\right] \\\\tag {78}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[133,295,480,314]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left. \\\\mathbf {a d} \\\\left(\\\\mathbf {x} _ {n} ^ {c}\\\\right) = \\\\left[ r g \\\\left(\\\\mathbf {x} _ {n, 1}, \\\\mathbf {x} _ {n, 2}\\\\right) \\\\dots r g \\\\left(\\\\mathbf {x} _ {n, 1 2}, \\\\mathbf {x} _ {n, 1 1}\\\\right) \\\\right] \\\\right. \\\\tag {79}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[135,316,480,391]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf {t o} \\\\left(\\\\mathbf {x} _ {n} ^ {c}\\\\right) = \\\\left[ \\\\begin{array}{l} {\\\\left[ r g \\\\left(\\\\mathbf {x} _ {n, o}, \\\\mathbf {x} _ {n, 1}\\\\right) \\\\dots r g \\\\left(\\\\mathbf {x} _ {n, o}, \\\\mathbf {x} _ {n, 1 2}\\\\right) \\\\right] ^ {T}} \\\\\\\\ {\\\\left[ a z \\\\left(\\\\mathbf {x} _ {n, o}, \\\\mathbf {x} _ {n, 1}\\\\right) \\\\dots a z \\\\left(\\\\mathbf {x} _ {n, o}, \\\\mathbf {x} _ {n, 1 2}\\\\right) \\\\right] ^ {T}} \\\\\\\\ {\\\\left[ e l \\\\left(\\\\mathbf {x} _ {n, o}, \\\\mathbf {x} _ {n, 1}\\\\right) \\\\dots e l \\\\left(\\\\mathbf {x} _ {n, o}, \\\\mathbf {x} _ {n, 1 2}\\\\right) \\\\right] ^ {T}} \\\\end{array} \\\\right] \\\\tag {80}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[84,400,482,477]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"collect, respectively, the functions modeling all barometric altitude, ground clearances, aircraft pairwise distances, and target observations available over the network at a given instant $n$ . Lastly, we concatenate these functions into an extended observation function $\\\\mathbf{h}^c(\\\\mathbf{x}_n^c)$ such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[106,484,480,517]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf {h} ^ {c} \\\\left(\\\\mathbf {x} _ {n} ^ {c}\\\\right) = \\\\left[ \\\\mathbf {b a} \\\\left(\\\\mathbf {x} _ {n} ^ {c}\\\\right) ^ {T} \\\\mathbf {g c} \\\\left(\\\\mathbf {x} _ {n} ^ {c}; \\\\mathcal {M}\\\\right) ^ {T} \\\\mathbf {a d} \\\\left(\\\\mathbf {x} _ {n} ^ {c}\\\\right) ^ {T} \\\\mathbf {t o} \\\\left(\\\\mathbf {x} _ {n} ^ {c}\\\\right) ^ {T} \\\\right] ^ {T}. \\\\tag {81}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[84,518,482,563]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2) PCRLB: The posterior Cramér-Rao lower bound of the state estimation error for the centralized filtering problem has the form \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[192,571,480,590]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathrm {E} \\\\left\\\\{\\\\left| \\\\left| \\\\mathbf {x} _ {n} ^ {c} - \\\\hat {\\\\mathbf {x}} _ {n} ^ {c} \\\\right| \\\\right| ^ {2} \\\\right\\\\} \\\\geq \\\\operatorname {t r} \\\\left[ \\\\mathbf {J} _ {n} ^ {- 1} \\\\right] \\\\tag {82}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[84,597,482,628]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"in which, as derived in [39], the Fisher matrix $\\\\mathbf{J}_{n + 1}$ at instant $n + 1$ is recursively computed from $\\\\mathbf{J}_n$ as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[158,636,482,656]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf {J} _ {n + 1} = \\\\mathbf {D} _ {n} ^ {2 2} - \\\\mathbf {D} _ {n} ^ {2 1} \\\\left(\\\\mathbf {J} _ {n} + \\\\mathbf {D} _ {n} ^ {1 1}\\\\right) ^ {- 1} \\\\mathbf {D} _ {n} ^ {1 2}. \\\\tag {83}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[84,662,482,709]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The coefficients $\\\\mathbf{D}_n^{11},\\\\mathbf{D}_n^{12},\\\\mathbf{D}_n^{21}$ , and $\\\\mathbf{D}_n^{22}$ in (83), induced by the dynamic and observation models, defined in model (75) and (76), respectively, are computed such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[127,717,482,738]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf {D} _ {n} ^ {1 1} = \\\\left[ \\\\mathbf {F} _ {n} ^ {c} \\\\right] ^ {T} \\\\left[ \\\\mathbf {Q} _ {n} ^ {c} \\\\right] ^ {- 1} \\\\left[ \\\\mathbf {F} _ {n} ^ {c} \\\\right] \\\\tag {84}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[127,739,482,760]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf {D} _ {n} ^ {1 2} = - \\\\left[ \\\\mathbf {F} _ {n} ^ {c} \\\\right] ^ {T} \\\\left[ \\\\mathbf {Q} _ {n} ^ {c} \\\\right] ^ {- 1} \\\\tag {85}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[127,763,480,784]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf {D} _ {n} ^ {2 1} = \\\\left[ \\\\mathbf {D} _ {n} ^ {1 2} \\\\right] ^ {T} = - \\\\left[ \\\\mathbf {Q} _ {n} ^ {c} \\\\right] ^ {- 1} \\\\mathbf {F} _ {n} ^ {c} \\\\tag {86}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[127,786,480,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf {D} _ {n} ^ {2 2} = \\\\left[ \\\\mathbf {Q} _ {n} ^ {c} \\\\right] ^ {- 1} + \\\\mathrm {E} \\\\left\\\\{\\\\left[ \\\\mathbf {H} _ {n + 1} ^ {c} \\\\right] ^ {T} \\\\left[ \\\\mathbf {R} _ {n} ^ {c} \\\\right] ^ {- 1} \\\\mathbf {H} _ {n + 1} ^ {c} \\\\right\\\\} \\\\tag {87}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[86,815,477,833]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\mathbf{H}_{n + 1}^{c} = \\\\nabla_{\\\\mathbf{x}_{n + 1}^{c}}\\\\left[\\\\mathbf{h}^{c}\\\\left(\\\\mathbf{x}_{n + 1}^{c}\\\\right)\\\\right]$ . Then, we compute $\\\\mathbf{H}_n^c$ as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,840,480,883]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} \\\\mathbf {H} _ {n} ^ {c} = \\\\nabla_ {\\\\mathbf {x} _ {n} ^ {c}} \\\\mathbf {h} ^ {c} \\\\left(\\\\mathbf {x} _ {n} ^ {c}\\\\right) \\\\\\\\ = \\\\left[ \\\\left[ \\\\mathbf {H} _ {n, b a} ^ {c} \\\\right] ^ {T} \\\\left[ \\\\mathbf {H} _ {n, g c} ^ {c} \\\\right] ^ {T} \\\\left[ \\\\mathbf {H} _ {n, a d} ^ {c} \\\\right] ^ {T} \\\\left[ \\\\mathbf {H} _ {n, t o} ^ {c} \\\\right] ^ {T} \\\\right] ^ {T} \\\\tag {88} \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[86,890,482,922]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\mathbf{H}_{n,ba}^{c} = \\\\nabla_{\\\\mathbf{x}_{n}^{c}}\\\\mathbf{b}\\\\mathbf{a}(\\\\mathbf{x}_{n}^{c}),\\\\mathbf{H}_{n,gc}^{c} = \\\\nabla_{\\\\mathbf{x}_{n}^{c}}\\\\mathbf{g}\\\\mathbf{c}(\\\\mathbf{x}_{n}^{c};\\\\mathcal{M}),\\\\mathbf{H}_{n,ad}^{c} =$ $\\\\nabla_{\\\\mathbf{x}_n^c}\\\\mathbf{ad}(\\\\mathbf{x}_n^c)$ , and $\\\\mathbf{H}_{n,to}^{c} = \\\\nabla_{\\\\mathbf{x}_n^c}\\\\mathbf{t}\\\\mathbf{o}(\\\\mathbf{x}_n^c)$ are the Jacobians related \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[496,44,893,119]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"to the barometric altitude, ground clearance, aircraft distance, and target observations, respectively. Note that the Jacobian matrix $\\\\mathbf{H}_{n,to}^{c}$ includes block diagonal Jacobian submatrices for all available target observations—slant-range, azimuth, and elevation—at instant $n$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[498,155,584,167]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"REFERENCES \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[498,176,893,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"[1] H. Oliveira, S. S. Dias, and M. G. S. Bruno, \\\\\\\"Cooperative terrain navigation using hybrid GMM/SMC message passing on factor graphs,\\\\\\\" IEEE Trans. Aerosp. Electron. Syst., vol. 56, no. 5, pp. 3958-3970, Oct. 2020. \\\",\\\"[2] B. J. Frey, F. R. Kschischang, H.-A. Loeliger, and N. Wiberg, \\\\\\\"Factor graphs and algorithms,\\\\\\\" in Proc. Annu. Allerton Conf. Commun. Control Comput., 1997, pp. 666-680. \\\",\\\"[3] H.-A. Loeliger, “An introduction to factor graphs,” IEEE Signal Process. Mag., vol. 21, no. 1, pp. 28–41, Jan. 2004. \\\",\\\"[4] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 498–519, Feb. 2001. \\\",\\\"[5] F. S. Cattivelli and A. H. Sayed, \\\\\\\"Diffusion strategies for distributed Kalman filtering and smoothing,\\\\\\\" IEEE Trans. Autom. Control, vol. 55, no. 9, pp. 2069-2084, Sep. 2010. \\\",\\\"[6] K. Dedecius and P. M. Djuric, \\\\\\\"Sequential estimation and diffusion of information over networks: A Bayesian approach with exponential family of distributions,\\\\\\\" IEEE Trans. Signal Process., vol. 65, no. 7, pp. 1795-1809, Apr. 2017. \\\",\\\"[7] M. G. S. Bruno and S. S. Dias, “A Bayesian interpretation of distributed diffusion filtering algorithms [Lecture Notes],” IEEE Signal Process. Mag., vol. 35, no. 3, pp. 118–123, May 2018. \\\",\\\"[8] N. Bergman, “A Bayesian approach to terrain-aided navigation,” IFAC Proc. Volumes, vol. 30, no. 11, pp. 1457–1462, Jul. 1997. \\\",\\\"[9] N. Bergman, L. Ljung, and F. Gustafsson, \\\\\\\"Terrain navigation using Bayesian statistics,\\\\\\\" IEEE Control Syst., vol. 19, no. 3, pp. 33-40, Jun. 1999. \\\",\\\"[10] B. Copp and K. Subbarao, \\\\\\\"Nonlinear adaptive filtering in terrain-referenced navigation,\\\\\\\" IEEE Trans. Aerosp. Electron. Syst., vol. 51, no. 4, pp. 3461-3469, Oct. 2015. \\\",\\\"[11] P.J. Nordlund and F. Gustafsson, \\\\\\\"Marginalized particle filter for accurate and reliable terrain-aided navigation,\\\\\\\" IEEE Trans. Aerosp. Electron. Syst., vol. 45, no. 4, pp. 1385-1399, Oct. 2009. \\\",\\\"[12] J. Dunífák, M. Soták, M. Vesely, O. Straka, and W. Hawkins, \\\\\\\"Design of Rao-Blackwellized point-mass filter with application in terrain aided navigation,\\\\\\\" IEEE Trans. Aerosp. Electron. Syst., vol. 55, no. 1, pp. 251-272, Feb. 2019. \\\",\\\"[13] N. Correal and N. Patwari, \\\\\\\"Wireless sensor networks: Challenges and opportunities,\\\\\\\" in Proc. MPRG/Virginia Tech Wireless Symp., 2001, pp. 1-9. \\\",\\\"[14] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, \\\\\\\"Wireless sensor networks: A survey,\\\\\\\" Comput. Netw., vol. 38, no. 4, pp. 393-422, 2002. \\\",\\\"[15] M. Perkins, N. Correal, and B. O'Dea, \\\\\\\"Emergent wireless sensor network limitations: A plea for advancement in core technologies,\\\\\\\" in Proc. SENSORS, 2002, pp. 1505-1509. \\\",\\\"[16] M. Z. Win, F. Meyer, Z. Liu, W. Dai, S. Bartoletti, and A. Conti, \\\\\\\"Efficient multisensor localization for the Internet of Things: Exploring a new class of scalable localization algorithms,\\\\\\\" IEEE Signal Process. Mag., vol. 35, no. 5, pp. 153-167, Sep. 2018. \\\",\\\"[17] U. J. Ferner, H. Wymeersch, and M. Z. Win, \\\\\\\"Cooperative anchorless localization for large dynamic networks,\\\\\\\" in Proc. IEEE Int. Conf. Ultra-Wideband, 2008, pp. 181-185. \\\",\\\"[18] H. Wymeersch, J. Lien, and M. Z. Win, \\\\\\\"Cooperative localization in wireless networks,\\\\\\\" Proc. IEEE, vol. 97, no. 2, pp. 427-450, Feb. 2009. \\\",\\\"[19] F. Meyer, Z. Liu, and M. Z. Win, \\\\\\\"Network localization and navigation using measurements with uncertain origin,\\\\\\\" in Proc. 21st Int. Conf. Inf. Fusion, 2018, pp. 1-7. \\\"]\",\n \"page_idx\": \"13\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[88,940,117,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1004 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[256,939,888,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL.60, NO.1 FEBRUARY 2024 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[104,46,498,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"[20] G. C. Fernandes, S. S. Dias, M. R. Maximo, and M. G. S. Bruno, \\\\\\\"Cooperative localization for multiple soccer agents using factor graphs and sequential Monte Carlo,\\\\\\\" IEEE Access, vol. 8, pp. 213168-213184, 2020. \\\",\\\"[21] F. Meyer, P. Braca, P. Willett, and F. Hlawatsch, \\\\\\\"A scalable algorithm for tracking an unknown number of targets using multiple sensors,\\\\\\\" IEEE Trans. Signal Process., vol. 65, no. 13, pp. 3478-3493, Jul. 2017. \\\",\\\"[22] F. Meyer et al., \\\\\\\"Message passing algorithms for scalable multitarget tracking,\\\\\\\" Proc. IEEE, vol. 106, no. 2, pp. 221-259, Feb. 2018. \\\",\\\"[23] G. Soldi et al., \\\\\\\"Heterogeneous information fusion for multitarget tracking using the sum-product algorithm,\\\\\\\" in Proc. IEEE Int. Conf. Acoust. Speech Signal Process., 2019, pp. 5471-5475. \\\",\\\"[24] F. Meyer, O. Hlinka, H. Wymeersch, E. Riegler, and F. Hlawatsch, \\\\\\\"Distributed localization and tracking of mobile networks including noncooperative objects,\\\\\\\" IEEE Trans. Signal Inf. Process. Netw., vol. 2, no. 1, pp. 57-71, Mar. 2016. \\\",\\\"[25] J. Zhang, J. Cui, Z. Wang, Y. Ding, and Y. Xia, \\\\\\\"Distributed joint cooperative self-localization and target tracking algorithm for mobile networks,\\\\\\\" Sensors, vol. 19, no. 18, 2019, Art. no. 3829. \\\",\\\"[26] P. Sharma, A. A. Saucan, D. J. Bucci, and P. K. Varshney, \\\\\\\"Decentralized Gaussian filters for cooperative self-localization and multi-target tracking,\\\\\\\" IEEE Trans. Signal Process., vol. 67, no. 22, pp. 5896-5911, Nov. 2019. \\\",\\\"[27] G. Battistelli, L. Chisci, C. Fantacci, A. Farina, and B. Vo, \\\\\\\"Average Kullback-Leibler divergence for random finite sets,\\\\\\\" in Proc. 18th Int. Conf. Inf. Fusion, 2015, pp. 1359-1366. \\\",\\\"[28] G. Koliander, Y. El-Laham, P. Djuric, and F. Hlawatsch, “Fusion of probability density functions,” Proc. IEEE, vol. 110, no. 4, pp. 404–453, Apr. 2022. \\\",\\\"[29] M. Kayaalp, Y. Inan, E. Telatar, and A. Sayed, \\\\\\\"On the arithmetic and geometric fusion of beliefs for distributed inference,\\\\\\\" IEEE Trans. Autom. Contr., early access, Nov. 6, 2023, doi: 10.1109/TAC.2023.3330405. \\\",\\\"[30] S. J. Julier and J. K. Uhlmann, “A non-divergent estimation algorithm in the presence of unknown correlations,” in Proc. Amer. Control Conf., 1997, pp. 2369–2373. \\\",\\\"[31] T. Kropfreiter, F. Meyer, S. Coraluppi, C. Carthel, R. Mendrzik, and P. Willett, \\\\\\\"Track coalescence and repulsion: MHT, JPDA, and BP,\\\\\\\" in Proc. IEEE 24th Int. Conf. Inf. Fusion, Sun City, South Africa, 2021, pp. 1-8. \\\",\\\"[32] T. Kropfreiter, F. Meyer, D. Crouse, S. Coraluppi, F. Hlawatsch, and P. Willett, \\\\\\\"To coalesce or to repel? An analysis of MHT, JPDA, and belief propagation multitarget tracking methods,\\\\\\\" Aug. 2023, arXiv:2308.06326. \\\",\\\"[33] S. S. Dias and M. G. S. Bruno, \\\\\\\"Distributed bernoulli filters for joint detection and tracking in sensor networks,\\\\\\\" IEEE Trans. Signal Inf. Process. over Netw., vol. 2, no. 3, pp. 260-275, Sep. 2016. \\\",\\\"[34] S. Blackman and R. Popoli, Design and Analysis of Modern Tracking Systems. Boston, MA, USA: Artech House, 1999. \\\",\\\"[35] N. Patwari, J. N. Ash, S. Kyperountas, A. O. Hero, R. L. Moses, and N. S. Correal, \\\\\\\"Locating the nodes: Cooperative localization in wireless sensor networks,\\\\\\\" IEEE Signal Process. Mag., vol. 22, no. 4, pp. 54-69, Jul. 2005. \\\",\\\"[36] H. Wymeersch, U. J. Ferner, and M. Z. Win, \\\\\\\"Cooperative Bayesian self-tracking for wireless networks,\\\\\\\" IEEE Commun. Lett., vol. 12, no. 7, pp. 505-507, Jul. 2008. \\\",\\\"[37] S. Zhang, E. Staudinger, S. Sand, R. Raulefs, and A. Dammann, \\\\\\\"Anchor-free localization using round-trip delay measurements for martian swarm exploration,\\\\\\\" in Proc. IEEE/ION Position Location Navigation Symp., 2014, pp. 1130-1139. \\\",\\\"[38] O. Hlinka, O. Sluckak, F. Hlawatsch, P. M. Djuric, and M. Rupp, \\\\\\\"Likelihood consensus and its application to distributed particle filtering,\\\\\\\" IEEE Trans. Signal Process., vol. 60, no. 8, pp. 4334-4349, Aug. 2012. \\\",\\\"[39] P. Tichavsky, C. H. Muravchik, and A. Nehorai, \\\\\\\"Posterior Cramér-Rao bounds for discrete-time nonlinear filtering,\\\\\\\" IEEE Trans. Signal Process., vol. 46, no. 5, pp. 1386-1396, May 1998. \\\"]\",\n \"page_idx\": \"14\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[514,47,635,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0506/9fdfaa5c626cddee685ec3068655de35200f669ddefb53e0ac64b2f3960c5f7d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[648,46,908,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hallysson Oliveira (Member, IEEE) received the bachelor's degree in computer engineering and the master's degree in computer science from Universidade Federal do Espirito Santo, Vitoria, Brazil, in 2002 and 2005, respectively, and the master's degree in aeronautical engineering and the Ph.D. degree in electrical and computer engineering from Instituto Tecnologico de Aeronáutica (ITA), São José dos Campos, Brazil, in 2007 and 2022, respectively. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[663,160,908,171]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From 2006 to 2021, he worked for Embraer \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,172,908,239]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"S.A. as a Research Software Engineer working in R&T projects related to different technologies and areas of application, such as image processing, pattern recognition, machine learning, sensor fusion, and autonomous navigation. Since 2021 he has been with ADASI working with software simulation and integration and currently leading the embedded software team. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,292,633,410]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0188/efdc6c15403c6898a71a2606bbb0333d91ff63c3094e1d5a79b8360f7c605c0f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[648,292,908,395]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Stiven S. Dias (Member, IEEE) received the bachelor's degree in computer engineering from Universidade Federal do Espirito Santo, Vitoria, Brazil, in 2006, the master's degree in aeronautical engineering, and the Ph.D. degree in electrical and computer engineering from Instituto Tecnologico de Aeronáutica (ITA), São José dos Campos, Brazil, in 2008 and 2014, respectively. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[648,395,908,417]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"He has been with Eindhoven University of Technology (TUE), Eindhoven, Netherlands, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,417,908,532]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"since 2021 as a Machine Learning Lecturer, where he has supervised several graduation projects mainly focused on anomaly detection and prediction applications. From 2008 to 2021, he worked for the Embraer Group, São Paulo, Brazil, as System Engineer Specialist designing coastal surveillance applications and as a Research Engineer developing advanced data fusion systems for embedded command and control (C2) systems. In 2023, he has changed his role at TUe to a Research Fellow and has rejoined Embraer S.A. as a Data Fusion Research Engineer to lead R&T projects. His main research interests include distributed estimation in sensor networks, Bayesian inference, and machine learning. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,585,633,700]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0011/a38bc944a8b427de2cc37234724fd45dc59c1bb7656f33af182a1f229d68a5d7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[648,584,908,664]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Marcelo G. S. Bruno (Senior Member, IEEE) received the bachelor's and master's degrees in electrical engineering from the University of São Paulo, São Paulo, Brazil, in 1992 and 1993 respectively, and the Ph.D. degree in electrical and computer engineering from Carnegie Mellon University, Pittsburgh, PA, USA, in 1998. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[648,664,908,710]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"He is currently affiliated with the Electrical Engineering Division, Instituto Tecnológico de Aeronáutica (ITA), São José dos Campos, Brazil, where he is an Associate Professor. His \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,710,908,868]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"research interests include statistical signal processing, particularly Markov random fields (Mrfs), hidden Markov models (HMMs), particle filters/sequential Monte Carlo methods (SMC), Markov Chain Monte Carlo (MCMC), statistics on manifolds, random finite sets (RFS), Bernoulli and multi-Bernoulli filters, nonparametric belief propagation, and their applications in target detection/tracking, image processing, machine learning, mobile robotics, and telecommunications. His current research is focused on distributed detection, estimation and filtering in sensor networks combining Bayesian methods with iterative consensus, diffusion and random information dissemination strategies. Particular areas under investigation include distributed filtering and estimation in Matrix Lie groups; joint cooperative localization and noncooperative detection and tracking in sensor networks; and manifold-constrained message passing on factor graphs. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,868,908,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Dr. Bruno served as an Associate Editor of IEEE Signal Processing Letters, IEEE TRANSACTIONS ON SIGNAL PROCESSING, and IEEE TRANSACTIONS ON SIGNAL AND INFORMATION PROCESSING OVER NETWORKS. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,939,750,950]\",\n \"code_body\": null,\n 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","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,285,257,301]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Series Editors ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,312,774,457]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Claudio Canuto, Politecnico di Torino, Turin, Italy \nGiulianella Coletti, Università di Perugia, Perugia, Italy \nGraziano Gentili, Università di Firenze, Florence, Italy \nAndrea Malchiodi, Scuola Normale Superiore, Pisa, Italy \nPaolo Marcellini, Università di Firenze, Florence, Italy \nEmilia Mezzetti, Università di Trieste, Trieste, Italy \nGioconda Moscariello, Università di Napoli 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","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,234,501,250]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"THE SERIES IS INDEXED IN SCOPUS ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,288,813,304]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"More information about this series at http://www.springer.com/series/10283 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,820,169]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Elisabetta Colombo • Barbara Fantechi • Paola Frediani • Donatella Iacono $\\bullet$ Rita Pardini Editors ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,273,646,361]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Birational Geometry and Moduli Spaces ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,852,371,900]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Springer ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,350,166]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Editors \nElisabetta Colombo \nDepartment of Mathematics \nUniversity of Milan \nMilano, Italy \nBarbara Fantechi \nSISSA - International School \nfor Advanced Studies \nTrieste, Italy \nPaola Frediani \nDepartment of Mathematics \nUniversity of Pavia \nPavia, Italy Donatella Iacono \nDepartment of Mathematics Universita degli Studi di Bari Bari, Italy \nRita Pardini \nDepartment of Mathematics \nUniversity of Pisa \nPisa, Italy $^ ©$ Springer Nature Switzerland AG 2020 \nThis work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. 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","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,791,876,858]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,872,813,899]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,596,701,657]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ISSN 2281-518X ISSN 2281-5198 (electronic) Springer INdAM Series ISBN 978-3-030-37113-5 ISBN 978-3-030-37114-2 (eBook) https://doi.org/10.1007/978-3-030-37114-2 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,239,112]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Preface ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,342,876,376]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This volume collects contributions from speakers at the INdAM Workshop “Birational Geometry and Moduli Spaces”, which was held in Rome on 11–15 June 2018. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,379,881,430]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The workshop was devoted to the interplay between birational geometry and moduli spaces, two central topics in Algebraic Geometry that have always attracted the interest of national and international researchers. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,433,881,666]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A longstanding problem in geometry is the classification of geometric objects. The starting point is a geometric object, such as a curve, a variety or a sheaf, defined by some equations or given by some conditions. Then, one can deform it to get a new geometric object related to the old one. This expresses the concept of a family of an object and the main purpose is the classification of these families, in such a way that the classifying space is a reasonable geometric space. This space is the so-called moduli space and its geometric points parametrise the objects that we are considering. One is interested in classifying geometric objects up to isomorphism, so that the moduli space has a variety structure, or geometric objects with their automorphisms, thus meaning that the moduli space has a stack structure. In order to understand the local and global structure of moduli spaces, it is important to handle well techniques from deformation theory. Indeed, deformation theory can be regarded as a tool to understand the local geometry of moduli spaces. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,666,879,755]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There is also interest in classifying geometric objects up to birational morphism. Many results are known for the birational study of algebraic surfaces, while for higher dimension there are many open problems that can be tackled via the minimal model program. Great progress has been made in recent years, supporting the development of new techniques and innovative ideas. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,758,879,810]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The workshop focused on these research aspects and their interactions and offered the possibility to disseminate the knowledge of advanced results and key techniques used to solve open problems in the area. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,812,879,882]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This volume covers many topics in the wide research area of birational geometry and moduli spaces, and includes both surveys and original research papers. In particular, it contains works on irreducible holomorphic symplectic manifolds, Severi varieties, degeneration of Calabi–Yau varieties, uniruled threefolds, toric Fano threefolds, mirror symmetry, canonical bundle formula, the Lefschetz principle, birational transformation, and deformation of diagrams of algebras. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[865,923,876,932]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,876,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,126,881,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We are indebted to Francesco Bastianelli and Antonio Rapagnetta for their help in organising the workshop and to Emilia Mezzetti for her support as a member of the Scientific Committee. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,180,874,214]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On behalf of the Scientific and Organising Committees, we would like to thank the Foundation Compositio Mathematica for their support. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,216,874,250]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We are all very grateful to the Istituto Nazionale di Alta Matematica “Francesco Severi” (INdAM) for the generous support and help that made all this possible. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,252,881,304]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finally, we would like to offer special thanks to all the participants and speakers of the INdAM Workshop, and especially to the authors who accepted the invitation to publish here. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,324,241,414]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Milano, Italy Trieste, Italy Pavia, Italy Bari, Italy Pisa, Italy ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[694,324,881,414]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Elisabetta Colombo Barbara Fantechi Paola Frediani Donatella Iacono Rita Pardini ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[817,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Preface ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,54,136,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,261,112]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Contents ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,342,881,394]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Negative Rational Curves and Their Deformations on Hyperkähler Manifolds. . \nEkaterina Amerik ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,405,881,457]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moduli Spaces of Cubic Threefolds and of Irreducible Holomorphic Symplectic Manifolds . 13 Chiara Camere ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,468,881,522]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Note on Severi Varieties of Nodal Curves on Enriques Surfaces . 29 \nCiro Ciliberto, Thomas Dedieu, Concettina Galati, \nand Andreas Leopold Knutsen ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,531,876,566]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Travel Guide to the Canonical Bundle Formula 37 Enrica Floris and Vladimir Lazic´ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,576,881,630]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model . 57 Anne-Sophie Kaloghiros ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,639,879,675]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras . 77 Emma Lepri and Marco Manetti ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,684,881,738]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties . 109 César Lozano Huerta and Alex Massarenti ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,747,879,800]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? . 133 Luigi Lunardon ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,810,876,846]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Examples of Irreducible Symplectic Varieties 151 Arvid Perego ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[854,920,879,933]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"vii ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,876,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An Example of Mirror Symmetry for Fano Threefolds. 173 Andrea Petracci ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,135,876,169]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Chern Numbers of Uniruled Threefolds . 189 Stefan Schreieder and Luca Tasin ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,148,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"viii ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[806,54,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Contents ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,88,403,112]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"About the Editors ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,342,881,394]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Elisabetta Colombo is Associate Professor of Geometry at the University of Milan. Her research field is complex algebraic geometry, and she studies mainly curves and abelian varieties and their moduli. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,417,881,469]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Barbara Fantechi is Full Professor in Geometry at SISSA-ISAS in Trieste. Her research interests include deformation theory, derived algebraic geometry and stacks. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,490,881,545]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Paola Frediani is Associate Professor of Geometry at the University of Pavia. Her research area is algebraic geometry, in particular moduli spaces of curves and abelian varieties and Hodge theory. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,567,879,621]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Donatella Iacono is a Researcher in Geometry at the University of Bari. Her research focuses on deformation theory and differential graded Lie algebras in algebraic geometry. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,642,881,696]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Rita Pardini is Full Professor of Geometry at the University of Pisa. Her research area is algebraic geometry, especially surfaces and their moduli, irregular varieties and coverings. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[861,921,876,932]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,758,141]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Negative Rational Curves and Their Deformations on Hyperkähler Manifolds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,177,300,193]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ekaterina Amerik ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,291,881,343]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract We survey some results about rational curves on hyperkähler manifolds, explaining how to prove a certain deformation-invariance statement for loci covered by rational curves with negative Beauville–Bogomolov square. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,361,740,379]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords Hyperkähler manifold $\\cdot$ Rational curve $\\cdot$ MBM classes ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,417,879,451]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This text is an expanded version of a talk which I gave at the INDAM workshop in Rome in June 2018. It largely follows the paper [2] and our other joint works. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,496,437,518]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 Hyperkähler Manifolds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,543,881,668]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A compact Kähler manifold $X$ is irreducible holomorphic symplectic (IHSM), or hyperkähler (see next section for a brief explanation of the term), if it is simplyconnected and $H ^ { 2 , 0 } ( X ) = \\mathbb { C } \\sigma$ for some nowhere degenerate (“symplectic”) form $\\sigma$ . Such a manifold is even-dimensional (we set $2 n \\ = \\ d i m ( X ) )$ and has trivial canonical bundle. Such manifolds can be seen as higher-dimensional analogues of K3 surfaces. The most-studied IHSM are actually closely related to K3 surfaces, namely those are the deformations of their punctual Hilbert schemes; we call them ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,897,690,941]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_1 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[804,91,876,139]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,719,881,779]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The results presented at the INDAM workshop and in this survey have been obtained jointly with M. Verbitsky. Partially supported by Brazilian-French network in Mathematics, partially supported by the Russian Academic Excellence Project ‘5-100’. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[865,899,876,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,797,863,878]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Amerik $\\boxtimes ,$ Université Paris-Sud, Laboratoire de Mathématiques d’Orsay, Orsay, France National Research University Higher School of Economics, Laboratory of Algebraic Geometry and Its Applications, Moscow, Russia e-mail: ekaterina.amerik@math.u-psud.fr ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,879,142]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IHSM of K3 type. By Bogomolov’s decomposition theorem, together with the abelian and the Calabi–Yau manifolds the hyperkähler manifolds are building blocks for the varieties with zero first Chern class. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,144,881,234]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The main feature that the hyperkähler manifolds share with surfaces is the presence of a natural integral quadratic form $q$ on their second cohomology: the Beauville–Bogomolov form. It can be defined by integrating the differential forms, but we omit the explicit definition and only mention the following Fujiki’s formula, which stresses the topological nature of $q$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[416,247,578,270]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0799/45e10753cbdff0ab0291e311cd409131ffc0081d11e0c3d87c84ac2d3eb9978f.jpg","list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n( q ( \\alpha ) ) ^ { n } = C \\alpha ^ { 2 n } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,286,881,412]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $C$ is a positive constant which only depends on the deformation type of $X$ . It is well-known that this form has the same signature as the intersection form on a surface would have, that is, $( 3 , b _ { 2 } - 3 )$ on $H ^ { 2 } ( X , \\mathbb { R } )$ and $( 1 , b _ { 2 } - 3 )$ on $H ^ { 1 , 1 } ( X , \\mathbb { R } )$ . We refer to [5] or [6] for details (another classical reference [13] is suggested by the referee). The cone of classes of strictly positive square in $H ^ { 1 , 1 } ( X , \\mathbb { R } )$ has two connected components. We call the positive cone $\\mathsf { P o s } ( X )$ the one containing the Kähler classes. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,459,353,480]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 Rational Curves ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,503,881,630]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On a K3 surface S, for every integral $( 1 , 1 )$ -class $z$ of square $^ { - 2 }$ , either $z$ or $- z$ is effective by Riemann–Roch formula. If $z$ is represented by an irreducible curve, we call it a $^ { - 2 }$ -curve. By adjunction, such a curve is smooth and rational. These curves are important for the understanding of the geometry of $s$ : for instance, by the work of Pyatetski-Shapiro and Shafarevich [14] in the algebraic setting, or Looijenga and Peters [9] in the Kähler one, the ample cone of $s$ is bounded, within the positive cone, by the orthogonals to $^ { - 2 }$ -curves. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,675,683,696]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.1 Lower Bound for the Dimension of a Family ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,719,881,827]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A sufficiently general (in particular non-projective) hyperkähler manifold has no curves at all, indeed the deformation theory implies that it has no integral classes of type (1, 1). When it does, one would like to understand the deformations of such a rational curve $C \\subset X$ inside and outside of $X$ . A lower bound for the dimension of the deformation space of maps $f : \\mathbb { P } ^ { 1 } \\to X$ is provided by the Euler characteristics of the pullback of the tangent space, computed by the Riemann–Roch formula ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[300,845,697,864]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0334/61b0683a9d5fce8fea3fe5f9b50e204e30761956234db923de2d69d805028f8a.jpg","list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\chi ( f ^ { * } T _ { X } ) = - K _ { X } C + d i m ( X ) = d i m ( X )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[794,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Amerik ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,54,132,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,90,876,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"as soon as $K _ { X }$ is trivial, so that $C$ deforms in a family of dimension $2 n - 3$ (where 3 accounts for the automorphisms of $\\mathbb { P } ^ { 1 }$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,126,881,324]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Z. Ran noticed in [15] that on a hyperkähler $X$ , there is a better bound: a rational curve deforms in a family of dimension at least $2 n - 2$ . We have learned the following beautiful proof of this fact from Eyal Markman. Recall first that a hyperkähler manifold comes together with a natural deformation of complex structure, called the twistor deformation. It appears as follows: by Yau’s theorem, on an irreducible holomorphic symplectic manifold there is a hyperkähler metric, that is a Riemannian metric $g$ with three complex structures $I , J , K$ , which are Kähler with respect to $g$ (i.e. the corresponding forms are Kähler) and multiply as quaternions. For any $( a , b , c )$ with $a ^ { 2 } + b ^ { 2 } + c ^ { 2 } = 1 , a I + b J + c K$ is also a Kähler structure, so that $X$ is a member of a family $\\mathcal { X } _ { t }$ , $t \\in S ^ { 2 } , \\mathcal { X }$ is in fact a complex manifold and its projection $\\pi$ to $S ^ { 2 } = \\mathbb { C P } ^ { 1 }$ is holomorphic. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,324,881,466]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now Markman’s idea is as follows: let $C \\subset X$ be a rational curve, that is the image of some (say generically injective) map $f : \\mathbb { P } ^ { 1 } \\to X$ . By Riemann–Roch again, it must move within $\\mathcal { X }$ in a family of dimension at least $- K _ { \\mathcal { X } } C + d i m ( \\mathcal { X } ) -$ $3 = 2 n - 2$ . But it is well-known that the neighbouring fibers $X _ { t }$ of $X$ in $\\mathcal { X }$ do not contain any curves at all, whereas a deformation of a curve in a fiber of $\\pi$ obviously must be contained in a fiber of $\\pi$ (the intersection number of a curve and a fiber is zero when the curve is contained in a fiber and strictly positive when not). So in fact all deformations of $C$ in $\\mathcal { X }$ remain in $X$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,513,307,534]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2 Minimality ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,557,881,719]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If there is only a finite number of rational curves through a general point of the locus they cover, then this locus is of dimension $2 n - 1$ , that is, a divisor (remark that $X$ itself is not uniruled). However in general this does not have to be true. Already on manifolds of K3 type examples show that the uniruled loci can have any dimension between $n$ and $2 n - 1$ . As for the dimension of the family of deformations of a rational curve $C$ , it can be arbitrarily large if one does not impose some minimality assumption, for example, that $C$ is of minimal degree among the rational curves through the general point of the corresponding locus. Behind such a minimality assumption is of course the famous bend-and-break lemma. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,729,876,764]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 2.2.1 (Mori) In any one-parameter family of rational curves through two distinct points $x$ and $y$ , there is a reducible member. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,773,712,791]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"So the minimality of the degree is equivalent to the following. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,800,881,836]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition A rational curve $C$ is called minimal if there is only a finite number of deformations of $C$ through two general points of the locus they cover. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,846,879,899]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By bend-and-break lemma, any irreducible uniruled subvariety $Z$ of $X$ is generically covered by a family of minimal rational curves, which are deformations of some minimal curve $C \\subset Z$ . We say that $Z$ is a locus of $C$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,719,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Negative Rational Curves and Their Deformations on Hyperkähler Manifolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[865,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,879,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $C$ is a minimal rational curve, let $Z$ be a locus of $C$ and consider the rational quotient $r : Z \\to Q$ . The restriction of the form $\\sigma$ to $Z$ at its general point has kernel along the fiber of $r$ , since there are no holomorphic $k$ -forms on a rationally connected manifold for $k > 0$ . One derives the following ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,169,879,223]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 2.2.2 The codimension $k$ of $Z$ is equal to the relative dimension of $r$ , and any minimal rational curve deforms in a family of dimension exactly $2 n - 2$ in $X$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,234,881,304]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Indeed since $\\sigma$ is non-degenerate on $X$ , its kernel in restriction to $Z$ is at most of rank $k$ , on the other hand we must have at least a $k - 1$ -parameter family of rational curves through a general point of $Z$ , so the fibers of rational quotient are at least $k$ -dimensional. This also shows that $Z$ is coisotropic. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,315,874,349]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Applying Markman’s trick to the local universal family instead of the twistor deformation, one gets the following ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,358,876,396]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 2.2.3 A minimal rational curve in $X$ deforms to a neighbouring IHSM $X _ { t }$ if and only if its cohomology class remains of type $( 1 , 1 )$ on $X _ { t }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,405,881,548]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is interesting for several reasons to understand the loci of minimal rational curves and their behaviour under global deformations. One encounters two problems here. First of all, under a global deformation a minimal curve can cease to be minimal. Secondly, the family of deformations of $C$ may have several components, $C$ being minimal in some of them but not in the others. So the geometry of uniruled loci is still not well-understood even on much studied IHSM. For instance one conjectures but does not know that a projective deformation of the Hilbert square of a K3 contains a rational surface. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,549,881,620]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The point of the present talk is to indicate that all such problems have a satisfactory solution in the case when the Beauville–Bogomolov square of $C$ is negative. Here the Beauville–Bogomolov square of a curve is a rational number, obtained by considering $H _ { 2 } ( X , \\mathbb { Z } )$ as an overlattice of $H ^ { 2 } ( X , \\mathbb { Z } )$ by duality. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,621,876,656]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The loci covered by minimal rational curves of negative square are especially interesting as these are loci of birational contractions, at least in the projective case. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,702,396,723]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 Deformation Spaces ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,741,448,761]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.1 Teichmüller and Torelli ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,785,879,893]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Deformations of IHSM are unobstructed by the work of Todorov [17]. Thus the local deformation space $D e f ( X )$ of $X$ is identified with a neighbourhood of zero in $H ^ { 1 } ( X , T _ { X } )$ , which is in turn identified with $H ^ { 1 } ( X , \\Omega _ { X } ^ { 1 } )$ by means of the symplectic form. The locus of deformations where a class $\\lambda$ remains of type $( 1 , 1 )$ is the hyperplane orthogonal to $\\lambda$ with respect to the Beauville–Bogomolov form. This local picture globalizes as follows. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[794,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Amerik ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,54,132,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $M$ be the underlying differentiable manifold and ${ \\mathsf { C o m p } } ( M )$ the space of complex structures of Kähler type (by Kodaira–Spencer stability theorem, this is an open subset in the space of all complex structures). We consider the Teichmüller space ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[289,175,703,199]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0163/d4cba24dccf1a57edd455e4e326d6e8bb23cd4d58550536d28c57537ff3125b2.jpg","list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathsf { T e i c h } = \\mathsf { T e i c h } ( M ) = \\mathsf { C o m p } ( M ) / \\mathsf { D i f f } ^ { 0 } ( M ) ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,214,881,342]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\mathsf { D i f f } ^ { 0 } ( M )$ is the group of isotopies: the action of the whole diffeomorphism group $\\mathsf { D i f f } ( M )$ on ${ \\mathsf { C o m p } } ( M )$ is “too bad” for taking any reasonable quotient. Instead we consider only the action of the mapping class group $\\Gamma =$ ${ \\dot { \\mathsf { D i f f } } } ( M ) / { \\mathsf { D i f f } } ^ { 0 } ( M )$ on Teich $( M )$ . Huybrechts in [7] proved that Teich $( M )$ has only finitely many connected components when $M$ carries a hyperkähler structure. By abuse of notation, we shall use the notation Teich $( M )$ for the component containing our chosen complex structure. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,342,881,396]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The space Teich $( M )$ is only generically Hausdorff. Verbitsky in [18] proved the Torelli theorem for hyperkähler manifolds, namely that the Hausdorff quotient Teich $_ { b } ( M )$ is isomorphic to the period domain. The latter is defined as follows: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[255,409,742,433]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0336/5d47af8b1a207f2c16557848d22273016fb4a11195f921055a3ab13c3d3f7996.jpg","list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathsf { P e r } ( M ) = \\{ l \\in \\mathbb { P } H ^ { 2 } ( M , \\mathbb { C } ) | q ( l , l ) = 0 , q ( l , \\bar { l } ) > 0 \\}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,448,879,504]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and the period map per from $\\mathsf { T e i c h } ( M )$ to ${ \\mathsf { P e r } } ( M )$ , realizing the up-tononhausdorffness isomorphism, is defined by sending the parameter point for a complex manifold $X$ to the complex line generated by the class of $\\sigma$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,504,879,576]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is often convenient (e.g. in Verbitsky’s proof) to see ${ \\mathsf { P e r } } ( M )$ as the grassmannian of positive (with respect to q) 2-planes in the real cohomology $G r _ { + + } ( 2 , H ^ { 2 } ( M , \\mathbb { R } ) )$ (the transition from the open subset of the complex quadric as above is simply by taking the real and the imaginary part). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,576,881,666]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Define the subspace $\\mathsf { T e i c h } _ { z } = \\mathsf { T e i c h } _ { z } ( M ) \\subset \\mathsf { T e i c h } ( M )$ as the locus where an integral class $z$ remains of type $( 1 , 1 )$ . It is easy to see that this is the inverse image by per of the hyperplane $z ^ { \\bot }$ (the $q$ -orthogonal to $z$ ). It is clear from the description in the following section that ${ \\mathsf { T e i c h } } _ { z }$ is not Hausdorff even generically and splits naturally in two “halves”. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,711,594,732]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.2 Kähler Chambers and MBM Classes ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,756,876,792]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The following theorem summarizes the main results of [1] (similar results to some of those have been independently obtained by Mongardi in [11]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,800,879,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3.2.1 The Kähler cone $K ( X )$ is a connected component of the complement to a union of hyperplanes $z ^ { \\bot }$ , $z \\in R \\cap H ^ { 1 , 1 } ( X )$ , in $\\mathsf { P o s } ( X )$ , where $R$ is the set of negative integral classes in $H ^ { 2 } ( X , \\mathbb { Z } )$ , called MBM classes, which can be characterized by the following equivalent conditions: z is MBM iff ${ \\mathsf { T e i c h } } _ { z }$ containes no twistor curve iff a rational multiple of z is represented by a rational curve on a “generic $^ { \\circ 1 }$ deformation of $X$ within ${ \\mathsf { T e i c h } } _ { z }$ (that is, such deformation that only the multiples of z are integral classes of type (1, 1)). The orthogonal hyperplane to an MBM class of type (1, 1) supports the wall of the Kähler cone of $X$ , or of $a$ birational model $X ^ { \\prime }$ , or the monodromy image of such a wall. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,719,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Negative Rational Curves and Their Deformations on Hyperkähler Manifolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[865,54,879,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"14","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,91,879,163]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,172,879,226]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The orthogonals to MBM classes of type (1, 1) thus divide the positive cone into chambers, called Kähler-Weil chambers by Markman. Each chamber is the Kähler cone of a birational model $X ^ { \\prime }$ or its monodromy translate. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,235,879,289]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3.2.2 (Markman) The points in $p e r ^ { - 1 } ( p e r ( X ) )$ , that is, the inseparable points of Teich one of which corresponds to $X$ , are in one-to-one correspondence with the Kähler-Weil chambers of $\\mathsf { P o s } ( X )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,298,881,370]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In particular, ${ \\mathsf { T e i c h } } _ { z }$ is non-separated even at its general point: at such a point $z$ is the only MBM class of type $( 1 , 1 )$ and $z ^ { \\bot }$ divides the positive cone in two chambers. Choosing an orientation, we naturally separate ${ \\mathsf { T e i c h } } _ { z }$ in two halves $\\mathtt { T e i c h } _ { z } ^ { \\pm }$ (the complex structures which have the Kähler cone “to the left” and “to the right” of $z ^ { \\bot }$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,372,881,570]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The next step is to take the part of, say, $\\mathtt { T e i c h } _ { z } ^ { + }$ corresponding to chambers adjacent to $z ^ { \\bot }$ (that is, such that $z ^ { \\bot }$ contains a wall, rather than a lower-dimensional face, of the chamber in question). We denote this part by $\\mathsf { T e i c h } _ { z } ^ { m i n }$ . Its separable quotient coincides with $\\mathtt { T e i c h } _ { z } ^ { + }$ itself, but fibers over some points of Per are slightly smaller than those in $\\mathtt { T e i c h } _ { z } ^ { + }$ : indeed we only keep the chambers adjacent to $z ^ { \\bot }$ , so that on the varieties parameterized by $\\mathsf { T e i c h } _ { z } ^ { m i n }$ , a multiple of the class $z$ is represented by a minimal rational curve (as $z ^ { \\bot }$ gives a part of the boundary of the Kähler cone with non-empty interior), whereas on the varieties in the complement $\\mathsf { T e i c h } _ { z } ^ { + } - \\mathsf { T e i c h } _ { z } ^ { m i n }$ this shall be true only for some monodromy transform of $z$ and only on some birational model, whereas $z$ itself may be e.g. the class of a reducible rational curve. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,572,876,608]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We are going to be concerned with deformations of rational curves in the class $z$ over Teichmin . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,653,305,672]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 Main Result ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,696,881,824]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $X$ is in $\\mathsf { T e i c h } _ { z } ^ { m i n }$ we consider the locus $Z$ covered by curves whose class is proportional to $z$ . When $X$ is projective, it follows from Kawamata’s base point free theorem that $Z$ is the contraction locus of a birational morphism. Indeed take a sufficiently general integral class in that part of $z ^ { \\bot }$ which is a wall of the Kähler cone of $X$ . By Lefschetz theorem it is the class of a line bundle $L$ which is nef since it is on the boundary of the Kähler cone, big since by definition its square is strictly positive, and therefore eventually base-point-free by Kawamata’a theorem. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,869,876,900]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1On an arbitrary deformation, an MBM class would be, up to a multiple, represented by a possibly reducible rational curve, but the converse is in general false. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[794,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Amerik ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,54,132,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"15","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,879,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The morphism given by the sections of the line bundle $L ^ { \\otimes m }$ for $m$ large enough contracts exactly the curves in the classes proportional to z. In particular $Z$ has only finitely many irreducible components, and also $Z$ is uniruled since all contraction loci are uniruled (see e.g. [8]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,163,881,252]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The finiteness (and, post factum, the uniruledness) carries over to the non-projective case, since the loci $Z _ { t } ~ \\subset ~ X _ { t }$ of curves in a class proportional to z on $X _ { t }$ are deformations of $Z \\subset X$ . We note that a recent work of Bakker and Lehn [4] seems to imply a much less obvious fact that $Z$ is contractible even if $X$ is non-projective. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,253,585,270]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our main results can be summarized as follows. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,280,881,424]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 4.1 Let $\\mathcal { X }$ be the universal family of IHSM with second Betti number greater than 5 over $\\mathsf { T e i c h } _ { z } ^ { m i n }$ and $\\mathcal { Z }$ the subvariety covered by curves of class proportional to z. All fibers $Z _ { t }$ of $\\mathcal { Z }$ are homeomorphic and stratified diffeomorphic, with the possible exception of t corresponding to the complex structures with the maximal Picard number. Moreover the homeomorphisms (and stratified diffeomorphisms) can be chosen in such a way that they respect rational curves, so that they induce maps between Zariski-open subsets of fibers of rational quotients on $Z _ { t }$ and $Z _ { t } ^ { \\prime } , t , t ^ { \\prime } \\in \\mathsf { T e i c h } _ { z } ^ { m i n }$ . These maps are bimeromorphisms. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,433,881,560]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 4.2 The complex manifolds $X _ { t }$ with maximal Picard number are isolated points of $\\mathsf { T e i c h } _ { z } ^ { m i n }$ . Indeed the image of such a complex structure by the period map is in the intersection of $h ^ { \\perp }$ for all $h \\in P i c ( X _ { t } )$ . The results of Bakker and Lehn seem to imply a strengthening of ours, eliminating the nonmaximality assumption and eventually giving a real analytic isomorphism of the loci $Z _ { t }$ inducing a biholomorphism of fibers of the rational quotient. This is a work in progress which we hope to describe in a forthcoming paper. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,605,284,626]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 Ergodicity ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,648,879,720]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our main tool is Verbitsky’s ergodicity theorem for the action of the mapping class group on the Teichmüller space ([19, 20]). An action on a space with a measure is ergodic if any invariant measurable subset or its complement has measure zero. Most of the orbits of such an action on a manifold are dense. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,729,876,767]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 5.1 (Verbitsky) Let $X$ be an IHSM with $b _ { 2 } ( X ) ~ \\geq ~ 5$ . There are three types of orbits of the action of the mapping class group $\\Gamma$ on Teich: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,774,876,810]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(1) closed orbits of complex structures with rational period plane, i.e. such that Picard number is maximal; ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,812,881,846]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(2) dense orbits of complex structures such that the period plane contains no rational vector; ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,719,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Negative Rational Curves and Their Deformations on Hyperkähler Manifolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[865,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"16","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,879,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"17","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(3) “intermediate orbits”: orbits of complex structures whose period planes contain a single rational vector v. The orbit closure then consists of all period planes containing v and is totally real, that is, neither contained in a complex subvariety nor contains a positive-dimensional one, even locally. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,171,881,349]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"17","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sketch of proof It is done in two steps. The mapping class group acts on the period domain and one first proves these statements for period planes (the orbit of a period plane is dense whenever the plane contains no rational vector, etc). This is obtained by Ratner theory which describes actions of groups generated by unipotents on homogeneous spaces $\\Gamma \\backslash G$ , where $\\Gamma$ is a lattice in a simple Lie group $G$ . Indeed another result by Verbitsky, a corollary of a general theorem by Sullivan, affirms that the image of the mapping class group $\\Gamma$ in $G = S O ^ { + } ( 3 , b _ { 2 } - 3 )$ (the automorphism group of the second cohomology of $X$ ) is a lattice. Let $G = S O ^ { + } ( 3 , b _ { 2 } - 3 )$ and $H = S O ^ { + } ( 1 , b _ { 2 } - 3 )$ , so that $G / H$ is fibered in $S O ( 2 )$ over Per and $H$ is generated by unipotents. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,351,881,512]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"17","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ratner theory (see [16], but also [12] for a very accessible exposition) states that $\\overline { { x H } }$ is again an orbit under a closed intermediate subgroup $s$ , also generated by unipotents and in which $\\Gamma \\cap S$ is a lattice (this particular result is known as Ratner’s orbit closure theorem and applies to actions of subgroups generated by unipotents on a quotient of a Lie group by a lattice). From the study of Lie group structure on $G$ we derive that the subgroup must be either $H$ itself (the orbit is closed), or the whole of $G$ (the orbit is dense), or the stabilizer of an extra vector $\\cong S O ^ { + } ( 2 , b _ { 2 } - 3 )$ (the third case). One concludes passing in an obvious way (via the double quotient) from an $H$ -action on $\\Gamma \\backslash G$ to a $\\Gamma$ -action on $G / H$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,515,881,890]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"17","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The other step is to prove that the period map commutes with taking orbit closures, in the following way. A point of Teich can be seen as a pair of a complex structure and a Kähler chamber. Introduce the space Teich $K$ which consists of pairs $( I , \\omega )$ where $I \\in$ Teich and $\\omega$ is a Kähler form of square 1. The period map is injective on $\\mathsf { T e i c h } _ { K }$ . In other words, $\\mathsf { T e i c h } _ { K }$ is naturally embedded in $\\mathsf { P e r } _ { K }$ , the homogeneous manifold of all pairs consisting of a period point $I \\in$ Per and an element $\\omega$ of square one in its positive cone (which indeed depends only on the period point, not on the complex structure itself). The latter is a homogeneous space, so we can try to apply Ratner theory to prove the following result, which clearly implies what we need: for any $I$ , the closure of the $\\Gamma$ -orbit of $( I , K ( I ) ) \\subset$ Teich $\\kappa \\subset \\mathsf { P e r } _ { K }$ contains the orbit of (PerI, Pos(I )) (here by an orbit of the subset we mean the union of its translates and $K ( I )$ denotes the Kähler cone of the complex structure $I$ ). Now one can construct orbits of one-parameter subgroups which are entirely contained in $( I , K ( I ) )$ and such that the closure of their projection to $\\mathsf { P e r } _ { K } / \\Gamma$ contains the projection of the positive cone. Indeed, one deduces from the non-maximality of the Picard number that $K ( I )$ has a “round part”. This is used to find many horocycles in $K ( I )$ tangent to the round part of the boundary. The horocycle is an orbit of a one-parameter unipotent subgroup. Applying Ratner theory to a sufficiently general horocycle of this type, one sees that the closure of its image in $\\mathsf { P e r } _ { K } / \\Gamma$ contains an entire $S O ( H ^ { 1 , 1 } ( I ) )$ -orbit ([20], Proposition 3.5), which is the positive cone $\\mathsf { P o s } ( I )$ . □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[794,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"17","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Amerik ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,54,132,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"17","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,91,881,219]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For our purposes, we are concerned with the space $\\mathsf { T e i c h } _ { z } ^ { m i n }$ which maps onto $\\mathsf { P e r } _ { z }$ , that is, the hyperplane $z ^ { \\bot }$ in Per. On these spaces there is an action of $\\Gamma _ { z }$ , the subgroup of the mapping class group whose action on the second cohomology fixes z. The first step of the argument, concerning the orbits of period planes, carries over verbatim, we only have to suppose $b _ { 2 } > 5$ rather than $b _ { 2 } \\geq 5$ as we are now working in a hyperplane. The nature of the considerations in the second step is such that they naturally carry over to $\\mathsf { T e i c h } _ { z } ^ { m i n }$ rather than ${ \\mathsf { T e i c h } } _ { z }$ or $\\mathtt { T e i c h } _ { z } ^ { + }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,226,879,301]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 5.2 Assume $b _ { 2 } ( M ) > 5$ and let $z \\in H ^ { 2 } ( M , \\mathbb { Z } )$ be an MBM class and $\\Gamma _ { z }$ the subgroup of the mapping class group consisting of all elements whose action on the second cohomology fixes z. Then $\\Gamma _ { z }$ acts on $\\mathsf { T e i c h } _ { z } ^ { \\mathrm { m i n } }$ ergodically, and there are the same three types of orbits of this action as above. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,310,876,346]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof It proceeds along the same lines. We introduce the spaces $\\mathsf { P e r } _ { K , z }$ consisting of pairs ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[298,360,699,384]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0694/7f84924342304367e0b2d8463dd48b7cca1f6ea309791e13464cf04938c51e51.jpg","list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\{ ( \\mathsf { P e r } ( I ) , \\omega \\in \\mathsf { P o s } ( I ) \\cap z ^ { \\perp } ) , q ( \\omega , \\omega ) = 1 \\}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,399,881,653]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and $\\mathsf { T e i c h } _ { K , z }$ consisting of pairs $( I \\in \\mathsf { T e i c h } _ { z } ^ { \\operatorname* { m i n } } , \\omega ) \\in \\mathsf { P e r } _ { K , z }$ where $\\omega$ belongs to the wall of the Kähler cone given by $z ^ { \\bot }$ . We denote such a wall by $K ( I ) _ { z }$ , though of course its elements are not Kähler forms in the complex structure $I$ , but rather semipositive limits of those. Since the complex structures in $\\mathsf { T e i c h } _ { z } ^ { \\mathrm { m i n } }$ which have the same period point are in one-to-one correspondence with the walls of the Kähler chambers in which the other MBM classes partition $z ^ { \\bot }$ , $\\mathsf { T e i c h } _ { K , z }$ again embeds naturally in $\\mathsf { P e r } _ { K , z }$ . We fix a complex structure $I$ with non-maximal Picard number. We need to prove that the closure of the $\\Gamma _ { z }$ -orbit of the subset $( I , K ( I ) _ { z } )$ contains the orbit of $( \\mathsf { P e r } ( I ) , \\mathsf { P o s } ( I ) \\cap z ^ { \\perp } )$ . This is done exactly in the same way as in Verbitsky’s theorem. We take a general three-dimensional subspace $W$ in $z ^ { \\bot }$ , the intersection of $W$ with our wall $K ( I ) _ { z }$ contains horocycles, and we deduce from Ratner orbit closure theorem and Proposition 3.5 of [20] that the closure of the projection of such a horocycle to $\\mathsf { P e r } _ { K , z } / \\Gamma _ { z }$ is large, containing an $S O ( H ^ { 1 , 1 } ( I ) \\cap$ $z ^ { \\bot }$ )-orbit, which is the projection of ${ \\mathsf { P o s } } ( I ) \\cap z ^ { \\perp }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,698,451,719]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 Proof of the Main Result ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,741,881,851]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We are concerned with the family of smooth manifolds $\\mathcal { X }$ over $\\mathsf { T e i c h } _ { z } ^ { \\mathrm { m i n } }$ and its analytic subset $\\mathcal { Z }$ , and we wish to show that $\\mathcal { Z }$ has no degenerations except possibly over points of maximal Picard number. It is well-known that an analytic subset of a complex manifold admits a “nice” stratification (Whitney stratification or Thom-Mather stratification; see [10]). Recall also the following first isotopy lemma by Thom (we refer to [10] for precise definitions and proofs). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,719,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Negative Rational Curves and Their Deformations on Hyperkähler Manifolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[865,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"18","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,879,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma ([10], Proposition 11.1) Let $f : Y B$ be a smooth mapping of smooth manifolds and W a closed subset of $Y$ admitting Whitney stratification, such that $f : W \\to B$ is proper. If the restriction of f to each stratum of $W$ is a submersion then $W$ is locally trivial over $B$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,171,881,334]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In our setting we deduce that the family $\\mathcal { Z }$ is locally trivial on the complement $B \\subset \\mathsf { T e i c h } _ { z } ^ { m i n }$ to a union (possibly countable, but finite in a neighbourhood of any point in the base) $P \\subset \\mathsf { T e i c h } _ { z } ^ { m i n }$ of proper analytic subsets. If $x \\in \\mathsf { T e i c h } _ { z } ^ { m i n }$ is a point with dense $\\Gamma$ -orbit, then its orbit hits $B$ and therefore $Z _ { x }$ is naturally stratified diffeomorphic to $Z _ { b }$ , $b \\in B$ . If the orbit of $x \\in \\mathsf { T e i c h } _ { z } ^ { m i n }$ does not hit $B$ , itself and its closure must be contained in $P$ . But we know about the orbit closures that even locally they are not contained in a proper complex subvariety, except possibly when $x$ corresponds to a manifold $X$ with maximal Picard number. Thus the degenerations in the family $\\mathcal { Z }$ can only happen over such points. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,334,879,460]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The same is true for other families over $\\mathsf { T e i c h } _ { z } ^ { m i n }$ coming from the geometry of IHSM over it. We obtain stratified diffeomorphisms preserving rational curves by applying the $\\Gamma$ -orbit argument to Barlet spaces parameterizing rational curves with cohomology class proportinal to $z$ , and then to the incidence varieties. Note that these spaces are not naturally embedded into smooth manifolds, so it is not quite the first isotopy lemma which applies, but there are versions of it suitable for our situation (e.g. [21], corollaire 5.1). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,462,881,515]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finally, at a general point of a fiber of the rational quotient, the holomorphic tangent space is generated by the tangents to rational curves. A diffeomorphism preserving the holomorphic tangent space is a biholomorphism. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,524,879,614]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note Added in Proof During the INDAM workshop, A. Rapagnetta attracted our attention to a recent work by Bakker and Lehn [4], which in the meantime allowed us to considerably strengthen the above result. The main point is that it indeed follows from Bakker and Lehn’s work that minimal rational curves can be contracted even in the non-projective case. An interested reader may consult our recent preprint [3]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,660,248,680]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[127,702,881,882]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Amerik, E., Verbitsky, M.: Rational curves on hyperkähler manifolds. Int. Math. Res. Not. 2015(23), 13009–13045 (2015) \n2. Amerik, E., Verbitsky, M.: MBM loci in families of hyperkahler manifolds and centers of birational contractions (2018). arXiv:1804.00463 \n3. Amerik, E., Verbitsky, M.: Contraction centers in families of hyperkähler manifolds (2019). arXiv:1903.04884 \n4. Bakker, B., Lehn, C.: A global Torelli theorem for singular symplectic varieties (2016). arXiv:1612.07894 \n5. Beauville, A.: Variétés dont la première classe de Chern est nulle. J. Differ. Geom. 18, 755–782 (1983) \n6. Huybrechts, D.: Compact hyperkähler manifolds: basic results. Invent. Math. 135, 63–113 (1999) \n7. Huybrechts, D.: Finiteness results for compact hyperkähler manifolds. J. Reine Angew. Math. 558, 15–22 (2003) \n8. Kawamata, Y.: On the length of an extremal rational curve. Invent. Math. 105(3), 609–611 (1991) \n9. Looijenga, E., Peters, C.: Torelli theorems for Kähler K3 surfaces. Compos. Math. 42(2), 145– 186 (1980/1981) \n10. Mather, J.: Notes on Topological Stability. Harvard University, Cambridge (1970) \n11. Mongardi, G.: A note on the Kähler and Mori cones of hyperkähler manifolds. Asian J. Math. 19(4), 583–591 (2015) \n12. Morris, D.W.: Ratner’s Theorems on Unipotent Flows. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (2005) \n13. O’Grady, K.: Higher-dimensional analogues of K3 surfaces. In: Current Developments in Algebraic Geometry. Mathematical Sciences Research Institute Publications, vol. 59, pp. 257– 293. Cambridge University Press, Cambridge (2012) \n14. Pjateckii-Shapiro, I.I., Shafarevich, I.R.: Torelli’s theorem for algebraic surfaces of type K3. Izv. Akad. Nauk SSSR Ser. Mat. 35, 530–572 (1971) \n15. Ran, Z.: Hodge theory and deformations of maps. Compos. Math. 97(3), 309–328 (1995) \n16. Ratner, M.: Raghunathan’s topological conjecture and distributions of unipotent flows. Duke Math. J. 63(1), 235–280 (1991) \n17. Todorov, A.: The Weil-Petersson geometry of the moduli space of $S U ( n \\geq 3 )$ ) (Calabi–Yau) manifolds. I. Commun. Math. Phys. 126(2), 325–346 (1989) \n18. Verbitsky, M.: Mapping class group and a global Torelli theorem for hyperkähler manifolds. Duke Math. J. 162(15), 2929–2986 (2013) \n19. Verbitsky, M.: Ergodic complex structures on hyperkähler manifolds. Acta Math. 215(1), 161– 182 (2015) \n20. Verbitsky, M.: Ergodic complex structures on hyperkahler manifolds: an erratum (2017). arXiv:1708.05802 \n21. Verdier, J.-L.: Stratifications de Whitney et théorème de Bertini-Sard. Invent. Math. 36, 295– 312 (1976) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,54,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[794,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"19","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Amerik ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,881,527]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"20","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,719,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"20","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Negative Rational Curves and Their Deformations on Hyperkähler Manifolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"20","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"11 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,667,168]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moduli Spaces of Cubic Threefolds and of Irreducible Holomorphic Symplectic Manifolds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,202,273,220]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Chiara Camere ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,286,881,412]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract In this survey, based on joint work of the author and S. Boissière and A. Sarti, we will describe an isomorphism between the moduli space of smooth cubic threefolds, as described by Allcock, Carlson and Toledo, and the moduli space of fourfolds of $K 3 ^ { [ 2 ] }$ -type with a special non-symplectic automorphism of order three; then, I will show some consequences of this isomorphism concerning degenerations of non-symplectic automorphisms. Finally we will explore possible generalizations of the problem to higher dimensions and other moduli spaces of cubic threefolds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,432,861,521]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords Irreducible holomorphic symplectic manifold $\\cdot$ Hyperkähler manifold $\\cdot$ Non-symplectic automorphism $\\cdot$ Ball quotient $\\cdot$ Cubic threefold $\\cdot$ Moduli space of cubic threefold $\\cdot$ Occult period map $\\cdot$ Pfaffian cubic threefold $\\cdot$ Degeneration of irreducible holomorphic symplectic manifold $\\cdot$ Degeneration of automorphism $\\cdot$ Fano variety of a cubic fourfold ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,558,309,579]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,602,879,656]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The aim of this survey is to illustrate the results of a joint work by the author and S. Boissière and A. Sarti [17]; as a consequence, almost all the material here is neither original nor new, with the exception of a part of Sect. 4. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,657,879,764]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Irreducible holomorphic symplectic (denoted $I H S$ ) manifolds are smooth compact Kähler manifolds $X$ which are simply connected and which carry a holomorphic symplectic two-form $\\omega _ { X }$ such that $H ^ { 2 , 0 } ( X ) = \\mathbb { C } \\omega _ { X }$ . These manifolds, which also carry a hyperkähler structure, are one of the building blocks of Beauville– Bogomolov decomposition theorem [8], and in particular they have $c _ { 1 } ( X ) _ { \\mathbb { R } } = 0$ . One of the main properties of these manifolds is the fact that the group $H ^ { 2 } ( X , \\mathbb { Z } )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,897,690,941]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_2 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,812,845,879]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Camere $( \\boxtimes )$ Max Planck Institute of Mathematics, Bonn, Germany Present Address: Dipartimento di Matematica, Università degli Studi di Milano, Milano, Italy e-mail: chiara.camere@mpim-bonn.mpg.de; camere@dima.unige.it; chiara.camere@unimi.it ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[804,91,876,139]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,899,876,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"21","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,881,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"can be endowed with the structure of an integral non-degenerate indefinite lattice of signature $( 3 , b _ { 2 } ( X ) - 3 )$ by means of the so-called Beauville–Bogomolov quadratic form $q _ { X }$ . The global Torelli theorem of Huybrechts [25], Markman [33], and Verbitsky [40] asserts that this lattice controls in many senses the geometry of the manifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,180,881,324]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"One deformation family of IHS manifolds is that of fourfolds of $K 3 ^ { [ 2 ] }$ -type: these are deformations of the Hilbert scheme of two points of a smooth $K 3$ surface. In particular, all these fourfolds $X$ have $b _ { 2 } ( X ) = 2 3$ , the maximal one for an IHS in dimension four, and the Beauville–Bogomolov lattice is isometric to $L : = L _ { K 3 } \\oplus$ $\\langle - 2 \\rangle$ , where $L _ { K 3 }$ is the $K 3$ lattice ${ \\cal U } ^ { \\oplus 3 } \\oplus { \\cal E } _ { 8 } ( - 1 ) ^ { \\oplus 2 }$ . Among the projective elements of this deformation family, Beauville and Donagi show in [11] that the Fano variety $F ( Y )$ of lines contained in a smooth cubic fourfold $Y \\subset \\mathbb { P } ^ { 5 }$ is a fourfold of $K 3 ^ { [ 2 ] }$ - type which always carries a polarization of degree six. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,324,881,503]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An automorphism $\\sigma \\in \\operatorname { A u t } ( X )$ of an IHS manifold $X$ induces an isometry $\\sigma ^ { * } \\in$ $O ( H ^ { 2 } ( X , \\mathbb { Z } ) )$ , more precisely one induced by a monodromy operator on it (see [33, Theorem 1.3]), and an isomorphism of Hodge structures on $H ^ { 2 } ( X , \\mathbb { C } )$ . As a consequence, the line $H ^ { 2 , 0 } ( X )$ is invariant under $\\sigma ^ { * }$ : if $\\sigma ^ { * } \\omega _ { X } = \\omega _ { X }$ , we say that $\\sigma$ is symplectic; else, if $\\sigma ^ { * } \\omega _ { X } = \\xi \\omega _ { X }$ with $\\xi$ a non-trivial root of unity, we say that $\\sigma$ is non-symplectic. From work of Beauville [7, Proposition 6] we know that the existence of such a non-symplectic automorphism forces the manifold $X$ to be projective (there exists an invariant ample class); moreover, the order $n$ of $\\sigma$ , if it is finite, is bounded by the inequality $\\varphi ( n ) \\leq b _ { 2 } ( X ) - \\rho ( X )$ , where $\\varphi$ denotes Euler’s totient function and $\\rho ( X )$ is the Picard number of $X$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,506,881,737]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Starting with the paper [14], we have been interested in classifying nonsymplectic automorphisms of prime order acting on fourfolds of $K 3 ^ { [ 2 ] }$ -type. Because of the properties just mentioned above, when $X$ is a projective fourfold of $K 3 ^ { [ 2 ] }$ -type and $\\sigma ~ \\in ~ \\operatorname { A u t } ( X )$ is non-symplectic of prime order $p$ , we have $2 \\leq p \\leq 2 3$ . The key idea of [14] is to classify the isometries of $L$ induced by such automorphisms, exactly as it is done in the case of $K 3$ surfaces by Nikulin [36] and by Artebani–Sarti–Taki [5]. As in the case of $K 3$ surfaces, in the case of fourfolds $X$ of $K 3 ^ { [ 2 ] }$ -type the map $\\operatorname { A u t } ( X ) \\to O ( H ^ { 2 } ( X , \\mathbb { Z } ) )$ is injective, so that $\\sigma ^ { * }$ is non-trivial if $\\sigma$ is. Building up on fundamental work contained in [13], we show that these actions are classified by a pair of primitive sublattices: the invariant lattice $T : = H ^ { 2 } ( X , \\mathbb { Z } ) ^ { \\sigma ^ { * } }$ and its orthogonal complement $s$ inside $L$ . This allows the partial classification contained in [14]: we could obtain all the cases with $p \\neq 5 , 2 3$ ; $p = 2 3$ was then solved in [15] and $p = 5$ in [39]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[143,738,756,756]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Nowadays, the state of the art of the classification is the following: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,765,881,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"• Most cases can be constructed starting from a non-symplectic automorphism $\\tau$ of the same order $p$ on a smooth $K 3$ surface $\\Sigma$ , either by considering the natural automorphism $\\sigma = \\tau ^ { [ 2 ] }$ on $\\Sigma ^ { [ 2 ] }$ (see for example [12] for further details), or by considering the (twisted) induced automorphism on a moduli space of stable (twisted) sheaves on $\\Sigma$ (see [35, 38] and [19] for further details). For $p = 2$ , there is the first (historically) remarkable exception to the above cases when $T = \\langle 2 \\rangle$ : this is Beauville’s famous non-natural non-symplectic involution defined on the Hilbert scheme of two points of a smooth quartic surface not containing a line [7]. This example is in fact a special instance of the nonsymplectic cover involution on O’Grady’s double EPW sextics (see [22, 37] and references therein). There are a few more cases which cannot come from $K 3$ surfaces and are degenerations of this one. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[792,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,54,141,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"22","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[143,90,879,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,180,881,267]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"• For $p = 3$ , two exceptions are given by $T = \\langle 6 \\rangle$ and $T = \\langle 6 \\rangle \\oplus E _ { 6 } ^ { * } ( 3 ) $ : in both cases the only examples that we can explicitly construct are obtained on Fano varieties of lines of cubic fourfolds endowed with additional symmetries. The first case is the main object of [17], whereas the second one has still to be further understood. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,270,881,378]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"• For $p = 2 3$ , the only possibility is $T = \\langle 4 6 \\rangle$ : in [15] we show that there exists only one fourfold of $K 3 ^ { [ 2 ] }$ -type carrying such a non-symplectic automorphism of order 23 inside the 20-dimensional family of $\\langle 4 6 \\rangle$ -polarized fourfolds of $K 3 ^ { [ 2 ] }$ - type once we fix the action on $L$ . Unfortunately, the proof is not constructive but uses in a crucial way the global Torelli theorem for IHS manifolds, and the explicit construction of such a fourfold remains an open question. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,387,881,530]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As already mentioned, the main object of [17], and of this survey as well, is the case $p = 3$ , $T = \\langle 6 \\rangle$ : we will see that in this case we obtain a relation between the moduli space of marked fourfolds of $K 3 ^ { [ 2 ] }$ -type with such a non-symplectic automorphism of order three and the classical GIT moduli space of smooth cubic threefolds. The existence of such a relation is not so surprising once one sees the first example of such a non-symplectic automorphism, perhaps more surprising is the fact that there is an isomorphism between these two moduli spaces; indeed, both are uniformized by a ten-dimensional complex ball quotient. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,531,881,620]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In Sect. 2 we will explain how to obtain the isomorphism and some of the immediate consequences, after illustrating how the two moduli spaces appearing are constructed, respectively in [16] and in [2]. More precisely, we will see that both moduli spaces are isomorphic to the complement of a hyperplane arrangement $\\mathcal { H }$ inside a ten-dimensional complex ball quotient. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,621,881,764]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Section 3 then deals with understanding what happens on hyperplanes inside $\\mathcal { H } \\mathrm { : \\Omega }$ this is completely understood in the case of cubic threefolds, since we obtain semistable points in the GIT quotient, corresponding either to nodal cubics or to the chordal cubic, as shown in [2]. What we investigate in Sect. 3 is what happens to the non-symplectic automorphism of order three and how it degenerates. The final outcome of this analysis will be that the two subarrangements are in fact birational to moduli spaces of $K 3 ^ { [ 2 ] }$ -fourfolds carrying a non-symplectic automorphism of order three of different type in the classification. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,765,879,854]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The picture that we obtained is thus the analogue in higher dimension of the isomorphism of the moduli space of cubic surfaces with the moduli space of $K 3$ surfaces with a special non-symplectic automorphism of order three, illustrated by Dolgachev, van Geemen and Kondo in [¯ 21] building on the construction of Allcock, Carlson and Toledo in [1] (see also [9] for a nice survey on this subject). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,855,876,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finally, in Sect. 4, we explore related problems. As we observed above, nonnatural automorphisms on Hilbert schemes of $K 3$ surfaces are extremely difficult to construct; in [17], we also proved that there exist Pfaffian cyclic cubic fourfolds $Y \\subset \\mathbb { P } ^ { 5 }$ such that $\\psi : F ( Y ) \\simeq \\Sigma ^ { [ 2 ] }$ for a certain $K 3$ surface $\\Sigma$ of degree 14 not containing a line, by means of Beauville–Donagi’s isomorphism [11]. On $\\Sigma ^ { [ 2 ] }$ we then obtain a non-natural automorphism by conjugation with $\\psi$ of the automorphism of order three, which we construct in Sect. 2, on the Fano variety of any cyclic cubic fourfold. Another interesting question is what happens to higher dimensional manifolds of $K 3 ^ { [ n ] }$ -type related to cyclic cubic fourfolds, for example Lehn–Lehn– Sorger–van Straten’s eightfolds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,571,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moduli Spaces of Cubic Threefolds and of IHS Manifolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"23","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,232]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"24","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,279,567,300]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"24","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 An Isomorphism of Moduli Spaces ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,324,881,394]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"24","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section we will review the main constructions of [2] and [16] of the moduli spaces, and then prove the isomorphism among the two moduli spaces. Before describing the two moduli spaces involved, we first of all recall the main example of a non-symplectic automorphism of order three which we will consider here. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,439,642,462]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"24","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.1 Fano Varieties of Cyclic Cubic Fourfolds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,483,879,666]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"24","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recall that, for any smooth cubic fourfold $Y \\subset \\mathbb { P } ^ { 5 }$ , Beauville and Donagi show in [11] that the Fano variety $F ( Y )$ of lines contained in $Y$ is of $K 3 ^ { [ 2 ] }$ -type; moreover, they prove that the Plücker polarization $H _ { F }$ has degree six, so that $\\operatorname* { P i c } ( F ( Y ) ) \\supset \\langle 6 \\rangle$ , and that the Abel–Jacobi map $A : H ^ { 4 } ( Y , \\mathbb { Z } ) \\ { \\stackrel { \\cdot } { \\to } } \\ H ^ { 2 } ( F ( Y ) , \\mathbb { Z } )$ maps $H _ { Y } ^ { 2 }$ , where $H _ { Y }$ is the ample class on $Y$ , to $H _ { F }$ and induces an anti-isometry on primitive cohomology (the orthogonal of the polarization), i.e. an isometry $A _ { 0 } : H _ { 0 } ^ { 4 } ( Y , \\mathbb { Z } ) ( - 1 ) \\to H _ { 0 } ^ { 2 } ( F ( Y ) , \\mathbb { Z } )$ . As a consequence, $H _ { 0 } ^ { 2 } ( F ( Y ) , \\mathbb { Z } ) \\cong S : =$ $U ^ { \\oplus 2 } \\oplus E _ { 8 } ( - 1 ) ^ { \\oplus 2 } \\oplus A _ { 2 } ( - 1 )$ as lattices, where $U$ denotes the standard hyperbolic plane and $E _ { 8 } ( - 1 )$ , $A _ { 2 } ( - 1 )$ are the negative definite lattices associated to the Dynkin diagrams $E _ { 8 }$ and $A _ { 2 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,666,881,900]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"24","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $Y \\subset \\mathbb { P } ^ { 5 }$ be the triple cover of $\\mathbb { P } ^ { 4 }$ branched along a smooth cubic threefold $C \\subset \\mathbb { P } ^ { 4 }$ ; such a fourfold is what is usually called a cyclic cubic fourfold. If $G ( X _ { 0 } :$ $\\dots : X _ { 4 } ) = 0$ is an equation defining $C$ , then $Y$ is the zero locus defined by $X _ { 5 } ^ { 3 } +$ $G = 0$ , and it is smooth if and only if $C$ is smooth. We choose $\\sigma \\in \\operatorname { A u t } ( Y )$ to be the cover automorphism obtained as restriction to $Y$ of the map $X _ { 5 } \\mapsto \\xi X _ { 5 }$ , for $\\xi$ a fixed (for the rest of the paper) primitive third root of unity. Let $\\sigma$ be the induced automorphism on $F ( Y )$ : in [14, Example 4.6] we show that it is a non-symplectic automorphism of order three, that the fixed locus is the Fano surface $F ( C )$ of lines contained in the cubic threefold $C$ and that the invariant lattice of $\\sigma$ is $T : = \\langle 6 \\rangle$ , embedded in $L$ in such a way that its orthogonal complement is isometric to $s$ (this is the so-called embedding of non-split type of an ample class of square six, see [23] for further details). It is clear that such an automorphism cannot be natural nor induced on a moduli space of stable sheaves on a K3 surface by a non-symplectic automorphism of the underlying K3 surface, since these can only exist on moduli spaces $X$ of stable sheaves on K3 surfaces with $\\rho ( X ) \\ge 2$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[792,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"24","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,54,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"24","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,879,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,171,703,192]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2 An “Occult Period Map” for Cubic Threefolds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,214,881,268]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We now start to recall Allcock–Carlson–Toledo’s construction of a period map for the moduli space of smooth cubic threefolds, which has been later on defined “occult” by Kudla and Rapoport in [27]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,270,881,468]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $C _ { 3 } ^ { s } : = | O _ { \\mathbb { P } ^ { 4 } } ( 3 ) | \\ / \\ P G L _ { 5 } ( \\mathbb { C } )$ be the GIT moduli space of stable cubic threefolds; it contains as an open subset the moduli space $C _ { 3 } ^ { s m }$ of smooth cubic threefolds. Classical work of Clemens and Griffiths gives a map ${ C } _ { 3 } ^ { s m } \\ \\ { \\mathcal { A } } _ { 5 }$ by associating to $C$ its Intermediate Jacobian $I J ( C ) : = H ^ { 2 , 1 } ( C , \\mathbb { C } ) ^ { * } / H ^ { 3 } ( C , \\mathbb { Z } )$ , which is a ppav of dimension 5; the problem is that these two moduli spaces are respectively of dimension 10 and 15, so there is no hope to get an isomorphism. This is the reason for which Allcock, Carlson and Toledo came up with a different idea to construct a period map for smooth cubic threefolds by using the period map of smooth cubic fourfolds, and in particular by showing that it gives an isomorphism with a 10-dimensional complex ball quotient. We will now review the main ideas of their construction. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,468,881,600]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $C \\in C _ { 3 } ^ { s m }$ and let $Y \\subset \\mathbb { P } ^ { 5 }$ be the associated cyclic cubic fourfold. Recall that a marking of any cubic fourfold $Y$ is an isometry $\\eta _ { 0 } : H _ { 0 } ^ { 4 } ( Y , \\mathbb { Z } ) \\to S ( - 1 )$ and that the period of the marked pair $( Y , \\eta _ { 0 } )$ is $[ \\eta _ { 0 } ( H ^ { 3 , 1 } ( Y ) ) ] \\in \\mathbb { P } ( S ( - 1 ) \\otimes \\mathbb { C } )$ . The key idea is now to use the restriction of the period map to cyclic cubic fourfolds to define a period map for cubic threefolds: when $Y$ is cyclic, its period is an eigenvector for the action of $\\sigma ^ { * }$ , so that $[ \\eta _ { 0 } ( H ^ { 3 , 1 } ( Y ) ) ] \\in \\mathbb { P } ( S ( - 1 ) _ { \\xi } )$ , where $S ( - 1 ) _ { \\xi }$ is the eigenspace of $\\eta _ { 0 } \\circ \\sigma ^ { * } \\circ \\eta _ { 0 } ^ { - 1 }$ relative to $\\xi$ inside $S ( - 1 ) \\otimes \\mathbb { C }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,602,879,693]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Fix as initial data a smooth cubic threefold $\\bar { C } \\in C _ { 3 } ^ { s m }$ , and let $\\bar { Y } , \\bar { \\sigma }$ and $\\bar { \\eta _ { 0 } }$ be respectively the cyclic cubic fourfold associated to $\\bar { C }$ , the cover automorphism as defined above and a chosen marking of $\\bar { Y }$ . We define $\\rho _ { 0 } : = \\bar { \\eta _ { 0 } } \\circ \\bar { \\sigma } \\circ \\bar { \\eta _ { 0 } } ^ { - 1 } \\in$ $O ( S ( - 1 ) )$ to be the abstract isometry induced by $\\bar { \\sigma }$ , and we use it to move the action around the locally trivial family of cubic fourfolds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,695,881,767]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A framing of $C \\in C _ { 3 } ^ { s m }$ is the equivalence class of a marking $\\eta _ { 0 }$ of $Y$ such that $\\eta _ { 0 } \\circ \\rho _ { 0 } = \\rho _ { 0 } \\circ \\eta _ { 0 }$ , up to the action of $\\mu _ { 6 } : = \\{ \\pm \\mathrm { i d } _ { S } , \\pm \\rho _ { 0 } , \\pm \\rho _ { 0 } ^ { 2 } \\}$ : indeed, $S ( - 1 )$ has a structure of $\\mathbb { Z } [ \\xi ]$ -module induced by $\\rho _ { 0 }$ , and $\\mu _ { 6 }$ is exactly the group of units in $\\mathbb { Z } [ \\xi ]$ . We define the moduli space of framed pairs up to isomorphism: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[277,783,715,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0527/6fa3f67a6e9a6877d8918ce4ea919605bf78359bb15d5000dc1ad7ad29a4ae4c.jpg","list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathcal { F } _ { 3 } ^ { s m } : = \\{ ( C , \\eta _ { 0 } ) | C \\in C _ { 3 } ^ { s m } , \\ \\eta _ { 0 } \\ \\mathrm { i s ~ a ~ f r a m e } \\} / \\simeq .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,821,881,894]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The group $\\Gamma _ { 0 } : = \\{ \\gamma \\in O ( S ( - 1 ) ) | \\gamma \\circ \\rho _ { 0 } = \\rho _ { 0 } \\circ \\gamma \\}$ acts on a framed pair by $( C , \\eta _ { 0 } ) \\mapsto ( C , \\gamma \\circ \\eta _ { 0 } )$ , but because of our definition of framings as equivalence classes, the action is not free. We thus introduce $\\mathbb { P } \\Gamma : = \\Gamma _ { 0 } / \\mu _ { 6 }$ , and we recover the moduli space of smooth cubic threefolds as the quotient $C _ { 3 } ^ { s m } \\simeq \\mathcal { F } _ { 3 } ^ { s m } / \\mathbb { P } \\Gamma$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,571,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moduli Spaces of Cubic Threefolds and of IHS Manifolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"25","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We can now study the map induced by the period map of cubic fourfolds on $\\mathcal { F } _ { 3 } ^ { s m } \\colon$ as we already observed, the period $[ \\eta _ { 0 } ( H ^ { 3 , 1 } ( Y ) ) ] \\in \\mathbb { P } ( S ( - 1 ) _ { \\xi } )$ and it is welldefined even for the framing, since composing with any element in $\\mu _ { 6 }$ corresponds to multiplying the period for a non-zero scalar. A standard argument shows that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[250,175,744,199]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0484/f80fbd1d3a1732d31e9ee59c08523959d4235c0cf3c50af9b351e0ef3276a1ef.jpg","list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n[ \\eta _ { 0 } ( H ^ { 3 , 1 } ( Y ) ) ] \\in \\mathbb { B } _ { 1 0 } \\simeq \\{ x \\in \\mathbb { P } ( S ( - 1 ) _ { \\xi } ) | ( x , \\bar { x } ) < 0 \\} ;\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,214,728,232]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"equivalently, the period domain is a ten-dimensional complex ball. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,240,881,297]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 1 The period map sending $( C , \\eta _ { 0 } ) \\in \\mathcal { F } _ { 3 } ^ { s m } t o \\left[ \\eta _ { 0 } ( H ^ { 3 , 1 } ( Y ) ) \\right] \\in \\mathbb { B } _ { 1 0 }$ defines an isomorphism onto the image and is equivariant with respect to the action of P\b, so that we have the following commutative diagram: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[375,309,626,399]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0713/79eefc3d726829e8adea7325e84cacda846c203ecbf1e503e4cd877826db8945.jpg","list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { l l } { \\mathcal { F } _ { 3 } ^ { s m } \\xrightarrow { \\quad \\simeq \\quad \\simeq \\quad \\gg \\mathbb { B } _ { 1 0 } \\setminus \\mathcal { H } \\quad } } \\\\ { \\bigg \\downarrow \\quad } \\\\ { C _ { 3 } ^ { s m } \\xrightarrow { \\quad \\geqslant \\quad ( \\mathbb { B } _ { 1 0 } \\setminus \\mathcal { H } ) / \\mathbb { P } \\Gamma } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,418,876,454]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\mathcal { H }$ is the union of hyperplanes in $\\mathbb { B } _ { 1 0 }$ orthogonal to classes $\\delta \\in S ( - 1 )$ with $\\delta ^ { 2 } = 2$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,501,744,522]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.3 Moduli Spaces of $( \\pmb { \\rho } , \\pmb { T } )$ -Polarized IHS Fourfolds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,545,881,653]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section, we recall the constructions of [16] and [17] of moduli spaces of IHS manifolds of $K 3 ^ { [ n ] }$ -type carrying a non-symplectic automorphism of prime order associated to a given pair of sublattices $( T , S )$ , but instead of working in full generality we illustrate everything in the case of interest here. We fix a primitive embedding $j : S \\hookrightarrow L$ and we call $\\theta \\in L$ a generator of $j ( S ) ^ { \\perp }$ , which is of degree six as we have already discussed. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,653,881,801]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A marking of a fourfold of $K 3 ^ { [ 2 ] }$ -type $X$ is an isometry $\\eta ~ : ~ H ^ { 2 } ( X , \\mathbb { Z } ) \\ \\to$ $L$ . Let $\\boldsymbol { \\mathcal { M } } _ { L }$ be the moduli space of marked pairs $( X , \\eta )$ of $K 3 ^ { [ 2 ] }$ -type modulo isomorphism. The choice of the initial data made in Sect. 2.2 produces also initial data for this construction: indeed, we consider $F ( { \\bar { Y } } )$ the Fano variety of the cyclic cubic fourfold $\\bar { Y }$ and we define a marking by extending $\\bar { \\eta _ { 0 } } \\circ A ^ { - 1 }$ to a map $\\bar { \\eta } ~ : { \\cal H } ^ { 2 } ( F ( \\bar { Y } ) , \\mathbb { Z } ) { \\cal L }$ which sends $H _ { F ( \\bar { Y } ) }$ to $\\theta$ . In this way we obtain a point of $\\boldsymbol { \\mathcal { M } } _ { L }$ ; let $\\mathcal { M } _ { L } ^ { + }$ be the connected component containing $( F ( { \\bar { Y } } ) , { \\bar { \\eta } } )$ . We also get a non-symplectic automorphism $\\bar { \\sigma } \\in \\mathop { \\mathrm { A u t } } ( F ( \\bar { Y } ) )$ of order three and a corresponding isometry $\\rho : = \\bar { \\eta } \\circ \\bar { \\sigma } ^ { \\ast } \\circ \\bar { \\eta } ^ { - 1 } \\in O ( L )$ of order three which satisfies the following additional properties: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[792,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Camere ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"26","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"18 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,879,129]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,139,881,211]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. $\\rho _ { \\lvert S } = \\rho _ { 0 }$ ; \n2. $\\dot { \\rho \\mathbf { \\rho } } \\in \\mathbf { M o n } ^ { 2 } ( L ) : = \\bar { \\eta } \\circ \\mathbf { M o n } ^ { 2 } ( F ( \\bar { Y } ) \\circ \\bar { \\eta } ^ { - 1 }$ is a monodromy operator on $L$ , so by [23, §3] it has real spinor norm $\\mathbf { s n } _ { L } ^ { \\mathbb { R } } ( \\rho ) = 1$ (see loc. cit. for the definition of $\\mathrm { s n } _ { L } ^ { \\mathbb { R } } .$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,219,881,274]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 1 A $\\left( \\rho , \\langle 6 \\rangle \\right)$ -polarized pair $( X , \\eta )$ is the data of a marked pair $( X , \\eta ) \\in$ $\\boldsymbol { M } _ { L } ^ { + }$ and $\\sigma ~ \\in ~ \\operatorname { A u t } ( X )$ respectively, for which there exists $\\iota : \\langle 6 \\rangle \\hookrightarrow \\operatorname { P i c } ( X )$ a primitive embedding of lattices such that $\\eta \\circ \\iota = j$ and such that $\\eta \\circ \\sigma \\circ \\eta ^ { - 1 } = \\rho$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,282,881,337]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This definition turns out to be the counterpart of the notion of a framing for a cubic threefold. We define $M _ { \\rho , \\langle 6 \\rangle }$ to be the subspace of isomorphism classes of $\\left( \\rho , \\langle 6 \\rangle \\right)$ -polarized pairs inside $\\boldsymbol { M } _ { L } ^ { + }$ . The stabilizer of $M _ { \\rho , \\langle 6 \\rangle }$ is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[287,352,708,376]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0593/19359532945b4fe42c96d38010c07c85a0da285e3e5a0babab372eee85ac63b8.jpg","list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf { M o n } ^ { 2 } ( L ) _ { \\rho } : = \\{ g \\in \\mathbf { M o n } ^ { 2 } ( L ) | g \\circ \\rho = \\rho \\circ g \\} ;\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,391,660,411]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"we define the quotient space $N _ { \\rho , \\langle 6 \\rangle } : = M _ { \\rho , \\langle 6 \\rangle } / \\operatorname { M o n } ^ { 2 } ( L ) _ { \\rho }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,412,881,465]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $\\Gamma _ { \\rho } : = r _ { S } ( { \\mathbf { M o n } } ^ { 2 } ( L ) _ { \\rho } )$ be the image of $\\mathsf { M o n } ^ { 2 } ( L ) _ { \\rho }$ via the restriction map $r _ { S } :$ $O ( L ) \\to O ( S )$ , which acts on the periods in $\\mathbb { P } ( S \\otimes \\mathbb { C } )$ . One of the key ingredients of our main result is the following ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,474,712,493]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 1 There is an isomorphism of groups $\\Gamma _ { \\rho } \\simeq \\Gamma _ { 0 } / \\{ \\pm \\mathrm { i d } _ { S } \\}$ . ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,500,881,627]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The period map of marked pairs of $K 3 ^ { [ 2 ] }$ -type restricts to a holomorphic map $\\begin{array} { r l r } { M _ { \\rho , \\langle 6 \\rangle } } & { { } \\to } & { { \\mathbb { B } } _ { 1 0 } } \\end{array}$ , since again $[ \\eta ( H ^ { 2 , 0 } ( X ) ) ] ~ \\in ~ \\mathbb { P } ( S _ { \\xi } )$ (this is nothing but the equivariance of the Abel–Jacobi map of a cyclic cubic fourfold with respect to the action of $\\sigma$ ). In order to describe the image we need crucially the notion of MBM class $\\delta \\ \\in \\ L$ introduced by Amerik and Verbitsky in [3], which we do not recall in detail here since, as we will see in Sect. 2.4, it is possible to characterize these classes lattice-theoretically. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,635,881,708]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2 The period map restricts to an isomorphism of the subspace of isomorphism classes of $\\left( \\rho , \\langle 6 \\rangle \\right)$ -polarized pairs $M _ { \\rho , \\langle 6 \\rangle }$ onto its image inside $\\mathbb { B } _ { 1 0 }$ , and it is equivariant with respect to the action of $\\mathsf { M o n } ^ { 2 } ( L ) _ { \\rho }$ , so that it descends to the quotients. We have the following commutative diagram: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[362,722,637,809]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"1004/3a4449a02372326ccef05c0351e404712027be078254b8c236a836260abf37d4.jpg","list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } { M _ { \\rho , \\langle 6 \\rangle } \\xrightarrow { \\quad \\simeq \\quad \\mathbb { B } _ { 1 0 } \\setminus \\mathcal { H } ^ { \\prime } } } & { { } } \\\\ { \\bigg \\downarrow } & { { } } \\\\ { N _ { \\rho , \\langle 6 \\rangle } \\xrightarrow { \\quad \\geqslant \\quad \\mathbb { P } _ { 6 } \\quad \\qquad } \\mathbb { V } ^ { \\prime } ) / \\Gamma _ { \\rho } } & { { } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[113,828,842,848]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\mathcal { H }$ is the union of hyperplanes in $\\mathbb { B } _ { 1 0 }$ orthogonal to MBM classes $\\delta \\in S$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,51,571,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moduli Spaces of Cubic Threefolds and of IHS Manifolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,52,879,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"27","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"19 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,382,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.4 The Isomorphism ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,132,879,186]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We now conclude the proof of the main result: we are left with comparing the two hyperplane arrangements, since it immediately follows from Lemma 1 that the two arithmetic subgroups $\\mathbb { P } \\Gamma$ and $\\Gamma _ { \\rho }$ have the same orbits on $\\mathbb { B } _ { 1 0 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,187,881,259]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is known from work of Bayer–Hassett–Tschinkel [6] and Mongardi [34] that MBM classes for fourfolds of $K 3 ^ { [ 2 ] }$ -type are exactly classes $\\delta \\in L$ either with $\\delta ^ { 2 } = - 2$ or with $\\delta ^ { 2 } = - 1 0$ and $\\mathrm { d i v } _ { L } ( \\delta ) = 2$ , where the divisibility of $x$ in a lattice $L$ is defined as the generator of the ideal $( x , L )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,267,719,286]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 2 All MBM classes $\\delta \\in S$ satisfy $\\delta ^ { 2 } = - 2$ , and $\\mathcal { H } = \\mathcal { H }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,295,881,331]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This follows easily from the fact that $\\mathrm { d i v } _ { S } ( \\delta )$ is a common divisor of $\\operatorname* { d e t } ( S ) = 3$ and of $\\operatorname* { d e t } ( L ) = 2$ , so it cannot be two. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,340,881,414]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3 There is an isomorphism between the moduli spaces of smooth cubic threefolds $C _ { 3 } ^ { s m }$ and the arithmetic quotient $N _ { \\rho , \\langle 6 \\rangle }$ of the subspace of isomorphism classes of $( \\rho , \\langle 6 \\rangle )$ -polarized pairs inside $\\mathcal { M } _ { L } ^ { + }$ , such that the following diagram commutes: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[314,427,681,513]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0724/2515834a508a5c10dcd1285223b35c055a819a788f447427c217e256bf785113.jpg","list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[123,534,879,570]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Whilst a priori we did not know whether any element of $N _ { \\rho , \\langle 6 \\rangle }$ was exactly the Fano variety of a cyclic cubic fourfold, this follows immediately from Theorem 3. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,579,876,615]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 1 Any $( X , \\eta ) \\in { \\cal M } _ { \\rho , \\langle 6 \\rangle }$ is of the form $( F ( Y ) , \\eta )$ for some $Y$ cyclic cubic fourfold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,624,876,660]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A second interesting consequence is that we obtain a more classically-flavoured period map also in the case of fourfolds of $K 3 ^ { [ 2 ] }$ -type. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,669,881,705]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 2 There is an open embedding of $N _ { \\rho , \\langle 6 \\rangle }$ inside ${ \\mathcal { A } } _ { 5 }$ given by sending $( X , \\eta )$ into ${ \\mathrm { A l b } } ( X ^ { \\sigma } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,713,790,732]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Indeed, this follows immediately from the fact that ${ \\mathrm { A l b } } ( F ( C ) ) \\cong I J ( C )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,777,328,798]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 Degenerations ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,821,881,894]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Degenerations of smooth cubic threefolds are well-studied; in particular it is proven in [2] that the hyperplane arrangement $\\mathcal { H }$ is the union of two sub-arrangements ${ \\mathcal { H } } _ { n }$ and $\\mathcal { H } _ { c }$ respectively corresponding to stable cubics with at least one nodal singularity and to the semistable chordal cubic. From a lattice theoretical viewpoint, the two cases correspond respectively to nodal and chordal roots $\\delta \\in S ( - 1 )$ of square 2. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[792,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Camere ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"28","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"20 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,90,879,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"29","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,126,881,288]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"29","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the case of cubic threefolds, it is nowadays very well-understood what happens when the period hits these hyperplanes; here, we want to answer the same question in the case of IHS manifolds. In particular the aim of this section is to explain what happens to the non-symplectic automorphism of order three when the period of $( X , \\eta )$ is in $\\mathcal { H } .$ This has been already explained by Dolgachev and Kondo in ¯ [20] for $K 3$ surfaces. We know that in the fiber of the period map of fourfolds of $K 3 ^ { [ 2 ] }$ -type over a point in the hyperplane arrangement $\\mathcal { H }$ we can still find smooth fourfolds; in order to understand what happens to the $( \\rho , T )$ -polarization we apply [33, Theorem 1.6] and obtain the following statement: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,295,881,388]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"29","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 1 Let $\\omega \\in \\delta ^ { \\perp } \\subset \\mathcal { H }$ and let $( X , \\eta )$ be any element in $\\mathcal { P } _ { 6 } ^ { - 1 } ( \\omega )$ ; then there exists $\\beta \\ \\in \\ \\operatorname { B i r } ( X )$ and $w \\in W _ { \\mathrm { e x c } } ( X )$ such that $\\eta ^ { - 1 } \\circ \\rho \\circ \\eta \\ : = \\ : w \\circ \\beta ^ { * }$ , where $W _ { \\mathrm { e x c } } ( X ) \\subset O ( H ^ { 2 } ( X , \\mathbb { Z } ) )$ is the subgroup generated by reflections in prime exceptional divisors, (i.e. reduced and irreducible effective divisors of negative degree). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,396,881,487]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"29","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As a consequence of Proposition 1, one might be tempted to expect that the isometry $\\rho$ no longer induces an automorphism but only a birational map of order three with a non-empty indeterminacy locus. In fact this is not the case, and the automorphism changes in the sense that it has a different action on the lattice. In order to explain this phenomenon we need to recall some definitions from [20]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,495,876,531]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"29","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 2 Given $\\omega \\in \\mathcal { H } .$ , the degeneracy sublattice of $\\omega$ is the primitive sublattice of $s$ spanned by MBM classes which are orthogonal to $\\omega$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,540,881,650]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"29","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The key remark is that for any MBM class in $s$ , we have $\\delta ^ { \\perp } \\cap \\mathbb { P } ( S _ { \\xi } ) = \\rho ( \\delta ) ^ { \\perp } \\cap$ $\\mathbb { P } ( S _ { \\xi } )$ . From now on we suppose $\\omega \\in H \\subset { \\mathcal { H } }$ for only one hyperplane $H = \\delta ^ { \\perp }$ ; in this case the degeneracy lattice is $R _ { \\delta } : = \\langle \\delta , \\rho ( \\delta ) \\rangle \\cong A _ { 2 } ( - 1 )$ ; this admits, up to isometry, two primitive embeddings into $s$ , according to the two cases $R _ { \\delta } ^ { \\perp } =$ $U \\oplus U ( a ) \\oplus E _ { 8 } ( - 1 ) ^ { \\oplus 2 }$ for $a = 1$ , 3; we speak respectively about chordal and nodal roots $\\delta$ in the two cases $a = 1 , 3$ , since the following holds: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,657,874,695]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"29","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 3 Let $\\delta \\in S$ have square $^ { - 2 }$ and let $\\omega \\in \\delta ^ { \\perp } \\subset \\mathcal { H }$ be very general; then, $R _ { \\delta } ^ { \\perp }$ is unimodular if and only if $\\omega \\in \\mathcal { H } _ { c }$ , if and only if $a = 1$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,704,879,740]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"29","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We now want to look at how the isometry $\\rho$ degenerates along the hyperplanes. We call $T _ { \\delta }$ the saturation of $\\langle 6 \\rangle \\oplus R _ { \\delta }$ in $L$ , so that $\\overline { { ( T _ { \\delta } ) _ { L } ^ { \\perp } } } \\subset S _ { \\delta } : = \\overline { { ( R _ { \\delta } ) _ { S } ^ { \\perp } } }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,749,879,785]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"29","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 4 The lattice $T _ { \\delta }$ is respectively isometric to $T _ { n } : = U ( 3 ) \\oplus \\langle - 2 \\rangle \\cong \\langle 6 \\rangle \\oplus$ $A _ { 2 } ( - 1 )$ for $\\omega \\in \\mathcal { H } _ { n }$ (case $a = 3$ ), and to $T _ { c } : = U \\oplus \\langle - 2 \\rangle$ for $\\omega \\in \\mathcal { H } _ { c }$ (case $a = 1$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,794,876,828]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"29","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We define $\\rho _ { \\delta }$ as the isometry of $L$ obtained by gluing $\\mathrm { i d } _ { T _ { \\delta } }$ and $\\rho _ { | S _ { \\delta } }$ : it is the degeneration of the isometry $\\rho$ to $\\delta ^ { \\perp }$ , and by construction we have $L ^ { \\rho _ { \\delta } } = T _ { \\delta }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,837,879,881]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"29","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 2 The isometry $\\rho _ { \\delta } \\in \\mathbf { M o n } ^ { 2 } ( L )$ and we have $\\rho _ { \\delta } = r _ { \\rho ( \\delta ) } \\circ r _ { \\delta } \\circ \\rho$ , where $r _ { v } \\in O ( L )$ denotes the reflection in $v$ , given by $\\begin{array} { r } { x \\mapsto x - 2 \\frac { ( x , v ) } { ( v , v ) } v } \\end{array}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,571,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"29","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moduli Spaces of Cubic Threefolds and of IHS Manifolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"29","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,876,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"30","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In order to distinguish the two cases, we denote by $\\rho _ { n }$ and $\\rho _ { c }$ the isometries $\\rho _ { \\delta }$ respectively when $\\delta$ is a nodal and a chordal root. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,171,375,190]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"30","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.1 The Nodal Locus ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,214,881,360]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"30","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A semistable nodal cubic threefold, whose period maps to the very general $\\omega \\in \\delta ^ { \\perp }$ with $\\delta$ a nodal root, is of the form $C : X _ { 0 } Q + M _ { 3 } = 0$ with an $A _ { 1 }$ -singularity in $p = [ 1 : 0 : \\ldots : 0 ]$ ; hence, the associated cubic fourfold has equation $Y : X _ { 0 } Q +$ $M _ { 3 } + X _ { 3 } ^ { 5 } = 0$ with an $A _ { 2 }$ -singularity in $p = [ 1 : 0 : \\ldots : 0 ]$ . Allcock, Carlson and Toledo extend $\\mathcal { P } _ { 3 }$ to semistable points by using the period of $\\Sigma : \\ d Q = M _ { 3 } + X _ { 5 } ^ { 3 } = 0$ : when the singularity of $C$ is $A _ { 1 }$ , this is a smooth $K 3$ surface, endowed with a nonsymplectic automorphism $\\tau$ of order three, with invariant lattice $U ( 3 ) \\subset H ^ { 2 } ( \\Sigma , \\mathbb { Z } )$ (see [4]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,361,881,450]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"30","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Indeed, it is a classical fact that $F ( Y )$ is singular in this situation and that the singular locus is contained in $F ( Y , p )$ , defined as the surface of lines contained in $Y$ which pass through one of the isolated singularities $p$ (compare with [30, 37] and [41]). The surface $\\Sigma$ is the minimal resolution of $F ( Y , p )$ in this case, and Lehn shows in [30] that $\\Sigma ^ { [ 2 ] } F ( Y )$ is a resolution of singularities. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,451,881,506]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"30","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moreover, the natural automorphism $\\tau ^ { \\left[ 2 \\right] }$ on $\\Sigma ^ { [ \\bar { 2 } ] }$ is induced by the original automorphism of $\\mathbb { P } ^ { 5 }$ which we had chosen at the beginning, and its invariant sublattice is exactly $T _ { n }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,513,881,551]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"30","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 3 There exists a marking $\\eta : H ^ { 2 } ( \\Sigma ^ { [ 2 ] } , \\mathbb { Z } ) \\to L$ such that $( \\Sigma ^ { [ 2 ] } , \\eta )$ is $( \\rho _ { n } , T _ { n } )$ -polarized. In particular, $\\eta ^ { - 1 } \\circ \\rho _ { n } \\circ \\eta = ( \\tau ^ { [ 2 ] } ) ^ { * }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,558,881,680]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"30","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We now choose a very general point $( \\Sigma ^ { [ 2 ] } , \\eta )$ as above and such that $\\operatorname { P i c } ( \\Sigma ^ { [ 2 ] } ) \\cong$ $T _ { n }$ , and we define $K ( T _ { n } ) : = \\eta ( \\mathcal { K } _ { \\Sigma ^ { [ 2 ] } } )$ . Following the construction of [16] and the techniques first developed in [26], we say that $( X , \\eta ) \\in \\mathcal { M } _ { \\rho _ { n } , T _ { n } } ^ { \\xi }$ is $K ( T _ { n } )$ -general if $\\eta ( \\mathcal { K } _ { X } ) \\cap ( T _ { n } ) _ { \\mathbb { R } } = K ( T _ { n } )$ ; the moduli space of $K ( T _ { n } )$ -general $( \\rho _ { T _ { n } } , T _ { n } )$ -polarized K ∩ = fourfolds of K3[2]-type is MξK(Tn),ρn a nd its quotient for the action of $\\mathbf { M o n } ^ { 2 } ( L , \\rho _ { n } )$ is NξK(Tn),ρn . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[143,680,763,696]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"30","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Making the above reasoning global, in [17] we obtain the following $\\Delta _ { 3 } ^ { s } = C _ { 3 } ^ { s } \\setminus C _ { 3 } ^ { s m }$ is birational to ${ \\cal N } _ { K ( T _ { n } ) , \\rho _ { n } } ^ { \\xi }$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,705,772,728]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"30","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,770,400,791]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"30","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.2 The Chordal Locus ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,815,881,887]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"30","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the case of the stable chordal locus $\\Xi _ { 3 } ^ { s }$ the picture is more complicated: associated to one semistable chordal cubic we find a hyperplane arrangement. The fact is, as Allcock, Carlson and Toledo show, that the variation of Hodge structure along a 1- parameter family hitting $\\mathcal { H } _ { c }$ depends on the path chosen, so that from one semistable point in the moduli space one gets in fact an entire hyperplane arrangement. Again, they show that the variety realizing the degeneration of the Hodge structure is a $K 3$ surface $\\Sigma$ carrying a non-symplectic automorphism $\\tau$ of order three having fixed lattice isometric to $U$ (see [4]). The final result, though through a less geometrical proof, is the following ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[792,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"30","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Camere ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"30","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"22 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,881,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,189,731,210]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 5 The stable chordal locus $\\Xi _ { 3 } ^ { s }$ is birational to $\\surd _ { K ( T _ { c } ) , \\rho _ { c } } ^ { \\xi }$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,217,881,292]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It remains an open question to understand the geometry of the (singular) Fano variety of lines of the cyclic fourfold corresponding to the chordal cubic threefold, and which relation, if any, it has with the Hilbert schemes of $K 3$ surfaces in NξK(Tc),ρc . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,292,879,363]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorems 4 and 5 in some sense shed also new light on why smooth cubic threefolds are degenerating to $K 3$ surfaces: in dimension four the picture becomes much more uniform and the degeneration is explained as degeneration of nonsymplectic automorphisms of order three on fourfolds of $K 3 ^ { [ 2 ] }$ -type. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,408,669,429]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 Further Developments and Open Questions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,447,542,468]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4.1 Pfaffian Cyclic Cubic Fourfolds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,490,881,618]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Other natural loci to study inside the arithmetic quotient $N _ { \\rho , \\left. 6 \\right. } = M _ { \\rho , \\left. 6 \\right. } / \\mathrm { M o n } ^ { 2 } ( L ) _ { \\rho }$ are Heegner divisors, and in particular those coming from the intersection among the family of cyclic cubic fourfolds and the so-called Hassett’s divisors $\\mathcal { H } _ { d } \\in C _ { 4 } ^ { s m }$ (see [24] for the exact definition). In [17] we have studied the closely-related question about the existence of a cyclic Pfaffian cubic fourfold not containing a plane (indeed, the closure of the Pfaffian locus is $\\mathcal { H } _ { 1 4 }$ , and cubics containing a plane are exactly the elements of $\\mathcal { H } _ { 8 }$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,618,881,707]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recall the following construction from classical Pfaffian geometry, as described in [11]: let $V$ be a six-dimensional vector space and let ${ \\mathcal P } f \\subset \\mathbb P ( \\Lambda ^ { 2 } V ^ { * } )$ be the hypersurface of degenerate skew-symmetric 2-forms. Any Pfaffian cubic fourfold $Y$ is then of the form $\\mathcal P f \\cap \\mathbb { P } ( W )$ for $W \\subset \\Lambda ^ { 2 } V ^ { * }$ six-dimensional. The dual $K 3$ surface of $Y$ is $\\Sigma : = G ( 2 , V ) \\cap \\mathbb { P } ( W ^ { \\perp } )$ , of degree 14. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,717,881,770]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 4 There exists a smooth cyclic Pfaffian cubic fourfold $Y \\subset \\mathbb { P } ^ { 5 }$ defined over $\\mathbb { Q }$ which does not contain a plane, and such that the dual $K 3$ surface $\\Sigma$ does not contain a line. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,780,879,852]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The interest of Proposition 4 is that Beauville and Donagi proved in [11] that under these special assumptions, the Fano variety of a Pfaffian cubic fourfold is isomorphic to the Hilbert scheme of two points on $\\Sigma$ . As a consequence, the automorphism is defined on $\\Sigma ^ { [ 2 ] }$ and we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,861,876,896]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 3 There exists a non-natural non-symplectic automorphism of order three on $\\Sigma ^ { [ 2 ] }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,571,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moduli Spaces of Cubic Threefolds and of IHS Manifolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"31","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,180]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"See [17] for a geometrical description. We remark that more in general, one obtains a non-natural automorphism on a Hilbert square whenever there exists a $K 3$ surface $\\Sigma$ such that $\\Sigma ^ { [ 2 ] } \\cong F ( Y )$ for a cyclic cubic fourfold Y , but the question of studying the intersection, which happens in positive codimension, among Hassett’s divisors and the locus of cyclic cubic fourfolds remains open. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,225,589,246]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4.2 Higher Dimensional IHS Manifolds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,267,881,358]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Fourfolds of $K 3 ^ { [ 2 ] }$ -type are not the only IHS manifolds that can be associated to cubic fourfolds: a recent example are the Lehn–Lehn–Sorger–van Straten $( L L S \\nu S )$ eightfolds of $K 3 ^ { [ 4 ] }$ -type, defined as contraction $Z$ of the Hilbert scheme of generalized twisted cubic curves contained in a smooth cubic fourfold $Y$ which does not contain a plane [31]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,360,881,504]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In [18, §6] we show that there is a non-symplectic automorphism $\\zeta$ of order three on the LLSvS eightfold $Z$ associated to a smooth cyclic cubic fourfold not containing a plane; we can thus define the isometry $\\rho _ { Z } \\ { \\in \\ \\mathrm { { M o n } } ^ { 2 } } ( L _ { K 3 } \\oplus \\langle - 6 \\rangle )$ associated to his action (starting from a fixed initial example as above) the invariant lattice of $\\zeta$ is $\\langle 2 \\rangle$ and its orthogonal complement is again $s$ inside $L _ { K 3 } \\oplus \\langle - 6 \\rangle$ . Exactly the same theory illustrated in 2.3 following [16] leads to the construction of the moduli space ${ \\cal M } _ { \\rho _ { Z } , \\langle 2 \\rangle }$ of $( \\rho _ { Z } , \\langle 2 \\rangle )$ -polarized eightfolds of $K 3 ^ { [ 4 ] }$ -type; in particular, [16, Theorem 1.2] implies the following ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,512,883,549]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 4 Let $\\mathcal { H } ^ { \\prime }$ be the hyperplane arrangement in $\\mathbb { P } ( S \\otimes \\mathbb { C } )$ consisting of hyperplanes orthogonal to MBM classes in S. The period map ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[373,564,624,587]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0371/8e6dff16679b107355845dfe60157d0d48cd0e8edf9a24e2bed58f082fe634f0.jpg","list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathcal { P } _ { 2 } : M _ { \\rho _ { Z } , \\langle 2 \\rangle } \\to \\mathbb { B } _ { 1 0 } \\setminus \\mathcal { H } ^ { \\prime \\prime }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,602,876,621]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is an isomorphism, equivariant with respect to the action of $\\mathbf { M o n } ^ { 2 } ( L _ { K 3 } \\oplus \\langle - 6 \\rangle ) _ { \\rho _ { Z } }$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,629,881,756]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It follows from [34] that in dimension eight the MBM classes that can be contained in $s$ are classes $\\delta \\in S$ either such that $\\delta ^ { 2 } = - 2$ or such that $\\delta ^ { 2 } = - 6$ or $\\delta ^ { 2 } = - 2 4$ and $\\mathrm { d i v } = 3$ . As a consequence, $\\mathcal { H } ^ { \\prime }$ is strictly bigger than $\\mathcal { H } _ { \\mathrm { ~ ~ } }$ , though this was to be expected because the construction of the LLSvS eighfold is only possible in the open subset of cubic fourfolds not containing a plane. Another difference which arises with the increase in dimension is that the group of isometries induced by monodromy operators is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,770,876,794]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0061/4e263f94e28fb64d45c164706e03ebd86e2a02736cfc49c6f0d37b63eb9f6bf6.jpg","list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf { M o n } ^ { 2 } ( L _ { K 3 } \\oplus \\langle - 6 \\rangle ) = \\{ g \\in O ( L _ { K 3 } \\oplus \\langle - 6 \\rangle ) | \\operatorname { s n } ^ { \\mathbb { B } } ( g ) = 1 , \\bar { g } = \\mathrm { i d } \\in O ( q _ { A _ { L _ { K 3 } \\oplus \\langle - 6 \\rangle } } \\} ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,809,537,827]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"so a priori it could be of index two inside $\\mathbb { P } \\Gamma$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[143,828,624,845]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In any case, ideally one obtains a finite rational map ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[302,860,694,884]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0506/e9d7b8fae03d0c7c2d02951f9077fb4365d06d06e4614df11c36466eb2a5f8dd.jpg","list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\left. M _ { \\rho _ { Z } , \\langle 2 \\rangle } / \\left. M o n ^ { 2 } ( L _ { K 3 } \\oplus \\langle - 6 \\rangle ) _ { \\rho _ { Z } } \\right. \\right. - \\left. C _ { 3 } ^ { s m } . \\right.\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[792,54,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Camere ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,54,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"32","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,214]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moreover, as it was explained in another talk of this workshop by Lahoz, forthcoming work of Bayer, Lahoz, Macrì, Neuer, Perry and Stellari includes the construction of many other families of higher dimensional polarized manifolds of $K 3 ^ { [ n ] }$ -type associated to cubic fourfolds (see [32, Theorem 5.11]), so it is expected that there is a “tower” of rational maps from moduli spaces of manifolds of $K 3 ^ { [ n ] }$ -type with a non-symplectic automorphism of order three having invariant lattice of rank one to the moduli space of smooth cubic threefolds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,261,334,280]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4.3 Another Case ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,306,881,468]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the recent work [29], the authors use the same techniques of Allcock, Carlson and Toledo in order to study the moduli space $\\mathcal { W }$ of pairs $( C , H )$ consisting of a smooth cubic threefold $C \\in C _ { 3 } ^ { s m }$ and of a hyperplane $H$ . In particular, if $G =$ 0 is an equation of $C$ and $L \\ = \\ 0$ is an equation of $H$ , they consider the cubic fourfold $Y$ defined by the equation $G + X _ { 5 } ^ { 2 } L \\ = \\ 0$ , and they consider the open subset $\\mathcal { W } ^ { 0 } \\subset \\mathcal { W }$ given by pairs for which $Y$ is smooth, so that they can study the restriction of the period map $\\mathcal { P } _ { 4 }$ to it. Again, this construction allows to relate the GIT compactification $\\bar { \\mathcal W }$ of $\\mathcal { W } ^ { 0 }$ and the Baily–Borel compactification of the period domain. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,471,881,650]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Again, if one considers the Fano variety of lines $F ( Y )$ , this has a non-symplectic involution $\\iota$ induced by $X _ { 5 } ~ \\mapsto ~ - X _ { 5 }$ , as observed in [10]. The computations contained in [29] on the transcendental part of $H ^ { 4 } ( Y , \\mathbb { Z } )$ , which is isomorphic to $U ^ { \\oplus 2 } \\oplus D _ { 4 } ^ { \\oplus 3 }$ , shed new light on this involution: since $F ( Y )$ is general, by dimension count, in the family carrying this kind of involution, we have that the orthogonal complement of the invariant lattice $M$ of $\\iota$ is $N : = U ^ { \\oplus 2 } \\oplus D _ { 4 } ( - 1 ) ^ { \\oplus 3 }$ and that $M$ is a 2-elementary overlattice of $\\langle 6 \\rangle \\oplus E _ { 6 } ( - 2 )$ . A standard computation shows that $M \\cong \\langle 2 \\rangle \\oplus \\langle - 2 \\rangle ^ { \\oplus 6 }$ : we remark that in [14, Figure 2] this case is erroneously omitted, though it cannot be realized by a natural automorphism, because $N$ has no primitive embedding in $L _ { K 3 }$ (see [28, Theorem 3.4]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,651,879,776]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Fix a primitive embedding $j _ { M } : M \\subset L$ such that $j _ { M } ( M ) ^ { \\perp } \\cong N$ ; the fourfolds of $K 3 ^ { [ 2 ] }$ -type with such a non-symplectic involution fixing $M$ are exactly all the lattice polarized fourfolds $( X , \\eta ) \\in \\mathcal { M } _ { L }$ such that there exists $i : M \\subset H ^ { 2 } ( X , \\mathbb { Z } )$ primitive with $\\eta \\circ i = j _ { M }$ and such that $i ( M ) \\cap { \\mathcal { K } } _ { X } \\neq \\emptyset$ , as explained in [26]. We denote by $\\textstyle { \\mathcal { M } } _ { M } \\subset { \\mathcal { M } } _ { L }$ the subspace of these lattice-polarized pairs and by $M _ { M , K }$ the connected component containing a pair $( F ( Y ) , \\eta )$ with $Y$ smooth as above such that $\\operatorname { P i c } ( F ( Y ) ) \\cong M$ , where $K : = \\eta ( \\mathcal { K } _ { F ( Y ) } \\cap \\operatorname { P i c } ( F ( Y ) ) )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,777,879,828]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Forthcoming work of the author will try to answer the question whether $\\mathcal { W }$ and $M _ { M , K }$ are isomorphic, and, if the answer is positive, to check whether the isomorphism extends to the compactifications. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,851,879,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acknowledgements The author wants to express her gratitude to S. Boissière and A. Sarti for reading a draft of this survey and to the Max Planck Institute of Mathematics for providing a great working environment. The comments of the anonymous referee, which the author gratefully appreciated, have greatly helped to improve and clarify the exposition. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,571,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moduli Spaces of Cubic Threefolds and of IHS Manifolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"33","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,250,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"34","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,138,881,899]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"34","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Allcock, D., Carlson, J.A., Toledo, D.: The complex hyperbolic geometry of the moduli space of cubic surfaces. J. Algebraic Geom. 11(4), 659–724 (2002) \n2. Allcock, D., Carlson, J.A., Toledo, D.: The moduli space of cubic threefolds as a ball quotient. Mem. Am. Math. Soc. 209(985), xi $+ 7 0$ (2011) \n3. Amerik, E., Verbitsky, M.: Rational curves on hyperkähler manifolds. Int. Math. Res. Not. 2015(23), 13009–13045 (2015) \n4. Artebani, M., Sarti, A.: Non-symplectic automorphisms of order 3 on $K 3$ surfaces. Math. Ann. 342(4), 903–921 (2008) \n5. Artebani, M., Sarti, A., Taki, S.: $K 3$ surfaces with non-symplectic automorphisms of prime order. Math. Z. 268(1–2), 507–533 (2011). With an appendix by Shigeyuki Kondo¯ \n6. Bayer, A., Hassett, B., Tschinkel, Y.: Mori cones of holomorphic symplectic varieties of K3 type. Ann. Sci. Éc. Norm. Supér. (4) 48(4), 941–950 (2015) \n7. Beauville, A.: Some remarks on Kähler manifolds with $c _ { 1 } = 0$ . In: Classification of Algebraic and Analytic Manifolds. Progress in Mathematics, vol. 39, pp. 1–26. Katata (1982) \n8. Beauville, A.: Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18(4), 755–782 (1983) \n9. Beauville, A.: Moduli of cubic surfaces and Hodge theory (after Allcock, Carlson, Toledo). In: Géométries à courbure négative ou nulle, groupes discrets et rigidités. Séminar Congress, vol. 18, pp. 445–466 (2009) \n10. Beauville, A.: Antisymplectic involutions of holomorphic symplectic manifolds. J. Topol. 4(2), 300–304 (2011) \n11. Beauville, A., Donagi, R.: La variété des droites d’une hypersurface cubique de dimension 4. C. R. Acad. Sci. Paris Sér. I Math. 301(14), 703–706 (1985) \n12. Boissière, S.: Automorphismes naturels de l’espace de Douady de points sur une surface. Can. J. Math. 64(1), 3–23 (2012) \n13. Boissière, S., Nieper-Wißkirchen, M., Sarti, A.: Smith theory and irreducible holomorphic symplectic manifolds. J. Topol. 6(2), 361–390 (2013) \n14. Boissière, S., Camere, C., Sarti, A.: Classification of automorphisms on a deformation family of hyper-Kähler four-folds by $p$ -elementary lattices. Kyoto J. Math. 56(3), 465–499 (2016) \n15. Boissière, S., Camere, C., Mongardi, G., Sarti, A.: Isometries of ideal lattices and hyperkähler manifolds. Int. Math. Res. Not. 2016(4), 963–977 (2016) \n16. Boissière, S., Camere, C., Sarti, A.: Complex ball quotients from manifolds of $K 3 ^ { [ n ] }$ -type. J. Pure Appl. Algebra, 223(3), 1123–1138 (2019). https://doi.org/10.1016/j.jpaa.2018.05.017 \n17. Boissière, S., Camere, C., Sarti, A.: Cubic threefolds and hyperkähler manifolds uniformized by the 10-dimensional complex ball. Ann. 373(3–4), 1429–1455 (2019). https://doi.org/10. 1007/s00208-018-1766-9 \n18. Camere, C., Cattaneo, A.: Non-symplectic automorphisms of odd prime order on manifolds of $K 3 ^ { [ n ] }$ -type. Manuscripta Math. (2019). https://doi.org/10.1007/s00229-019-01163-4 \n19. Camere, C., Kapustka, G., Kapustka, M., Mongardi, G.: Verra four-folds, twisted sheaves, and the last involution. Int. Math. Res. Not. 2019(21), 6661–6710 (2018). https://doi.org/10.1093/ imrn/rnx327 \n20. Dolgachev, I.V., Kondo, S.: Moduli of ¯ $K 3$ surfaces and complex ball quotients. In: Arithmetic and Geometry Around Hypergeometric Functions. Progress in Mathematics, vol. 260, pp. 43– 100. Birkhäuser, Basel (2007) \n21. Dolgachev, I., van Geemen, B., Kondo, S.: A complex ball uniformization of the moduli space ¯ of cubic surfaces via periods of $K 3$ surfaces. J. Reine Angew. Math. 588, 99–148 (2005) \n22. Ferretti, A.: Special subvarieties of EPW sextics. Math. Z. 272(3–4), 1137–1164 (2012) \n23. Gritsenko, V., Hulek, K., Sankaran, G.K.: Moduli of K3 surfaces and irreducible symplectic manifolds. In: Handbook of Moduli. Vol. I. Advanced Lectures in Mathematics, vol. 24, pp. 459–526. International Press, Somerville (2013) \n24. Hassett, B.: Special cubic fourfolds. Compos. Math. 120(1), 1–23 (2000) \n25. Huybrechts, D.: Compact hyper-Kähler manifolds: basic results. Invent. Math. 135(1), 63–113 (1999) \n26. Joumaah, M.: Non-symplectic involutions of irreducible symplectic manifolds of $K 3 ^ { [ n ] }$ -type. Math. Z. 283(3–4), 761–790 (2016) \n27. Kudla, S., Rapoport, M.: On occult period maps. Pac. J. Math. 260(2), 565–581 (2012) \n28. Laza, R.: Maximally algebraic potentially irrational cubic fourfolds (2018). arXiv:1805.04063 \n29. Laza, R., Pearlstein, G., Zhang, Z.: On the moduli space of pairs consisting of a cubic threefold and a hyperplane. Adv. Math. 340, 684–722 (2018) \n30. Lehn, C.: Twisted cubics on singular cubic fourfolds—on Starr’s fibration. Math. Z. 290(1), 379–388 (2018) \n31. Lehn, C., Lehn, M., Sorger, C., van Straten, D.: Twisted cubics on cubic fourfolds. J. Reine Angew. Math. 731, 87–128 (2017) \n32. Macrì, E., Stellari, P.: Lectures on non-commutative K3 surfaces, Bridgeland stability, and moduli spaces. In: Birational Geometry of Hypersurfaces. Lecture Notes of the Unione Matematica Italiana. Springer, Berlin (2019). arXiv:1807.06169 \n33. Markman, E.: A survey of Torelli and monodromy results for holomorphic-symplectic varieties. In: Complex and Differential Geometry. Springer Proceedings in Mathematics, vol. 8, pp. 257–322 (2011) \n34. Mongardi, G.: A note on the Kähler and Mori cones of hyperkähler manifolds. Asian J. Math. 19(4), 583–591 (2015) \n35. Mongardi, G., Wandel, M.: Induced automorphisms on irreducible symplectic manifolds. J. Lond. Math. Soc. (2) 92(1), 123–143 (2015) \n36. Nikulin, V.V.: Integral symmetric bilinear forms and some of their applications. Math. USSR Izv. 14, 103–167 (1980) \n37. O’Grady, K.G.: Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics. Duke Math. J. 134(1), 99–137 (2006) \n38. Ohashi, H., Wandel, M.: Non-natural non-symplectic involutions on symplectic manifolds of $K 3 ^ { [ 2 ] }$ -type (2013). arXiv:1305.6353 \n39. Tari, K.: Automorphismes des variétés de Kummer généralisées, Ph.D. thesis, Université de Poitiers, 2015 \n40. Verbitsky, M.: Mapping class group and a global torelli theorem for hyperkähler manifolds. Duke Math. J. 162(15), 2929–2986 (2013). https://doi.org/10.1215/00127094-2382680 \n41. Yamagishi, R.: Singularities of Fano varieties of lines on singular cubic fourfolds (2017). arXiv:1712.03546 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[792,54,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"34","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Camere ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,54,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"34","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[116,87,881,605]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"35","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,571,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"35","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moduli Spaces of Cubic Threefolds and of IHS Manifolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"35","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,669,141]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Note on Severi Varieties of Nodal Curves on Enriques Surfaces ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,175,742,211]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ciro Ciliberto, Thomas Dedieu, Concettina Galati, and Andreas Leopold Knutsen ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,286,881,430]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract Let $| L |$ be a linear system on a smooth complex Enriques surface $s$ whose general member is a smooth and irreducible curve of genus $p$ , with $L ^ { 2 } > 0$ , and let $V _ { | L | , \\delta } ( S )$ be the Severi variety of irreducible $\\delta$ -nodal curves in $| L |$ . We denote by $\\pi : X S$ the universal covering of $s$ . In this note we compute the dimensions of the irreducible components $V$ of $V _ { | L | , \\delta } ( S )$ . In particular we prove that, if $C$ is the curve corresponding to a general element $[ C ]$ of $V$ , then the codimension of $V$ in $| L |$ is $\\delta$ if $\\pi ^ { - 1 } ( C )$ is irreducible in $X$ and it is $\\delta - 1$ if $\\pi ^ { - 1 } ( C )$ consists of two irreducible components. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,450,792,466]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords Severi varieties $\\cdot$ Moduli $\\cdot$ Nodal curves $\\cdot$ Enriques surfaces ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,487,728,506]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2010 Mathematics Subject Classification 14H20, 14H10, 14J28 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,897,692,941]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_3 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[804,91,876,139]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,834,649,878]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. L. Knutsen Department of Mathematics, University of Bergen, Bergen, Norway e-mail: andreas.knutsen@math.uib.no ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,648,699,690]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Ciliberto Dipartimento di Matematica, Università di Roma Tor Vergata, Roma, Italy e-mail: cilibert@axp.mat.uniroma2.it ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,768,867,825]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Galati $\\bigstar$ ) Dipartimento di Matematica e Informatica, Università della Calabria, Arcavacata di Rende (CS), Italy e-mail: galati@mat.unical.it ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,701,856,758]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"T. Dedieu Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS IMT, Toulouse, France e-mail: thomas.dedieu@math.univ-toulouse.fr ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,899,876,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"36","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,312,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,133,881,186]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $s$ be a smooth complex projective surface and $L$ a line bundle on $s$ such that the complete linear system $| L |$ contains smooth, irreducible curves (such a line bundle, or linear system, is often called a Bertini system). Let ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[334,202,660,238]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0418/5cb3f1506ca78496fb8afbf95dd857eeae6f02832c660a8112e588861c9393f3.jpg","list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\np : = p _ { a } ( L ) = \\frac { 1 } { 2 } L \\cdot ( L + K _ { S } ) + 1 ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,253,514,270]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"be the arithmetic genus of any curve in $| L |$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,271,876,307]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For any integer $0 ~ \\le ~ \\delta ~ \\le ~ p$ , consider the locally closed, functorially defined subscheme of $| L |$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[375,324,624,345]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0357/ab71e43db2d15c7057a95ca1386c99f61257397280bee695140da81ed312c141.jpg","list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nV _ { | L | , \\delta } ( S ) \\mathrm { o r s i m p l y } V _ { | L | , \\delta }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,360,881,414]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"parameterizing irreducible curves in $| L |$ having only $\\delta$ nodes as singularities; this is called the Severi variety of $\\delta$ -nodal curves in $| L |$ . We will let $g : = p - \\delta$ , the geometric genus of the curves in $V _ { | L | , \\delta }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,415,879,468]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is well-known that, if $V _ { | L | , \\delta }$ is non-empty, then all of its irreducible components $V$ have dimension $\\mathrm { d i m } ( V ) \\geq \\mathrm { d i m } | L | - \\delta .$ . More precisely, the Zariski tangent space to $V _ { | L | , \\delta }$ at the point corresponding to $C$ is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[330,484,667,509]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0010/8bb758794d0ee67e3dcbb90dd9a46c92e50520d44f96cecbd8dd4fbc81455b20.jpg","list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nT _ { [ C ] } V _ { | L | , \\delta } \\simeq H ^ { 0 } ( L \\otimes \\mathcal { T } _ { N } ) / < C > ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,524,881,632]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\mathcal { T } _ { N } = \\mathcal { T } _ { N | S }$ is the ideal sheaf of subscheme $N$ of $s$ consisting of the $\\delta$ nodes of $C$ (see, e.g., [4, $\\ S 1 ]$ ). Thus, $V _ { | L | , \\delta }$ is smooth of dimension dim $| L | - \\delta$ at $[ C ]$ if and only if the set of nodes $N$ imposes independent conditions on $| L |$ . In this case, $V _ { | L | , \\delta }$ is said to be regular at $[ C ]$ . An irreducible component $V$ of $V _ { | L | , \\delta }$ will be said to be regular if the condition of regularity is satisfied at any of its points, equivalently, if it is smooth of dimension $\\dim | L | - \\delta$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,633,881,794]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The existence and regularity problems of $V _ { | L | , \\delta } ( S )$ have been studied in many cases and are the most basic problems one may ask on Severi varieties. We only mention some of known results. In the case $S \\simeq \\mathbb { P } ^ { 2 }$ , Severi proved the existence and regularity of $V _ { | L | , \\delta } ( S )$ in [13]. The description of the tangent space is due to Severi and later to Zariski [15]. The existence and regularity of $V _ { | L | , \\delta } ( S )$ when $s$ is of general type has been studied in [4] and [3]. Further regularity results are provided in [10]. More recently Severi varieties on K3 surfaces have received a lot of attention for many reasons. In this case Severi varieties are known to be regular (cf. [14]) and are nonempty on general K3 surfaces by Mumford and Chen (cf. [2, 12]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,795,881,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As far as we know, Severi varieties on Enriques surfaces have not been studied yet, apart from [8, Thm. 4.12] which limits the singularities of a general member of the Severi variety $V _ { | L | } ^ { g }$ of irreducible genus $g$ curves in $| L |$ , and gives a sufficient condition for the density of the latter in the Severi variety $V _ { | L | , p - g }$ of $( p - g )$ -nodal curves. In particular, the existence problem is mainly open and we intend to treat it in a forthcoming article. The result of this paper is Proposition 1, which answers the regularity question for Severi varieties of nodal curves on Enriques surfaces. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[740,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Ciliberto et al. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"37","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"30 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,763,111]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 Regularity of Severi Varieties on Enriques Surfaces ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,133,879,186]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $s$ be a smooth Enriques surface, i.e. a smooth complex surface with nontrivial canonical bundle $\\omega _ { S } \\not \\cong O _ { S }$ , such that $\\omega _ { S } ^ { \\otimes 2 } \\simeq { \\cal O } _ { S }$ and $H ^ { 1 } ( O _ { S } ) = 0$ . We denote linear (resp. numerical) equivalence by $\\sim ( \\mathrm { r e s p . } \\equiv )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,187,881,331]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $L$ be a line bundle on $s$ such that $L ^ { 2 } \\ > \\ 0$ . It is well-known that $| L |$ contains smooth, irreducible curves if and only if it contains irreducible curves (see [5, Thm. 4.1 and Prop. 8.2]); in other words, on Enriques surfaces the Bertini linear systems are the linear systems that contain irreducible curves. Moreover, by [6, Prop. 2.4], this is equivalent to $L$ being nef and not of the form $L \\sim P + R$ , with $| P |$ an elliptic pencil and $R$ a smooth rational curve such that $P \\cdot R = 2$ (in which case $p = 2 ,$ ). If $| L |$ is a Bertini linear system, the adjunction formula, the Riemann– Roch theorem, and Mumford vanishing yield that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[318,345,678,369]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0092/d3d819fb5bdb8cac17274936d8635a2afe56709562c7efaa8707d4787ad7f0a6.jpg","list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nL ^ { 2 } = 2 ( p - 1 ) \\mathrm { a n d } \\mathrm { d i m } | L | = p - 1\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,385,275,402]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(see, e.g., [5, 7]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[143,403,728,421]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $K _ { S }$ be the canonical divisor. It defines an étale double cover ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[435,438,562,456]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"1019/5d1003755db4fd418dbb82d15fb62f642d48daea415321f12be5728f18310cb9.jpg","list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\pi : X \\longrightarrow S\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,474,881,546]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $X$ is a smooth, projective $K 3$ surface (that is, $\\omega _ { X } \\simeq O _ { X }$ and $H ^ { 1 } ( O _ { X } ) = 0 ;$ ), endowed with a fixed-point-free involution $\\iota$ , which is the universal covering of $s$ . Conversely, the quotient of any- $K 3$ surface by a fixed-point-free involution is an Enriques surface. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,548,881,620]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $C \\subset S$ be a reduced and irreducible curve of genus $g \\ge 2$ . We will henceforth denote by $\\nu _ { C } : { \\widetilde { C } } \\to C$ the normalization of $C$ and define $\\eta _ { \\cal C } : = { \\cal O } _ { \\cal C } ( K _ { \\cal S } ) =$ $O _ { C } ( - K _ { S } )$ , a nontrivial 2-torsion element in $\\operatorname { P i c } ^ { 0 } C$ , and $\\eta _ { \\widetilde { C } } : = \\nu _ { C } ^ { * } \\eta _ { C }$ . The fact that $\\eta _ { C }$ is nontrivial follows from the cohomology of the restriction sequence ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[232,636,758,656]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0746/e008f4d9bc83d24c3fec86e6de875445b9a2d2e770bb4b5a1388f6b8a1a8092b.jpg","list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n0 \\longrightarrow O _ { S } ( K _ { S } - C ) \\longrightarrow O _ { S } ( K _ { S } ) \\longrightarrow \\eta _ { C } \\longrightarrow 0 ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,677,881,714]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"which yields $h ^ { 0 } ( \\eta _ { C } ) = h ^ { 1 } ( K _ { S } - C ) = h ^ { 1 } ( C ) = 0$ , the latter vanishing as $C$ is big and nef. One has the fiber product ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[382,729,615,825]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0528/b3df77c5d4ed74a669184ebace98cdba893f2d192aed18400449e36787d1a32f.jpg","list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l r } & { } & { ( \\pi ^ { - 1 } C ) \\times _ { C } \\widetilde { C } \\longrightarrow \\widetilde { C } } \\\\ & { } & { \\downarrow } \\\\ & { } & { ( \\pi ^ { - 1 } C ) \\xrightarrow [ ] { } { \\times _ { C } \\widetilde { C } } \\longrightarrow { \\cal C } , } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,52,412,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Severi Varieties on Enriques Surfaces ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"38","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[612,88,876,539]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 1 $\\\\eta _ { \\\\tilde { C } } = \\\\nu _ { C } ^ { * } ( \\\\eta _ { C } ) \\\\neq 0$ \",\"Fig. 2 $\\\\eta _ { \\\\tilde { C } } = \\\\nu _ { C } ^ { * } ( \\\\eta _ { C } ) = 0$ \"]","image_footnote":"[]","img_path":"images/0534/faac3f0347d90c1ac08b8026dbe25ba5e2be076d355ba4c2ab0c965008a9b639.jpg","list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[118,567,881,623]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where π| 1 and the upper horizontal map are the double coverings induced respectively by- - $\\eta _ { C }$ and $\\eta _ { \\widetilde { C } }$ . By standard results on coverings of complex manifolds (cf. [1, Sect. I.17]), two cases may happen:- - ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,630,881,704]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"• $\\eta _ { \\widetilde { C } } \\nsimeq O _ { \\widetilde { C } }$ and $\\pi ^ { - 1 } C$ is irreducible, as in Fig. 1; $\\eta _ { \\widetilde { C } } \\simeq O _ { \\widetilde { C } }$ and $\\pi ^ { - 1 } C$ consists of two irreducible components conjugated by the involution $\\iota$ . These two components are not isomorphic to $C$ , as $\\eta _ { C }$ is nontrivial, as in Fig. 2 (each component of $\\tilde { C }$ is a partial normalization of $C$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,713,881,785]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As mentioned in the Introduction, it is well-known that any irreducible component of a Severi variety on a $K 3$ surface is regular when nonempty (see, e.g., [4, Ex. 1.3]; see also [8, $\\ S 4 . 2 ]$ ). The corresponding result on Enriques surfaces is the following. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,785,879,821]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"First note that, in the above notation, the dimension of the Severi variety of genus\u0003 \u0004 $g = p _ { g } ( C )$ curves in $| L | = | C |$ at the point $[ C ]$ satisfies the inequalities ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[179,837,767,888]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0002/c245b900dc7bed6c46ab74675404cfb34f8c79576bf895ba0f1e6a34ecf380e7.jpg","list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\ng - 1 \\leq \\dim _ { { [ C ] } } \\bigl ( V _ { | L | } ^ { g } \\bigr ) \\leq h ^ { 0 } ( \\omega _ { \\tilde { C } } \\otimes \\eta _ { \\tilde { C } } ) = \\left\\{ g - 1 \\begin{array} { l l } { \\mathrm { i f } \\eta _ { \\tilde { C } } \\not \\simeq O _ { \\tilde { C } } } \\\\ { g } & { \\mathrm { i f } \\eta _ { \\tilde { C } } \\simeq O _ { \\tilde { C } } } \\end{array} \\right.\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,54,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[740,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"39","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Ciliberto et al. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,142]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(see [8, ineq. (2.6) and Lem. 2.3]). Our result implies that the second inequality in (3) is in fact an equality when $C$ is nodal, and gives a concrete geometric description of the situation in both cases. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,150,881,225]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 1 Let $L$ be a Bertini linear system, with $L ^ { 2 } > 0$ , on a smooth Enriques surface $S$ . Then the Severi variety $V _ { | L | , \\delta } ( S )$ is smooth and every irreducible component $V \\subseteq V _ { | L | , \\delta } ( S )$ has either dimension $g \\mathrm { ~ - ~ } 1$ or $g$ ; in the former case the component is regular. Furthermore, with the notation introduced above, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,232,876,270]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. for any curve $C$ in a $( g - 1 )$ -dimensional irreducible component $V$ , $\\pi ^ { - 1 } C$ is irreducible (whence an element in $V _ { | \\pi ^ { * } L | , 2 \\delta } ( X ) )$ ; ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,270,881,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. for any $g$ -dimensional component $V$ , there is a line bundle $L ^ { \\prime }$ on $X$ with $( L ^ { \\prime } ) ^ { 2 } =$ $2 ( p - d ) - 2$ and $L ^ { \\prime } \\cdot \\iota ^ { * } L ^ { \\prime } = 2 d$ for some integer $d$ satisfying ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[432,319,587,358]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0288/2241f5f8760b7d9848a9fe07a9a32b6838ae862918284b550be5e29695b24063.jpg","list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\frac { p - 1 } { 2 } \\leq d \\leq \\delta ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,370,879,462]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"such that $\\pi ^ { * } L \\simeq L ^ { \\prime } \\otimes \\iota ^ { * } L ^ { \\prime }$ , and the curves parametrized by $V \\subseteq V _ { | L | , \\delta } ( S )$ are the birational images by $\\pi$ of the curves in $V _ { | L ^ { \\prime } | , \\delta - d } ( X )$ intersecting their conjugates by $\\iota$ transversely (in 2d points). In other words, for any $[ C ] \\in V$ , we have $\\pi ^ { - 1 } C = Y + \\iota ( Y )$ , with $[ Y ] \\in V _ { | L ^ { \\prime } | , \\delta - d } ( X )$ and $[ \\iota ( Y ) ] \\in V _ { | \\iota ^ { * } L ^ { \\prime } | , \\delta - d } ( X )$ intersecting transversely. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,462,876,500]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Furthermore, if $L ^ { \\prime } \\simeq \\iota ^ { * } L ^ { \\prime }$ , which is the case if $s$ is general in moduli, then $\\begin{array} { r } { d = \\frac { p - 1 } { 2 } } \\end{array}$ and $L \\sim 2 M$ , for some $M \\in \\operatorname { P i c } S$ such that $M ^ { 2 } = d$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,507,881,615]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We will henceforth refer to components of dimension $g - 1$ as regular and the ones of dimension $g$ as nonregular. Note however that from a parametric perspective the Severi variety has the expected dimension and is smooth in both cases, as the fact that (3) is an equality indicates; we do not dwell on this here, and refer to [8] for a discussion of the differences between the parametric and Cartesian points of view (the latter is the one we adopted in this text). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,617,881,669]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note that Proposition 1 does not assert that the Severi variety $V _ { | L | , \\delta }$ is necessarilynon-empty: in such a situation, $V _ { | L | , \\delta }$ does not have any irreducible component and the statement is empty. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,677,881,734]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Pick any curve $C$ in an irreducible component $V$ of $V _ { | L | , \\delta } ( S )$ . Let $f : \\widetilde { S } S$ be the blow-up of $s$ at $N$ , the scheme of the δ nodes of - $C$ , denote by e the (total) exceptional divisor and by - $\\widetilde { C }$ the strict transform of $C$ . Thus $f _ { | \\widetilde { C } } = \\nu _ { C }$ and we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[318,749,678,773]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0339/0caaee88fbc3f402e816aa356fb2677a6f93be4fdf90e2615f11532335421a62.jpg","list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { \\widetilde { S } } \\sim f ^ { * } K _ { S } + \\mathfrak { e } \\mathrm { ~ a n d ~ } \\widetilde { C } \\sim f ^ { * } C - 2 \\mathfrak { e } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,791,394,807]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From the restriction sequence ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[232,822,763,849]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0577/e2ee572338439d2dcdbe19fdf8f41126cd202252c9bfaeb4632a0d1cfc8b2b7a.jpg","list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n0 \\longrightarrow { \\cal O } _ { \\widetilde { S } } ( \\mathfrak { c } ) \\longrightarrow { \\cal O } _ { \\widetilde { S } } ( \\widetilde { C } + \\mathfrak { c } ) \\longrightarrow \\omega _ { \\widetilde { C } } ( \\eta _ { \\widetilde { C } } ) \\longrightarrow 0 \\longrightarrow 0 \\longrightarrow 0 \\longrightarrow 0 .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,52,412,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Severi Varieties on Enriques Surfaces ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"40","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,191,106]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"we find ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[150,121,849,232]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0063/85f1c7aba00ced4bd9583d643d0931cf6b442681bb43488d29eb54e13dccfd9c.jpg","list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\dim T _ { [ C ] } V _ { | L | , \\delta } ( S ) = \\dim | L \\otimes \\mathcal { T } _ { N } | = h ^ { 0 } ( L \\otimes \\mathcal { T } _ { N } ) - 1 = h ^ { 0 } ( f ^ { * } L - \\mathfrak { e } ) - 1 } \\\\ & { \\qquad = h ^ { 0 } ( O _ { \\widetilde { S } } ( \\widetilde { C } + \\mathfrak { e } ) ) - 1 = h ^ { 0 } ( \\omega _ { \\widetilde { C } } ( \\eta _ { \\widetilde { C } } ) ) } \\\\ & { \\qquad = \\left\\{ \\begin{array} { l l } { g - 1 , } & { \\mathrm { i f ~ } \\eta _ { \\widetilde { C } } \\not \\cong O _ { \\widetilde { C } } , } \\\\ { g , } & { \\mathrm { i f ~ } \\eta _ { \\widetilde { C } } \\simeq O _ { \\widetilde { C } } . } \\end{array} \\right. } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[123,244,879,280]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the upper case, by (1), we have that $V _ { | L | , \\delta }$ is smooth at $[ C ]$ of dimension $g - 1 = p - \\delta - 1 = \\dim | L \\otimes \\mathcal { T } _ { N } |$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,282,881,426]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Assume next that we are in the lower case. Then, by the discussion prior to the proposition, we have $\\pi ^ { - 1 } C = Y + \\iota ( Y )$ for an irreducible curve $Y$ on $X$ , such that $\\pi$ maps both $Y$ and $\\iota ( Y )$ birationally, but not isomorphically, to $C$ . In particular, $Y$ and $\\iota ( Y )$ have geometric genus $p _ { g } ( Y ) = p _ { g } ( \\iota ( Y ) ) = p _ { g } ( C ) = p - \\delta = g$ . Set $L ^ { \\prime } : =$ $O _ { X } ( Y )$ and $2 d : = Y \\cdot \\iota ( Y )$ . Note that $d$ is an integer because, if $y = \\iota ( x ) \\in Y \\cap \\iota ( Y )$ , then $\\iota ( y ) = x \\in Y \\cap \\iota ( Y )$ . Since $Y \\simeq \\iota ( Y )$ and $\\pi$ is étale, both $Y$ and $\\iota ( Y )$ are nodal with $\\delta - d$ nodes and they intersect transversely at $2 d$ points, which are pairwise conjugate by $\\iota$ , and therefore map to $d$ nodes of $C$ . Hence $d \\leq \\delta$ . We have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[223,442,774,462]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0892/5edff3eaff0bbee0d27868c9384c5acc798c5203f477f482200070673a1fff53.jpg","list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\np _ { a } ( Y ) = p _ { a } ( \\iota ( Y ) ) = g + \\delta - d = p - \\delta + \\delta - d = p - d ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,480,193,496]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"whence ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[391,512,608,534]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0943/aa64b6913dfda5f8ba5c87ace187ea3c7acaf2f6b9542199447597c63dadf740.jpg","list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n( L ^ { \\prime } ) ^ { 2 } = 2 ( p - 1 - d ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,552,471,569]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By the Hodge index theorem, we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[198,584,797,620]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0017/ca4ccb7c9352563f8a82820575e165e78b8fb93f19d59f699d6f9053c44ec009.jpg","list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n4 ( p - 1 - d ) ^ { 2 } = \\left( ( L ^ { \\prime } ) ^ { 2 } \\right) ^ { 2 } = ( L ^ { \\prime } ) ^ { 2 } ( \\iota ^ { * } L ^ { \\prime } ) ^ { 2 } \\leq \\left( L ^ { \\prime } \\cdot \\iota ^ { * } L ^ { \\prime } \\right) ^ { 2 } = 4 d ^ { 2 } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,635,309,651]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"whence $p - 1 \\leq 2 d$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,651,881,761]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By the regularity of Severi varieties on $K 3$ surfaces, any irreducible component of $V _ { | L ^ { \\prime } | , \\delta - d } ( X )$ has dimension $\\dim | L ^ { \\prime } | - ( \\delta - d ) \\ : = \\ : p _ { g } ( Y ) \\ : = \\ : g$ . Hence, $V$ is $g$ -dimensional; more precisely, the curves parameterized by $V$ are the (birational) images by $\\pi$ of the curves in an irreducible component of $V _ { | L ^ { \\prime } | , \\delta - d } ( X )$ intersecting their conjugates by $\\iota$ transversely (in $2 d$ points). By (4), it also follows that dim $V =$ dim $T _ { [ C ] } V _ { | L | , \\delta } ( S )$ , so that $[ C ]$ is a smooth point of $V _ { | L | , \\delta } ( S )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,761,881,851]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To prove the final assertion of the proposition, observe that, by the regularity of Severi varieties on $K 3$ surfaces, we may deform $Y$ and $\\iota ( Y )$ on $X$ to irreducible curves $Y ^ { \\prime }$ and $\\iota ( Y ^ { \\prime } )$ with any number of nodes $\\leq \\delta - d$ and intersecting transversally in $2 d$ points; in particular, we may deform $Y$ and $\\iota ( Y )$ to smooth curves $Y ^ { \\prime }$ and $\\iota ( Y ^ { \\prime } )$ . Thus, $C ^ { \\prime } : = \\pi ( Y ^ { \\prime } )$ is a member of $V _ { | L | , d }$ , whence of geometric genus $p - d$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"34 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[740,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"41","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Ciliberto et al. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since d $\\operatorname* { l i m } | Y ^ { \\prime } | = p _ { a } ( Y ^ { \\prime } ) = p _ { g } ( C ^ { \\prime } ) = p _ { a } ( C ^ { \\prime } ) - d = p - d$ , the component of $V _ { | L | , d }$ containing $[ C ^ { \\prime } ]$ has dimension $\\mathrm { d i m } | L | - d + 1 = p - d$ . We thus have di $\\mathrm { m } | { \\cal L } \\otimes { \\mathcal { T } } _ { N ^ { \\prime } } | = \\dim | { \\cal L } | - d + 1$ , where $N ^ { \\prime }$ is the set of $d$ nodes of $C ^ { \\prime }$ , hence $N ^ { \\prime }$ does not impose independent conditions on $| L |$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,163,881,270]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Assume now that $L ^ { \\prime } \\simeq \\iota ^ { * } L ^ { \\prime }$ , which—as is well-known (see, e.g., [9, $\\ S 1 1 ]$ )—is the case occurring for generic $s$ , as then Pic $X$ is precisely the invariant part under $\\iota$ of $H _ { 2 } ( X , \\mathbb { Z } )$ . Then $2 d = L ^ { \\prime } \\cdot \\iota ^ { * } L ^ { \\prime } = ( L ^ { \\prime } ) ^ { 2 } = 2 ( p - 1 - d )$ , so that $p - 1 = 2 d$ . Since $L ^ { 2 } = 2 ( p - 1 ) = 4 d$ and $N ^ { \\prime }$ does not impose independent conditions on $| L |$ , by [11, Prop. 3.7] there is an effective divisor $D \\subset S$ containing $N ^ { \\prime }$ satisfying $L - 2 D \\geq 0$ and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[341,285,656,322]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0983/f35e67c5f67d551b8cf4ab47a5ddbbbfef7837e5cb2b7ff86a6bc3f43b391b41.jpg","list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nL \\cdot D - d \\leq D ^ { 2 } \\leq \\frac { ( i ) } { 2 } L \\cdot D \\stackrel { ( i i ) } { \\leq } d ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,334,879,408]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"with equality in (i) or (ii) only if $L \\equiv 2 D$ ; moreover, since $L - 2 D \\ge 0$ , the numerical equivalence $L \\equiv 2 D$ implies the linear equivalence $L \\sim 2 D$ . Now since $N ^ { \\prime } \\subset D$ , we must have $L \\cdot D = C ^ { \\prime } \\cdot D \\geq 2 d$ , hence the inequalities in (6) are all equalities, and thus $D ^ { 2 } = d$ and $L \\sim 2 D$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,415,881,469]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The following corollary is a straightforward consequence of Prop. 1 and the fact that the nodes on curves in a regular component in a Severi variety (on any surface and in particular on a K3 surface) can be independently smoothened. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,480,879,534]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 1 If a Severi variety $V _ { | L | , \\delta }$ on an Enriques surface has a regular (resp., nonregular) component, then for any $0 \\leq \\delta ^ { \\prime } \\leq \\delta$ (resp., $d \\leq \\delta ^ { \\prime } \\leq \\delta$ , with $d$ as in Prop. 1), also $V _ { | L | , \\delta ^ { \\prime } }$ contains a regular (resp., nonregular) component. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,581,248,599]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[127,621,881,893]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Barth, W., Hulek, K., Peters, C., van de Ven, A.: Compact Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3rd edn., Folge, vol. 4. Springer, Berlin (2004) \n2. Chen, X.: Rational curves on $K 3$ surfaces. J. Algebraic Geom. 8, 245–278 (1999) \n3. Chiantini, L., Ciliberto, C.: On the Severi varieties of surfaces in $\\mathbb { P } ^ { 3 }$ . J. Algebraic Geom. 8, 67–83 (1999) \n4. Chiantini, L., Sernesi, E.: Nodal curves on surfaces of general type. Math. Ann. 307, 41–56 (1997) \n5. Cossec, F.R.: Projective models of Enriques surfaces. Math. Ann. 265, 283–334 (1983) \n6. Cossec, F.R.: On the Picard group of Enriques surfaces. Math. Ann. 271, 577–600 (1985) \n7. Cossec, F.R., Dolgachev, I.V.: Enriques Surfaces. I. Progress in Mathematics, vol. 76. Birkhäuser, Boston (1989) \n8. Dedieu, T., Sernesi, E.: Equigeneric and equisingular families of curves on surfaces. Publ. Mat. 61, 175–212 (2017) \n9. Dolgachev, I.V., Kondo, S.: Moduli of $K 3$ surfaces and complex ball quotients. In: Arithmetic and Geometry Around Hypergeometric Functions. Progress in Mathematics, vol. 260, pp. 43– 100. Birkhäuser, Basel (2007) \n10. Flamini, F.: Some results of regularity for Severi varieties of projective surfaces. Commun. Algebra 29, 2297–2311 (2001) \n11. Knutsen, A.L.: On kth-order embeddings of $K 3$ surfaces and Enriques surfaces. Manuscr. Math. 104, 211–237 (2001) \n12. Mori, S., Mukai, S.: The uniruledness of the moduli space of curves of genus 11. In: Proceedings of Algebraic Geometry. Tokyo/Kyoto. Lecture Notes in Mathematics, vol. 1016, pp. 334–353. Springer, Berlin (1983) \n13. Severi, F., Vorlesungen über algebraische Geometrie. Teubner, Leipzig (1921) \n14. Tannenbaum, A.: Families of curves with nodes on K3 surfaces. Math. Ann. 260, 239–253 (1982) \n15. Zariski, O.: Dimension theoretic characterization of maximal irreducible sistems of plane nodal curves. Am. J. Math. 104, 209–226 (1982) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,412,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Severi Varieties on Enriques Surfaces ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"42","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,91,881,240]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"43","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,54,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"43","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[740,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"43","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Ciliberto et al. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,744,139]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Travel Guide to the Canonical Bundle Formula ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,175,444,193]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Enrica Floris and Vladimir Lazic´ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,291,879,325]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract We survey known results on the canonical bundle formula and its applications in algebraic geometry. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,345,876,361]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords Canonical bundle formula $\\cdot$ Moduli divisor $\\cdot$ Minimal Model Program ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,399,309,420]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,444,881,496]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Minimal Model Program (MMP) predicts that every projective pair with mild singularities is birationally built out of three classes of pairs: those whose log canonical classes are ample, numerically trivial or anti-ample. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,498,879,588]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"More precisely, let $( X , \\Delta )$ be a log canonical pair. Then there should exist a birational contraction $\\varphi \\colon ( X , \\Delta ) \\quad \\longrightarrow \\quad ( X _ { \\mathrm { m i n } } , \\Delta _ { \\mathrm { m i n } } )$ together with a fibration $f \\colon ( X _ { \\mathrm { m i n } } , \\Delta _ { \\mathrm { m i n } } ) \\ \\ X _ { \\mathrm { c a n } }$ so that $K _ { X _ { \\operatorname* { m i n } } } + \\Delta _ { \\operatorname* { m i n } } \\sim _ { \\mathbb { Q } } f ^ { * } A$ , for a suitable ample $\\mathbb { Q }$ -divisor $A$ on $X _ { \\mathrm { { c a n } } }$ . Note that the Iitaka dimension of $K _ { X } + \\Delta$ restricted to a general fibre of the composed map $f \\circ \\varphi$ is zero. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,588,876,623]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is a natural and important question to determine whether singularities of the MMP are preserved in this process. The singularities of $( X _ { \\mathrm { m i n } } , \\Delta _ { \\mathrm { m i n } } )$ are the same as those of $( X , \\Delta )$ . On the other hand, it remains an open problem whether there exists a boundary divisor $\\Delta _ { \\mathrm { { c a n } } }$ on $X _ { \\mathrm { { c a n } } }$ such that $( X _ { \\mathrm { { c a n } } } , \\Delta _ { \\mathrm { { c a n } } } )$ is log canonical and $K _ { X _ { \\mathrm { m i n } } } + \\Delta _ { \\mathrm { m i n } } \\sim _ { \\mathbb { Q } } f ^ { * } ( X _ { \\mathrm { c a n } } + \\Delta _ { \\mathrm { c a n } } ) ;$ ; in other words, whether singularities of $( X , \\Delta )$ descend to the canonical model $X _ { \\mathrm { { c a n } } }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,897,690,941]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_4 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,768,815,825]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Floris $( \\boxtimes )$ 0 Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348, Futuroscope Chasseneuil Cedex, France e-mail: enrica.floris@univ-poitiers.fr ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[804,91,876,139]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,836,724,878]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"V. Lazic´ Fachrichtung Mathematik, Universität des Saarlandes, Saarbrücken, Germany e-mail: lazic@math.uni-sb.de ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,675,881,750]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lazic was supported by the DFG-Emmy-Noether-Nachwuchsgruppe “Gute Strukturen in der ´ höherdimensionalen birationalen Geometrie”. During the initial stage of this project Floris was funded by the Max Planck Institute for Mathematics in Bonn. We would like to thank A. Golota for pointing out the reference [17] and for useful conversations and the referee for his/her careful reading. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,899,876,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"44","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,879,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,162,881,270]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"When the singularities of $( X , \\Delta )$ are klt, it is known by a work of Ambro and Kawamata that such a divisor exists; this result has already had numerous consequences in birational geometry. However, the proof is not constructive: more precisely, one loses control of the coefficients of $\\Delta$ at the last step. It is desirable that the singularities of $( X _ { \\mathrm { { c a n } } } , \\Delta _ { \\mathrm { { c a n } } } )$ reflect in a canonical way the singularities of $( X , \\Delta )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[143,270,610,286]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In general, with notation as above, it is known that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[353,306,640,325]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"1018/93228e2e75606fecb04b90befe10517ae58ef01df6bd9a23f71ddd8799193147.jpg","list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nA \\sim _ { \\mathbb { Q } } K _ { X _ { \\mathrm { c a n } } } + B _ { X _ { \\mathrm { c a n } } } + M _ { X _ { \\mathrm { c a n } } } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,340,881,412]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $B _ { X _ { \\mathrm { c a n } } }$ —the discriminant—is closely related to the singularities of $f$ , and the divisor $M _ { X _ { \\mathrm { c a n } } }$ —the moduli divisor—conjecturally carries information on the birational variation of the fibres of $f$ . A formula of this form is called the canonical bundle formula. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,414,881,468]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This paper is an attempt to give an account of all the known results on the canonical bundle formula, and to serve as a guide to those wishing to study this important subject. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,513,410,534]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 Lc-Trivial Fibrations ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,557,879,593]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We work over $\\mathbb { C }$ . We denote ${ \\mathsf { b y } } \\equiv , \\sim$ and ${ \\sim } _ { \\mathbb { Q } }$ the numerical, linear and $\\mathbb { Q }$ -linear equivalence of divisors respectively. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,594,881,683]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For a Weil $\\mathbb { Q }$ -divisor $D = \\sum d _ { i } D _ { i }$ , for a real number $r$ we denote $D ^ { \\leq r } : =$ $\\textstyle \\sum _ { d _ { i } \\leq r } d _ { i } D _ { i }$ . If $f \\colon X \\ \\to \\ Y$ is a proper surjective morphism between normal varieties and $D$ is a Weil $\\mathbb { R }$ -divisor on $X$ , then $D _ { v }$ and $D _ { h }$ denote the vertical and the horizontal part of $D$ with respect to $f$ . In this setup, we say that $D$ is $f$ -exceptional if codimY Supp $f ( D ) \\geq 2$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,684,879,737]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section we introduce the main topic of this survey—lc-trivial fibrations. In this section we define them and give some examples which will accompany us through the paper. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,738,879,773]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We first need to introduce singularities of pairs. This is by now a standard topic in higher dimensional birational geometry, and good references are [27] and [26]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,774,881,827]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A pair $( X , \\Delta )$ consists of a normal variety $X$ and a Weil\u0003 $\\mathbb { Q }$ -divisor $\\Delta$ such that\u0004 $K _ { X } + \\Delta$ is $\\mathbb { Q }$ -Cartier. A pair $( X , \\Delta )$ is log smooth if $X$ is smooth and the support of $\\Delta$ is a simple normal crossings divisor. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,828,879,882]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A log resolution of a pair $( X , \\Delta )$ is a birational morphism $f \\colon Y \\to X$ such that the exceptional locus $\\operatorname { E x c } ( f )$ is a divisor and the pair $\\left( Y , \\operatorname { S u p p } ( f _ { * } ^ { - 1 } \\Delta + \\operatorname { E x c } ( f ) \\right)$ is log smooth. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[706,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Floris and V. Lazic´ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,54,141,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"45","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,881,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $( X , \\Delta )$ be a pair and if $\\pi \\colon Y X$ is a birational morphism with $Y$ normal, we can write ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[284,138,710,168]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0003/b91d006a5f4a05012ff26f47da28110b52d79ca533f33856578bf96bd12e4eac.jpg","list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { Y } \\sim _ { \\mathbb { Q } } \\pi ^ { * } ( K _ { X } + \\Delta ) + \\sum a ( E _ { i } , X , \\Delta ) \\cdot E _ { i } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,180,881,270]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $E _ { i } ~ \\subseteq ~ Y$ are distinct prime divisors and the numbers $a ( E _ { i } , X , \\Delta ) \\in \\mathbb { Q }$ are called discrepancies. The order of vanishing at the generic point of each $E _ { i }$ defines a geometric valuation on $\\mathbb { C } ( X )$ . The pair $( X , \\Delta )$ is $k l t$ , respectively log canonical, if $a ( E , X , \\Delta ) > - 1$ , respectively $a ( E , X , \\Delta ) \\geq - 1$ , for every geometric valuation $E$ over $X$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,271,881,325]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Much of what we say in this paper can be generalised to pairs $( X , \\Delta )$ , where $\\Delta$ is allowed to have real coefficients. We stick to rational divisors mostly for reasons of clarity and simplicity. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,370,523,391]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.1 Definition and First Examples ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,414,876,450]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The objects for which we can write a canonical bundle formula are called lc-trivial fibrations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,459,876,496]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 2.1 Let $( X , \\Delta )$ be a pair. A morphism $f \\colon ( X , \\Delta ) \\ \\ Y$ to a normal projective variety $Y$ is a klt-trivial, respectively $l c$ -trivial, fibration if: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,504,881,578]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(a) $f$ is a surjective morphism with connected fibres, \n(b) $( X , \\Delta )$ has klt, respectively log canonical, singularities over the generic point of $Y$ , \n(c) there exists a $\\mathbb { Q }$ -Cartier $\\mathbb { Q }$ -divisor $D$ on $Y$ such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[428,594,605,615]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0247/7674071a4942d50a218e901c1182f42569f0f3fde4cfa5b00797c67cf5dd4db7.jpg","list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { X } + \\Delta \\sim _ { \\mathbb { Q } } f ^ { * } D ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,630,881,698]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(d) there exists a log resolution $\\pi ^ { \\prime } \\colon X ^ { \\prime } \\to X$ of $( X , \\Delta )$ such that, if $\\mathcal { E }$ is the set of all geometric valuations over $X$ which are defined by a prime divisor $E$ on $X ^ { \\prime }$ such that $a ( E , X , \\Delta ) > - 1$ , and if we denote $\\Xi ^ { \\prime } = \\sum _ { E \\in { \\mathcal { E } } } a ( E , X , \\Delta ) \\cdot E$ , then ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[389,713,646,734]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0643/0f5aa705ab313d8e8912b7e405c4d0eeb0a73eeef8ed7081e2175704007fcb78.jpg","list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf { r k } ( f \\circ \\pi ^ { \\prime } ) _ { * } \\mathcal { O } _ { X ^ { \\prime } } ( [ \\Xi ^ { \\prime } ] ) = 1 .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,750,881,786]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Terminology 2.2 In [2], klt-trivial fibrations as in Definition 2.1 are called lc-trivial fibrations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,797,879,833]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 2.3 We make a few comments on the condition (d) in Definition 2.1. For simplicity, assume that the pair $( X , \\Delta )$ is klt. Note that then ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[364,848,635,870]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0082/1e0b1e4e8f91b949478ef068aab22b91588ce0f464416353c012ad821b714e7b.jpg","list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\Xi ^ { \\prime } \\sim _ { \\mathbb { Q } } K _ { X ^ { \\prime } } - \\pi ^ { \\prime * } ( K _ { X } + \\Delta ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,51,505,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Travel Guide to the Canonical Bundle Formula ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,52,879,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"46","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"39 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,876,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The divisor $\\lceil \\Xi ^ { \\prime } \\rceil$ is effective on the generic fibre of the morphism $f \\circ \\pi ^ { \\prime }$ by (b), hence $\\mathbf { r k } ( f \\circ \\pi ^ { \\prime } ) _ { * } \\mathcal { O } _ { X ^ { \\prime } } ( \\lceil \\Xi ^ { \\prime } \\rceil ) \\geq 1 .$ . Therefore, the point of (d) is the opposite inequality. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,127,881,234]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The most important case to keep in mind is that when the divisor $\\Delta$ is effective on the generic fibre: indeed, in that case the divisor $\\lceil \\Xi ^ { \\prime } \\rceil$ is an effective exceptional divisor on the generic fibre of $f \\circ \\pi ^ { \\prime }$ , and the condition (d) is immediate. However, in order to be able to study lc-trivial fibrations by applying basic operations of birational geometry in Sect. 2.2, it is crucial to allow divisors $\\Delta$ with negative coefficients. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,234,881,325]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"One more thing to notice is that if (d) holds for a log resolution $\\pi ^ { \\prime }$ , then it holds on any log resolution $\\pi ^ { \\prime \\prime } \\colon X ^ { \\prime \\prime } \\to X$ which factors through $\\pi ^ { \\prime }$ . Indeed, define the divisor $\\Xi ^ { \\prime \\prime }$ on $X ^ { \\prime \\prime }$ analogously as in Definition 2.1, and let $\\theta \\colon X ^ { \\prime \\prime } X ^ { \\prime }$ be the induced morphism. Since $X ^ { \\prime }$ and $X ^ { \\prime \\prime }$ are smooth, there exists an integral effective divisor $E$ such that $K _ { X ^ { \\prime \\prime } } \\sim \\theta ^ { * } K _ { X ^ { \\prime } } + E$ . Thus, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[170,339,826,391]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0067/f9de043feebd0fa0a7c0778ac2207454d73e3c125db7c138f34bdf38279d6902.jpg","list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { f _ { * } \\pi _ { * } ^ { \\prime \\prime } \\mathcal { O } _ { X ^ { \\prime \\prime } } ( \\lceil \\Xi ^ { \\prime \\prime } \\rceil ) = f _ { * } \\pi _ { * } ^ { \\prime \\prime } \\mathcal { O } _ { X ^ { \\prime \\prime } } ( \\lceil \\theta ^ { * } \\Xi ^ { \\prime } + E \\rceil ) = f _ { * } \\pi _ { * } ^ { \\prime \\prime } \\mathcal { O } _ { X ^ { \\prime \\prime } } ( \\lceil \\theta ^ { * } \\Xi ^ { \\prime } \\rceil + E ) } \\\\ & { \\qquad \\subseteq f _ { * } \\pi _ { * } ^ { \\prime \\prime } \\mathcal { O } _ { X ^ { \\prime \\prime } } ( \\theta ^ { * } \\lceil \\Xi ^ { \\prime } \\rceil + E ) = f _ { * } \\pi _ { * } ^ { \\prime } \\mathcal { O } _ { X ^ { \\prime } } ( \\lceil \\Xi ^ { \\prime } \\rceil ) , } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,405,840,423]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where we used that $\\lceil \\theta ^ { * } \\Xi ^ { \\prime } \\rceil \\leq \\theta ^ { * } \\lceil \\Xi ^ { \\prime } \\rceil$ . Therefore, $\\mathrm { r k } ( f \\circ \\pi ^ { \\prime \\prime } ) _ { * } \\mathcal { O } _ { X ^ { \\prime \\prime } } ( [ \\Xi ^ { \\prime \\prime } ] ) = 1$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,424,876,459]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In general, one can show that (d) is independent of the choice of the resolution by using [14, Lemma 2.7]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,468,876,504]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now we can formulate the canonical bundle formula associated to an lc-trivial fibration. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,513,881,567]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 2.4 Let $f \\colon ( X , \\Delta ) \\to Y$ be an lc-trivial fibration such that $K _ { X } + \\Delta \\sim _ { \\mathbb { Q } }$ $f ^ { * } D$ for some $\\mathbb { Q }$ -Cartier $\\mathbb { Q }$ -divisor $D$ on $Y$ . If $P \\subseteq Y$ is a prime divisor, the log canonical threshold of $f ^ { * } P$ with respect to $( X , \\Delta )$ is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[129,584,861,605]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\gamma _ { P } = \\operatorname* { s u p } \\{ t \\in \\mathbb { R } \\mid ( X , \\Delta + t f ^ { * } P )$ is log canonical over the generic point of $P$ }. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,621,881,675]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The condition that $( X , \\Delta + t f ^ { * } P )$ is log canonical over the generic point of \b $P$ means that for every geometric valuation $E$ over $X$ which surjects onto $P$ , we have $a ( E , X , \\Delta + t f ^ { * } P ) \\geq - 1 .$ . The discriminant of $f$ is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[394,692,603,714]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0353/273f1a6aed8b5d9d93e60297e9cf431d523c61012550f61429123a2a7a7e748b.jpg","list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { B _ { Y } = \\sum _ { P } ( 1 - \\gamma _ { P } ) P . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,731,879,767]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Fix $\\varphi \\in \\mathbb { C } ( X )$ and the smallest positive integer $r$ such that $\\begin{array} { r } { K _ { X } + \\Delta + \\frac { 1 } { r } \\operatorname { d i v } \\varphi = f ^ { * } D } \\end{array}$ . Then there exists a unique Weil $\\mathbb { Q }$ -divisor $M _ { Y }$ , the moduli part of $f$ , such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[296,780,701,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0908/a973c4f0cbdc1c11231df3236029e9b41719bcb74042a76755d49ee7d603cac0.jpg","list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { K _ { X } + \\Delta + \\frac 1 r \\operatorname { d i v } \\varphi = f ^ { * } ( K _ { Y } + B _ { Y } + M _ { Y } ) . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,821,685,839]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This formula is the canonical bundle formula associated to $f$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,846,881,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 2.5 The definition of the discriminant as above first appeared in [23, Theorem 2]. The discriminant is a Weil $\\mathbb { Q }$ -divisor on $Y$ , and it is effective if $\\Delta$ is effective. We notice that the discriminant is defined as an actual divisor, while the moduli part is defined only up to $\\mathbb { Q }$ -linear equivalence: it depends on the choice of $D$ . Further, if $G$ is a $\\mathbb { Q }$ -divisor on $Y$ , then $f \\colon ( X , \\Delta + f ^ { * } G ) \\to Y$ is an lc-trivial fibration with discriminant $B _ { Y } + G$ and moduli divisor $M _ { Y }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"40 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[706,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"47","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Floris and V. Lazic´ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,879,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,162,879,216]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note that if we are interested in proving properties of the moduli divisor of an lc-trivial fibrations as above, we may always assume that the pair $( X , \\Delta )$ is log canonical by [12, Remark 3.6], although we may not assume that $\\Delta$ is effective. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,223,879,261]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 2.6 Assume that $X$ is smooth, that $\\Delta = 0$ , that $Y$ is a curve, that $f ^ { - 1 } ( P )$ is smooth and that $f ^ { * } P = m f ^ { - 1 } ( P )$ is a multiple fibre. Then $\\begin{array} { r } { \\gamma _ { P } = \\frac { 1 } { m } } \\end{array}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,268,881,378]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 2.7 This example is historically the first example of a canonical bundle formula. Let $f \\colon X \\to C$ be an elliptic fibration, that is, $X$ is a smooth surface, $C$ is a smooth curve and a general fibre of $f$ is a smooth elliptic curve. We assume furthermore that $f$ is relatively minimal: there are no $( - 1 )$ -curves contained in the fibres of $f$ . Kodaira in [24, Theorem 6.2] classified the singular fibres of $f$ . Kodaira’s canonical bundle formula reads as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[364,394,630,414]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"1023/a66b8185be1265725ffd0b9be6f35b123f34b1db532124960a3973172faffff4.jpg","list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { X } \\sim f ^ { * } ( K _ { C } + B _ { C } + M _ { C } ) ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,432,881,486]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $B _ { C }$ is defined in terms of the classification of the singular fibres and by [24, 32] we have $1 2 M _ { C } = j ^ { * } \\mathcal { O } _ { \\mathbb { P } ^ { 1 } } ( 1 )$ , with $j \\colon C \\ \\to \\ \\mathbb { P } ^ { 1 }$ being the $j$ -invariant. For a detailed account on Kodaira’s canonical bundle formula see [4, Chapter V, $\\ S 7 \\mathrm { - } \\ S 1 3 ]$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,492,881,569]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 2.8 Let $\\ b X = \\mathbb P ^ { 1 } \\ b X \\mathbb P ^ { 1 }$ and let $D$ be a reduced divisor of bidegree $( d , k )$ with $d \\geq 2 .$ Let $\\begin{array} { r } { \\Delta = \\frac { 2 } { d } D } \\end{array}$ and let $f \\colon ( X , \\Delta ) \\to \\mathbb { P } ^ { 1 }$ be the projection onto the second factor. Then $f$ is an lc-trivial fibration. Indeed, $K _ { X } + \\Delta$ has bidegree $( - 2 , - 2 ) +$ $\\textstyle { \\frac { 2 } { d } } ( d , k ) = ( 0 , - 2 + { \\frac { 2 k } { d } } )$ and therefore is the pullback of a divisor from $\\mathbb { P } ^ { 1 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,612,435,633]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2 Base Change Property ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,656,881,710]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this subsection we investigate how canonical bundle formulas behave under base change. This will help improve the properties of the moduli part of a canonical bundle formula, at least on a sufficiently high birational model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,711,876,746]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $f \\colon ( X , \\Delta ) \\ \\to \\ Y$ is a klt-trivial fibration (respectively lc-trivial), and if we consider a base change diagram ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[398,765,601,855]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0331/59564a581e7264aa550e72c51003e72a949a1703a2882a2b2053349f92fb20b4.jpg","list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l r } & { ( X ^ { \\prime } , \\Delta ^ { \\prime } ) \\xrightarrow { \\tau } ( X , \\Delta ) } & \\\\ & { f ^ { \\prime } \\downarrow } & \\\\ & { Y ^ { \\prime } \\xrightarrow { \\rho } ( \\begin{array} { c c } { \\rho } & { \\sqrt { f } } \\\\ { \\rho } & { \\rho } \\end{array} ) } & \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,505,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"48","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Travel Guide to the Canonical Bundle Formula ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,876,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"49","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\rho$ is a proper generically finite morphism, $X ^ { \\prime }$ is the normalisation of the fibre product and $\\Delta ^ { \\prime }$ is defined so that we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[373,142,621,163]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0635/70142ecbf933f57f99715597ebde0dbbea79ac019740c9142e909802c8e5c0b4.jpg","list_items":null,"page_idx":"49","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { X ^ { \\prime } } + \\Delta ^ { \\prime } = \\tau ^ { * } ( K _ { X } + \\Delta ) ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,180,879,234]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"49","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"then $f ^ { \\prime } \\colon ( X ^ { \\prime } , \\Delta ^ { \\prime } ) \\to Y ^ { \\prime }$ is also a klt-trivial (respectively lc-trivial) fibration. In the rest of the paper, we implicitly refer to this klt-trivial fibration when writing $M _ { Y ^ { \\prime } }$ and $B _ { Y ^ { \\prime } }$ for the moduli part and discriminant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,235,876,270]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"49","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $\\rho$ is birational, then $\\rho _ { * } M _ { Y ^ { \\prime } } = M _ { Y }$ and $\\rho _ { * } B _ { Y ^ { \\prime } } = B _ { Y }$ ; in other words, these collections of divisors form $^ b$ -divisors, see for instance [2, Section 1.2]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[143,271,810,288]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"49","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The following is the base change property of canonical bundle formulas. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,297,881,351]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"49","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2.9 Let $f \\colon ( X , \\Delta ) \\to Y$ be an lc-trivial fibration. Then there exists $a$ proper birational morphism $Y ^ { \\prime } Y$ such that for every proper birational morphism $\\pi \\colon Y ^ { \\prime \\prime } \\to Y ^ { \\prime }$ we have: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,360,785,396]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"49","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(i) $K _ { Y ^ { \\prime } } + B _ { Y ^ { \\prime } }$ is a $\\mathbb { Q }$ -Cartier divisor and $K _ { Y ^ { \\prime \\prime } } + B _ { Y ^ { \\prime \\prime } } = \\pi ^ { * } ( K _ { Y ^ { \\prime } } + B _ { Y ^ { \\prime } } ) ,$ (ii) $M _ { Y ^ { \\prime } }$ is a nef $\\mathbb { Q }$ -Cartier divisor and $M _ { Y ^ { \\prime \\prime } } = \\pi ^ { * } M _ { Y ^ { \\prime } }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,405,881,495]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"49","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The first version of Theorem 2.9 is [23, Theorem 2], which essentially shows the nefness of the moduli part; this is also the point of the proof where the condition (d) in Definition 2.1 is used. Theorem 2.9 has been proved in this form for klt-trivial fibrations by Ambro [2, Theorem 0.2]. For lc-trivial fibrations, it was proved by Kollár [25, Theorem 8.3.7] and [18, Theorem 3.6], with an alternative proof in [11]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,496,879,549]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"49","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the context of the previous theorem, we say that $M _ { Y ^ { \\prime } }$ descends to $Y ^ { \\prime }$ , and we call any such $Y ^ { \\prime }$ , where additionally $B _ { Y ^ { \\prime } }$ has simple normal crossings support, an Ambro model for $f$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,551,881,818]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"49","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We give a brief sketch of the proof of the nefness of the moduli divisor in the previous theorem; very good references are [2, Lemma 5.2] and especially [25, Theorem 8.5.1 and $\\ S 8 . 1 0 ]$ , where many more details are given. By the proof of [25, Theorem 8.5.1], it suffices to show the claim after making a suitable generically finite base change and taking the cyclic cover of $X$ associated to $\\sqrt [ r ] { \\varphi }$ . The base change as in (1) that we are aiming for is a composition of a log resolution with a Kawamata cover such that, on an open subset of $U ^ { \\prime } \\subseteq Y ^ { \\prime }$ whose complement has codimension at least 2 in $Y ^ { \\prime }$ , the local systems $R ^ { i } f _ { * } ^ { \\prime } \\mathbb { C } _ { X ^ { \\prime } } | _ { U ^ { \\prime } }$ have unipotent monodromies; this is the content of [25, 8.10.7–8.10.10]. We may also assume that $M _ { Y ^ { \\prime } }$ is a Cartier divisor. If $f$ is a klt-trivial fibration, then by [25, Theorem 8.5.1] the line bundle $\\mathcal { O } _ { Y ^ { \\prime } } ( M _ { Y ^ { \\prime } } )$ is a quotient of the locally free sheaf $f _ { * } ^ { \\prime } \\omega _ { X ^ { \\prime } / Y ^ { \\prime } }$ . Then one applies [22, Theorem 5], which asserts that $f _ { * } ^ { \\prime } \\omega _ { X ^ { \\prime } / Y ^ { \\prime } }$ is the canonical extension of the bottom piece of the Hodge filtration of $R ^ { \\dim X - \\dim Y } f _ { * } ^ { \\prime } \\mathbb { C } _ { X ^ { \\prime } } | _ { U ^ { \\prime } }$ , and hence its quotients are nef by the same result. A similar statement holds also for lc-trivial fibrations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,821,879,873]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"49","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recently, a stronger statement was shown in [17, Theorem 1.1]. The result shows that [22, Theorem 5] can be improved to show that not only any quotient of $f _ { * } ^ { \\prime } \\omega _ { X ^ { \\prime } / Y ^ { \\prime } }$ is nef, but moreover, it carries a singular metric whose Lelong numbers are all zero. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[706,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"49","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Floris and V. Lazic´ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"49","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"42 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,90,879,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This is much stronger than being nef, as it implies, in particular, that the multiplier ideal associated to this metric is trivial. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,126,533,142]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We summarise this in the following result. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,151,881,205]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2.10 Let $f \\colon ( X , \\Delta ) \\to Y$ be a klt-trivial fibration. Then there exists $a$ proper birational morphism $Y ^ { \\prime } Y$ such that for every proper birational morphism $\\pi \\colon Y ^ { \\prime \\prime } \\to Y ^ { \\prime }$ we have: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,214,881,270]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(i) $K _ { Y ^ { \\prime } } + B _ { Y ^ { \\prime } }$ is a $\\mathbb { Q }$ -Cartier divisor and $K _ { Y ^ { \\prime \\prime } } + B _ { Y ^ { \\prime \\prime } } = \\pi ^ { * } ( K _ { Y ^ { \\prime } } + B _ { Y ^ { \\prime } } ) .$ , (ii) $M _ { Y ^ { \\prime } }$ is a $\\mathbb { Q }$ -Cartier divisor carrying a singular metric whose all Lelong numbers are zero, and $M _ { Y ^ { \\prime \\prime } } = \\pi ^ { * } M _ { Y ^ { \\prime } }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,277,881,351]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The proof of part (ii) of the theorem is the same as the sketch of the proof of Theorem 2.9(ii) above, since a $\\mathbb { Q }$ -divisor carries a singular metric whose all Lelong numbers are zero if and only if its pullback by a proper surjective map carries a singular metric whose all Lelong numbers are zero by [7, Corollary 4]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,396,451,415]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.3 Inversion of Adjunction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,439,881,584]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In order to appreciate the following result and to see why base change property is important, let us go back to the construction of the canonical bundle formula. Recall that the discriminant divisor was constructed in terms of local log canonical thresholds, that is, log canonical thresholds over the generic point of a prime divisor; in particular, with notation from Definition 2.4, for some prime divisor $P$ on $Y$ , the pair $( X , \\Delta + \\gamma _ { P } f ^ { * } P )$ does not have to be globally log canonical. However, the following inversion of adjunction [2, Theorem 3.1] states that this is precisely what happens on an Ambro model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,593,879,665]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2.11 Let $f \\colon ( X , \\Delta ) \\ \\to \\ Y$ be an lc-trivial fibration, and assume that $Y$ is an Ambro model for $f$ . Then $( Y , B _ { Y } )$ has klt, respectively log canonical, singularities in a neighbourhood of a point $y ~ \\in ~ Y$ if and only if $( X , \\Delta )$ has $k l t$ , respectively log canonical, singularities in a neighbourhood of $f ^ { - 1 } ( y )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,674,876,710]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note that Theorem 2.11 is stated for klt-trivial fibrations in [2], but the proof extends verbatim to the lc-trivial case by using Theorem 2.9. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,711,876,746]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We finish this subsection with the following nice result [3, Proposition 3.1] which we will apply in the proof of Theorem 2.16 below. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,755,879,810]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2.12 Let $f \\colon ( X , \\Delta ) \\to Y$ be a klt-trivial fibration, and assume that $Y$ is an Ambro model for $f$ . Then for every base change by a proper generically finite morphism $w \\colon W \\to Y$ we have $K _ { W } + B _ { W } \\sim _ { \\mathbb { Q } } w ^ { * } ( K _ { Y } + B _ { Y } )$ and $M _ { W } \\sim _ { \\mathbb { Q } } w ^ { * } M _ { Y }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,505,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"50","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Travel Guide to the Canonical Bundle Formula ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,567,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.4 Coefficients of the Moduli Divisor ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,132,876,169]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Often in applications one needs to bound the denominators of $M _ { Y }$ . For instance, such bounds were used in [10, Theorem 1.2]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,178,876,213]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2.13 For each nonnegative integer b there exists an integer N depending on $b$ such that the following holds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,214,881,288]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $f \\colon ( X , \\Delta ) \\ \\to \\ Y$ be a klt-trivial fibration with a general fibre $F$ , and let $r$ be the smallest positive integer such that $r ( K _ { F } + \\Delta | _ { F } ) \\sim 0 .$ . Let $E F$ be the associated $r$ -th cyclic cover and let $\\overline { { E } }$ be a resolution of singularities of $E$ . If $\\dim { H ^ { \\dim \\overline { { E } } } } ( \\overline { { E } } , \\mathbb { C } ) = \\bar { b }$ , then the divisor $N M _ { Y }$ is integral. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,297,876,333]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The result was proved in [19, Theorem 3.1] when $\\Delta = 0$ , but the same proof works for klt-trivial fibrations [10, Theorem 5.1]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,334,876,369]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A more refined result holds when a general fibre is a rational curve, [9, Theorem 1.6(2)]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,378,874,414]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2.14 Fix a positive integer r. For a prime number $q$ set $s ( q ) = \\operatorname* { m a x } \\{ s \\mid$ $q ^ { s } \\leq 2 r \\}$ and define ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[416,427,580,472]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0342/b57e001ad36d49cdbb1bfad8fc3ab7e1be31d1f824e27fb151953355237bad97.jpg","list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nN = \\prod _ { q p r i m e } q ^ { s ( q ) } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,484,881,540]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(a) Let $f \\colon ( X , \\Delta ) \\ \\to \\ Y$ be an lc-trivial fibration whose general fibre $F$ is a rational curve, and assume that $r$ is the smallest positive integer such that $r ( K _ { F } + \\Delta | _ { F } ) \\sim 0 .$ . Then the divisor $N M _ { Y }$ is integral. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,540,881,594]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(b) Assume $r$ is odd. Then there exists an lc-trivial fibration $f \\colon ( X , \\Delta ) \\to Y$ such that if v is the smallest integer for which the divisor $v M _ { Y }$ is integral, then $v =$ $N / r$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,639,510,660]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.5 Goodness of Moduli Divisors ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,683,881,773]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now we come to the central topic of this survey, already announced in the introduction: the descent of singularities. Since we already know the nefness of the moduli divisor by Theorem 2.9, if it were additionally big, then this would allow to conclude in many cases. Bigness is too much to ask; however, the following result of Ambro [3, Theorem 3.3 and Proposition 4.4] turns out to be almost as good. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[706,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Floris and V. Lazic´ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"51","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"44 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,144]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2.15 Let $f \\colon ( X , \\Delta ) \\ \\to \\ Y$ be a klt-trivial fibration between normal projective varieties such that $\\Delta$ is effective over the generic point of Y . Then there exists a diagram ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[334,160,669,247]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0740/541e9d4cad5c4759ad43625a780c5daa43b37865df1e0a5e3f14798ed275b1aa.jpg","list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { c c c } { { ( X , \\Delta ) } } & { { } } & { { ( X ^ { + } , \\Delta ^ { + } ) } } \\\\ { { f \\Bigg \\downarrow } } & { { } } & { { \\Bigg \\downarrow f ^ { + } } } \\\\ { { Y \\longleftarrow \\quad } } & { { W \\longleftrightarrow { Y ^ { + } } } } & { { } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,268,211,285]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"such that: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,294,881,385]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(i) $f ^ { + } \\colon ( X ^ { + } , \\Delta ^ { + } ) \\to Y ^ { + }$ is a klt-trivial fibration, \n(ii) $\\tau$ is generically finite and surjective, and $\\rho$ is surjective, \n(iii) if $M _ { Y }$ and $M _ { Y ^ { + } }$ are the moduli divisors of $f$ and $f ^ { + }$ respectively, then $M _ { Y ^ { + } }$ is big and, after possibly a birational base change, we have $\\tau ^ { * } M _ { Y } = \\rho ^ { * } M _ { Y ^ { + } }$ , \n$( i \\nu )$ there exists a non-empty open set $U \\subseteq W$ and an isomorphism ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[291,399,751,487]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0515/f685aa053a04842dc32fb72bf44ac243c403958fd93a56062cc32085289ca2d8.jpg","list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } { ( X , \\Delta ) \\times _ { Y } U \\xrightarrow { \\cong } } & { { } \\xrightarrow { \\cong \\qquad ( X ^ { + } , \\Delta ^ { + } ) \\times _ { Y ^ { + } } U } } \\\\ { \\qquad } & { { } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[129,503,441,521]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(v) if there exists an isomorphism ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[359,537,683,557]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0836/32e71a41d6f5b71c4d6ec6b5c0d5e663967328eebc90a2709dcc3b156ba2ed40.jpg","list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\Phi \\colon ( X , \\Delta ) \\times _ { Y } U ( F , \\Delta _ { F } ) \\times U\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[157,573,879,591]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"over a non-empty open subset $U \\subseteq Y$ , then $\\Phi$ extends to an isomorphism over ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[293,605,749,630]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0949/737c4824ed20eb8bb59693e4115e16392a654c68d3fe56c0c4bd145774246909.jpg","list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nY ^ { 0 } = Y \\setminus \\big ( \\mathrm { S u p p } B _ { Y } \\cup \\mathrm { S i n g } ( Y ) \\cup f ( \\mathrm { S u p p } \\Delta _ { v } ^ { \\le 0 } ) \\big ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[148,645,858,663]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note that in (v) one does not need that $\\Delta$ is effective on the generic fibre of $f$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,665,874,701]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For us, the most important part of this result is (iii). Its immediate consequence is the descent of klt singularities [3, Theorem 0.2]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,708,881,782]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2.16 Let $( X , \\Delta )$ be a projective klt pair with $\\Delta$ effective, and let $f \\colon X \\to$ $Y$ be a surjective morphism to a normal projective variety such that $K _ { X } + \\Delta \\sim _ { \\mathbb { Q } }$ $f ^ { * } D$ for some $\\mathbb { Q }$ -Cartier $\\mathbb { Q }$ -divisor $D$ on $Y$ . Then there exists an effective $\\mathbb { Q }$ -divisor $\\Delta \\gamma$ on $Y$ such that the pair $( Y , \\Delta _ { Y } )$ is klt and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[364,798,633,819]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0834/8179d8ed163b308051bffd1ac6b952d578ae43084ab60771f87b76446fe71220.jpg","list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { X } + \\Delta \\sim _ { \\mathbb { Q } } f ^ { * } ( K _ { Y } + \\Delta _ { Y } ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,51,505,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Travel Guide to the Canonical Bundle Formula ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,52,879,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"52","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"45 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,879,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof We use the notation from Theorem 2.15, which clearly applies in our situation. We have the base change diagram ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[387,144,608,231]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0936/d84ae0b5ad4dbbf60dd8688ce6d46f6c7771033bb03d9206ae88957936de44a2.jpg","list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { c } { { ( W , \\Delta _ { W } ) \\xrightarrow { w } > ( X , \\Delta ) } } \\\\ { { f _ { W } \\Bigg \\downarrow } } \\\\ { { W \\xrightarrow [ \\tau ] { } \\qquad \\forall . } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,252,881,325]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We may additionally assume that $\\tau$ factors through an Ambro model $\\pi \\colon Y ^ { \\prime } \\to Y$ of $f$ , and denote by $\\sigma : W \\to Y ^ { \\prime }$ the induced morphism. By replacing $W$ by a suitable log resolution, by an easy argument with the Stein factorisation of $\\sigma$ together with Theorem 2.12 we may assume that $W$ is an Ambro model of $f _ { W }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,325,879,415]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now, $\\tau ^ { * } D \\ \\sim _ { \\mathbb { Q } } \\ K _ { W } \\ + \\ B _ { W } \\ + \\ M _ { W }$ , and by the inversion of adjunction, Theorem 2.11, the pair $( W , B _ { W } )$ is klt. Since the divisor $M _ { Y ^ { + } }$ is nef and big, by using a version of Kodaira’s trick [27, Proposition 2.61] together with Bertini’s theorem, we may find an effective $\\mathbb { Q }$ -divisor $E _ { W }$ on $W$ such that $M _ { W } \\ \\sim _ { \\mathbb { Q } } \\ E _ { W }$ and such that the pair $( W , B _ { W } + E _ { W } )$ is klt. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,415,876,454]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By Theorem 2.12 we have $K _ { W } + B _ { W } \\sim _ { \\mathbb { Q } } \\sigma ^ { * } ( K _ { Y ^ { \\prime } } + B _ { Y ^ { \\prime } } )$ and $M _ { W } \\sim _ { \\mathbb { Q } } \\sigma ^ { * } M _ { Y ^ { \\prime } }$ Hence, if we denote $\\begin{array} { r } { E _ { Y ^ { \\prime } } = \\frac { 1 } { \\deg \\sigma } \\sigma _ { * } E } \\end{array}$ , we have $M _ { Y ^ { \\prime } } \\sim _ { \\mathbb { Q } } E _ { Y ^ { \\prime } }$ and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[298,471,699,492]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0672/81052eaf079ca0c706dbc63f61ec79ef8913114c8cb6ec79a87b1aec22ee9e34.jpg","list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { W } + B _ { W } + E \\sim _ { \\mathbb { Q } } \\sigma ^ { * } ( K _ { Y ^ { \\prime } } + B _ { Y ^ { \\prime } } + E _ { Y ^ { \\prime } } ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,507,756,527]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Then the pair $( Y ^ { \\prime } , B _ { Y ^ { \\prime } } + E _ { Y ^ { \\prime } } )$ is klt by [27, Proposition 5.20]. Setting ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[328,543,667,563]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0126/b8cdff66a61c9733f11302b4bdeba41be663b58b08d5ed4805266d4e03744cfb.jpg","list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\Delta _ { Y } = \\pi _ { * } ( B _ { Y ^ { \\prime } } + E _ { Y ^ { \\prime } } ) = B _ { Y } + \\pi _ { * } E ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[123,579,876,617]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"we have $K _ { Y } + \\Delta _ { Y } \\sim _ { \\mathbb { Q } } D$ , and $( Y , \\Delta _ { Y } )$ is a klt pair. Since $\\Delta$ is effective, the divisor $B _ { Y }$ is effective, thus $\\Delta \\gamma$ is effective. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,624,881,696]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finally, we mention that Theorem 2.15(i)–(iii) was generalised to lc-trivial fibrations where $\\Delta \\geq 0$ and $( X , \\Delta )$ is log canonical in [18, Lemma 1.1]. The proof involves running a careful MMP in order to reduce to a situation where one has a klt-trivial fibration. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,705,881,813]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 2.17 Theorem 2.15(iv) does not hold for lc-trivial fibrations. For instance, let $\\boldsymbol { X } \\ : = \\ : \\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 }$ , let $f$ be the second projection, let $\\delta$ be the diagonal and set $\\begin{array} { r } { \\Delta = \\delta + \\frac 1 2 \\{ 0 \\} \\times \\mathbb { P } ^ { 1 } + \\frac 1 2 \\{ \\infty \\} \\times \\mathbb { P } ^ { 1 } } \\end{array}$ . Then $f \\colon ( X , \\Delta ) \\to \\mathbb { P } ^ { 1 }$ is an lc-trivial fibration with discriminant supported on $\\{ 0 \\} \\cup \\{ \\infty \\}$ . By considering log resolutions, one calculates\u0003 \u0004 that the discriminant is equal to $\\textstyle { \\frac { 1 } { 2 } } 0 + { \\frac { 1 } { 2 } } \\infty$ , hence the moduli divisor is torsion. Indeed, we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[296,828,699,852]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0920/1b9cca47006faf1fb1e6d7980d7db7536527d6eb7edbe8d816130a49fc7e2a1a.jpg","list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { K _ { X } + \\Delta \\sim _ { \\mathbb { Q } } f ^ { * } \\big ( K _ { \\mathbb { P } ^ { 1 } } + \\frac { 1 } { 2 } 0 + \\frac { 1 } { 2 } \\infty + M _ { \\mathbb { P } ^ { 1 } } \\big ) , } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,867,544,887]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"but the divisor $K _ { X } + \\Delta$ has bi-degree $( 0 , - 1 )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[706,51,879,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Floris and V. Lazic´ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"53","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"46 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,879,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"However, $f$ is not birational to a product. Indeed, it induces a family of rational curves with 4 marked points parametrised by $\\mathbb { P } ^ { 1 } \\colon$ for $t \\in \\mathbb { P } ^ { 1 }$ , set $p _ { 1 } ( t ) = 0$ , $p _ { 2 } ( t ) =$ 1, $p _ { 3 } ( t ) = p _ { 4 } ( t ) = t$ . This family of marked curves is not trivial, therefore the fibration cannot be locally a product. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,207,514,226]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 B-Semiampleness Conjectures ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,250,881,358]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We saw above in Theorem 2.16 that klt singularities descend along a klt-trivial fibration. However, even if one had the full analogue of Theorem 2.15 in the log canonical setting one could not follow the same strategy to show that log canonical singularities descend. Moreover, one sees that the use of Kodaira’s trick in the proof of Theorem 2.16 obliterated the connection of the coefficients of the divisor $\\Delta \\gamma$ to that of the divisor $\\Delta$ . In order to remedy the situation, something more is needed. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,360,881,412]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The main conjecture on the canonical bundle formula predicts something much stronger: that the moduli part is semiample on an Ambro model of the fibration. We discuss it in this section. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,414,876,450]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There are two versions of the conjecture; the stronger one was proposed in [31, Conjecture 7.13.3]. We start with the stronger, effective version. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,459,881,548]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Effective B-Semiampleness Conjecture Fix positive integers d and r. Then there exists an integer m depending only on $d$ and $r$ , such that for any $l c$ -trivial fibration $f \\colon ( X , B ) \\to Y$ with the generic fibre $F$ , if dim $F = d$ and $r$ is the smallest positive integer such that $r ( K _ { F } + B | _ { F } ) \\sim 0$ , there exists an Ambro model $Y ^ { \\prime }$ of $f$ such that $m M _ { Y ^ { \\prime } }$ is base point free. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,558,879,611]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"More generally, any conjecture as above, in which $m$ depends on some invariants of the generic fibre of the fibration, goes under the name of effective $^ b$ - semiampleness. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,612,879,683]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the original statement [31, Conjecture 7.13.3], the constant m depended on $\\mathrm { d i m } X$ and $r$ . The main result of [10] is that it suffices to show the Effective B-Semiampleness Conjecture in the case where $Y$ is a curve. This led to the formulation of the Effective B-Semiampleness Conjecture above. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,684,874,719]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The conjecture is widely open. We list below the cases where it is known. They all make use of some notion of moduli space for the fibres. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,729,876,764]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3.1 The Effective $B$ -Semiampleness Conjecture holds in the following cases: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,773,881,881]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(1) if general fibres are elliptic curves by [24, 32]; we have $m = 1 2$ ; \n(2) if general fibres are rational curves [31, Theorem 8.1]; \n(3) if general fibres are $K 3$ surfaces or abelian varieties of dimension d by [13, Theorem 1.2]; then we have $m = 1 9 k$ , respectively $m = k ( d + 1 )$ , where $k$ is a weight associated to the Baily-Borel-Satake compactification of the period domain. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,503,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"54","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Travel Guide to the Canonical Bundle Formula ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[145,90,820,106]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the weaker version of the conjecture we lose control on the constant $m$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,115,876,153]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"B-Semiampleness Conjecture Let $f \\colon ( X , B ) \\to Y$ be an lc-trivial fibration. Then there exists an Ambro model $Y ^ { \\prime }$ of $f$ such that $M _ { Y ^ { \\prime } }$ is semiample. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,162,881,232]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"More is known about this conjecture than about its effective version above, although the progress has been limited to the cases where either the bases $Y$ or the fibres of $f$ are low dimensional. We summarise the situation for low dimensional fibres in the following result. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,241,876,279]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3.2 Apart from the cases listed in Theorem 3.1, the B-Semiampleness Conjecture holds in the following cases: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,286,881,360]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(1) if the fibres are surfaces of Kodaira dimension 0 by [13, Lemma 4.1 and Corollary 6.4] and [29, Part I, (5.15.9)(ii)]; \n(2) if $f$ is a klt-trivial fibration and the generic fibre is a uniruled surface not isomorphic to $\\mathbb { P } ^ { 2 }$ by [8, Theorem 1.7]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,367,876,402]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The B-Semiampleness Conjecture holds for another important family of fibrations: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,412,879,468]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3.3 Let $f \\colon ( X , \\Delta ) \\to Y$ be an lc-trivial fibration, and assume that the \nmoduli part $M _ { Y }$ descends to Y . If $M _ { Y } \\equiv 0$ , then $M _ { Y } \\sim _ { \\mathbb { Q } } 0$ . In particular, if dim $Y = 1$ , then $M _ { Y }$ is semiample. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,475,881,530]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3.3 is [3, Theorem 3.5] for klt-trivial fibrations and [10, Theorem 1.3] for lc-trivial fibrations. Theorem 1.2 in [10] states that Effective B-Semiampleness Conjecture is true for klt-trivial fibrations with numerically trivial moduli part. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,531,879,585]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Another partial result is [5, Theorem 3.2]. They prove that a small perturbation of the moduli part in a specific direction is semiample, under some effectivity hypotheses for $K _ { Y } + B _ { Y }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,630,441,650]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.1 Restrictions to Divisors ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,674,881,728]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As we saw in the previous subsection, the progress on the B-Semiampleness Conjecture has been concentrated on the cases of either the low dimension of the base $Y$ or the low dimension of the generic fibre of the fibration $f$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,729,881,800]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In [12, Theorem A] we obtained the following result towards the conjecture valid in every dimension. Note that the phrase the $B$ -semiampleness Conjecture in dimension $n$ means that we consider the conjecture in the case when the dimension of the base $Y$ is $n$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,809,817,827]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3.4 Assume the $B$ -Semiampleness Conjecture in dimension $n - 1$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,828,881,881]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $( X , \\Delta )$ be a log canonical pair and let $f \\colon ( X , \\Delta ) \\ \\to \\ Y$ be an lc-trivial fibration to an $n$ -dimensional variety $Y$ , where the divisor $\\Delta$ is effective over the generic point of Y . Assume that $Y$ is an Ambro model for $f$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[706,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Floris and V. Lazic´ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,54,141,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"55","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,879,144]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Then for every birational model $\\pi \\colon Y ^ { \\prime } \\to Y$ and for every prime divisor $T$ on $Y ^ { \\prime }$ with the normalisation $T ^ { \\nu }$ and the induced morphism $\\nu \\colon T ^ { \\nu } \\to Y ^ { \\prime }$ , the divisor $\\nu ^ { * } \\pi ^ { * } M _ { Y }$ is semiample on $T ^ { \\nu }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,153,879,205]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As a corollary, combining with Theorem 3.3, we obtain that the restriction of the moduli part to every prime divisor on every sufficiently high birational model of $Y$ is semiample if $Y$ is a surface. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,208,881,441]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We comment on the proof of Theorem 3.4, as it will be useful in the following subsection. We first apply a base change to $Y$ and modify $( X , \\Delta )$ by blowing up suitably, but we try to remember $( X , \\Delta )$ along the proof. We then run a suitable relative MMP over $Y$ , which contracts many “bad” components of $\\Delta$ (in particular, those with negative coefficients); as a result, we obtain a new lc-trivial fibration $g \\colon ( W , \\Delta _ { W } ) \\to Y$ with $\\Delta _ { W } \\geq 0$ and with the same moduli divisor $M _ { Y }$ . Choosing a minimal log canonical centre $s$ of $( W , \\Delta _ { W } )$ which surjects onto $T$ , we obtain an induced $k l t$ -trivial fibration $g | _ { S } \\colon ( S , \\Delta _ { S } ) \\to T ^ { \\prime }$ , where $T ^ { \\prime }$ is obtained from the Stein factorisation of the morphism $S \\to T$ . Then we first show that $M _ { Y } | _ { T ^ { \\prime } }$ is almost $M _ { T ^ { \\prime } }$ . Even though at this step we may not deduce equality between these two divisors, we can control their difference in a very precise manner. After a suitable further blowup of $Y$ , we can force this difference to disappear and we conclude by induction on the dimension. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,487,375,507]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.2 Reduction Result ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,531,879,584]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As we mentioned above, in the setup of lc-trivial fibrations $f \\colon ( X , \\Delta ) \\to Y$ one does not assume that $\\Delta$ is effective. Furthermore, often it is much more difficult to work with lc-trivial fibrations than with klt-trivial fibrations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,587,876,621]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In [12] the B-Semiampleness Conjecture is reduced to the following conjecture with much weaker hypotheses. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,630,881,684]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conjecture 3.5 Let $( X , \\Delta )$ be a log canonical pair and let $f \\colon ( X , \\Delta ) \\to Y$ be $a$ klt-trivial fibration to an $n$ -dimensional variety Y . If $Y$ is an Ambro model of $f$ and if the moduli divisor $M _ { Y }$ is big, then $M _ { Y }$ is semiample. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,693,419,711]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We show in [12, Theorem E]: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,720,876,756]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3.6 Assume Conjecture 3.5 in dimensions at most n. Then the $B$ - Semiampleness Conjecture holds in dimension $n$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,765,738,783]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The proof is similar to the proof of Theorem 3.4 sketched above. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,505,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"56","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Travel Guide to the Canonical Bundle Formula ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,348,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"57","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.3 Generalisation ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,132,879,222]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"57","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the papers [16, 20] the authors consider slc-trivial fibrations. Those are completely analogous to lc-trivial fibrations, the difference being that the ambient space $X$ is not irreducible, but the pair $( X , \\Delta )$ is slc on the generic fibre of the fibration; such a setup appears occasionally in inductive proofs. The precise statement is [16, Definition 4.1]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,223,881,294]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"57","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Then one can define, as in the case of lc-trivial fibrations, the moduli divisor and the discriminant. Then [16, Theorem 1.2] proves the analogue of Theorem 2.9 for slc-trivial fibrations, and [20, Theorem 1.3] shows the analogue of Theorem 3.3 in this context. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,340,403,361]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"57","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 Parabolic Fibrations ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,385,879,420]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"57","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finally, in this section we discuss a more general situation than that of lc-trivial fibrations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,421,881,546]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"57","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $g \\colon ( X , \\Delta ) \\to Z$ be a surjective morphism, where $( X , \\Delta )$ is a klt projective pair and \u0003 $Z$ is a projective variety. Assume that \u0004 $g _ { * } \\mathcal { O } _ { X } \\big ( m ( K _ { X } + \\Delta ) \\big ) \\neq 0$ for some positive integer $m$ , and consider the relative Iitaka fibration $f \\colon X \\to Y$ associated to $K _ { X } + \\Delta$ . Possibly by blowing up further, one may assume that $( X , \\Delta )$ is log smooth and that $f$ is a morphism. Then if $F$ is a general fibre of $f$ , we have $\\kappa \\big ( F , ( K _ { X } + \\Delta ) | _ { F } \\big ) = 0$ , however $K _ { X } + \\Delta$ is not necessarily a pullback from $Y$ . One still wonders if there is a canonical bundle formula for the map $f$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,548,876,582]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"57","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The resulting formula is the canonical bundle formula of Fujino and Mori [19]. We first need a definition, which is justified from the setup above.\u0003 \u0004 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,591,879,648]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"57","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 4.1 A parabolic fibration is a fibration $f \\colon ( X , \\Delta ) \\to Y$ , where $( X , \\Delta )$ is a projective klt pair, $Y$ is a smooth projective variety and if $F$ is the generic fibre of $f$ , then $\\kappa \\big ( F , ( K _ { X } + \\Delta ) | _ { F } \\big ) = 0$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,656,876,690]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"57","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The following is [19, Section 4], the canonical bundle formula of Fujino and Mori. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,699,879,735]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"57","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 4.2 Let $f \\colon ( X , \\Delta ) \\to Y$ be a parabolic fibration, where $\\Delta$ is effective. Then there is a commutative diagram ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[428,752,569,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0253/4c866d0d811e56238dad330e6f3e6a516b17bf61e123e8c40e9129cd7f46c02e.jpg","list_items":null,"page_idx":"57","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { c } { { X ^ { \\prime } \\hfill ~ \\frac { \\tau ^ { \\prime } } { \\tau } X } } \\\\ { { f ^ { \\prime } \\hfill \\sqrt { \\hfill } } } \\\\ { { Y ^ { \\prime } \\hfill ~ \\frac { \\displaystyle \\vphantom { \\Bigg | } \\tau Y , } { \\tau Y , } } } \\end{array} \\} f \\hfill ~ \n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,864,876,900]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"57","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\tau$ and $\\tau ^ { \\prime }$ are birational, $X ^ { \\prime }$ and $Y ^ { \\prime }$ are smooth, and $f ^ { \\prime }$ has connected fibres, such that the following holds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"57","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"50 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[706,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"57","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Floris and V. Lazic´ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,876,127]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"58","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There exist effective $\\mathbb { Q }$ -divisors $B ^ { + }$ and $B ^ { - }$ on $X ^ { \\prime }$ without common components, a $\\mathbb { Q }$ -divisor $\\Delta ^ { \\prime } \\geq 0$ on $X ^ { \\prime }$ and $\\mathbb { Q }$ -divisors $B _ { Y ^ { \\prime } }$ and $M _ { Y ^ { \\prime } }$ on $Y$ such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[259,142,735,165]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0843/9c19159b852b3966883dfd7e24124ed04baf7fa748c5922d4259beb364b119d6.jpg","list_items":null,"page_idx":"58","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { X ^ { \\prime } } + \\Delta ^ { \\prime } + B ^ { - } \\sim _ { \\mathbb { Q } } f ^ { \\prime * } ( K _ { Y ^ { \\prime } } + B _ { Y ^ { \\prime } } + M _ { Y ^ { \\prime } } ) + B ^ { + } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,181,391,199]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"58","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"with the following properties: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[127,207,881,243]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"58","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(i) the pair $( { X } ^ { \\prime } , { \\Delta } ^ { \\prime } )$ is klt and log smooth, and there exists an effective exceptional divisor $E$ on $X ^ { \\prime }$ such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[364,259,678,282]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0553/a9fc79ed289bb35cdac51eebe2c39469b4ccbcbaba25bec1b98106f91b68c09d.jpg","list_items":null,"page_idx":"58","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { X ^ { \\prime } } + \\Delta ^ { \\prime } \\sim _ { \\mathbb { Q } } \\tau ^ { \\prime * } ( K _ { X } + \\Delta ) + E ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,297,883,406]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"58","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(ii) $f _ { * } ^ { \\prime } \\mathcal { O } _ { X ^ { \\prime } } ( \\lfloor n B ^ { + } \\rfloor ) \\simeq \\mathcal { O } _ { Y ^ { \\prime } }$ for all $n \\in \\mathbb { N }$ , \n(iii) $B ^ { - }$ is $f ^ { \\prime }$ -exceptional and $\\tau ^ { \\prime }$ -exceptional, \n$( i \\nu )$ the induced map $f ^ { \\prime } \\colon ( X ^ { \\prime } , \\Delta ^ { \\prime } + B ^ { - } - B ^ { + } ) \\to Y ^ { \\prime }$ is a klt-trivial fibration, and $B _ { Y ^ { \\prime } }$ and \u0003 $M _ { Y ^ { \\prime } }$ are the corresponding discriminant and moduli divisors,\u0004 \u0003 \u0004 \u0003 \n(v) the pair $( Y ^ { \\prime } , B _ { Y ^ { \\prime } } )$ is klt, $B _ { Y ^ { \\prime } } \\geq 0$ and $M _ { Y ^ { \\prime } }$ is nef, \n$( \\nu i )$ for every $n \\in \\mathbb { N }$ sufficiently divisible we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[164,420,876,444]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0092/711db7d2151f1e4cad393e4a8d76e2268f01bc4664da3f2694a172cffa488ba3.jpg","list_items":null,"page_idx":"58","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nH ^ { 0 } \\big ( X , n ( K _ { X } + \\Delta ) \\big ) \\simeq H ^ { 0 } \\big ( X ^ { \\prime } , n ( K _ { X ^ { \\prime } } + \\Delta ^ { \\prime } ) \\big ) \\simeq H ^ { 0 } \\big ( Y ^ { \\prime } , n ( K _ { Y ^ { \\prime } } + B _ { Y ^ { \\prime } } + M _ { Y ^ { \\prime } } ) \\big ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,459,881,513]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"58","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There are several non-trivial parts of this formula which do not follow from considerations in the previous sections: the existence of divisors $B ^ { + }$ and $B ^ { - }$ with the properties (ii) and (iii) above, as well as the fact that $B _ { Y ^ { \\prime } }$ is effective. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,515,881,603]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"58","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Part (vi) follows immediately from (i), (ii) and (iii). We sketch how (iv) follows from (i), (ii) and (iii), following [1, Lemma 4.2]. Let $F ^ { \\prime }$ be a general fibre of\u0003 $f ^ { \\prime }$ , and we define the divisor\u0004 $\\Xi ^ { \\prime }$ with respect to $f ^ { \\prime } \\colon ( X ^ { \\prime } , \\Delta ^ { \\prime } + B ^ { - } - B ^ { + } ) \\to Y ^ { \\prime }$ as in Definition 2.1(d). We may assume that $\\Xi ^ { \\prime } = - \\Delta ^ { \\prime } - B ^ { - } + B ^ { + }$ , and $B ^ { - } | _ { F ^ { \\prime } } = 0$ by (iii). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,603,876,641]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"58","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We have $( K _ { X ^ { \\prime } } + \\Delta ^ { \\prime } + B ^ { - } - B ^ { + } ) | _ { F ^ { \\prime } } \\sim _ { \\mathbb { Q } } 0$ by construction and $\\kappa \\big ( F ^ { \\prime } , ( K _ { X ^ { \\prime } } +$ $\\Delta ^ { \\prime } ) | _ { F ^ { \\prime } } ) = 0$ by (i), hence ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[293,657,701,680]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0165/941c11613f306845af51c5651e8fb215ffe8d7feef117c90c83126016e63639c.jpg","list_items":null,"page_idx":"58","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\kappa ( F ^ { \\prime } , B ^ { + } | _ { F ^ { \\prime } } ) = \\kappa \\big ( F ^ { \\prime } , ( B ^ { + } - B ^ { - } ) | _ { F ^ { \\prime } } \\big ) = 0 .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,695,879,732]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"58","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since there exists a positive integer $b$ such that $\\lceil B ^ { + } \\rceil | _ { F ^ { \\prime } } \\leq b B ^ { + } | _ { F ^ { \\prime } }$ , this implies $\\kappa \\left( F ^ { \\prime } , \\lceil B ^ { + } \\rceil | _ { F ^ { \\prime } } \\right) = 0$ . Therefore,\u0003 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[218,746,776,831]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0140/518eb59d144e683f55fa675c84c05e422bd574b3355f4506e633b02e55cdee65.jpg","list_items":null,"page_idx":"58","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\kappa \\big ( F ^ { \\prime } , \\lceil \\Xi ^ { \\prime } \\rceil | _ { F ^ { \\prime } } \\big ) = \\kappa \\big ( F ^ { \\prime } , \\lceil - \\Delta ^ { \\prime } - B ^ { - } + B ^ { + } \\rceil | _ { F ^ { \\prime } } \\big ) } \\\\ & { \\qquad \\leq \\kappa \\big ( F ^ { \\prime } , \\lceil - \\Delta ^ { \\prime } \\rceil | _ { F ^ { \\prime } } + \\lceil - B ^ { - } \\rceil | _ { F ^ { \\prime } } + \\lceil B ^ { + } \\rceil | _ { F ^ { \\prime } } \\big ) } \\\\ & { \\qquad \\leq \\kappa \\big ( F ^ { \\prime } , \\lceil B ^ { + } \\rceil | _ { F ^ { \\prime } } \\big ) = 0 . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,845,642,863]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"58","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This shows part (d) of Definition 2.1, and the rest is easy. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,51,505,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"58","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Travel Guide to the Canonical Bundle Formula ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"58","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,879,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"59","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In order to apply the previous result, we recall that for a log canonical pair $( X , \\Delta )$ , the ring ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[289,139,710,181]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0231/d195f4cce33a933c5ba0c15726082bd2dd8fd0c88c959dd3ec7334704bba9e08.jpg","list_items":null,"page_idx":"59","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nR ( X , K _ { X } + \\Delta ) = \\bigoplus _ { n \\in \\mathbb { N } } H ^ { 0 } \\big ( X , \\lfloor n ( K _ { X } + \\Delta ) \\rfloor \\big )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,196,881,256]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"59","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is the canonical ring of $( X , \\Delta )$ . We also recall that for a graded ring $R = \\bigoplus _ { \\mathrm { ~ \\tiny ~ R ~ } } R _ { n }$ , the $d$ -th Veronese subring of $R$ is defined as $R ^ { ( d ) } : = \\bigoplus _ { n \\in \\mathbb { N } } R _ { d n } .$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,256,881,328]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"59","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An immediate consequence of Theorem 4.2 is the following result [19, Theorem 5.2], which has widespread use in the Minimal Model Program. It often allows to pass from a pair $( X , \\Delta )$ with $\\kappa ( X , K _ { X } + \\Delta ) \\geq 0$ to a pair $( { X } ^ { \\prime } , { \\Delta } ^ { \\prime } )$ on which $K _ { X ^ { \\prime } } + \\Delta ^ { \\prime }$ is big. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,336,881,391]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"59","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 4.3 Let $( X , \\Delta )$ be a projective klt pair with $\\kappa ( K _ { X } + \\Delta ) = \\ell \\geq 0 .$ . Then there exist an $\\ell$ -dimensional klt pair $( { X } ^ { \\prime } , { \\Delta } ^ { \\prime } )$ with $\\kappa ( X ^ { \\prime } , K _ { X ^ { \\prime } } + \\Delta ^ { \\prime } ) = \\ell$ and positive integers $d$ and $d ^ { \\prime }$ such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[302,406,697,429]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0971/ba45c7011bbe9528c5fa28e9ef14ede116aa2307de3080269fe0c38d59a51e23.jpg","list_items":null,"page_idx":"59","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nR ( X , K _ { X } + \\Delta ) ^ { ( d ) } \\simeq R ( X ^ { \\prime } , K _ { X ^ { \\prime } } + \\Delta ^ { \\prime } ) ^ { ( d ^ { \\prime } ) } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,445,876,481]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"59","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The proof follows immediately from Theorem 4.2(vi), by combining it with the\u0004 proof of Theorem 2.16; see also the proof of Theorem 4.4 below. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,483,881,590]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"59","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Assume now that $( X , \\Delta )$ has simple normal crossings and let $\\Delta ^ { + }$ and $\\Delta ^ { - }$ be effective divisors without common components such that $\\Delta = \\Delta ^ { + } - \\Delta ^ { - }$ . Then if $\\kappa \\left( F , ( K _ { X } + \\Delta ^ { + } ) | _ { F } \\right) = 0$ for a general fibre $F$ of $f$ , and if there exists a good model of $( F , \\Delta ^ { + } | _ { F } )$ , then it was shown in [15, Theorem 3.13] that the moduli b-divisor is b-nef and good in the sense of [3, Definition 3.2]; this is an application of MMP techniques from [18] and [3, Theorem 3.3]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,591,881,680]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"59","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We finish the paper with the following result, which can sometimes be used in order to avoid running a Minimal Model Program; for instance, compare the proofs of [21, Lemma 4.4] and [28, Theorem 5.3]. Note that $\\nu ( X , L )$ denotes the numerical dimension of a divisor $L$ on a projective variety $X$ , see for instance [28, $\\ S 2 . 2 ]$ for basic properties and related references. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,689,881,744]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"59","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 4.4 Let $( X , \\Delta )$ be a projective klt pair and let $f \\colon ( X , \\Delta ) \\ \\to \\ Y$ be $a$ parabolic fibration such that $\\nu \\big ( F , ( K _ { X } + \\Delta ) | _ { F } \\big ) = 0$ for $a$ general fibre $F$ of $f$ . Then there exists a commutative diagram ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[428,758,571,849]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0297/34fa4a108075e61e5bd2eacc5c6d0e26854bf285eef3075ff45df24761205874.jpg","list_items":null,"page_idx":"59","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { c } { { X ^ { \\prime } \\xrightarrow { \\pi ^ { \\prime } } \\times X } } \\\\ { { f ^ { \\prime } \\Biggl \\downarrow } } \\\\ { { Y ^ { \\prime } \\xrightarrow { } Y , } } \\end{array} \\downarrow { } f\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[706,51,879,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"59","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Floris and V. Lazic´ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"59","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"52 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,881,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $X ^ { \\prime }$ and $Y ^ { \\prime }$ are smooth, $f ^ { \\prime }$ has connected fibres, $\\pi$ and $\\pi ^ { \\prime }$ are birational, and such that, if we write ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[337,142,658,163]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0955/d8b8f377c8bbf9f04a7e88057bf289988fffeaff7f789d391de4a1bcdd3e8056.jpg","list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { X ^ { \\prime } } + \\Delta ^ { \\prime } \\sim _ { \\mathbb { Q } } \\pi ^ { \\prime * } ( K _ { X } + \\Delta ) + E ^ { \\prime } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,180,608,198]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\Delta ^ { \\prime }$ and $E ^ { \\prime }$ have no common components, then: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[129,208,246,225]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(i) we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[314,241,724,264]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0136/38422ac441c437e60567f406b2b700a711b908590e4afd766882d839a4c5d84a.jpg","list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { X ^ { \\prime } } + \\Delta ^ { \\prime } + B ^ { - } \\sim _ { \\mathbb { Q } } f ^ { \\prime * } ( K _ { Y ^ { \\prime } } + \\Delta _ { Y ^ { \\prime } } ) + B ^ { + } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[159,279,881,315]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where the pair $( Y ^ { \\prime } , \\Delta _ { Y ^ { \\prime } } )$ is klt, and the divisors $B ^ { + }$ and $B ^ { - }$ are effective and have no common components, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,315,692,352]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(ii) $B ^ { - }$ is $\\pi ^ { \\prime }$ -exceptional and $f ^ { \\prime }$ -exceptional, (iii) we have $f _ { * } ^ { \\prime } { \\mathcal { O } } _ { X ^ { \\prime } } ( \\lfloor \\ell B ^ { + } \\rfloor ) \\simeq { \\mathcal { O } } _ { Y ^ { \\prime } }$ for all positive integers $\\ell$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[111,360,808,379]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moreover, if $K _ { X } + \\Delta$ is pseudoeffective, then \u0004 \u0003 $K _ { Y ^ { \\prime } } + \\Delta _ { Y ^ { \\prime } }$ is pseudoeffective.\u0004 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,387,701,405]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof By [6, Corollaire 3.4] and [30, Corollary V.4.9] we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[280,420,717,444]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0084/1febfd3b0e665f2973bd8b9762e8f794e5ed3a4e78508be12a0d0d9933cdd552.jpg","list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { \\kappa \\Big ( F , ( K _ { X } + \\Delta ) | _ { F } \\Big ) = \\nu \\Big ( F , ( K _ { X } + \\Delta ) | _ { F } \\Big ) = 0 . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,459,881,495]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By Theorem 4.2 there exists a diagram as in the theorem such that (ii) and (iii) hold, as well as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[259,510,735,534]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0970/79d349015cc553890e431e6ae3cfe7f175a587c49860f5ea063032fdb813b606.jpg","list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { X ^ { \\prime } } + \\Delta ^ { \\prime } + B ^ { - } \\sim _ { \\mathbb { Q } } f ^ { \\prime * } ( K _ { Y ^ { \\prime } } + B _ { Y ^ { \\prime } } + M _ { Y ^ { \\prime } } ) + B ^ { + } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,548,881,621]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $( Y ^ { \\prime } , B _ { Y ^ { \\prime } } )$ is klt and $M _ { Y ^ { \\prime } }$ is the moduli part of the associated klt-trivial fibration\u0003 \u0004 $f ^ { \\prime } \\colon ( X ^ { \\prime } , \\Delta ^ { \\prime } + B ^ { - } - B ^ { + } ) \\to Y ^ { \\prime }$ . Then analogously as in the proof of Theorem 2.16 one shows that there exists an effective $\\mathbb { Q }$ -divisor $\\Delta _ { Y ^ { \\prime } } \\sim _ { \\mathbb { Q } } B _ { Y ^ { \\prime } } + M _ { Y ^ { \\prime } }$ such that the pair $( Y ^ { \\prime } , \\Delta _ { Y ^ { \\prime } } )$ is klt, which gives (i). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,621,881,747]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finally, if $F ^ { \\prime }$ is a general fibre of $f ^ { \\prime }$ , we have $\\nu \\big ( F ^ { \\prime } , ( K _ { X ^ { \\prime } } + \\Delta ^ { \\prime } ) | _ { F ^ { \\prime } } \\big ) = 0$ by (2) and by [28, Lemma 2.3]. Therefore, there exists a good model of $( F ^ { \\prime } , \\Delta ^ { \\prime } | _ { F ^ { \\prime } } )$ by [6, Corollaire 3.4] and [30, Corollary V.4.9], hence $( K _ { X ^ { \\prime } } + \\Delta ^ { \\prime } ) | _ { F ^ { \\prime } }$ is geometrically abundant in the sense of [30, Definition V.2.23]. Then by [30, Lemma V.2.27] for an ample divisor\u0003 $A$ on $Y ^ { \\prime }$ and for any positive rational number\u0003 $\\varepsilon$ , the divisor\u0004 $K _ { X ^ { \\prime } } +$ $\\Delta ^ { \\prime } + \\varepsilon f ^ { \\prime * } A$ is geometrically abundant. In particular, $\\kappa ( X ^ { \\prime } , K _ { X ^ { \\prime } } + \\Delta ^ { \\prime } + \\varepsilon f ^ { \\prime * } A ) \\ge 0$ , and hence by (i) and by (iii) we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[198,761,799,815]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0360/0a4f5ed4d27a6b4dff6bee6f897f0eb56b0ee59484bd385aa542511b663fa0d6.jpg","list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\kappa \\big ( Y ^ { \\prime } , K _ { Y ^ { \\prime } } + \\Delta _ { Y ^ { \\prime } } + \\varepsilon A ) = \\kappa \\big ( X ^ { \\prime } , f ^ { \\prime * } ( K _ { Y ^ { \\prime } } + \\Delta _ { Y ^ { \\prime } } + \\varepsilon A ) + B ^ { + } \\big ) } \\\\ & { ~ } \\\\ & { ~ = \\kappa ( X ^ { \\prime } , K _ { X ^ { \\prime } } + \\Delta ^ { \\prime } + B ^ { - } + \\varepsilon f ^ { \\prime * } A ) \\ge 0 . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,830,881,866]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since this holds for any positive rational number $\\varepsilon$ , we conclude that $K _ { Y ^ { \\prime } } + \\Delta _ { Y ^ { \\prime } }$ is pseudoeffective, as desired. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,51,505,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Travel Guide to the Canonical Bundle Formula ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,52,879,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"60","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"53 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,250,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"61","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,136,883,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"61","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Ambro, F.: Nef dimension of minimal models. Math. Ann. 330(2), 309–322 (2004) \n2. Ambro, F.: Shokurov’s boundary property. J. Differ. Geom. 67(2), 229–255 (2004) \n3. Ambro, F.: The moduli $^ b$ -divisor of an lc-trivial fibration. Compos. Math. 141(2), 385–403 (2005) \n4. Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. In: Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 4. Springer, Berlin (1984) \n5. Birkar, C., Chen, Y.: Images of manifolds with semi-ample anti-canonical divisor. J. Algebraic Geom. 25(2), 273–287 (2016) \n6. Druel, S.: Quelques remarques sur la décomposition de Zariski divisorielle sur les variétés dont la première classe de Chern est nulle. Math. Z. 267(1–2), 413–423 (2011) \n7. Favre, C.: Note on pull-back and Lelong number of currents. Bull. Soc. Math. France 127(3), 445–458 (1999) \n8. Filipazzi, S.: On a generalized canonical bundle formula and generalized adjunction. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2018, to appear). arXiv:1807.04847 \n9. Floris, E.: Bounds on the denominators in the canonical bundle formula. Ann. Inst. Fourier (Grenoble) 63(5), 1951–1969 (2013) \n10. Floris, E.: Inductive approach to effective b-semiampleness. Int. Math. Res. Not. IMRN 2014(6), 1465–1492 (2014) \n11. Floris, E.: Nefness: generalization to the lc case. C. R. Math. Acad. Sci. Paris 352(4), 333–337 (2014) \n12. Floris, E., Lazic, V.: On the B-semiampleness conjecture. Épijournal Géom. Algébrique ´ 3, Article No. 12 (2019) \n13. Fujino, O.: A canonical bundle formula for certain algebraic fiber spaces and its applications. Nagoya Math. J. 172, 129–171 (2003) \n14. Fujino, O.: Theory of non-lc ideal sheaves: basic properties. Kyoto J. Math. 50(2), 225–245 (2010) \n15. Fujino, O.: Some remarks on the minimal model program for log canonical pairs. J. Math. Sci. Univ. Tokyo 22(1), 149–192 (2015) \n16. Fujino, O.: Fundamental properties of basic slc-trivial fibrations (2018). arXiv:1804.11134 \n17. Fujino, O., Fujisawa, T.: On semipositivity theorems (2017). arXiv:1701.02039 \n18. Fujino, O., Gongyo, Y.: On the moduli b-divisors of lc-trivial fibrations. Ann. Inst. Fourier 64(4), 1721–1735 (2014) \n19. Fujino, O., Mori, S.: A canonical bundle formula. J. Differ. Geom. 56(1), 167–188 (2000) \n20. Fujino, O., Fujisawa, T., Liu, H.: Fundamental properties of basic slc-trivial fibrations, II (2018). arXiv:1808.10604 \n21. Gongyo, Y., Lehmann, B.: Reduction maps and minimal model theory. Compos. Math. 149(2), 295–308 (2013) \n22. Kawamata, Y.: Characterization of abelian varieties. Compos. Math. 43(2), 253–276 (1981) \n23. Kawamata, Y.: Subadjunction of log canonical divisors. II. Am. J. Math. 120(5), 893–899 (1998) \n24. Kodaira, K.: On compact complex analytic surfaces. II. Ann. Math. (3) 77, 563–626 (1963) \n25. Kollár, J.: Kodaira’s canonical bundle formula and adjunction. In: Corti, A. (ed.) Flips for 3- Folds and 4-Folds. Oxford Lecture Series in Mathematics and Its Applications, vol. 35, Oxford University Press, Oxford, pp. 134–162 (2007) \n26. Kollár, J.: Singularities of the minimal model program. Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge (2013). With a collaboration of Sándor Kovács \n27. Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge (1998) \n28. Lazic, V., Peternell, Th.: On Generalised Abundance, I. Publ. Res. Inst. Math. Sci. (2018, to ´ appear). arXiv:1808.00438 \n29. Mori, S.: Classification of higher-dimensional varieties. In: Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proceedings of Symposia in Pure Mathematics, vol. 46, pp. 269–331. American Mathematical Society, Providence (1987) \n30. Nakayama, N.: Zariski-Decomposition and Abundance. MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo (2004) \n31. Prokhorov, Yu.G., Shokurov, V.V.: Towards the second main theorem on complements. J. Algebraic Geom. 18(1), 151–199 (2009) \n32. Ueno, K.: Classification of algebraic varieties. I. Compos. Math. 27, 277–342 (1973) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,54,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"61","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[706,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"61","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Floris and V. Lazic´ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,881,213]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"62","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,505,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"62","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Travel Guide to the Canonical Bundle Formula ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"62","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,747,166]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"63","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[123,202,362,220]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"63","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Anne-Sophie Kaloghiros ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,288,879,358]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"63","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract It is known that a maximal intersection log canonical Calabi–Yau surface pair is crepant birational to a toric pair. This does not hold in higher dimension: this article presents some examples of maximal intersection Calabi–Yau pairs that admit no toric model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,378,738,394]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"63","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords Calabi–Yau $\\cdot$ Fano varieties $\\cdot$ Minimal model program ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,432,496,453]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"63","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 Introduction and Motivation ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[123,477,876,512]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"63","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Calabi–Yau (CY) pair $( X , D _ { X } )$ consists of a normal projective variety $X$ and a reduced sum of integral Weil divisors $D _ { X }$ such that $K _ { X } + D _ { X } \\sim _ { \\mathbb { Z } } 0$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,513,881,602]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"63","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The class of CY pairs arises naturally in a number of problems and comprises examples with very different birational geometry. Indeed, on the one hand, a Gorenstein Calabi–Yau variety $X$ can be identified with the CY pair $( X , 0 )$ . On the other hand, if $X$ is a Fano variety, and if $D _ { X }$ is an effective reduced anticanonical divisor, then $( X , D _ { X } )$ is also a CY pair. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,611,234,629]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"63","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 1 ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,639,881,732]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"63","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(a) A pair $( X , D _ { X } )$ is (t, dlt) (resp. $\\displaystyle ( \\mathbf { t } , \\mathbf { l } \\mathbf { c } ) )$ if $X$ is $\\mathbb { Q }$ -factorial, terminal and $( X , D _ { X } )$ divisorially log terminal (resp. log canonical). \n(b) A birational map $( X , D _ { X } ) \\stackrel { \\varphi } { \\quad - - } ( Y , D _ { Y } )$ is volume preserving or crepant\b birational if $a _ { E } ( K _ { X } + D _ { X } ) = a _ { E } ( K _ { Y } + D _ { Y } )$ for every geometric valuation $E$ with centre on $X$ and on $Y$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,743,881,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"63","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The dual complex of a dlt pair $( Z , D z = \\sum D _ { i } )$ is the regular cell complex obtained by attaching an $( | I | - 1 )$ -dimensional cell for every irreducible component of a non-empty intersection $\\cap _ { i \\in I } D _ { i }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,834,769,879]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"63","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A.-S. Kaloghiros $( \\boxtimes )$ Department of Mathematics, Brunel University London, Uxbridge, UK e-mail: anne-sophie.kaloghiros@brunel.ac.uk; annesophie.kaloghiros@brunel.ac.uk ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,897,690,941]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"63","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_5 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[804,91,876,139]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"63","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,899,876,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"63","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,876,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"64","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The dual complex encodes the combinatorics of the lc centres of a dlt pair and [5]- shows that its PL homeomorphism class is a volume preserving birational invariant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,126,879,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"64","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By [3, Theorem 1.9], a (t, lc) CY pair $( X , D _ { X } )$ has a volume preserving (t, dlt) modification $( \\widetilde { X } , D _ { \\tilde { X } } ) ( X , D _ { X } )$ , and the birational map between two such modifications is volume preserving. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,180,876,216]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"64","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abusing notation, I call dual complex the following volume preserving birational invariant of a (t, lc) CY pair $( X , D _ { X } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,223,876,261]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"64","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 2 ${ \\mathcal { D } } ( X , D _ { X } )$ is the PL homeomorphism class of the dual complex of a volume preserving (t, dlt) modification of $( X , D _ { X } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,270,881,342]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"64","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As the underlying varieties of CY pairs range from CY to Fano varieties, they can have very different birational properties. However, $X$ being Fano is not a volume preserving birational invariant of the pair $( X , D _ { X } )$ . Following [14], I consider the following volume preserving birational invariant notion: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,351,881,385]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"64","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3 A (t, lc) CY pair $( X , D _ { X } )$ has maximal intersection if dim $\\phantom { } _ { 1 } \\mathscr { D } ( X , D _ { X } )$ $= \\dim X - 1$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,394,881,468]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"64","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In other words, $( X , D _ { X } )$ has maximal intersection if there is a volume preserving (t, dlt) modification of $( X , D _ { X } )$ with a 0-dimensional log canonical centre. Maximal intersection CY pairs have some Fano-type properties; Kollár and $\\mathrm { X u }$ show the following: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,475,756,495]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"64","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 1 Let $( X , D _ { X } )$ be a dlt maximal intersection CY pair, then: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,503,881,561]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"64","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. [14, Proposition 19] $X$ is rationally connected, \n2. [14, Theorem 21] there is a volume preserving map $( X , D _ { X } ) \\mathrel { - } \\stackrel { \\varphi } { } ( Z , D _ { Z } )$ such that $D _ { Z }$ fully supports a big and semiample divisor. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,570,881,642]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"64","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark $I$ The expression “Fano-type” should be understood with a pinch of salt. Having maximal intersection is a degenerate condition: a general (t, lc) CY pair $( X , D _ { X } )$ with $X$ Fano and $D _ { X }$ a reduced anticanonical section need not have maximal intersection. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,651,881,686]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"64","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 4 A toric pair $( X , D _ { X } )$ is a (t, lc) CY pair formed by a toric variety and the reduced sum of toric invariant divisors. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,689,760,705]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"64","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A toric model is a volume preserving birational map to a toric pair. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,716,211,732]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"64","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 2 ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,743,881,815]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"64","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(a) If $X$ is a normal toric variety and $D _ { X }$ the reduced sum of toric invariant divisors, $( X , D _ { X } )$ is log canonical [4, Corollary 11.4.25], so that $( X , D _ { X } )$ is a toric pair precisely when $X$ is terminal and $\\mathbb { Q }$ -factorial. \n(b) If a CY pair $( X , D _ { X } )$ has a toric model, $X$ is rational. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[744,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"64","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A.-S. Kaloghiros ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,141,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"64","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[141,108,765,127]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"65","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 1 A CY pair with a toric model has maximal intersection. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,163,876,199]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"65","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 3 In dimension 2, the converse holds: maximal intersection CY surface pairs are precisely those with a toric model [7, Proposition 1.3]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,208,881,298]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"65","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The characterisation of CY pairs with a toric model is an open and difficult problem. A characterisation of toric pairs was conjectured by Shokurov and is proved in [1], but it is not clear how to refine it to get information on the existence of a toric model. A motivation to better understand the birational geometry of CY pairs and their relation to toric pairs comes from mirror symmetry. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,300,881,408]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"65","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The mirror conjecture extends from a duality between Calabi–Yau varieties to a correspondence between Fano varieties and Landau-Ginzburg models, i.e. noncompact Kähler manifolds endowed with a superpotential. Most known constructions of mirror partners rely on toric features such as the existence of a toric model or of a toric degeneration. In an exciting development, Gross, Hacking and Keel conjecture the following construction for mirrors of maximal intersection CY pairs. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,415,879,487]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"65","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conjecture 1 ([7, Section 0.4] [8, Conjecture 1.9]) Let $( Y , D _ { Y } )$ be a simple normal crossings maximal intersection CY pair and denote by \u000e $U$ the complement \u000f $Y \\setminus D _ { Y }$ . Assume that $D _ { Y }$ supports an ample divisor, let $R$ be the ring $k [ { \\mathrm { P i c } } ( Y ) ^ { \\times } ] ,$ $\\Omega$ the canonical volume form on $U$ and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,498,851,531]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0867/5b003161e76c67b2060fc053d7b234f011a484f598aff3ffe5c409abb7152469.jpg","list_items":null,"page_idx":"65","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nU ^ { \\mathrm { t r o p } } ( \\mathbb { Z } ) = \\{ \\substack { \\mathrm { d i v i s o r i a l ~ v a l u a t i o n s } \\upsilon : k ( U ) \\setminus \\{ 0 \\} \\mathbb { Z } \\mathrm { ~ w i t h ~ } \\upsilon ( \\Omega ) < 0 } \\} \\cup \\{ 0 \\} .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,546,879,600]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"65","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Then, the free $R$ -module $V$ with basis $U ^ { \\mathrm { t r o p } } ( \\mathbb { Z } )$ has a natural finitely generated $R$ - algebra structure whose structure constants are non-negative integers determined by counts of rational curves on $U$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[143,602,690,618]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"65","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Denote by $K$ the torus $\\operatorname { K e r } \\{ \\operatorname { P i c Y } \\to \\operatorname { P i c } ( U ) \\}$ . The fibration ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[339,636,658,657]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0675/33c5f4dfcf39a537e9a6b19ceaa30009ff90052d9633bb8865129f69c0cd2df2.jpg","list_items":null,"page_idx":"65","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\np \\colon \\operatorname { S p e c } ( V ) \\to \\operatorname { S p e c } ( R ) = T _ { \\operatorname { P i c } ( Y ) }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,672,879,708]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"65","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is a $T _ { K }$ -equivariant flat family of affine maximal intersection log CY varieties. The quotient ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[384,726,612,747]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0146/477cacd943c482502b25c241ef98c37229fb3dcb940d25065ea0bb10b1273b5a.jpg","list_items":null,"page_idx":"65","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname { S p e c } ( V ) / T _ { K } \\to T _ { \\operatorname { P i c } ( U ) }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,762,571,780]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"65","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"only depends on $U$ and is the mirror family of $U$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,789,879,825]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"65","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Versions of Conjecture 1 are proved for cluster varieties in [9], but relatively few examples are known. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,827,881,899]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"65","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This note presents examples of maximal intersection CY pairs that do not admit a toric model and for which one can hope to construct the mirror partner proposed in Conjecture 1. Table 1 summarises the properties of the Examples of (t, lc) CY pairs constructed in Sects. 3 and 4. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,52,854,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"65","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"65","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,111,879,331]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0832/ed83a89cfa1b694271cb8e4c7388662f31b055540605ac00b85f89cfaa45c5ed.jpg","list_items":null,"page_idx":"66","sub_type":null,"table_body":"<table><tr><td rowspan=1 colspan=1>Boundary</td><td rowspan=1 colspan=1>Reference</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>Sing(Dx)</td><td rowspan=1 colspan=1>dim D(X,Dx)</td></tr><tr><td rowspan=5 colspan=1>Normal</td><td rowspan=1 colspan=1>Example 2</td><td rowspan=1 colspan=1>Smooth X4 C P4</td><td rowspan=1 colspan=1>Cusp T4,4.4</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=2 colspan=1>Example 3</td><td rowspan=1 colspan=1>Nodal X4 C P4</td><td rowspan=1 colspan=1>Cusp T3,3.4 and</td><td rowspan=2 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>#Sing(X)=3</td><td rowspan=1 colspan=1>3odps</td></tr><tr><td rowspan=1 colspan=1>Example 4</td><td rowspan=1 colspan=1>Smooth X4 C P4</td><td rowspan=1 colspan=1>Cusp T3.4.4</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>Example 8</td><td rowspan=1 colspan=1>Smooth X4 C P4</td><td rowspan=1 colspan=1>Two cusps T2.3.6</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=6 colspan=1>Non-normal</td><td rowspan=2 colspan=1>Example 5</td><td rowspan=2 colspan=1>Smooth X3 C P4</td><td rowspan=1 colspan=1>Nodal plane cubic curve</td><td rowspan=2 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>Double pinch point at node</td></tr><tr><td rowspan=2 colspan=1>Example 6</td><td rowspan=1 colspan=1>Nodal X4 C P4</td><td rowspan=1 colspan=1>LUL'</td><td rowspan=2 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>#Sing(X)=6</td><td rowspan=1 colspan=1>Double pinch point at L ∩ L'</td></tr><tr><td rowspan=2 colspan=1>Example 7</td><td rowspan=1 colspan=1>Nodal X4 C P4</td><td rowspan=1 colspan=1>LUL'</td><td rowspan=2 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>#Sing(X）=6</td><td rowspan=1 colspan=1>LnL'=0</td></tr></table>","table_caption":"[\"Table 1 Examples of (t, lc) CY pairs that do not admit a toric model \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[118,363,635,384]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"66","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 Auxiliary Results on Threefold CY Pairs ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,406,881,534]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"66","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The examples in Sect. 3 are threefold maximal intersection CY pairs whose underlying varieties are birationally rigid. In particular, such pairs admit no toric model; this shows that the results in [7] on maximal intersection surface CY pairs do not extend to higher dimensions. In this section, I first recall some results on birational rigidity of Fano threefolds. Then, I introduce the (t, dlt) modifications suited to the construction outlined in Conjecture 1 and discuss the singularities of the boundary $D _ { X }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,579,394,600]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"66","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.1 Birational Rigidity ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,623,879,659]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"66","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $X$ be a terminal $\\mathbb { Q }$ -factorial Fano threefold. When $X$ has Picard rank 1, $X$ is a Mori fibre space, i.e. an end product of the classical MMP. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,669,879,710]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"66","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 5 A birational map $Y / S \\stackrel { \\varphi } { \\mathrm { \\tiny ~ - - } } \\mathrm { ~ \\nabla ~ } Y ^ { \\prime } / S ^ { \\prime }$ between Mori fibre spaces $Y / S$ and $Y ^ { \\prime } / S ^ { \\prime }$ is square if it fits into a commutative square ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[398,726,599,807]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0829/b0aaca31191cdf991d02203f69613254d38a4b95a6b4f832b72c0d68878a092a.jpg","list_items":null,"page_idx":"66","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { l } { Y \\mathrm { ~ --- ~ } - \\mathrm { ~ ! ~ } \\underline { { \\varphi } } \\mathrm { ~ --- ~ } - \\mathrm { ~ ! ~ } Y ^ { \\prime } } \\\\ { \\Big \\downarrow } \\\\ { \\begin{array} { r } { \\underline { { \\begin{array} { r } { \\begin{array} { r } { \\begin{array} { r } { \\begin{array} { r } { \\end{array} } } \\end{array} } \\end{array} } } } \\\\ { S \\mathrm { ~ --- ~ } - \\mathrm { ~ ! ~ } \\underline { { g } } \\mathrm { ~ --- ~ } - \\mathrm { ~ } \\underline { { \\succ } } \\mathrm { ~ ^ { \\prime } ~ } } \\end{array} } } \\end{array} \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,831,879,873]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"66","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $g$ is birational and the restriction $Y _ { \\eta } \\stackrel { \\varphi _ { \\eta } } { \\ - - \\ + \\ } Y _ { \\eta } ^ { \\prime }$ is biregular, where $\\eta$ is the function field of the base $k ( S )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[744,54,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"66","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A.-S. Kaloghiros ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"66","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"60 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,94,879,154]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"67","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Mori fibre space $Y / S$ is (birationally) rigid if for every birational map $Y / S \\stackrel { \\varphi } { - - } $ $Y ^ { \\prime } / S ^ { \\prime }$ to another Mori fibre space, there is a birational self map $Y / S \\stackrel { \\alpha } { \\longrightarrow } X / S$ such that $\\varphi \\circ \\alpha$ is square. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,162,876,198]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"67","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In particular, if $X$ is a rigid Mori fibre space, then $X$ is non-rational and no (t, lc) CY pair $( X , D _ { X } )$ admits a toric model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,199,879,253]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"67","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Non-singular quartic hypersurfaces $X _ { 4 } ~ \\subset ~ \\mathbb { P } ^ { 4 }$ are probably the most famous examples of birationally rigid threefolds [11]. Some mildly singular quartic hypersurfaces are also known to be birationally rigid, in particular, we have: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,261,879,316]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"67","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 1 ([2, 18]) Let $X _ { 4 } \\subset \\mathbb { P } ^ { 4 }$ be a quartic hypersurface with no worse than ordinary double points. If $| \\mathrm { S i n g } ( X ) | \\leq 8$ , then $X$ is $\\mathbb { Q }$ -factorial (in particular, $X$ is a Mori fibre space) and is birationally rigid. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,361,517,382]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"67","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2 Singularities of the Boundary ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,405,881,498]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"67","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"I now state some results on the singularities of the boundary of a threefold (t, lc) CY pair. Let $( X , D _ { X } )$ be a threefold (t, lc) CY pair and $( \\widetilde { X } , \\overbrace { D _ { \\widetilde { X } } } )$ a (t, dlt) modification. A stratum of $( \\widetilde { X } , D _ { \\widetilde { X } } )$ is an irreducible component of a non-empty intersection of\u0003 \u0004 components of $D _ { \\tilde { X } }$ . Given a stratum $W$ , there is a divisor $\\mathrm { D i f f } _ { W } D _ { \\tilde { X } }$ on $W$ such that $( W , \\mathrm { D i f f } _ { W } D _ { \\tilde { X } } )$ is a lc CY pair and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[328,513,671,539]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0203/2f1e8cb4d65ef07b4e148bc4c0b7d80cc8892f4692ea581278ee737e533f34fe.jpg","list_items":null,"page_idx":"67","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { W } + \\cal { D } \\mathrm { i f f } _ { W } { \\cal D } _ { \\tilde { X } } \\sim _ { \\mathbb { Q } } \\left( K _ { \\tilde { X } } + { \\cal D } _ { \\tilde { X } } \\right) _ { | W } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,552,876,590]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"67","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"When $K _ { \\tilde { X } } + D _ { \\tilde { X } }$ is Cartier and $D _ { \\tilde { X } }$ reduced, $\\mathrm { D i f f } _ { W } D _ { \\tilde { X } }$ is the sum of the restrictions of the components of $D _ { \\tilde { X } }$ that do not contain $W$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,590,879,660]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"67","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In particular, for any irreducible component $s$ of $D _ { \\tilde { X } }$ , the link of S in ${ \\mathcal { D } } ( X , D _ { X } )$ is the dual complex $\\mathcal { D } ( S , \\mathrm { D i f f } _ { S } D _ { \\tilde { X } } )$ . Therefore, if $( X , D _ { X } )$ has maximal intersection, so does $( S , \\mathrm { D i f f } _ { S } D _ { \\tilde { X } } )$ . By the results of [7], $( S , { \\mathrm { D i f f } } _ { S } D _ { X } )$ then has a toric model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,662,881,750]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"67","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As $X$ has terminal singularities, $X$ is normal and Cohen–Macaulay. Any Cartier component $s$ of the boundary $D _ { X }$ is Cohen–Macaulay and satisfies Serre’s condition $S _ { 2 }$ . By [15, Proposition 16.9], $( S , { \\mathrm { D i f f } } _ { S } D _ { X } )$ is semi log canonical (slc). As a special case, if $X$ is Gorenstein and $D _ { X }$ is irreducible, this shows that $D _ { X }$ has slc singularities. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,752,881,804]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"67","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"I am particularly interested in producing examples of (t, lc) CY pairs for which- - the mirror partners proposed in Conjecture 1 can be constructed. The discussion in-[10] motivates the following definition: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,812,881,852]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"67","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 6 A (t, dlt) modification - $( \\widetilde { X } , D _ { \\tilde { X } } ) ( X , D _ { X } )$ is good if $( \\widetilde { X } , D _ { \\widetilde { X } } )$ is a log smooth variety (in the sense of log geometry), i.e. if $( \\widetilde { X } , D _ { \\widetilde { X } } )$ is a toroidal pair. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,860,879,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"67","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 4 The properties of (t, dlt) pairs and those of toroidal pairs are somehow different. On the one hand, $( \\widetilde { X } , D _ { \\widetilde { X } } )$ being dlt implies that the pair has simple normal crossings at the generic point of each stratum, which is not in general satisfied by- - toroidal pairs. On the other hand, as $\\widetilde { X }$ has terminal singularities and is analytically isomorphic to a toric variety, by [4, Theorem 11.4.8], $\\widetilde { X }$ only has finite quotient singularities, i.e. $\\widetilde { X }$ is an orbifold (in particular, $\\widetilde { X }$ has no Gorenstein singularities). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,51,872,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"67","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model 61 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[854,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"67","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,879,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"68","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,169,881,279]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"68","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Normal Singularities Let $p \\in \\mathrm { S i n g } ( D _ { i } )$ be an isolated singularity lying on a single component of the boundary. Since $K _ { X } + D _ { X } \\sim _ { \\mathbb { Z } } 0$ , the indices of $K _ { X }$ and of $D _ { X }$ at $p$ are the same. After taking a canonical cover, assume that $X$ is Gorenstein at $p$ , and without loss of generality (for the purpose of this discussion) assume that $D _ { X } = D _ { i }$ is irreducible and that $\\mathrm { S i n g } ( D _ { X } ) = \\{ p \\}$ . Then, since $D _ { X }$ is Cartier, by [15, Proposition 16.4], we have $K _ { D _ { X } } \\sim _ { \\mathbb { Q } } ( K _ { X } + D _ { X } ) _ { | D _ { X } }$ , so that $D _ { X }$ is Gorenstein. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,280,881,334]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"68","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $f \\colon ( { \\widetilde { X } } , D _ { \\tilde { X } } ) \\ \\ ( X , D _ { X } )$ is a good modification, $\\widetilde { X }$ is smooth (because it is Gorenstein) and up to composing with the minimal resolution of $D _ { \\tilde { X } }$ , we may assume that $f _ { | D _ { \\tilde { X } } } \\colon D _ { \\tilde { X } } \\to D _ { X }$ is a resolution. Write: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[268,349,728,372]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0237/778a246c4b729342a440a488777a0f0ad1d0d81e9961b44de560e7fd016079e2.jpg","list_items":null,"page_idx":"68","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { D _ { \\tilde { X } } } = ( K _ { \\tilde { X } } + D _ { \\tilde { X } } ) _ { } { \\mid { \\cal D } _ { \\tilde { X } } } = ( f _ { \\mid { \\cal D } _ { \\tilde { X } } } ) ^ { \\ast } ( K _ { \\cal D _ { X } } ) - E _ { \\mid { \\cal D } _ { \\tilde { X } } }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[123,388,879,426]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"68","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $E$ is defined by $K _ { \\tilde { X } } + f _ { * } ^ { - 1 } D _ { X } + E = f ^ { * } ( K _ { X } + D _ { X } )$ . We see that $p$ is canonical if $E \\cap D _ { \\tilde { X } } = \\beta$ , and elliptic otherwise. Indeed, let ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[410,442,589,469]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0356/31a5eac5a88e503d5f5cae3196a929be71e057dd00ed9859efd2427dd0d7e7d6.jpg","list_items":null,"page_idx":"68","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nD _ { \\tilde { X } } \\stackrel { q } { \\to } \\overline { { { D } } } _ { X } \\stackrel { \\mu } { \\to } D _ { X }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,484,881,629]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"68","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"be the factorisation of $f _ { | D _ { \\tilde { X } } }$ through the minimal resolution of $( p \\in D _ { X } )$ . Then, $q$ is either an isomorphism or an isomorphism at the generic point of each component of $E _ { | D _ { \\tilde { X } } }$ by [3, Lemma 2.5] because $f$ is volume preserving. We have: $K _ { \\overline { { D } } _ { X } } \\ =$ $\\mu ^ { * } K _ { D _ { X } } - Z$ , where the effective cycle $Z = q _ { * } ( E _ { D _ { \\tilde { X } } } )$ is either empty (and $p$ is canonical) or a reduced sum of $\\mu$ -exceptional curves (and $p$ is elliptic). In the second case, $Z \\sim - K _ { D _ { \\tilde { X } } }$ is the fundamental cycle of $( p \\in D _ { X } )$ . If $Z$ is irreducible, it is reduced and has genus 1; if not, every irreducible component of $Z$ is a smooth rational curve of self-intersection $^ { - 2 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,629,881,701]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"68","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"When $p$ is elliptic, $Z$ is reduced and $p$ is a Kodaira singularity [12, Theorem 2.9], i.e. a resolution is obtained by blowing up points of the singular fibre in a degeneration of elliptic curves; further, in Arnold’s terminology, the singularity $p$ is uni or bimodal. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,702,881,809]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"68","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Further, $p \\in D _ { i }$ is a hypersurface singularity (resp. a codimension 2 complete intersection, resp. not a complete intersection) when $- 3 \\leq Z ^ { 2 } \\leq - 1$ (resp. $\\bar { Z } ^ { 2 } =$ $^ { - 4 }$ , resp. $Z ^ { 2 } \\leq - 5 )$ [16]. When $- 4 \\leq Z ^ { 2 } \\leq - 1$ , normal forms are known for $p \\in D _ { i }$ : Table 2 lists normal forms of slc hypersurface singularities, while normal forms of codimension 2 complete intersections elliptic singularities are given in [20]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[744,54,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"68","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A.-S. Kaloghiros ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,54,141,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"68","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,112,881,453]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0388/bf0fb97d5d67f8ac311e4c2e4a267065c8feb139c7057062de6167784b9e3abc.jpg","list_items":null,"page_idx":"69","sub_type":null,"table_body":"<table><tr><td rowspan=1 colspan=1>Type</td><td rowspan=1 colspan=1>Name</td><td rowspan=1 colspan=1>Symbol</td><td rowspan=1 colspan=2>Equationf ∈C[x,y,z]</td><td rowspan=1 colspan=1>multof</td></tr><tr><td rowspan=1 colspan=1>Terminal</td><td rowspan=1 colspan=1>Smooth</td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=2>x</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=5 colspan=1>Canonical</td><td rowspan=5 colspan=1>du Val</td><td rowspan=1 colspan=1>An</td><td rowspan=1 colspan=1>x²+y²+zn+1</td><td rowspan=1 colspan=1>n≥1</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>D</td><td rowspan=1 colspan=1>x²+z（y²+n-2）</td><td rowspan=1 colspan=1>n≥4</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>E6</td><td rowspan=1 colspan=1>x²+y³+4</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>x²+y³+yz³</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>E8</td><td rowspan=1 colspan=1>x²+y³+5</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=4 colspan=1>1c</td><td rowspan=3 colspan=1>Simple elliptic</td><td rowspan=1 colspan=1>X1.0</td><td rowspan=1 colspan=1>x²+y4+z4+λxyz</td><td rowspan=1 colspan=1>4≠64</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>J2.0</td><td rowspan=1 colspan=1>x²+y³++λxyz</td><td rowspan=1 colspan=1>≠432</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>T3.3.3</td><td rowspan=1 colspan=1>x³+y³+³+λxyz</td><td rowspan=1 colspan=1>x≠-27</td><td rowspan=1 colspan=1>3</td></tr><tr><td rowspan=1 colspan=1>Cusp</td><td rowspan=1 colspan=1>Tp.q,r</td><td rowspan=1 colspan=1>xP+yq+z+xyz</td><td rowspan=1 colspan=1>++<1</td><td rowspan=1 colspan=1>2or3</td></tr><tr><td rowspan=7 colspan=1>slc</td><td rowspan=1 colspan=1>Normal crossing</td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=1>x²+y²</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>Pinch point</td><td rowspan=1 colspan=1>D</td><td rowspan=1 colspan=1>x²+y²</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=5 colspan=1>Degenerate cusp</td><td rowspan=1 colspan=1>T2.0.8</td><td rowspan=1 colspan=1>x²+y²+z²</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>T2.q，8</td><td rowspan=1 colspan=1>x²+y²（z²+y9-2)</td><td rowspan=1 colspan=1>q≥3</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>T.80.8</td><td rowspan=1 colspan=1>xyz</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>3</td></tr><tr><td rowspan=1 colspan=1>Tp.0.8</td><td rowspan=1 colspan=1>xyz+xp</td><td rowspan=1 colspan=1>p≥3</td><td rowspan=1 colspan=1>③</td></tr><tr><td rowspan=1 colspan=1>Tp.q.</td><td rowspan=1 colspan=1>xyz+xP+yq</td><td rowspan=1 colspan=1>q≥p≥3</td><td rowspan=1 colspan=1>3</td></tr></table>","table_caption":"[\"Table 2 Dimension 2 slc hypersurface singularities [17] \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[118,483,801,524]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"69","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 Examples of Rigid Maximal Intersection Threefold CY Pairs ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,549,881,621]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"69","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"All the examples below are (t, lc) CY pairs $( X , D _ { X } )$ which admit no toric model. Except for Example 5, all underlying varieties $X$ are birationally rigid quartic hypersurfaces by Proposition 1; the underlying variety in Example 5 is a smooth cubic threefold, and therefore non-rational. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,666,562,689]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"69","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.1 Examples with Normal Boundary ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[143,731,854,767]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"69","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 2 Consider the CY pair $( X , D _ { X } )$ where $X$ is the nonsingular quartic hypersurface ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[255,782,742,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0452/d6facd7c312fad3dfa673b87e49ba44604203b16cb5f6ecffd8b0f5e73418d10.jpg","list_items":null,"page_idx":"69","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nX = \\{ x _ { 1 } ^ { 4 } + x _ { 2 } ^ { 4 } + x _ { 3 } ^ { 4 } + x _ { 0 } x _ { 1 } x _ { 2 } x _ { 3 } + x _ { 4 } ( x _ { 0 } ^ { 3 } + x _ { 4 } ^ { 3 } ) = 0 \\}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,821,580,840]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"69","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and $D _ { X }$ is its hyperplane section $X \\cap \\{ x _ { 4 } = 0 \\}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,794,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"69","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[769,855,879,872]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"69","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(continued) ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"69","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[148,109,364,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 2 (continued) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,126,854,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The quartic surface $D _ { X }$ has a unique singular point $p = ( 1 { : } 0 { : } 0 { : } 0 { : } 0 )$ , and using the notation of Table 1, $p$ is locally analytically equivalent to a $T _ { 4 , 4 , 4 }$ cusp ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[350,192,644,217]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0985/07950970fe2f04c55b0cbcff23008e88ce58cfd24d7a54e1aaa454fca84cc1b6.jpg","list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n0 \\in \\{ x ^ { 4 } + y ^ { 4 } + z ^ { 4 } + x y z = 0 \\} .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[148,232,801,250]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$D _ { X }$ is easily seen to be rational: the projection from the triple point $p$ is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[414,264,580,291]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0787/1fc1e846e1539c6e71b852b5a53a47449a9b73a0c16db2b9688817bd159ac451.jpg","list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nD _ { X } \\ { \\mathrm { ~ -- } } \\to \\mathbb { P } _ { x _ { 1 } , x _ { 2 } , x _ { 3 } } ^ { 2 } ;\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[148,306,854,343]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"this map is the blowup of the 12 points $\\{ x _ { 1 } ^ { 4 } + x _ { 2 } ^ { 4 } + x _ { 3 } ^ { 4 } = x _ { 1 } x _ { 2 } x _ { 3 } = 0 \\}$ , of which 4 lie on each coordinate line $L _ { i } = \\{ x _ { i } = 0 \\}$ , for $i = 1 , 2 , 3$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,345,851,379]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"I treat this example in detail and construct explicitly a good (t, dlt) modification of the pair $( X , D _ { X } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,381,851,433]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $f \\colon X _ { p } \\ \\to \\ X$ be the blowup of $p$ , then $X _ { p }$ is non-singular, the exceptional divisor $E$ satisfies $( E , { \\cal O } _ { E } ( E ) ) \\ = \\ ( { \\mathbb P } ^ { \\hat { 2 } } , { \\cal O } _ { \\mathbb P ^ { 2 } } ( - 1 ) )$ , and if $D$ denotes the proper transform of $D _ { X }$ , we have: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[348,450,649,472]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0386/17249a1e7cbe376503dafc6402674c1b8353b935ed3d3cc90e4bc29daa799873.jpg","list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { X _ { p } } + D + E = f ^ { * } ( K _ { X } + D ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,489,851,545]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Explicitly, the blowup $\\mathcal { F } \\to \\mathbb { P } ^ { 4 }$ of $\\mathbb { P } ^ { 4 }$ at $p$ is the rank 2 toric variety⎞ $\\mathrm { T V } ( I , A )$ , where $ I ~ = ~ ( u , x _ { 0 } ) \\cap ( x _ { 1 } , . . . , x _ { 4 } )$ is the irrelevant ideal of $\\mathbb { C } [ u , x _ { 0 } , \\ldots , x _ { 4 } ]$ and $A$ is the action of ⎝ $\\mathbb { C } ^ { * } \\times \\mathbb { C } ^ { * }$ with weights:⎠ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[375,560,621,621]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0112/51a94e6c18c3cc4bfec4c4aee0f8319a6a66c1645622f525d68e706585275154.jpg","list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\left( \\begin{array} { c c c c c c } { { u \\ x _ { 0 } \\ s _ { 1 } } } & { { s _ { 2 } } } & { { s _ { 3 } } } & { { s _ { 4 } } } \\\\ { { 1 \\ 0 \\ - 1 \\ - 1 \\ - 1 \\ - 1 \\ - 1 } } \\\\ { { 0 \\ 1 } } & { { 1 } } & { { 1 } } & { { 1 } } & { { 1 } } \\end{array} \\right) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[175,638,378,654]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The equation of $X _ { p }$ is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[207,671,785,695]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0298/44c29223727eb4e66aa090544dd14d46cb94d9802f2c2e3b6f550df87e5dc57e.jpg","list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nX _ { p } = \\{ u ^ { 2 } \\big ( u ( s _ { 1 } ^ { 4 } + s _ { 2 } ^ { 4 } + s _ { 3 } ^ { 4 } ) + x _ { 0 } s _ { 1 } s _ { 2 } s _ { 3 } \\big ) + s _ { 4 } ( x _ { 0 } ^ { 3 } + u ^ { 3 } s _ { 4 } ^ { 3 } ) = 0 \\} ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,710,851,785]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"while $E ~ = ~ \\{ u ~ = ~ 0 \\}$ and $D = \\{ u ( s _ { 1 } ^ { 4 } + s _ { 2 } ^ { 4 } + s _ { 3 } ^ { 4 } ) + x _ { 0 } s _ { 1 } s _ { 2 } s _ { 3 } = 0 \\} .$ . By construction, $E$ is the projective plane with coordinates $s _ { 1 } , s _ { 2 } , s _ { 3 }$ . Note that $( X _ { p } , D + E )$ is not dlt because $D \\cap E \\ = \\ \\{ x _ { 0 } s _ { 1 } s _ { 2 } s _ { 3 } \\ = \\ 0 \\}$ consists of 3 concurrent lines $C _ { 1 } , C _ { 2 } , C _ { 3 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[170,785,733,803]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Consider $g _ { 1 } \\colon X _ { 1 } \\to X _ { p }$ the blowup of the nonsingular curve ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[350,819,646,840]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0964/0117b1c9a90f5d10a019d4b47fe616c8675f18df885ca0d33ab285663c60a29b.jpg","list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nC _ { 1 } = \\{ u = s _ { 1 } = s _ { 4 } = 0 \\} \\subset X _ { p } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[769,860,879,875]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(continued) ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[742,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A.-S. Kaloghiros ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"70","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"64 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[148,109,364,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 2 (continued) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,126,851,163]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The exceptional divisor of $g _ { 1 }$ is a surface $E _ { 1 } \\simeq \\mathbb { P } ( N _ { C _ { 1 } / X _ { p } } )$ , and since $C _ { 1 } \\simeq$ $\\mathbb { P } ^ { 1 }$ , the restriction sequence of normal bundles gives ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[227,181,765,202]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0312/30ae614d69127b32fb96e9fcfc2743e8e4c33c50f277b529017838d35228edc3.jpg","list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nN _ { C _ { 1 } / X _ { p } } \\simeq N _ { C _ { 1 } / E } \\oplus ( N _ { E / X _ { p } } ) _ { | C _ { 1 } } \\simeq O _ { C _ { 1 } } ( 1 ) \\oplus O _ { E } ( - 1 ) _ { | C _ { 1 } } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[148,217,380,234]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"so that $E _ { 1 } = \\mathbb { F } _ { 2 }$ . Further, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[300,252,697,274]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0894/99b97ce164b72e04cebd88174923cd77299e262b45484ac0b8fd8cc875957ec6.jpg","list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { X _ { 1 } } + D + E + E _ { 1 } = g _ { 1 } ^ { * } ( K _ { X _ { p } } + D + E )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,288,854,379]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where, abusing notation, I denote by $D$ and $E$ the proper transforms of the divisors $D$ and $E$ . The “restricted pair” on $E _ { 1 }$ is a surface CY pair $( E _ { 1 } , ( D +$ $E ) _ { | E _ { 1 } } )$ by adjunction. By construction, $E \\cap E _ { 1 }$ is the negative section $\\sigma$ . The curve $\\Gamma = D \\cap E _ { 1 }$ is irreducible, and since $( D + E ) _ { | E _ { 1 } }$ is anticanonical, we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,432,851,506]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The divisors $D , E , E _ { 1 }$ meet in two points, the dual complex $\\mathcal { D } ( X _ { 1 } , D + E +$ $E _ { 1 }$ ) is not simplicial because it is a sphere $S ^ { 2 }$ whose triangulation is given by 3 vertices on an equator. While not strictly necessary, we consider a further blowup to obtain a (t, dlt) pair with simplicial dual complex. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[175,507,610,524]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Denote by $C _ { 2 }$ the proper transform of the curve ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[407,542,589,560]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0920/5ba46d063c71fa2b7d364697e757e87caa4c44c7587d4f798d76e95dff7143a5.jpg","list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\{ u = s _ { 2 } = s _ { 4 } = 0 \\} .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[148,578,546,596]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Then $C _ { 2 } \\subset E \\cap D$ is rational, and as above ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[241,612,756,633]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0005/383c8c8809693750ae425b78614b085992f854f83f371ef381efb887c4e45210.jpg","list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nN _ { C _ { 2 } / X _ { 1 } } \\simeq N _ { C _ { 2 } / E } \\oplus ( N _ { E / X _ { 2 } } ) _ { | C _ { 2 } } = O _ { C _ { 2 } } ( 1 ) \\oplus O _ { C _ { 2 } } ( - 2 ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[143,650,851,684]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $g _ { 2 } \\colon X _ { 2 } \\to X _ { 1 }$ be the blowup of $C _ { 2 }$ , then the exceptional divisor of $g _ { 2 }$ is a Hirzebruch surface ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[378,702,621,723]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0344/5c26de794ef924895665ae03730af0614863adbc044995ddcbcf55dd58c9f716.jpg","list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nE _ { 2 } \\simeq \\mathbb { P } _ { \\mathbb { P } ^ { 1 } } ( N _ { C _ { 2 } / X _ { 1 } } ) \\simeq \\mathbb { F } _ { 3 } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,738,788,758]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Still denoting by $D , E , E _ { 1 }$ the strict transforms of $D , E , E _ { 1 }$ , we have: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[246,774,751,795]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0339/701ad4e8408e838ed7785ef1117721b50213cc77c96d143eab0b53394548c861.jpg","list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { X _ { 2 } } + D + E + E _ { 1 } + E _ { 2 } = g _ { 2 } ^ { * } ( K _ { X _ { 1 } } + D + E + E _ { 1 } ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,810,669,830]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The pair $( X _ { 2 } , D + E + E _ { 1 } + E _ { 2 } )$ is dlt; the composition ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[209,843,788,867]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0053/a1bad00ec56cdc30be8b9001711b935340c47fcdc7325535206cd445f15c1369.jpg","list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\ng _ { 2 } \\circ g _ { 1 } \\circ f \\colon ( \\widetilde { X } , D _ { \\widetilde { X } } ) = ( X _ { 2 } , D + E + E _ { 1 } + E _ { 2 } ) \\to ( X , D _ { X } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[769,879,879,896]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(continued) ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[116,51,858,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[854,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"71","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"65 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[145,109,421,139]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 2 (continued) is a good (t, dlt) modification. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,142,851,177]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The “restrictions” of $( \\widetilde { X } , D _ { \\widetilde { X } } )$ to the component of the boundary are the following surface anticanonical pairs: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,186,854,351]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$- ~ \\mathrm { O n } D \\colon ( E + E _ { 1 } + E _ { 2 } ) _ { | D }$ is a cycle of $( - 3 )$ -curves, the morphism $D \\to D _ { X }$ is the familiar resolution of the $T _ { 4 , 4 , 4 }$ cusp singularity; On $E$ : $( D + E _ { 1 } + E _ { 2 } ) _ { | E }$ is the triangle of coordinate lines with selfintersections (1, 1, 1); On $E _ { 1 }$ $\\vec { \\iota } _ { 1 } \\colon ( D + E + E _ { 2 } ) _ { | E _ { 1 } }$ is an anticanonical cycle with self-intersections $( 5 , - 3 , - 1 )$ ; On $E _ { 2 }$ : $( D + E + E _ { 1 } ) _ { | E _ { 2 } }$ is an anticanonical cycle with self-intersections $( 5 , - 3 , 0 )$ (as above, $E _ { | E _ { 2 } } \\sim \\sigma$ is a negative section, ${ \\cal E } _ { 1 | E _ { 2 } } \\sim f$ a fibre of $\\mathbb { F } _ { 3 } \\to \\mathbb { P } ^ { 1 }$ , and $D _ { | E _ { 2 } } \\sim 4 f + \\sigma )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,360,854,430]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It follows that the dual complex ${ \\mathcal { D } } ( X , D _ { X } )$ is $\\mathrm { P L }$ homeomorphic to a tetrahedron and $( X , D _ { X } )$ has maximal intersection. Note that $( 0 ~ \\in ~ D _ { X } )$ ) is a maximal intersection lc point, and since $D _ { X }$ is a rational surface, it has a toric model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,487,498,506]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example $^ 3$ Let $X$ be the hypersurface ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[232,521,763,543]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0633/c506f8addaaae4368cdd8be5c2869e94ac650f29037386907f2b3c2f82da959b.jpg","list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nX = \\{ x _ { 3 } ( x _ { 0 } ^ { 3 } + x _ { 1 } ^ { 3 } ) + x _ { 2 } ^ { 4 } + x _ { 0 } x _ { 1 } x _ { 2 } x _ { 3 } + x _ { 4 } ( x _ { 3 } ^ { 3 } + x _ { 4 } ^ { 3 } ) = 0 \\} ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,560,558,578]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and $D _ { X }$ its hyperplane section $X \\cap \\{ x _ { 4 } = 0 \\}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[170,578,799,596]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The quartic $X$ has 3 ordinary double points at the intersection points ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[405,609,589,635]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0441/15e945254390b90ceb01847104f15bbd5289c5ad2bf65ec26af7012910df57b8.jpg","list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nL \\cap \\{ x _ { 0 } ^ { 3 } + x _ { 1 } ^ { 3 } = 0 \\} ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,650,851,704]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $L$ is the line $\\{ x _ { 2 } ~ = ~ x _ { 3 } ~ = ~ x _ { 4 } ~ = ~ 0 \\}$ . The singular locus of $D _ { X }$ is $\\operatorname { S i n g } ( X ) \\cup \\{ p \\}$ , where $p = ( 0 { : } 0 { : } 0 { : } 1 { : } 0 )$ is a $T _ { 3 , 3 , 4 }$ cusp, i.e. locally analytically equivalent to ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[350,717,644,740]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0865/138d4b0bae52c1ad10d7794280a656b4abf35a9cfa3aec1fe086f0ba7e1f0049.jpg","list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n0 \\in \\{ x ^ { 3 } + y ^ { 3 } + z ^ { 4 } + x y z = 0 \\} .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[148,758,751,776]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The quartic surface $D _ { X }$ is rational; the projection of $D _ { X }$ from $p$ is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[414,789,580,815]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0761/cdbe416a30b891190e06650011274519b96d00d89749a3d23a0f9d44833f9411.jpg","list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nD _ { X } \\ { - } { \\to } \\ \\mathbb { P } _ { x _ { 0 } , x _ { 1 } , x _ { 2 } } ^ { 2 } ;\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,828,854,867]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"this map is defined outside of the 12 points (counted with multiplicity) defined by $\\{ x _ { 2 } ^ { 4 } \\overset { \\cdot } { = } x _ { 0 } ^ { 3 } + x _ { 1 } ^ { 3 } + x _ { 0 } x _ { 1 } x _ { 2 } = 0 \\}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[744,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A.-S. Kaloghiros ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"66 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[769,882,879,899]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"72","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(continued) ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[148,109,364,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"73","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 3 (continued) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,132,854,186]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"73","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $\\tilde { X } \\stackrel { f } { \\to } X$ is the composition of the blowups at the ordinary double points and at $p , \\widetilde { X }$ is smooth and $D _ { \\tilde { X } }$ is non-singular, so that $f$ is a good (t, dlt) modification. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,187,854,330]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"73","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The minimal resolution of $p \\in D _ { X }$ is a rational curve with self intersection-$C ^ { 2 } = - 3$ . Explicitly, taking the blowup of $X$ at $p$ , the proper transform is a rational surface $D$ . The exceptional curve is the preimage of a nodal cubic in $\\mathbb { P } ^ { 2 }$ blown up at 12 points counted with multiplicities. Note that $( \\widetilde { X } , D + E )$ is not dlt, but in order to obtain a (t, dlt) modification, we just need to blowup the node of $D \\cap E$ which is a nonsingular point of $\\widetilde { X } , \\bar { D ( \\mathbf { \\Phi } ) }$ and $E$ . The (t, dlt) modification of $( X , D _ { X } )$ in a neighbourhood of $p$ is good and the associated dual complex is 2-dimensional. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,331,851,367]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"73","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The pair $( X , D _ { X } )$ has maximal intersection; but as in the previous examples, $X$ is rigid, so that $( X , D _ { X } )$ can have no toric model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,423,681,441]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"73","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example $^ { 4 }$ Let $X$ be the nonsingular quartic hypersurface ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[218,456,781,480]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0918/4a562063a0074e5e55a6d6305169e71add7b6f6251999a1a8968f2608f7c5e8d.jpg","list_items":null,"page_idx":"73","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nX = \\{ x _ { 0 } ^ { 3 } x _ { 3 } + x _ { 1 } ^ { 4 } + x _ { 2 } ^ { 4 } + x _ { 0 } x _ { 1 } x _ { 2 } x _ { 3 } + x _ { 4 } ( x _ { 3 } ^ { 3 } + x _ { 4 } ^ { 3 } ) = 0 \\} \\subset \\mathbb { P } ^ { 4 }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,495,558,513]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"73","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and $D _ { X }$ its hyperplane section $X \\cap \\{ x _ { 4 } = 0 \\}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,515,854,549]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"73","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The surface $D _ { X }$ has a unique singular point $p = ( 0 { : } 0 { : } 0 { : } 1 { : } 0 )$ , which is a cusp $T _ { 3 , 4 , 4 }$ , i.e. is locally analytically equivalent to ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[350,564,644,587]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0594/db663062178195ec0a970bcc6780137c32adbbc31beb7c975ce78166a2865d47.jpg","list_items":null,"page_idx":"73","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n0 \\in \\{ x ^ { 3 } + y ^ { 4 } + z ^ { 4 } + x y z = 0 \\} .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,603,854,732]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"73","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As in Example 2, $X$ is non-singular, and finding a good (t, dlt) modification of $( X , D _ { X } )$ will amount to taking a minimal resolution of the singular point of $D _ { X }$ . Let $X _ { p } X$ be the blowup of $X$ at $p ; X _ { p }$ is non-singular and if $D$ denotes the proper transform of $D _ { X }$ , and $E$ the exceptional divisor, $D \\cap E$ consists of 2 rational curves of self intersection $^ { - 3 }$ and $^ { - 4 }$ . These curves are the proper transforms of $\\{ x _ { 0 } = 0 \\}$ and of $\\{ x _ { 0 } ^ { 2 } + x _ { 1 } x _ { 2 } = 0 \\}$ under the blow up of P2 x ,x ,x at the points ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[346,749,651,773]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0795/731dfece77a55abca32d78604fced71cb3d51e689a4a6198d4dcd118203e21c2.jpg","list_items":null,"page_idx":"73","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\{ x _ { 1 } ^ { 4 } + x _ { 2 } ^ { 4 } = x _ { 0 } ( x _ { 1 } x _ { 2 } + x _ { 0 } ^ { 2 } ) = 0 \\} .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,788,854,860]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"73","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The dual complex consists of 3 vertices that are joined by edges and span 2 distinct faces: ${ \\mathcal { D } } ( X , D _ { X } )$ is PL homeomorphic to a sphere $S ^ { 2 }$ whose triangulation is given by 3 vertices on an equator. The CY pair $( X , D _ { X } )$ has maximal intersection but no toric model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,822,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"73","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"73","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,615,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.2 Examples with Non-normal Boundary ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[148,151,826,171]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 5 This example is due to R. Svaldi. Consider the cubic threefold ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[268,184,731,210]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0035/1006d3a7c27ecf1f37f1439e8058fa1151a6b4ece51d0d3ba7a9ddd23c45ef5f.jpg","list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nX = \\{ x _ { 0 } x _ { 1 } x _ { 2 } + x _ { 1 } ^ { 3 } + x _ { 2 } ^ { 3 } + x _ { 3 } q + x _ { 4 } q ^ { \\prime } = 0 \\} \\subset \\mathbb { P } ^ { 4 }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,225,851,261]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $q , q ^ { \\prime }$ are homogeneous polynomials of degree 2 in $x _ { 0 } , \\cdots , x _ { 4 }$ . If the quadrics $q$ and $q ^ { \\prime }$ are general and if ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[302,277,694,298]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0630/132e170903f0011cc983f0181d16a86b16a03fd5890d712e93ca4232ae8d4416.jpg","list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n( q ( 1 , 0 , 0 , 0 , 0 ) , q ^ { \\prime } ( 1 , 0 , 0 , 0 , 0 ) ) \\neq ( 0 , 0 ) ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,313,790,333]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"then $X$ and $S = \\{ x _ { 3 } = 0 \\} \\cap X$ and $T = \\{ x _ { 4 } = 0 \\} \\cap X$ are nonsingular. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,334,851,385]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $D _ { X }$ be the anticanonical divisor $S + T$ . The curve $C = S \\cap T = \\Pi \\cap X$ for $\\Pi = \\{ x _ { 3 } = x _ { 4 } = 0 \\}$ is a nodal cubic. It follows that both $( S , C )$ and $( T , C )$ are log canonical, and therefore so is \u0014 $( X , D _ { X } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[141,387,851,421]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since $s$ and $T$ are smooth, $\\mathrm { S i n g } ( D _ { X } ) = S \\cap T = C$ , and if $p$ is the node of $C$ , we have:\u0014 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[195,435,788,490]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0449/a475fd5adc6bd29ae7062a496a5742c0ed0f7fdcad6cd3d2103379a19e5b6079.jpg","list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { ( p \\in D _ { X } ) \\sim 0 \\in \\big \\{ ( x y + x ^ { 3 } + y ^ { 3 } + z ) ( x y + x ^ { 3 } + y ^ { 3 } + t ) = 0 \\big \\} } \\\\ & { \\sim 0 \\in \\big \\{ ( x y + z ) ( x y + t ) = 0 \\big \\} \\sim 0 \\in \\big \\{ ( x y + z ) ( x y - z ) = 0 \\big \\} , } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,504,851,540]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\sim$ denotes analytic equivalence. Thus, $p \\in D _ { X }$ is a double pinch point, i.e. $p$ is locally analytically equivalent to $0 \\in \\{ x ^ { 2 } y ^ { 2 } - z ^ { 2 } = 0 \\}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,540,849,576]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We now construct a good (t, dlt) modification of $( X , D _ { X } )$ . Let $f \\colon X _ { C } \\to X$ be the blowup of $X$ along $C$ ; ${ \\mathrm { ; ~ } } \\mathrm { S i n g } ( X _ { C } )$ is an ordinary double point. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,578,851,648]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Indeed, let $\\Pi = \\{ x _ { 3 } = x _ { 4 } = 0 \\}$ , then $f$ is the restriction to $X$ of the blowup $\\mathcal { F } \\to \\mathbb { P } ^ { 4 }$ of $\\mathbb { P } ^ { 4 }$ along . Note that ⎛ $\\mathcal { F }$ is the rank 2 toric variety⎞ $\\mathrm { T V } ( I , A )$ , where $I = ( u , x _ { 0 } , x _ { 1 } , x _ { 2 } ) \\cap ( x _ { 3 } , x _ { 4 } )$ is the irrelevant ideal of $\\mathbb { C } [ u , x _ { 0 } , \\ldots , x _ { 4 } ]$ and $A$ is the action of $\\mathbb { C } ^ { * } \\times \\mathbb { C } ^ { * }$ with weights:⎝ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[384,665,615,725]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0175/3a0b1b16bf18169ce219d2705048b94746410682bb89c64d764649660e2708ca.jpg","list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\left( { \\begin{array} { l l l l l } { u \\ x _ { 0 } \\ x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \\ x _ { 4 } } \\\\ { 1 \\ 0 \\ \\ 0 \\ 0 \\ - 1 \\ - 1 } \\\\ { 0 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 } \\end{array} } \\right) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[148,741,353,758]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The equation of $X _ { C }$ is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[300,774,692,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0874/9dc62af1ffecd0ab8b062a965dbaa6356428f58a3525327af25c4d362c1b3c44.jpg","list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\{ x _ { 0 } x _ { 1 } x _ { 2 } + x _ { 1 } ^ { 3 } + x _ { 2 } ^ { 3 } + u ( x _ { 3 } q + x _ { 4 } q ^ { \\prime } ) = 0 \\} ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[742,52,876,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A.-S. Kaloghiros ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[769,816,879,833]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(continued) ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"74","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"68 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[145,109,521,141]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"75","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 5 (continued) so that $X _ { C }$ has a unique singular point at ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[182,157,813,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0813/2176defa2b9e7dd03dba3f4616500402dcb8ce18f50a999629c525880f1541f3.jpg","list_items":null,"page_idx":"75","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nx _ { 0 } - 1 = u = x _ { 1 } = x _ { 2 } = x _ { 3 } q ( 1 , 0 , 0 , 0 , 0 ) + x _ { 4 } q ^ { \\prime } ( 1 , 0 , 0 , 0 , 0 ) = 0 ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[148,195,854,231]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"75","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and this is a threefold ordinary double point. In addition, denoting by $E _ { f } =$ $\\{ u = 0 \\} \\cap X _ { C }$ the exceptional divisor, we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,246,671,270]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0185/5f9f825085eb0d74cac8cd09d8629c24eed54196477241231f113141f20b0f63.jpg","list_items":null,"page_idx":"75","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { X _ { C } } + \\tilde { S } + \\tilde { T } + E _ { f } = K _ { X } + S + T ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,285,640,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"75","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"so that the pair $( X _ { C } , \\tilde { S } + \\tilde { T } + E _ { f } )$ is a (t, lc) CY pair. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,307,854,451]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"75","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The pair $( X _ { C } , \\tilde { S } + \\tilde { T } + E _ { f } )$ is not dlt as the boundary has multiplicity 3 along the fibre $F$ over the node of $S \\cap T$ . The blowup of $F$ is not $\\mathbb { Q }$ - factorial, therefore in order to obtain a good (t, dlt) modification, we consider the divisorial contraction $g \\colon \\tilde { X } X _ { C }$ centred along $F$ . This is obtained by (a) blowing up the node, (b) then blowing up the proper transform of $F$ , (c) flopping a pair of lines with normal bundle $( - 1 , - 1 )$ and (d) contracting the proper transform of the $\\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 }$ above the node to a point $\\textstyle { \\frac { 1 } { 2 } } ( 1 , 1 , 1 )$ . The exceptional divisor of $g$ is denoted by $E _ { g }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,453,851,507]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"75","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The pair $( \\tilde { X } , \\tilde { S } + \\tilde { T } + \\tilde { E } _ { f } + E _ { g } )$ is the desired (t, dlt) modification of $( X , D _ { X } )$ , and it has maximal intersection. The dual complex is PL homeomorphic to a tetrahedron. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,564,567,582]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"75","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example $6$ Let $X$ be the quartic hypersurface ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,596,722,620]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0094/1d4fe569bb971885ddfb5fbea8b2009cde0b0682dec2dec2127ac1f1b8a791e5.jpg","list_items":null,"page_idx":"75","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nX = \\{ x _ { 1 } ^ { 2 } x _ { 2 } ^ { 2 } + x _ { 1 } x _ { 2 } x _ { 3 } l + x _ { 3 } ^ { 2 } q + x _ { 4 } f _ { 3 } = 0 \\} \\subset \\mathbb { P } ^ { 4 } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[148,636,851,689]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"75","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $l$ (resp. $q$ ) is a general linear (resp. quadratic) form in $x _ { 0 } , \\cdots , x _ { 3 }$ , and $f _ { 3 }$ a general homogeneous form of degree 3 in $x _ { 0 } , \\cdots , x _ { 4 }$ . Let $D _ { X }$ be the hyperplane section\u0014 $X \\cap \\{ x _ { 4 } = 0 \\}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,690,851,726]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"75","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As $l , q$ and $f _ { 3 }$ are general, $X$ has 6 ordinary double points. Indeed, denote by $L = \\{ x _ { 1 } = x _ { 3 } = x _ { 4 } = 0 \\}$ and $L ^ { \\prime } = \\{ x _ { 2 } = x _ { 3 } = x _ { 4 } = 0 \\}$ , then ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[154,741,845,764]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0503/4fecc342b72742e893dc59ad574982ee30fb998c0bb7bb7e6ec40a287e95e46a.jpg","list_items":null,"page_idx":"75","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathrm { S i n g } ( X ) = \\left\\{ L \\cap \\{ f _ { 3 } = 0 \\} \\right\\} \\cup \\left\\{ L ^ { \\prime } \\cap \\{ f _ { 3 } = 0 \\} \\right\\} = \\{ q _ { 1 } , q _ { 2 } , q _ { 3 } \\} \\cup \\{ q _ { 1 } ^ { \\prime } , q _ { 2 } ^ { \\prime } , q _ { 3 } ^ { \\prime } \\}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[769,780,879,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"75","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(continued) ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,51,804,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"75","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[854,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"75","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"69 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[148,109,364,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"76","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example $6$ (continued) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,124,851,159]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"76","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"which consists of 3 points on each of the lines. In the neighbourhood of each point $q _ { i }$ (resp. $q _ { i } ^ { \\prime } ,$ ) for $i = 1 , 2 , 3$ , the equation of $X$ is of the form ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[410,177,585,196]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0682/5799c199bded8fe3bbc71ebe70dbeab68dc2fed4eda8f1690b9a334296a039b3.jpg","list_items":null,"page_idx":"76","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n0 \\in \\{ x y + z t = 0 \\}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[143,213,851,249]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"76","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(and $D _ { X } = \\{ t = 0 \\} ,$ ) so that all singular points of $X$ are ordinary double points. The quartic hypersurface $X$ is birationally rigid by Proposition 1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[141,250,851,285]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"76","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The surface $D _ { X }$ is non-normal as it has multiplicity 2 along $L$ and $L ^ { \\prime }$ . The point $p = L \\cap L ^ { \\prime }$ is locally analytically equivalent to ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[396,298,599,322]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0532/3d2120e1dd3b64de3101c9fb3efee3db69e5d451bd75b277a48fa6031bfc1d01.jpg","list_items":null,"page_idx":"76","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n0 \\in \\{ x ^ { 2 } y ^ { 2 } + z ^ { 2 } = 0 \\} ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,339,851,375]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"76","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"so that $p \\in D _ { X }$ is a double pinch point. We conclude that the surface $D _ { X }$ has slc singularities, and hence $( X , D _ { X } )$ is a (t, lc) CY pair. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[170,376,653,393]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"76","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We construct a good (t, dlt) modification as follows. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,394,851,593]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"76","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"First, since $\\operatorname { S i n g } ( X ) \\cap L$ (resp. $\\operatorname { S i n g } ( X ) \\cap L ^ { \\prime } )$ is non-empty, the blowup of $X$ along $L$ (resp. along $L ^ { \\prime }$ ) is not $\\mathbb { Q }$ -factorial. In order to remain in the (t, dlt) category, we consider the divisorial extraction $f \\colon X _ { L } \\ \\to \\ X$ centered on $L$ (resp. $L ^ { \\prime }$ ).This is obtained by (a) blowing up the 3 nodes lying on $L$ , (b) blowing up the proper transform of $L$ , (c) flopping 3 pairs of lineswith normal bundle $( - 1 , - 1 )$ and (d) contracting the proper transforms of the three exceptional divisors $\\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 }$ lying above the nodes to points $\\textstyle { \\frac { 1 } { 2 } } ( 1 , 1 , 1 )$ . The exceptional divisor of $f$ is denoted by $E$ . Let $p \\colon { \\widetilde { X } } \\to X$ denote the morphism obtained by composing the divisorial extraction centered on $L$ with that centered on $L ^ { \\prime }$ (in any order), and let - $E , E ^ { \\prime }$ denote the exceptional divisors of the divisorial extractions. Then-","text_format":null,"text_level":null,"type":"text"},{"bbox":"[332,606,665,630]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0962/e2ab4adecb5b3e51fe085770d6eed7d68835f8810e560b0f9c5d078eb2b0024f.jpg","list_items":null,"page_idx":"76","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { \\widetilde { X } } + \\widetilde { D } + E + E ^ { \\prime } = p ^ { * } ( K _ { X } + D )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,647,854,702]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"76","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is a (t, dlt) modification of $( X , D _ { X } )$ and it has maximal intersection. The dual complex ${ \\mathcal { D } } ( X , D _ { X } )$ is PL homeomorphic to a sphere $S ^ { 2 }$ whose triangulation is given by 3 vertices on an equator. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,764,817,807]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"76","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 Further Results on Quartic Threefold CY Pairs: Beyond Maximal Intersection ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,830,879,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"76","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This section concentrates on $( t , l c )$ CY pairs $( X , D _ { X } )$ , where $X$ is a factorial quartic hypersurface in $\\mathbb { P } ^ { 4 }$ and $D$ is an irreducible hyperplane section of $X , ~ \\mathrm { I }$ give some more detail on the possible dual complexes of such pairs. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[744,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"76","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A.-S. Kaloghiros ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,54,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"76","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,232]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"77","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As explained in Sect. 2.2, $D _ { X }$ is slc because $( X , D _ { X } )$ is lc. In order to completely study the dual complexes of such (t, lc) CY pairs, one needs a good understanding of the normal forms of slc singularities that can lie on $D _ { X }$ . In the case of a general Fano $X$ , this step would require additional work, but here, $D _ { X }$ is a quartic surface in $\\mathbb { P } ^ { 3 }$ and the study of singularities of such surfaces has a rich history. I recall some results directly relevant to the construction of degenerate CY pairs $( X , D _ { X } )$ . The classification of singular quartic surfaces in $\\mathbb { P } ^ { 3 }$ can be broken in three independent cases. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,243,883,421]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"77","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(a) Quartic surfaces with no worse than rational double points: the minimal resolution is a $K 3$ surface. Possible configurations of canonical singularities were studied by several authors using the moduli theory of $K 3$ surfaces; there are several thousands possible configurations. The pair $( X , D _ { X } )$ is (t, dlt) and the dual complex of $( X , D _ { X } )$ is reduced to a point. \n(b) Non-normal quartic surfaces were classified by Urabe [19]; there are a handful of cases recalled in Theorem 2. \n(c) Non-canonical quartic surfaces with isolated singularities. These are studied by Wall [21] and Degtyarev [6] among others; their results are recalled in Theorem 3. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,430,740,450]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"77","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2 ([19]) A non-normal quartic surface $D \\subset \\mathbb { P } ^ { 3 }$ is one of: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,459,876,493]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"77","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. the cone over an irreducible plane quartic curve with a singular point of type $A _ { 1 }$ or $A _ { 2 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,495,765,513]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"77","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. a ruled surface over a smooth elliptic curve $G$ , $D = \\varphi _ { \\mathcal { L } } ( Z )$ , where: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,521,881,629]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"77","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(a) $M$ is a line bundle of degree 2 over $G , ~ N$ is a non-trivial line bundle of degree 0 over $G$ , $\\pi \\colon Z = \\mathbb { P } ( O _ { G } \\oplus N ) \\to G$ is the $\\mathbb { P } ^ { 1 }$ -bundle associated to $O _ { G } \\oplus N$ , $C _ { 1 }$ and $C _ { 2 }$ are the sections of $\\pi$ associated to $O _ { G } \\oplus N \\to O _ { G }$ and to $O _ { G } \\oplus N \\to N$ , and $\\mathcal { L } = O _ { Z } ( C _ { 1 } ) \\otimes \\pi ^ { * } M$ , $\\varphi \\underline { { c } }$ is the map associated to the linear system $| { \\mathcal { L } } |$ . Denoting by $L _ { i }$ the image by $\\varphi \\mathcal { L }$ of $C _ { i }$ , we have $\\mathrm { S i n g } ( D ) = L _ { 1 } \\cup L _ { 2 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[143,630,874,666]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"77","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(b) $M$ is a line bundle of rank 2 over $G , E$ is a rank 2 vector bundle over $G$ which fits in a non-splitting exact sequence ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[396,683,662,702]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0370/efa623d708bc826a55266989dcaada46b6b9d41fb6741167ab5dda584f73fc09.jpg","list_items":null,"page_idx":"77","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n0 O _ { G } E O _ { G } 0 ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[182,719,881,792]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"77","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\pi \\colon Z = \\mathbb { P } ( E ) \\to G$ is the $\\mathbb { P } ^ { 1 }$ -bundle associated to $E , C$ is the section of $\\pi$ associated to $E O _ { G }$ ${ \\bf \\varepsilon } , { \\mathcal { L } } = O _ { Z } ( C ) \\otimes \\pi ^ { * } M , \\varphi$ $\\varphi \\underline { { c } }$ is the map associated to the linear system $| \\mathcal { L } |$ . Denoting by $L$ the image by $\\varphi \\mathcal { L }$ of $C$ , we have $\\mathrm { S i n g } ( D ) = L$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,800,473,819]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"77","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3. a rational surface $D \\subset \\mathbb { P } ^ { 3 }$ which is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[138,827,881,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"77","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(a) the image of a smooth $S \\subset \\mathbb { P } ^ { 5 }$ under the projection from a line disjoint from S; $D$ has no isolated singular point and – $S = v _ { 2 } ( \\mathbb { P } ^ { 2 } )$ , where $v _ { 2 }$ is the Veronese embedding; $D$ is the Steiner Roman surface; ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,783,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"77","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"77","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[177,90,879,165]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"78","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$- \\ S = \\varphi ( \\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 } )$ , where $\\varphi$ is the embedding defined by $| l _ { 1 } + 2 l _ { 2 } | f o r l _ { 1 , 2 }$ the rulings of $\\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 }$ ; \n$- \\ S \\ = \\ \\varphi ( \\mathbb { F } _ { 2 } )$ , where $\\varphi$ is the embedding defined by $\\vert \\sigma + 3 f \\vert$ for $\\sigma$ the negative section and $f$ the fibre of $\\mathbb { F } _ { 2 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,171,881,226]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"78","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(b) the image of a surface $\\hat { D } \\subset \\mathbb { P } ^ { 4 }$ with canonical singularities under the projection from a point not lying on it; $\\hat { D }$ is a degenerate dP4 surface which is the blowup of $\\mathbb { P } ^ { 2 }$ in 5 points in almost general position. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,228,881,334]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"78","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(c) a rational surface embedded by a complete linear system on its normalisation $\\hat { D }$ ; the non-normal locus of $D$ is a line $L$ and $D$ may have isolated singularities outside $L$ . The minimal resolution of the normalisation of $D$ is a blowup of $\\mathbb { P } ^ { 2 }$ in 9 points. The normalisation of $D$ has at most two rational triple points lying on the inverse image of the non-normal locus; their images on $D$ are also triple points. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,343,419,361]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"78","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 5 $D$ is not slc in case 1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,370,881,426]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"78","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 1 Let $( X , D _ { X } )$ be a $( t , \\ l c )$ quartic CY pair with non-normal boundary. Then, $( X , D _ { X } )$ has maximal intersection except in the cases described in 2.(a) and $( b )$ of Theorem 2. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,462,567,481]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"78","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 7 Consider the pair $( X , D _ { X } )$ where: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[168,495,824,519]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0955/b533738050f72bc3daecd4a17f3f169de6c8d759454747f4215e020b83070d2d.jpg","list_items":null,"page_idx":"78","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nX = \\{ x _ { 0 } ^ { 2 } x _ { 3 } ^ { 2 } + x _ { 1 } ^ { 2 } x _ { 2 } x _ { 3 } + x _ { 2 } ^ { 2 } q ( x _ { 0 } , x _ { 1 } ) + x _ { 4 } f _ { 3 } = 0 \\} , D _ { X } = X \\cap \\{ x _ { 4 } = 0 \\} ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[148,534,851,570]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"78","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $q$ is a general quadratic form in $( x _ { 1 } , x _ { 2 } )$ and $f _ { 3 }$ a general cubic in $x _ { 0 } , \\cdots , x _ { 4 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,572,854,680]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"78","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since $q$ and $f _ { 3 }$ are general, the quartic hypersurface $X$ has 6 ordinary double points. Indeed, denote by $L ~ = ~ \\{ x _ { 0 } ~ = ~ x _ { 1 } ~ = ~ x _ { 4 } ~ = ~ 0 \\}$ , and by $L ^ { \\prime } = \\{ x _ { 2 } = x _ { 3 } = x _ { 4 } = 0 \\}$ , then ${ \\mathrm { S i n g } } ( X )$ consists of points of intersection of $\\{ f _ { 3 } = 0 \\}$ with $L \\cup L ^ { \\prime }$ ; there are 3 such points $\\{ q _ { 1 } , q _ { 2 } , q _ { 3 } \\}$ on $L$ and 3 points $\\{ q _ { 1 } ^ { \\prime } , q _ { 2 } ^ { \\prime } , q _ { 3 } ^ { \\prime } \\}$ on $L ^ { \\prime }$ because $f _ { 3 }$ is general. In the neighbourhood of each point $q _ { i }$ (resp. $q _ { i } ^ { \\prime } )$ for $i = 1 , 2 , 3$ , the equation of $X$ is of the form ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[410,696,585,716]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0399/ae44ea1fa0cba127f9f565bd503bf5e11bca49107c1652f730a1970f199ecabd.jpg","list_items":null,"page_idx":"78","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n0 \\in \\{ x y + z t = 0 \\}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[148,732,851,786]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"78","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(and $D _ { X } = \\{ t = 0 \\} )$ ) so that all singular points of $X$ are ordinary double points. The nodal quartic - $X$ is terminal and $\\mathbb { Q }$ -factorial because it has less than 9 ordinary double points; $X$ is birationally rigid by [2, 18]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,788,854,876]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"78","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Taking the divisorial extraction of the lines $L$ and $L ^ { \\prime }$ is enough to produce a dlt modification $( \\widetilde { X } , D _ { \\widetilde { X } } + E + E ^ { \\prime } )$ of $( X , D _ { X } )$ . As the lines $L$ , $L ^ { \\prime }$ are skew, $E$ and $E ^ { \\prime }$ are disjoint and $( X , D _ { X } )$ does not have maximal intersection. The dual complex has two 1-strata, The quartic surface $D _ { X }$ is an example of case 2.(a) in Theorem 2. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[744,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"78","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A.-S. Kaloghiros ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"78","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"72 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,876,129]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"79","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3 ([21]) A normal quartic surface $D \\subset \\mathbb { P } ^ { 3 }$ with at least one noncanonical singular point is one of: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,136,651,190]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"79","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. D has a single elliptic singularity and $D$ is rational, or \n2. D is a cone, or \n3. D is elliptically ruled and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,198,881,307]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"79","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(a) $D$ has a double point $p$ with tangent cone $z ^ { 2 }$ , the projection away from $p$ is the double cover of $\\mathbb { P } ^ { 2 }$ branched over a sextic curve $\\Gamma$ . The curve $\\Gamma$ is the union of 3 conics in a pencil that also contains a double line. When this line is a common chord, $D$ has two $T _ { 2 , 3 , 6 }$ singularities; when this line is $a$ common tangent, $D$ has one singularity of type $E _ { 4 , 0 }$ . In the first case, $D$ may have an additional $A _ { 1 }$ singular point. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,307,881,399]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"79","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(b) $D$ is $\\{ ( x _ { 0 } x _ { 3 } + q ( x _ { 1 } , x _ { 2 } ) ) ^ { 2 } + f _ { 4 } ( x _ { 1 } , x _ { 2 } , x _ { 3 } ) = 0 \\}$ and $\\{ f _ { 4 } ~ = ~ 0 \\}$ is four concurrent lines. Depending on whether $L = \\{ x _ { 3 } = 0 \\}$ is one of these lines or not and on whether the point of concurrence lies on $L , D$ has either two $T _ { 2 , 4 , 4 }$ singular points or one trimodal elliptic singularity. The surface may have additional canonical points $A _ { n }$ for $n = 1 , 2 , 3$ or $2 A _ { 1 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,435,681,454]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"79","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 8 Let $X$ be the nonsingular quartic hypersurface ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[198,468,801,492]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0849/500a21be1e3f9a828c1460cd808377d10714fbed25184d62e0044dabfd42f252.jpg","list_items":null,"page_idx":"79","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nX = \\{ x _ { 0 } ^ { 2 } x _ { 3 } ^ { 2 } + x _ { 0 } x _ { 1 } ^ { 3 } + x _ { 3 } x _ { 2 } ^ { 3 } + x _ { 0 } x _ { 1 } x _ { 2 } x _ { 3 } + x _ { 4 } ( x _ { 0 } ^ { 3 } + x _ { 3 } ^ { 3 } + x _ { 4 } ^ { 3 } ) = 0 \\}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[143,507,806,527]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"79","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and $D _ { X }$ its hyperplane section $X \\cap \\{ x _ { 4 } = 0 \\}$ . The surface $D _ { X }$ is normal, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[271,542,722,564]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0014/cdfc160e95ff0ab2b5a453f512a75d88a2d4ab7fd21bb16b0df9014e5a218b72.jpg","list_items":null,"page_idx":"79","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname { S i n g } ( D _ { X } ) = \\{ p , p ^ { \\prime } \\} = \\{ ( 1 { : } 0 { : } 0 { : } 0 { : } 0 ) , ( 0 { : } 0 { : } 0 { : } 1 { : } 0 ) \\} ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[148,579,849,615]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"79","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and each singular point is simple elliptic $J _ { 2 , 0 } ~ = ~ T _ { 2 , 3 , 6 }$ , i.e. is locally analytically equivalent to- $0 \\in \\{ x ^ { \\bar { 2 } } + y ^ { 3 } + z ^ { 6 } + x y z = 0 \\}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,617,854,743]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"79","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Here $X$ is nonsingular and $D _ { X }$ is irreducible and normal, and as I explain below, finding a good (t, dlt) modification amounts to constructing a minimal resolution of $D _ { X }$ . Let - ${ \\widetilde { X } } \\to X$ be the composition of the weighted blowups at $p = ( 1 { : } 0 { : } 0 { : } 0 { : } 0 )$ with weights $( 0 , 2 , 1 , 3 , 1 )$ and at $p ^ { \\prime } = ( 0 { : } 0 { : } 0 { : } 1 { : } 0 )$ with weights $( 3 , 1 , 2 , 0 , 1 )$ , and denote by $E$ and $E ^ { \\prime }$ the corresponding exceptional divisors. Note that $\\widetilde { X }$ is terminal and - $\\mathbb { Q }$ -factorial by [13, Theorem 3.5] and has no worse than cyclic quotient singularities. The morphism ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[364,758,635,785]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0901/54404990ccfb008908285a3461a5cd14cf4dcecaa2271a63fb9ab70a73c602c9.jpg","list_items":null,"page_idx":"79","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n( \\widetilde { X } , D + E + E ^ { \\prime } ) \\stackrel { f } { \\to } ( X , D )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,801,854,857]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"79","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is volume preserving and the intersection of $D$ with each exceptional divisor is a smooth elliptic curve $C _ { 6 } \\subset \\mathbb { P } ( 1 , 1 , 2 , 3 )$ not passing through the singular points of $E$ and $E ^ { \\prime }$ ; $f$ is a good (t, dlt) modification. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,51,858,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"79","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[769,873,879,888]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"79","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(continued) ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"79","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[148,109,364,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"80","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 8 (continued) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,126,854,177]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"80","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The dual complex ${ \\mathcal { D } } ( X , D _ { X } )$ is 1-dimensional, it has 3 vertices and 2 edges; $( X , D _ { X } )$ does not have maximal intersection. The quartic surface $D _ { X }$ is an example of case 3(a) in Theorem 3. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,213,879,267]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"80","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 2 Let $( X , D _ { X } )$ be a $( t , \\ l c )$ quartic CY pair. Assume that $D _ { X }$ is normal, has non-canonical singularities but is not a cone. Then $( X , D _ { X } )$ has maximal intersection except in cases 3.(a) and (b) of Theorem 3. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,277,879,367]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"80","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark $6$ When $\\dim { \\mathcal { D } } ( X , D _ { X } ) \\ = \\ 1$ , $D _ { X }$ either has two $T _ { 2 , 3 , 6 }$ or two $T _ { 2 , 4 , 4 }$ singularities. Indeed, as is explained in Sect. 2.2, singular points $p \\in D$ are Kodaira singularities, and in particular are at worst bimodal. The description of cases 3.(a) and (b) of Theorem 3 immediately implies the result, because a surface singularity of type $E _ { 4 , 0 }$ is trimodal. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,414,248,433]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"80","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[125,454,881,878]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"80","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Brown, M., McKernan, J., Svaldi, R., Zong, H.: A geometric characterisation of toric varieties. Duke Math. J. 167, 923–968 (2018) \n2. Cheltsov, I.: Factorial threefold hypersurfaces. J. Algebraic Geom. 19, 781–791 (2010) \n3. Corti, A., Kaloghiros, A.-S.: The Sarkisov program for Mori fibred Calabi–Yau pairs. Algebr. Geom. 3, 370–384 (2016) \n4. Cox, D., Little, J., Schenck, H., Toric Varieties. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011) \n5. de Fernex, T., Kollár, J., Xu, C.: The dual complex of singularities. Adv. Stud. Pure Math. 74, 103–130 (2017) \n6. Degtyarev, A.: Classification of surfaces of degree four having a nonsimple singular point, Math. USSR Izv. 35, 607–627 (1990) \n7. Gross, M., Hacking, P., Keel, S.: Mirror symmetry for log Calabi–Yau surfaces I. Inst. Hautes Etudes Sci. Publ. Math. 122, 65–168 (2015) \n8. Gross, M., Hacking, P., Keel, S.: Birational geometry of cluster algebras. Algebr. Geom. 2, 137–175 (2015) \n9. Gross, M., Hacking, P., Keel, S., Kontsevich, M.: Canonical bases for cluster algebras. J. Am. Math. Soc. 31, 497–608 (2018) \n10. Gross, M., Siebert, B.: Intrinsic mirror symmetry and punctured Gromov–Witten invariants. Proc. Sympos. Pure Math. 97(2), 199–230 (2018) \n11. Iskovskih, V., Manin, Ju.: Three-dimensional quartics and counterexamples to the Lüroth problem. Mat. Sb. 86, 140–166 (1971) \n12. Karras, U.: On pencils of curves and deformations of minimally elliptic singularities. Math. Ann. 47, 43–65 (1980) \n13. Kawakita, M.: Divisorial contractions in dimension three which contract divisors to smooth points. Invent. Math. 145, 105–119 (2001) \n14. Kollár, J., Xu, C.: The dual complex of Calabi–Yau pairs. Invent. Math. 205, 527–557 (2016) \n15. Kollár, J., et al.: Flips and abundance. Astérisque 211, 249–254 (1992) \n16. Laufer, H.: On minimally elliptic singularities. Am. J. Math. 99, 1257–1295 (1977) \n17. Liu, W., Rollenske, S.: Two-dimensional semi-log-canonical hypersurfaces. Le Matematiche 67, 185–202 (2012) \n18. Mella, M.: On the birational geometry of quartic 3-folds II: the importance of being $\\mathbb { Q }$ -factorial. Math. Ann. 330, 107–126 (2004) \n19. Urabe, T.: Classification of non-normal quartic surfaces. Tokyo J. Math. 9, 265–295 (1986) \n20. Wall, C.: Classification of unimodal isolated singularities of complete intersections. Proc. Sympos. Pure Math. 40, 625–640 (1983) \n21. Wall, C.: Sextic curves and quartic surfaces with higher singularities (1999). Preprint. https:// www.liverpool.ac.uk/\\~ctcw/hsscqs.ps ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[744,54,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"80","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A.-S. Kaloghiros ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,54,141,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"80","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,91,881,226]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"81","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[113,52,865,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"81","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"81","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,589,141]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"82","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,175,439,193]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"82","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Emma Lepri and Marco Manetti ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,286,881,394]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"82","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract In this paper we study classical deformations of diagrams of commutative algebras over a field of characteristic 0. In particular we determine several homotopy classes of DG-Lie algebras, each one of them controlling this above deformation problem: the first homotopy type is described in terms of the projective model structure on the category of diagrams of differential graded algebras, the others in terms of the Reedy model structure on truncated Bousfield-Kan approximations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,396,879,448]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"82","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The first half of the paper contains an elementary introduction to the projective model structure on the category of commutative differential graded algebras, while the second half is devoted to the main results. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,468,870,486]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"82","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords Model categories $\\cdot$ Deformation theory $\\cdot$ Differential graded algebras ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,522,309,543]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"82","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,567,881,728]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"82","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $\\mathbb { K }$ be a fixed field, $s$ a Noetherian commutative $\\mathbb { K }$ -algebra and $X = { \\mathrm { S p e c } } ( S )$ the associated affine scheme. It is well known that every deformation of $X$ , in the category of schemes over $\\mathbb { K }$ , is affine, hence the deformation theory of $X$ is the same of the deformation theory of $s$ inside the category $\\mathbf { A l g } _ { \\mathbb { K } }$ of unitary commutative $\\mathbb { K }$ - algebras. Similarly, the deformation theory of a separated Noetherian scheme $X$ over $\\mathbb { K }$ is the same as the deformation theory of a diagram in $\\mathbf { A l g } _ { \\mathbb { K } }$ . More precisely, if $N$ is the nerve of an affine open cover $\\{ U _ { i } \\}$ of $X$ , it is not difficult to prove that the deformations of $X$ (up to isomorphism) are the same as the deformations (up to isomorphism) of the diagram ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[234,746,758,767]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0140/a83cf2affd42a316cda131d9e7f9d1d0421ee086f7f610f8bec45002b6434948.jpg","list_items":null,"page_idx":"82","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nS _ { \\bullet } \\colon N \\to \\mathbf { A l g } _ { \\mathbb { K } } , \\qquad S _ { i _ { 0 } , \\dots , i _ { n } } = \\Gamma ( U _ { i _ { 0 } } \\cap \\cdots \\cap U _ { i _ { n } } , O _ { X } ) ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,897,692,941]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"82","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_6 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,818,865,878]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"82","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Lepri $\\cdot$ M. Manetti $( \\boxtimes$ ) Università degli Studi di Roma La Sapienza, Dipartimento di Matematica “Guido Castelnuovo”, Roma, Italy e-mail: manetti@mat.uniroma1.it ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[804,91,876,139]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"82","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,899,876,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"82","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,90,876,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"83","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $N$ is considered as a poset and as a small category in the obvious way, inside the category of diagrams $\\operatorname { F u n } ( N , \\mathbf { A } \\mathbf { l g } _ { \\mathbb { K } } ) = \\{ N \\to \\mathbf { A } \\mathbf { l g } _ { \\mathbb { K } } \\}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,126,881,232]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"83","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is well known (see e.g. [5, 13, 15]) that, if $\\mathbb { K }$ has characteristic 0, then every (commutative) deformation of an algebra $S \\in \\mathbf { A l g } _ { \\mathbb { K } }$ is isomorphic to $H ^ { 0 } ( R ^ { \\prime } )$ , where $R ^ { \\prime }$ is obtained by perturbing the differential of a fixed Tate resolution $R S$ [20]. This easily implies that the deformations of $s$ are controlled, in the sense of [12], by the differential graded Lie algebra of derivations of $R$ . A short introduction to differential graded algebras is given here in Sect. 2. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,234,881,322]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"83","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is possible to prove that the above strategy generalises to arbitrary Noetherian separated schemes, where Tate resolution is replaced by the algebraic analogue of Palamodov’s resolvent [16, 17]. This is possible because the nerve $N$ of a covering is a direct Reedy category, i.e., there exists a degree function deg $N \\to \\mathbb { N }$ such that every non identity arrow increases degree. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,324,879,412]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"83","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The aim of this paper is to study deformations of diagrams ${ \\mathcal { D } } \\to \\mathbf { A l g } _ { \\mathbb { K } }$ for a general small category $\\mathcal { D }$ . The first result is to extend the above strategy by detecting what is the correct notion of Tate resolution of a diagram $\\left[ = \\right.$ the correct notion of Palamodov’s resolvent for a diagram). In doing this it is extremely convenient to work in the framework of model structures, briefly recalled in Sect. 3. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,414,881,469]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"83","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The category $\\mathbf { A l g } _ { \\mathbb { K } }$ can be considered in an obvious way as a full subcategory of $\\mathbf { C D G A } _ { \\mathbb { K } }$ (resp.: $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ ), the category of commutative differential graded algebras (resp.: in non-positive degrees). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,469,879,524]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"83","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By a classical result of Bousfield and Gugenheim [1] the category $\\mathbf { C D G A } _ { \\mathbb { K } }$ admits a model structure where weak equivalences are the quasi-isomorphisms and fibrations are the surjective maps, cf. [4]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,524,879,612]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"83","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The category $\\mathbf { C D } \\mathbf { \\bar { G } A } _ { \\mathbb { K } } ^ { \\leq 0 }$ carries a similar model structure, where weak equivalences are the quasi-isomorphisms and fibrations are the surjective maps in negative degrees. Due to the lack of appropriate references, in Sect. 4 we provide an elementary proof of this fact, based on the properties of free and semifree extensions. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,615,881,720]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"83","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Section 5 is devoted to some technical lemmas that are probably well known to experts. In Sect. 6 we prove the main result of this paper (Theorem 4), namely that the deformation theory of a diagram $S _ { \\bullet } \\colon { \\mathcal { D } } \\ \\to \\ \\mathbf { A l g } _ { \\mathbb { K } }$ is controlled by the differential graded Lie algebra of derivations of a cofibrant replacement of $S _ { \\bullet }$ in the model category of diagrams $\\mathrm { F u n } ( \\mathcal { D } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } )$ ), equipped with the projective model structure. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,723,879,794]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"83","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Unfortunately, for general index categories $\\mathcal { D }$ , cofibrant replacements in the projective model structure are difficult to describe from the constructive point of view. For this reason, in the last sections we propose a different approach by describing a countable family of functors between small categories (Definition 15) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[300,812,692,831]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0430/ed3658eddca4cd81894bcfd932989c3b8d328ed11bff178b5cff3926e24207fa.jpg","list_items":null,"page_idx":"83","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\epsilon _ { k } \\colon N ( \\mathcal { D } ) _ { \\leq k } \\to \\mathcal { D } , \\qquad k = 2 , 3 , \\dots , \\infty ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[687,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"83","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Lepri and M. Manetti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,54,141,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"83","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,487,106]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"84","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"such that for every $k$ in the above range: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,115,879,151]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"84","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. every diagram $S _ { \\bullet } \\colon { \\mathcal { D } } \\to \\mathbf { A l g } _ { \\mathbb { K } }$ has the same isomorphism classes of deformations as $S _ { \\bullet } \\circ \\epsilon _ { k }$ ; ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,153,879,228]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"84","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. $N ( \\mathcal { D } ) _ { \\leq k }$ is a Reedy category (see Sect. 8) and the projective model structure on the category $\\mathrm { F u n } ( N ( \\mathcal { D } ) _ { \\leq k } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } )$ is the same as the Reedy model structure, hence with cofibrations described constructively in terms of latching objects and cofibrations in $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,235,879,289]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"84","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In our construction the functor $\\epsilon _ { \\infty }$ is the forgetful functor from the simplex category of $\\mathcal { D }$ (see Sect. 7), and $\\epsilon _ { k }$ is its restriction to the full subcategory of $p$ - simplexes, with $p \\leq k$ . The composition map ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[248,306,749,333]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0308/87e12017b2fa82feaa4f7e04ee21639f08f98a31665504af3d714015be6f0a45.jpg","list_items":null,"page_idx":"84","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathrm { F u n } ( \\mathcal { D } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } ) \\xrightarrow { - \\circ \\epsilon _ { \\infty } } \\mathrm { F u n } ( N ( \\mathcal { D } ) _ { \\leq \\infty } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,348,876,384]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"84","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is called Bousfield-Kan approximation and plays an important role in the homotopy theory of diagrams [3]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[138,385,760,402]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"84","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Putting together all the above facts, the main result of this paper is: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,411,876,447]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"84","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 1 $\\ Y = \\ \"$ Theorem $^ { 4 + }$ Corollary 1) Let $\\mathcal { D }$ be a small category, $S _ { \\bullet } \\colon { \\mathcal { D } } $ $\\mathbf { A l g } _ { \\mathbb { K } }$ a diagram of unitary commutative algebras. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,456,881,510]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"84","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Let $R _ { \\bullet } S _ { \\bullet }$ be a cofibrant replacement in $\\begin{array} { r } { \\mathrm { F u n } ( \\mathcal { D } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } , } \\end{array}$ ) with respect to the projective model structure. Then the DG-Lie algebra $L = \\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { \\bullet } , R _ { \\bullet } )$ controls the deformations of $S _ { \\bullet }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,513,881,584]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"84","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. For every $k ~ = ~ 2 , \\ldots , \\infty$ , let $\\epsilon _ { k } \\colon N ( \\mathcal { D } ) _ { \\leq k } \\ \\to \\ \\mathcal { D }$ be the functor defined in Definition 15 and let $R _ { \\bullet k } ~ ~ S _ { \\bullet } \\circ \\epsilon _ { k }$ be a Reedy cofibrant replacement in $\\mathrm { F u n } ( N ( \\mathcal { D } ) _ { \\leq k } , { \\bf C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } ,$ • • ). Then the DG-Lie algebra $L _ { k } \\ = \\ \\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { \\bullet k } , R _ { \\bullet k } )$ controls the deformations of $S _ { \\bullet }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,630,400,650]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"84","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1.1 Notation and Setup ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,672,881,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"84","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Throughout this paper we will work over a fixed field $\\mathbb { K }$ of characteristic 0. Unless otherwise specified, every (graded) vector space is assumed over $\\mathbb { K }$ and the symbol $\\otimes$ denotes the tensor product over $\\mathbb { K }$ . If $V = \\oplus _ { n \\in \\mathbb { Z } } V ^ { n }$ is a graded vector space, we denote by $\\overline { { a } }$ the degree of a non-zero homogeneous element $a$ : in other words ${ \\overline { { a } } } = n$ whenever $a \\neq 0$ and $a \\in V ^ { n }$ . It is implicitly assumed that if a mathematical formula contains the degree symbols $\\overline { { a } }$ , $\\overline { { b } }$ , . . . then all the elements $a , b , \\ldots$ involved are homogeneous and different from 0. As usual, for every complex of vector spaces $V$ , we shall denote by $Z ^ { n } ( V )$ , $B ^ { n } ( V )$ and $H ^ { n } ( V )$ the space of $n$ -cocycles, the space of $n$ -coboundaries and the nth cohomology group, respectively. We denote by Set the category of sets, by Grp the category of groups, by $\\mathbf { A l g } _ { \\mathbb { K } }$ the category of unitary commutative $\\mathbb { K }$ -algebras and by $\\mathbf { A r t _ { \\mathbb { K } } } \\subset \\mathbf { A l g _ { \\mathbb { K } } }$ the full subcategory of local Artin algebras with residue field $\\mathbb { K }$ . Finally, in order to avoid an excessive length we assume that the reader has a basic knowledge of differential graded Lie algebras and of the associated deformation functors: for instance, the papers [12, 14] contain everything needed for the comprehension of this paper. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,562,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"84","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,52,879,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"84","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"79 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,879,144]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"85","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,174,669,196]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"85","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 Commutative Differential Graded Algebras ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,219,881,292]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"85","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the first four sections of this paper we shall give a short survey, addressed to a wide mathematical audience, of some homotopical algebra that we use in the second part of the paper. We begin by recalling the definition and the first properties of unitary commutative differential graded algebras (DG-algebras for short) over $\\mathbb { K }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,300,881,376]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"85","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 1 A unitary commutative graded algebra is a graded vector space $A =$ $\\textstyle \\bigoplus _ { n \\in \\mathbb { Z } } A ^ { n }$ with a product $A ^ { i } \\times A ^ { j } \\longrightarrow A ^ { i + j }$ which is $\\mathbb { K }$ -linear, associative and graded commutative, i.e., such that $a b = ( - 1 ) ^ { \\overline { { { a } } } \\overline { { { b } } } } b a$ for every $a , b \\in A$ . Moreover there exists a unit $1 \\in A ^ { 0 }$ such that $1 a = a 1 = a$ for every $a \\in A$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,385,879,457]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"85","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A morphism of unitary commutative graded algebras is a morphism of graded vector spaces that commutes with products and preserves the units. We denote by $\\mathbf { C G A } _ { \\mathbb { K } }$ the category of unitary commutative graded algebras. In the above definition it is allowed that $1 = 0$ , and this happens if and only if $A = 0$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,457,881,584]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"85","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The usual construction of polynomials extends without difficulties to the graded case. Given a unitary commutative graded algebra $A$ and a set $\\{ x _ { i } \\} , ~ i ~ \\in ~ I$ , of indeterminates, each one equipped with a degree ${ \\overline { { x _ { i } } } } \\ \\in \\ \\mathbb { Z }$ , the polynomial algebra $A [ \\{ x _ { i } \\} ]$ is defined as the graded vector space generated by the monomials in $x _ { i }$ with coefficients in $A$ , subject to the relations $\\overline { { { x _ { i } } { x _ { j } } } } ~ = ~ ( - 1 ) ^ { \\overline { { { x _ { i } } } } \\overline { { { x _ { j } } } } } x _ { j } x _ { i }$ and $\\begin{array} { r l } { a x _ { i } } & { { } = } \\end{array}$ $( - 1 ) ^ { \\overline { { x _ { i } } } \\overline { { a } } } x _ { i } a$ , $a \\in A$ . For instance, if $\\overline { { x } } = 0$ and $\\overline { { y } } = 1$ , then $x y = y x$ , $y ^ { 2 } = 0$ and therefore $\\mathbb { K } [ x , y ] = \\mathbb { K } [ x ] \\oplus \\mathbb { K } [ x ] y$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,584,876,620]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"85","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Given $A \\in \\mathbf { C G A } _ { \\mathbb { K } }$ , a derivation of degree $k$ of $A$ is a linear map $\\alpha \\colon A \\to A$ such that $\\alpha ( A ^ { n } ) \\subset A ^ { n + k }$ for every $n$ , satisfying the (graded) Leibniz identity: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[346,635,651,656]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0283/9befbcd1f13b4468aa2c6a7b18c0c2f151b20db3ac8408aff206b10f11fcd4f4.jpg","list_items":null,"page_idx":"85","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\alpha ( a b ) = \\alpha ( a ) b + ( - 1 ) ^ { k \\overline { { { a } } } } a \\alpha ( b ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,672,733,692]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"85","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The vector space of derivations of degree $k$ is denoted $\\mathrm { D e r } _ { \\mathbb { K } } ^ { k } ( A , A )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,693,881,731]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"85","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $A \\ = \\ \\mathbb { K } [ \\{ x _ { i } \\} ]$ , by the Leibniz identity every derivation $\\alpha \\in { \\mathrm { D e r } } _ { \\mathbb { K } } ^ { k } ( A , A )$ is uniquely defined by the values $\\alpha ( x _ { i } ) \\in A ^ { \\overline { { x _ { i } } } + k }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,740,879,794]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"85","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 2 A commutative differential graded algebra (DG-algebra for short) is a graded commutative algebra $A$ equipped with a derivation $d \\in \\mathrm { D e r } _ { \\mathbb { K } } ^ { 1 } ( A , A )$ , called differential, such that $d ^ { 2 } = 0$ . In other words: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,801,669,858]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"85","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. $d ( A ^ { n } ) \\subseteq A ^ { n + 1 }$ , \n2. $d ^ { 2 } = 0$ , \n3. (Graded Leibniz identity) $d ( a b ) = d ( a ) b + ( - 1 ) ^ { \\overline { { { a } } } } a d ( b ) .$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,866,881,902]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"85","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A morphism of commutative differential graded algebras is a morphism of commutative graded algebras that commutes with differentials. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"85","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"80 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[687,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"85","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Lepri and M. Manetti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We denote by $\\mathbf { C D G A } _ { \\mathbb { K } }$ the category of commutative differential graded algebras. Notice that $d ( 1 ) = 0$ , and that $\\mathbb { K }$ and 0 are respectively the initial and the final object in the category $\\mathbf { C D G A } _ { \\mathbb { K } }$ . It is easy to see that this category is complete and cocomplete. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,169,881,261]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3 (Free Extensions) Let $\\begin{array} { r } { A \\in \\begin{array} { r } { \\mathbf { C D G A } _ { \\mathbb { K } } } \\end{array} } \\end{array}$ and $x _ { i }$ , $\\textit { i } \\in \\textit { I }$ , a set of indeterminates of degree ${ \\overline { { x _ { i } } } } ~ \\in ~ \\mathbb { Z }$ . Consider a parallel set of indeterminates $d x _ { i }$ , with $\\overline { { d x _ { i } } } = \\overline { { x _ { i } } } + 1$ and the polynomial extension $A \\to A [ \\{ x _ { i } , d x _ { i } \\} ]$ . The differential $d$ on $A$ can be extended to a differential on $A [ \\{ x _ { i } , d x _ { i } \\} ]$ by setting $d ( x _ { i } ) = d x _ { i }$ and $d ( d x _ { i } ) = 0$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,270,881,342]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The name free extension is motivated by the following property: for every morphism $f \\colon A \\ \\to \\ B$ in $\\mathbf { C D G A } _ { \\mathbb { K } }$ and every subset $\\{ b _ { i } \\} \\subset B$ with $b _ { i } ~ \\in ~ B ^ { \\overline { { x _ { i } } } }$ for every $i$ , there exists a unique morphism of DG-algebras $g \\colon A [ \\{ x _ { i } , d x _ { i } \\} ] B$ extending $f$ and such that $g ( x _ { i } ) = b _ { i }$ for every $i$ . Clearly $g ( d x _ { i } ) = d ( b _ { i } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,351,881,387]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 1 Every free extension of $_ { D G }$ -algebras is a quasi-isomorphism, i.e., the inclusion $A \\to A [ \\{ x _ { i } , d x _ { i } \\} ]$ induces an isomorphism in cohomology. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,394,881,466]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Since every element of $A [ \\{ x _ { i } , d x _ { i } \\} ]$ is a polynomial in a finite number of indeterminates, we can assume the set of variables finite, say $x _ { 1 } , \\ldots , x _ { n }$ , and proceed by induction on $n$ . Therefore it is sufficient to show that the inclusion $A \\to A [ x , d x ]$ is a quasi-isomorphism. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,468,876,504]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $x$ has even degree, then $( d x ) ^ { 2 } = 0$ and every homogeneous element of the quotient $A [ x , d x ] / A$ is of type ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[298,515,699,567]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"1016/33408f49e3b6513cf57157a77ffe3fa7a90f7b6fdb133156c4e4b0fae65f645e.jpg","list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nv = \\sum _ { i = 0 } ^ { n } x ^ { i + 1 } a _ { i } + x ^ { i } d x b _ { i } , \\qquad a _ { i } , b _ { i } \\in A .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,578,881,596]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $d v = 0$ then $d ( b _ { i } ) = ( i + 1 ) a _ { i }$ and therefore (notice the assumption char $( \\mathbb { K } ) = 0$ ) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[387,608,610,659]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"1008/428784783f2c8d85ca28ad12e30622708d3b042e825f084c769491b75a940fd9.jpg","list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nv = d \\left( \\sum _ { i = 0 } ^ { n } \\frac { x ^ { i + 1 } } { i + 1 } b _ { i } \\right) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[123,672,881,710]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $x$ has odd degree, then $x ^ { 2 } = 0$ and every homogeneous element of the quotient $A [ x , d x ] / A$ is of type ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[275,722,722,773]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0356/ffa50fea34f3505cae393ca117998e3c68a7b86d33a1fb75eb1fd50c3215bcc1.jpg","list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nu = \\sum _ { i = 0 } ^ { n } ( d x ) ^ { i + 1 } a _ { i } + x ( d x ) ^ { i } b _ { i } , \\qquad a _ { i } , b _ { i } \\in A .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,783,649,801]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $d u = 0$ then $d a _ { i } + b _ { i } = 0$ for every $i$ , and we can write ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[382,813,615,866]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0701/ea04aeb7b83ff1b5681e35d7ad403866f5d7ce311ae2cc940b0f9bd5d8c83e3c.jpg","list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nu = d \\left( \\sum _ { i = 0 } ^ { m } x ( d x ) ^ { i } a _ { i } \\right) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[113,879,826,899]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Thus we have proved that $A [ x , d x ] / A$ is an acyclic complex of vector spaces. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,562,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[861,882,879,896]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"86","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"□ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,91,879,147]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"87","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 4 Let $f \\colon A \\to B$ be a morphism in $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ , with $A = A ^ { 0 } \\in \\mathbf { A l g } _ { \\mathbb { K } }$ We shall say that $f$ is flat, or that $B$ is a flat $A$ -algebra if $B$ is a complex of flat $A$ -modules. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,156,879,211]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"87","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Clearly the above definition extends the usual notion of flat morphism of algebras. It is worth pointing out that there also exists a good notion of flatness for every morphism in $\\mathbf { C D G } \\mathbf { \\bar { A } } _ { \\mathbb { K } } ^ { \\leq 0 }$ [15]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,211,881,271]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"87","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For every $A \\in \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ we shall denote by $\\mathbf { C D G A } _ { A } ^ { \\leq 0 }$ the undercategory of maps $A B$ : the morphisms in $\\mathbf { C D G A } _ { A } ^ { \\leq 0 }$ are the commutative triangles. The following lemma is completely standard, see e.g. [19, Lemma A.4 and Theorem A.10]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[113,279,799,300]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"87","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 2 Let $A \\in \\mathbf { A r t } _ { \\mathbb { K } }$ and let $f \\colon B \\to C$ be a morphism in $\\mathbf { C D G A } _ { A } ^ { \\leq 0 }$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,307,881,435]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"87","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. if $C$ is flat over A and the induced map $B \\otimes _ { A } \\mathbb { K } \\to C \\otimes _ { A } \\mathbb { K }$ is an isomorphism, then $f$ is also an isomorphism; \n2. if $B$ , $C$ are flat $A$ -algebras and the induced map $B \\otimes _ { A } \\mathbb { K } \\to C \\otimes _ { A } \\mathbb { K }$ is a quasiisomorphism, then $f$ is also a quasi-isomorphism; \n3. if $B$ is flat over $A$ and $H ^ { i } ( B \\otimes _ { A } \\mathbb { K } ) = 0$ for every $i \\ < 0$ , then $H ^ { 0 } ( B )$ is a flat $A$ -algebra and the natural map $H ^ { 0 } ( B ) \\to H ^ { 0 } ( B \\otimes _ { A } \\mathbb { K } )$ induces an isomorphism $H ^ { 0 } ( B ) \\otimes _ { A } \\mathbb { K } = H ^ { 0 } ( B \\otimes _ { A } \\mathbb { K } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,478,710,500]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"87","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 A Very Short Introduction to Model Structures ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,522,881,576]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"87","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We briefly recall the definition of model category and some few basic results about them; the reader may consult [6, 7] for a deeper and more complete exposition of the subject. Throughout this section M will denote a fixed category. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,585,876,623]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"87","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 5 (Lifting Properties) Consider two morphisms $i \\colon A \\to B , f \\colon C \\to$ $D$ in M. If for every solid commutative diagram ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[432,638,562,707]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0901/d6c590c3afdb16ac4843d49db5aaec57e6ac985e149065895cf8c44695948ecf.jpg","list_items":null,"page_idx":"87","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[118,726,879,782]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"87","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"there exists the dotted arrow that makes both triangles commute, we shall say that the map $i$ has the left lifting property with respect to $f$ , and the map $f$ has the right lifting property with respect to $i$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,791,881,845]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"87","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For instance, in the category of sets, every injective map $i$ has the left lifting property with respect to any surjective map $f$ . The same holds in the category of vector spaces. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[687,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"87","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Lepri and M. Manetti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"87","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"82 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,881,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"88","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 6 (Retracts) A morphism $f$ in $\\mathbf { M }$ is called a retract of a morphism $g$ in $\\mathbf { M }$ if there exists a commutative diagram: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[389,142,608,265]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0604/35b67bdee57017e081f39e3bae7cc5b317786d8c7b09d7960285b3262a2a2cdb.jpg","list_items":null,"page_idx":"88","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[120,288,881,324]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"88","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 7 A model structure on $\\mathbf { M }$ is the data of three classes of maps: weak equivalences, fibrations and cofibrations, which satisfy the following axioms: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,333,881,384]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"88","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(M1) (2-out-of-3) If $f$ and $g$ are morphisms in $\\mathbf { M }$ such that the composition $g f$ is defined, and two out of the three $f , g$ and $g f$ are weak equivalences, so is the third. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,387,876,421]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"88","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(M2) (Retracts) If $f$ and $g$ are maps in $\\mathbf { M }$ such that $f$ is a retract of $g$ , and $g$ is a weak equivalence, a cofibration or a fibration, then so is $f$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,423,881,475]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"88","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(M3) (Lifting) A trivial fibration is map which is both a fibration and a weak equivalence; a trivial cofibration is map which is both a cofibration and a weak equivalence. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[157,486,879,540]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"88","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(a) Trivial fibrations have the right lifting property with respect to cofibrations. (b) Trivial cofibrations have the left lifting property with respect to fibrations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[113,549,783,567]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"88","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(M4) (Factorisation) Every morphism $g$ in $\\mathbf { M }$ admits two factorisations: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[177,576,820,612]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"88","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(CW, F): $g = q j$ , where $j$ is a trivial cofibration and $q$ is a fibration, (C, FW): $g = p i$ , where $i$ is a cofibration and $p$ is a trivial fibration. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,621,879,656]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"88","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 8 A model category is a complete and cocomplete category equipped\u0018 with a model structure. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,665,881,755]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"88","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In particular every model category has an initial object $\\amalg _ { \\varnothing }$ and a final object $\\Pi _ { \\emptyset }$ ; an object $X$ is called cofibrant in the morphism $\\operatorname { U } _ { \\varnothing } \\to X$ is a cofibration; it is called fibrant if the morphism $X \\to \\Pi _ { \\emptyset }$ is a fibration. A cofibrant replacement of an object $Y$ is a trivial fibration $X Y$ with $X$ cofibrant. The factorisation axiom guarantees that cofibrant replacements always exist. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,756,881,827]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"88","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For notational simplicity we shall denote by $\\mathcal { W }$ , $\\mathcal { F }$ and $c$ the classes of weak equivalences, fibrations and cofibrations, respectively. We shall denote by ${ \\mathcal { F W } } =$ $\\mathcal { F } \\cap \\mathcal { W }$ the class of trivial fibrations and by $C W = C \\cap \\mathcal { W }$ the class of trivial cofibrations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,837,879,873]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"88","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 3 If $j$ has the left (right) lifting property with respect to $f$ , and $i$ is a retract of $j$ , then i has the left (right) lifting property with respect to $f$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,882,387,900]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"88","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For a proof, see [6, 7.2.8]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,562,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"88","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"88","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,881,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"89","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 1 (Retract Argument) Let g be a map which can be factored as $g =$ pi ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,135,806,171]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"89","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. If g has the left lifting property with respect to p then g is a retract of i. \n2. If g has the right lifting property with respect to i then g is a retract of $p$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,178,387,198]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"89","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For a proof, see [7, 1.1.9]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,205,480,223]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"89","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 4 Let M be a model category: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,232,881,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"89","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. A map in M that has the left lifting property with respect to all trivial fibrations is a cofibration. \n2. A map in M that has the right lifting property with respect to all trivial cofibrations is a fibration. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,313,398,331]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"89","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For a proof, see [7, 1.1.10]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,342,881,412]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"89","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 1 It follows from the previous lemma that isomorphisms belong to all three classes of maps. Furthermore, two of the three classes $\\mathcal { W }$ , $c$ , $\\mathcal { F }$ determine the third. Pay attention to the fact that, for example, if $\\mathcal { W }$ and $\\mathcal { F }$ are two classes satisfying (M1) and (M2), in general they do not extend to a model structure. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,423,419,439]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"89","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The following lemma is clear. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,448,879,486]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"89","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 5 If $h = g f$ and both $f , g$ have the left (right) lifting property with respect to $p$ , then h has the left (right) lifting property with respect to $p$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,495,874,530]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"89","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 2 The previous lemmas show that the three classes of fibrations, cofibrations and weak equivalences are closed by composition. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,539,881,647]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"89","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Model categories were introduced by Quillen [18] under the name of (complete and cocomplete) closed model categories. Nowadays many authors (e.g. [6, 7]) assume that the (C,FW) and (CW,F) factorisations are functorial. Since in algebraic geometry it is often important to resolve minimally algebraic structures, we prefer here to adopt the original Quillen’s assumption, and require only the existence of factorisations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,693,421,713]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"89","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.1 Pre-model Structures ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,737,881,881]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"89","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In several concrete cases, a convenient way to describe model structures is in terms of pre-model structures. Since two out of the three classes ${ \\mathcal { W } } , C , { \\mathcal { F } }$ determine the third, it is typical to try to construct a model structure by establishing two out of the three classes and seeing whether they extend to a model structure. Often it happens that two classes have an easy description and the third is more complicated, but it has a nice subclass sufficiently large to ensure axiom (M4). The notion of left premodel structure applies when one has fixed the weak equivalences, fibrations and two more classes, as in the next definition. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[687,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"89","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Lepri and M. Manetti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"89","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"84 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,879,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"90","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 9 A left pre-model structure on $\\mathbf { M }$ is the data of four classes of maps: $\\mathcal { W } , \\mathcal { F } , C ^ { \\prime } , C \\mathcal { W } ^ { \\prime }$ such that: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,135,881,244]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"90","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. (2-out-of-3) The maps in $\\mathcal { W }$ satisfy the 2-out-of-3 property; \n2. (Retracts) The classes $\\mathcal { W }$ and $\\mathcal { F }$ are closed under retracts; \n3. $C W ^ { \\prime } \\subseteq \\mathcal { W }$ ; \n4. (Lifting) The maps in $C ^ { \\prime }$ have the left lifting property with respect to the maps in $\\mathcal { F } \\cap \\mathcal { W }$ ; the maps in $c w ^ { \\prime }$ have the left lifting property with respect to the maps in $\\mathcal { F }$ ; ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,243,667,261]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"90","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5. (Factorisation) Every map $g$ in $\\mathbf { M }$ has two factorisations: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,270,589,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"90","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(a) $g = q j$ , where $j$ is in $C ^ { \\prime }$ and $q$ is in $\\mathcal { F } \\cap \\mathcal { W }$ , (b) $g = p i$ , where $i$ is in $c w ^ { \\prime }$ and $p$ is in $\\mathcal { F }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,315,881,385]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"90","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2 Given a left pre-model structure ${ \\mathcal { W } } , { \\mathcal { F } } , C ^ { \\prime }$ , $c w ^ { \\prime }$ there exists a unique model structure where the weak equivalences are the maps in $\\mathcal { W }$ and the fibrations are the maps in $\\mathcal { F } .$ Notably, the cofibrations are the retracts of $C ^ { \\prime }$ , and the trivial cofibrations are the retracts of $C \\mathcal W ^ { \\prime }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,394,881,466]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"90","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof We set $c$ the retracts of $C ^ { \\prime }$ and check that $c , \\mathcal { F } , \\mathcal { W }$ satisfy the model category axioms (Definition 7). Axioms (M1) and (M2) follow immediately from the definition of pre-model structure and of $c$ . As above, for notational simplicity we shall denote by $C W = C \\cap \\mathcal { W }$ and $\\mathcal { F W } = \\mathcal { F } \\cap \\mathcal { W }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,468,881,546]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"90","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We first show that $c w ^ { \\prime } \\subset C$ . Let $g \\colon A \\ \\to \\ B$ be a morphism in $c w ^ { \\prime }$ and consider a factorisation $g \\colon A \\ { \\overset { C ^ { \\prime } } { \\to } } \\ X \\ { \\overset { \\mathcal { F W } } { \\to } } \\ B$ . Since $g$ has the left lifting property with respect to maps in $\\mathcal { F W }$ it follows by the retract argument that $g$ is a retract of an element of $C ^ { \\prime }$ . Since $C W ^ { \\prime } \\subset \\mathcal { W }$ by assumption, we have $C W ^ { \\prime } \\subset C W$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,548,879,624]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"90","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We now show that every map in $c w$ is a retract of a map in $c w ^ { \\prime }$ . Let $f \\colon A \\longrightarrow$ $B$ be a map in $c w$ , using the factorisation axiom $f \\colon A \\ { \\xrightarrow { C { \\mathcal { W } } ^ { \\prime } } } \\ X \\ { \\xrightarrow { \\mathcal { F } } } \\ B$ , and by the 2-out-of-3 axiom $X \\longrightarrow B$ is in $\\mathcal { F W } .$ . Therefore by the lifting axiom and the retract argument (1), $A \\longrightarrow B$ is a retract of $A \\longrightarrow X$ , so it is the retract of a map in $c w ^ { \\prime }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,626,879,716]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"90","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By definition of pre-model structure, maps in $C ^ { \\prime }$ have the left lifting property with respect to maps in $\\mathcal { F W }$ ; then by Lemma 3 maps in $c$ have the left lifting property with respect to maps in $\\mathcal { F W }$ . Similarly, maps in $c w ^ { \\prime }$ have the left lifting property with respect to maps in $\\mathcal { F } _ { : }$ , so we have that maps in $c w$ have the left lifting property with respect to maps in $\\mathcal { F }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,716,874,752]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"90","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The factorisation axiom (M4) is clear since we have already proved that $C \\subset C ^ { \\prime }$ and $C W ^ { \\prime } \\subseteq C W$ . □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,761,881,833]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"90","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is plain that one can also give the analogous notion of right pre-model structure, simply working in the opposite category and exchanging the role of $c$ and $\\mathcal { F } .$ . Finally, the reader should be aware that some authors use the name of pre-model structure for a completely different concept. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,562,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"90","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"90","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,553,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"91","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 Model Structure on DG-Algebras ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,132,879,205]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"91","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is well known that the category $\\mathbf { C D G A } _ { \\mathbb { K } }$ admits a model structure where weak equivalences are the quasi-isomorphisms and fibrations are the surjective maps [1, 4]. Consequently, by Lemma 4 the cofibrations are the morphisms that have the left lifting property with respect to the class of surjective quasi-isomorphisms. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,205,881,349]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"91","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Similarly, the category $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ of DG-algebras concentrated in non-positive degree admits a model structure where weak equivalences are the quasiisomorphisms and fibrations are the surjective maps in negative degree. It is worth noticing that the existence of the model structure on $\\mathbf { C } \\mathbf { \\bar { D } G } \\mathbf { A } _ { \\mathbb { K } } ^ { \\leq 0 }$ is an immediate consequence of [11, Proposition 4.5.4.6] applied to the standard (cofibrantly generated) model structure on the category of non-positively graded DG-vector spaces. Moreover, by Lurie’s result also follows that the model structure on $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ is combinatorial and cofibrantly generated. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,351,881,403]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"91","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section, following the ideas of [8], we give an elementary proof of the above mentioned model structure on $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ A≤0K , which relies on the notion of semifree extension. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,409,881,466]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"91","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 10 (Semifree Extension) Let $A \\in \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ , $I$ be a set, and let $x _ { i } , i \\in I$ be indeterminates of non-positive degree $\\overline { { x _ { i } } } \\in \\mathbb { Z } _ { \\leq 0 }$ . Any inclusion of DG-algebras of type ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[423,480,571,500]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0022/6d255f8454464f4dfca207a5d9434b30c26879217c01d9c49183dba423285c2b.jpg","list_items":null,"page_idx":"91","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nA \\longrightarrow A [ \\{ x _ { i } \\} ] ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,513,760,531]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"91","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"regardless of the differential on $A [ \\{ x _ { i } \\} ]$ , is called a semifree extension. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,540,876,576]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"91","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recall that a differential on $A [ \\{ x _ { i } \\} ]$ is determined by the differential on $A$ and by the values $d ( x _ { i } )$ . Every free extension is also semifree. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,584,876,638]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"91","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3 There exists a model structure on the category $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ , where weak equivalences are the quasi-isomorphisms and fibrations are the maps surjective in negative degree. Moreover: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,648,653,702]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"91","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. cofibrations are the retracts of semifree extensions, \n2. trivial cofibrations are the retracts of free extensions, \n3. trivial fibrations are the surjective quasi-isomorphisms. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,711,881,819]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"91","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The proof that every trivial fibration is surjective is a simple argument in basic homological algebra. In fact, if $f \\colon A \\ \\longrightarrow \\ B$ is a quasi-isomorphism which is surjective in negative degree, for every $x \\in B ^ { 0 }$ , since $d x = 0$ and $f \\colon H ^ { 0 } ( A ) \\to$ $H ^ { 0 } ( B )$ is bijective, there exist $y \\in A ^ { 0 }$ and $z \\in B ^ { - 1 }$ such that $f ( y ) = x + d z $ . Since $f \\colon A ^ { \\ - 1 } \\longrightarrow B ^ { - 1 }$ is surjective there exists $u \\in A ^ { - 1 }$ such that $f ( u ) = z$ and therefore $x = f ( y - d u )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,819,879,854]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"91","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now the proof of Theorem 3 follows, according to Theorem 2, from the fact that the four classes: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,864,489,900]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"91","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. quasi-isomorphisms, \n2. $\\mathcal { F }$ maps surjective in negative degree, \n3. $C ^ { \\prime }$ semifree extensions, \n4. $c w ^ { \\prime }$ free extensions. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"91","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"86 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[687,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"91","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Lepri and M. Manetti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,362,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,135,881,189]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"form a left pre-model structure. We have already proved in Lemma 1 that $c w ^ { \\prime } \\subset$ $\\mathcal { W }$ . The 2-out-of-3 axiom for $\\mathcal { W }$ is clear, and the retract axiom for $\\mathcal { W }$ , $\\mathcal { F }$ is also obvious: the retract of a injective (surjective) map is also injective (surjective). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,198,879,234]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 2 The maps in $C ^ { \\prime }$ , i.e., the semifree extensions, have the left lifting property with respect to all trivial fibrations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,243,651,262]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Consider the following solid commutative diagram ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[410,280,587,352]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0720/3040397b5c7fc224650f7bf519e1f0d653502ebb9350059b16d6afd4af642dd6.jpg","list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n{ \\underset { A \\left[ \\left\\{ X _ { i } \\right\\} \\right] } { \\overset { A } { \\longrightarrow } } } { \\underset { \\substack { \\overset { . . . . } { \\longrightarrow } } } { \\overset { \\alpha } { \\longrightarrow } } } C \n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,372,876,409]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $g$ is a trivial fibration and $f$ a semifree extension. For every integer $n \\geq - 1$ consider the DG-subalgebra of $A [ \\{ x _ { i } \\} ]$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[380,426,619,445]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0121/3e070fbdd1357698a346bf4f1bc201ff4ae9fe088de68ef20dcd1919be56fac5.jpg","list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nA _ { n } = A [ \\{ x _ { i } \\mid { \\overline { { x _ { i } } } } \\geq - n \\} ] .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,462,881,500]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We have $A _ { - 1 } = A$ , $\\cup _ { n } A _ { n } = A [ \\{ x _ { i } \\} ]$ and therefore, setting $\\gamma _ { - 1 } = \\alpha$ it is sufficient to prove by induction that for every $n \\geq 0$ we have a commutative diagram ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[421,515,576,590]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0570/8e224e1d7af8586ff240bcf6cddc67042aa189b4e4e4609ffaaeb35061fdb8b9.jpg","list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{c} \\begin{array} { c } { { A _ { n - 1 } \\xrightarrow { \\gamma _ { n - 1 } } C } } \\\\ { { \\downarrow \\underbrace { \\gamma _ { n } } \\end{array} \\downarrow g } _ { \\beta } } \\\\ { { A _ { n } \\xrightarrow { \\beta _ { n } } D } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,609,881,663]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where the left vertical arrow is the inclusion $A _ { n - 1 } \\subset A _ { n }$ . If $n = 0$ , then for every $x _ { i }$ with $\\overline { { x _ { i } } } = 0$ we have $d x _ { i } = 0$ . Since $g$ is surjective there exists $c _ { i } \\in C$ such that $g ( c _ { i } ) = \\beta ( x _ { i } )$ . We define $\\gamma _ { 0 }$ by setting $\\gamma _ { 0 } ( x _ { i } ) = c _ { i }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,665,876,699]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Assume now $n > 0$ and $\\gamma _ { n - 1 }$ already defined. If ${ \\overline { { x _ { i } } } } = - n$ , then ${ \\overline { { d ( x _ { i } ) } } } = - n + 1$ and therefore $d x _ { i } \\in A _ { n - 1 }$ . We have that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[179,714,817,738]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0126/6d85afc4794f03c6287f198c78be4429351e2e2eb12ba5ad624ed305b433eaa2.jpg","list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nd \\gamma _ { n - 1 } ( d x _ { i } ) = \\gamma _ { n - 1 } ( d ^ { 2 } x _ { i } ) = 0 , \\qquad g \\gamma _ { n - 1 } ( d x _ { i } ) = \\beta ( d x _ { i } ) = d \\beta ( x _ { i } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,753,879,791]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and since $g$ is injective in cohomology we have $\\gamma _ { n - 1 } ( d x _ { i } ) = d y _ { i }$ with $y _ { i } ~ \\in ~ C$ . Setting $z _ { i } = \\beta ( x _ { i } ) - g ( y _ { i } )$ we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[250,807,749,828]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0303/e46d5271feb501f725b0b9bf9d88053b37082638834fdfdaef43f67d414c8c86.jpg","list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nd z _ { i } = \\beta ( d x _ { i } ) - g ( d y _ { i } ) = \\beta ( d x _ { i } ) - g \\gamma _ { n - 1 } ( d x _ { i } ) = 0 .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[116,843,881,881]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since $g$ is a surjective quasi-isomorphism there exists $c _ { i } \\in C$ such that $d c _ { i } = 0$ and $g ( c _ { i } ) = z _ { i }$ . We can now define $\\gamma _ { n } ( x _ { i } ) = y _ { i } + c _ { i }$ . □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,562,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"92","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,879,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 3 Maps in $c w ^ { \\prime }$ have the left lifting property with respect to all fibrations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,135,651,154]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Consider the following solid commutative diagram ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[387,168,610,240]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0757/b474cc5e2158b36d4595f3378cce96508a1c044a586ba8c1534505a05ca698ca.jpg","list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { A \\xrightarrow { d } C \\qquad } \\\\ { i \\xrightarrow { i } \\qquad _ { i \\cdots \\cdots } \\qquad \\downarrow g } \\\\ { A [ \\{ x _ { i } , d x _ { i } \\} ] \\xrightarrow { d } \\qquad \\beta \\xrightarrow { d } D } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,259,881,315]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"with $g$ surjective in negative degrees. If $A = 0$ there is nothing to prove; otherwise, since $A [ \\{ x _ { i } , d x _ { i } \\} ] \\in \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\le 0 }$ , every $x _ { i }$ has negative degree and there exists $c _ { i } \\in C$ such that $g ( c _ { i } ) = \\beta ( x _ { i } )$ . We set $h ( x _ { i } ) = c _ { i }$ , $h ( d x _ { i } ) = d c _ { i }$ , and $h | _ { A } = \\alpha$ . □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,321,881,360]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 4 Every map in $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ can be factored as a free extension followed by a fibration. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,366,881,423]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Let $f : A B$ be a map in $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ . For every homogeneous element $b \\in B$ of strictly negative degree we add two indeterminates $x _ { b }$ and $d x _ { b }$ to $A$ , with $x _ { b }$ of degree $\\overline { { b } }$ and $d x _ { b }$ of degree $\\overline { { b } } + 1$ , obtaining the free extension ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[403,439,594,465]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0849/48a557bb97573fd53bc715929257a907ad70e7bccc8d5e56ba6881d38b58be21.jpg","list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nA \\xrightarrow { i } A [ \\{ x _ { b } , d x _ { b } \\} ]\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,481,626,501]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We define $\\pi : A [ \\{ x _ { b } , d x _ { b } \\} ] B$ in the following way: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,509,343,563]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. $\\pi$ is equal to $f$ on $A$ , \n2. $\\pi ( x _ { b } ) = b$ , \n3. $\\pi ( d x _ { b } ) = d b$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,572,658,590]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The map $\\pi$ is obviously a fibration and the composition ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[369,606,628,632]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0847/00ed1198a12a6f874a42c70d69622bd1e8cbe969025fb71d3a35de0ae413e731.jpg","list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nA \\ { \\overset { i } { \\longrightarrow } } \\ A [ \\{ x _ { b } , d x _ { b } \\} ] \\ { \\overset { \\pi } { \\longrightarrow } } \\ B\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,650,241,668]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is equal to $f$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,674,879,713]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 5 Every map in $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ can be factored as a semifree extension followed by a trivial fibration. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,720,881,774]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Since semifree extensions are closed by composition, according to Proposition 4 it is sufficient to prove that every morphism $f \\colon A \\ \\to \\ B$ factors as a composition of a semifree extension and a quasi-isomorphism. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,776,879,828]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We use the differential graded analog of the classical argument about the existence of Tate-Tyurina resolutions: we construct recursively a countable sequence of semifree extensions ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[343,848,653,866]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0502/33c18d0ad1c6488713dcd5a581ffee8514979d6f10ded8dc41dd6afa7a4274b6.jpg","list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nA = A _ { 0 } \\subset A _ { 1 } \\subset \\cdots \\subset A _ { n } \\subset \\cdots\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[687,52,876,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Lepri and M. Manetti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"88 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[858,651,879,666]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"93","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"□ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[116,88,831,108]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"94","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"together with morphisms of DG-algebras $f _ { n } \\colon A _ { n } \\to B$ such that $f _ { 0 } = f$ and: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,115,669,189]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"94","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. $A _ { n + 1 } = A _ { n } [ \\{ x _ { i } \\} ]$ , with ${ \\overline { { x _ { i } } } } = - n$ ; \n2. $f _ { n + 1 }$ extends $f _ { n }$ ; \n3. $f _ { n } \\colon Z ^ { i } ( A _ { n } ) \\to Z ^ { i } ( B )$ is surjective for every $i > - n$ ; \n4. $f _ { n } \\colon H ^ { i } ( A _ { n } ) \\to H ^ { i } ( B )$ is bijective for every $i > - n + 1$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[141,195,876,216]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"94","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $\\{ b _ { i } \\} , i \\in I$ , be a set of generators of $B ^ { 0 }$ as a $A ^ { 0 }$ -algebra; then we may define ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[246,232,751,252]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0686/c3087e339e50c919ecabbfb4d7048df602af50093becd8cbacda51e79b7c1d38.jpg","list_items":null,"page_idx":"94","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nA _ { 1 } = A [ \\{ x _ { i } \\} ] , \\qquad { \\overline { { x _ { i } } } } = 0 , \\quad d x _ { i } = 0 , \\quad f _ { 1 } ( x _ { i } ) = b _ { i } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,268,881,328]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"94","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Assume now $n > 0$ and $f _ { n } \\colon A _ { n } \\to B$ defined. By choosing a suitable set of generators of $Z ^ { - n } ( B )$ as $A _ { n } ^ { 0 }$ -module we can first consider a factorisation $f _ { n } \\colon A _ { n } \\subset$ $C { \\stackrel { g } { \\to } } B$ such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[209,345,785,366]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0614/36d217f64b352757a472fd8594b75d999c8bd9dd520546fee5b11b227e19204b.jpg","list_items":null,"page_idx":"94","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nC = A _ { n } [ \\{ x _ { i } \\} ] , \\qquad { \\overline { { x _ { i } } } } = - n , \\quad d x _ { i } = 0 , \\quad f _ { n } ( x _ { i } ) \\in Z ^ { - n } ( B ) ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,379,881,472]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"94","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and such that $g \\colon Z ^ { - n } ( C ) \\to Z ^ { - n } ( B )$ is surjective. If $g \\colon H ^ { - n + 1 } ( C ) \\to H ^ { - n + 1 } ( B )$ is bijective we can define $A _ { n + 1 } \\ = \\ C$ and $f _ { n + 1 } ~ = ~ g$ . Otherwise let $\\left\\{ c _ { i } \\right\\}$ be a set of elements in $Z ^ { - n + 1 } ( C )$ whose cohomology classes generate the kernel of $g \\colon H ^ { - n + 1 } ( C ) \\to H ^ { - n + 1 } ( B )$ , choose elements $b _ { i } ~ \\in ~ B ^ { - n }$ such that $d b _ { i } = g ( c _ { i } )$ and consider the factorisation ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[211,489,785,510]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0610/404290579577058c0a289e42838b10236d2a1bea34410806c268d06b45e81787.jpg","list_items":null,"page_idx":"94","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nA _ { n + 1 } = C [ \\{ x _ { i } \\} ] , \\qquad { \\overline { { x _ { i } } } } = - n , \\quad d x _ { i } = c _ { i } , \\quad f _ { n + 1 } ( x _ { i } ) = b _ { i } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,527,881,581]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"94","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is easy to verify that the map $f _ { n + 1 } \\colon A _ { n + 1 } \\ \\to \\ B$ has the required properties. Finally, since $H ^ { i } ( A _ { n } ) ~ = ~ H ^ { i } ( A _ { n + 1 } )$ for every $i \\ > \\ - n + 1$ , the colimit of the sequence $f _ { n + 1 } \\colon A _ { n + 1 } \\to B$ gives the required factorisation. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,585,881,647]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"94","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 3 Let $f$ be a morphism in $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ . We have already proved that ⎧ $f$ is a trivial fibration in $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ if and only if it is a trivial fibration in ⎪⎪⎨ $\\mathbf { C D G A } _ { \\mathbb { K } }$ . Since the truncation functor ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[209,657,785,737]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0897/9192c16da09704e76c167dafe714ceeba85699ecae5af773a29f3b02f49edcba.jpg","list_items":null,"page_idx":"94","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\tau \\colon \\mathbf { C D G A } _ { \\mathbb { K } } \\to \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } , \\qquad ( \\tau A ) ^ { n } = \\left\\{ \\begin{array} { l l } { A ^ { n } } & { \\quad n < 0 , } \\\\ { Z ^ { 0 } ( A ) } & { \\quad n = 0 , } \\\\ { 0 } & { \\quad n > 0 , } \\end{array} \\right.\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,750,881,812]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"94","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is right adjoint to the faithful natural inclusion $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } \\subset \\mathbf { C D G A } _ { \\mathbb { K } }$ and preserves trivial fibrations, by Lemma 4 it follows that $f$ is cofibration in $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ if and only if it is a cofibration in $\\mathbf { C D G A } _ { \\mathbb { K } }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,819,881,893]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"94","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The notions of semifree extension and left pre-model structure apply to many other contexts, for instance cochain complexes over a commutative ring, DGalgebras, DG-Lie algebras etc.: full details will appear in the forthcoming thesis of the first author. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,51,562,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"94","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[854,52,879,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"94","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"89 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,455,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 Modules and Derivations ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,132,879,187]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $( A , d _ { A } )$ be in $\\mathbf { C D G A } _ { \\mathbb { K } }$ , an $A$ -module is a differential graded vector space $( M , d _ { M } )$ together with an associative and distributive $\\mathbb { K }$ -linear left multiplication map $A \\times M \\to M$ , with the properties: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,193,731,232]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. $A ^ { i } M ^ { j } \\subset M ^ { i + j }$ , \n2. $d _ { M } ( a m ) = d _ { A } ( a ) m + ( - 1 ) ^ { \\overline { { { a } } } } a d _ { M } ( m ) \\mathrm { f o r e v e r y } a \\in A , m \\in M .$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,240,881,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A morphism of $A$ -modules is a morphism of differential graded vector spaces commuting with multiplications. Since $A$ is graded commutative, we can also define an associative right multiplication map $M \\times A \\to M$ by setting $m a = ( - 1 ) ^ { \\overline { { { a } } } \\overline { { { m } } } } a m$ , $a \\in A$ , $m \\in M$ . Notice that $a ( m b ) = ( a m ) b$ for every $a , b \\in A$ , $m \\in M$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,313,879,349]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The trivial extension of a DG-algebra $A$ by the $A$ -module $M$ is the direct sum of complexes $A \\oplus M$ equipped with the product: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[355,367,642,385]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0412/023b6f1bba25a92ad5376148cc8445e237ce99bd3d22c863145ed21505e771d8.jpg","list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n( a , m ) ( b , n ) = ( a b , m b + a n ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,402,876,439]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is immediate to see that $A \\oplus M \\in \\mathbf { C D G A } _ { \\mathbb { K } }$ , the projection $A \\oplus M \\to A$ is a morphism of DG-algebras and $M$ is a square-zero ideal of $A \\oplus M$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,439,881,512]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For a given graded vector space $M$ and an integer $n$ we shall denote by $M [ - n ]$ the same space with the degrees shifted by $- n$ , namely $M [ - n ] ^ { i } = M ^ { i - n }$ , and by $s ^ { n } \\colon M \\to M [ - n ]$ the tautological (bijective) map of degree $n$ . In other words, $s ^ { n }$ is the essentially the identity and its only effect is changing the degree: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[318,525,678,548]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"1012/99f1d4f829db06b194bea7a1806c93e94c8fa53197376c73c450ce5ada3555df.jpg","list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\ns ^ { n } \\colon M ^ { i } \\to M [ - n ] ^ { i + n } , \\qquad x \\mapsto s ^ { n } x .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,564,879,602]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $M$ is an $A$ -module, then $M [ - n ]$ is also an $A$ -module, where the differential and the product are defined accordingly to the Koszul sign rule: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[136,615,858,638]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0941/cd03942fde595c6d17f909bfa61bd94a4a4fa96abddd4bf846b779a0807116c9.jpg","list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nd ( s ^ { n } x ) = ( - 1 ) ^ { n } s ^ { n } d ( x ) , \\qquad a ( s ^ { n } x ) = ( - 1 ) ^ { n { \\overline { { a } } } } s ^ { n } ( a x ) , \\qquad ( s ^ { n } x ) a = s ^ { n } ( x a ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,656,876,692]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 11 Let $M$ be an $A$ -module. A $\\mathbb { K }$ -linear map $\\alpha \\colon A \\to M$ is a derivation of degree $j \\in \\mathbb Z$ if $\\alpha ( A ^ { n } ) \\subset M ^ { n + j }$ and it satisfies Leibniz’s law: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[350,705,646,728]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0173/74ce2498ffc6dc8b618f8366c2bd3ca4615ebc3fdcd90cfae910226cc1dcda38.jpg","list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\alpha ( a b ) = \\alpha ( a ) b + ( - 1 ) ^ { \\overline { { { a } } } j } a \\alpha ( b )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,746,881,840]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The vector space of derivations of degree $j$ from $A$ to $M$ is denoted $\\mathrm { D e r } _ { \\mathbb { K } } ^ { j } ( A , M )$ . The graded vector space $\\begin{array} { r } { \\mathrm { D e r } _ { \\mathbb K } ^ { * } ( A , M ) = \\bigoplus _ { j \\in \\mathbb Z } \\mathrm { D e r } _ { \\mathbb K } ^ { j } ( A , M ) } \\end{array}$ has a natural structure of $A$ -module, with multiplication $( a \\alpha ) ( x ) = a ( \\alpha ( x ) )$ and differential $( d \\alpha ) ( x ) =$ $d ( \\alpha ( x ) ) ~ - ~ ( - 1 ) ^ { \\overline { { { \\alpha } } } } \\alpha ( d x )$ . Observe that for every integer $n$ there is a natural isomorphism of $A$ -modules ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[309,858,687,879]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0567/7ae88f4da2f76a42d4cb7dace0f333051b2ce8b633df565986d7bf0717b0f689.jpg","list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathrm { D e r } _ { \\mathbb K } ^ { * } ( A , M [ - n ] ) \\to \\mathrm { D e r } _ { \\mathbb K } ^ { * } ( A , M ) [ - n ] .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[687,52,879,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Lepri and M. Manetti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"95","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"90 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,163]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Every morphism of DG-algebras $f \\colon A \\ \\to \\ B$ induces in the natural way an $A$ -module structure on $B$ . In this case the module of derivations will be denoted $\\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( A , B ; f )$ : a $\\mathbb { K }$ -linear map $\\alpha \\colon A \\ \\to \\ B$ is an $f$ -derivation of degree $k$ if $\\alpha ( A ^ { n } ) \\subset B ^ { n + k }$ and $\\alpha ( a b ) = \\alpha ( a ) f ( b ) + ( - 1 ) ^ { k \\overline { { { a } } } } f ( a ) \\alpha ( b )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,171,881,283]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 4 Let $f \\colon A \\to B$ be a morphism of DG-algebras, $I \\subset B$ a square-zero ideal and $\\pi \\colon B \\to B / I$ the quotient map. Then $I$ is a $B / I$ -module and then also an $A$ -module via the morphism $\\pi f$ . It is immediate to check that if $g \\colon A B$ is a morphism of graded algebras such that $\\pi g = \\pi f$ then $g - f \\colon A \\to I$ is a derivation of degree 0. Conversely, if $\\alpha \\in { \\mathrm { D e r } } _ { \\mathbb { K } } ^ { 0 } ( A , I )$ , then $f + \\alpha$ is a morphism of graded algebras, and it is a morphism of DG-algebras if and only if $\\alpha \\in Z ^ { 0 } ( \\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( A , I ) )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,291,876,327]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 6 Let $A \\in \\mathbf { C D G A } _ { \\mathbb { K } }$ be a cofibrant algebra and $f \\colon M \\to N$ a surjective quasi-isomorphism of $A$ -modules. Then the map ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[266,346,728,367]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0182/12317d3df4c1441fa73d72f8ecdbcba8f778012c253d66a85336466e4accf78a.jpg","list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nf _ { * } \\colon \\operatorname { D e r } _ { \\mathbb { K } } ^ { * } ( A , M ) \\to \\operatorname { D e r } _ { \\mathbb { K } } ^ { * } ( A , N ) , \\qquad \\alpha \\mapsto f \\alpha ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,387,432,405]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is a surjective quasi-isomorphism. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,412,879,448]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Since $f \\colon M [ - n ] \\ \\to \\ N [ - n ]$ is a surjective quasi-isomorphism for every integer $n$ it is sufficient to prove that: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,456,660,498]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. $f _ { * } \\colon \\operatorname { D e r } _ { \\mathbb { K } } ^ { 0 } ( A , M ) \\to \\operatorname { D e r } _ { \\mathbb { K } } ^ { 0 } ( A , N )$ is surjective; \n2. $f _ { * } \\colon H ^ { 0 } ( \\mathrm { D e r } _ { \\mathbb K } ^ { * } ( A , M ) ) \\to H ^ { 0 } ( \\mathrm { D e r } _ { \\mathbb K } ^ { * } ( A , N ) )$ is bijective. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,504,881,576]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let’s denote by $C ( P ) = P \\oplus P [ 1 ]$ the mapping cone of the identity of an $A$ - module $P$ , with the differential defined by the formula $d ( x + s ^ { - 1 } y ) = d x + y -$ $s ^ { - 1 } d y$ ; notice that $C ( P )$ is acyclic and the natural projection $C ( P ) P [ 1 ]$ is a morphism of- $A$ -modules. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[143,578,703,596]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For every linear map $\\alpha \\colon A P$ of degree 0 we shall denote ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[182,612,815,635]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0250/73fa32d07f0a51d4d5888ae03d3e3a0746b68945b49da44286489fb856655875.jpg","list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n{ \\widetilde \\alpha } \\colon A \\to A \\oplus C ( P ) , \\qquad { \\widetilde \\alpha } ( a ) = a + \\alpha ( a ) + s ^ { - 1 } ( \\alpha ( d a ) - d \\alpha ( a ) ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,654,879,708]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is straightforward to check that $\\widetilde { \\alpha }$ is a morphism of complexes and that every morphism of complexes $A \\to A \\oplus C ( P )$ lifting the identity on $A$ is obtained this way. Moreover, $\\alpha$ is a derivation if and only if - $\\widetilde { \\alpha }$ is a morphism in $\\mathbf { C D G A } _ { \\mathbb { K } }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,710,881,780]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since A ⊕ C(M) → A ⊕ C(N), $a + x + s ^ { - 1 } y \\mapsto a + f ( x ) + s ^ { - 1 } f ( y )$ , is a trivial fibration, the lifting of a derivation $\\alpha \\in \\mathrm { D e r } _ { \\mathbb { K } } ^ { 0 } ( A , N )$ is obtained by taking the lifting of the morphism of DG-algebras ${ \\widetilde { \\alpha } } \\colon A \\to A \\oplus C ( N )$ . This proves the first item. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,782,781,800]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $K$ is the kernel of $f$ , then we have an exact sequence of complexes ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[236,818,763,840]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0802/6e6ed6e857e990dcffb035fa8c0b751ed0ae93136fba22d749c0d283aac6b3cc.jpg","list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n0 \\to \\operatorname { D e r } _ { \\mathbb { K } } ^ { * } ( A , K ) \\to \\operatorname { D e r } _ { \\mathbb { K } } ^ { * } ( A , M ) \\to \\operatorname { D e r } _ { \\mathbb { K } } ^ { * } ( A , N ) \\to 0\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,858,879,896]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and in order to prove the second item it is sufficient to show that $\\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( A , K )$ is acyclic. By the shifting degree argument it is sufficient to prove that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,51,562,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[854,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"96","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"91 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,681,111]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"97","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$H ^ { 1 } ( \\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( A , K ) ) = 0 .$ . Given $\\beta \\in Z ^ { 1 } ( \\operatorname { D e r } _ { \\mathbb { K } } ^ { * } ( A , K ) )$ , the map ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[280,129,717,151]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0504/c1121281114b36bb118763e5e265df1056611a8afbb25af94b0b553d9ce43dde.jpg","list_items":null,"page_idx":"97","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n{ \\widehat { \\beta } } \\colon A \\to A \\oplus K [ 1 ] , \\qquad { \\widehat { \\beta } } ( a ) = a + s ^ { - 1 } \\beta ( a ) ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,171,881,226]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"97","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is a morphism of DG-algebras and the proof that $\\beta \\in B ^ { 1 } ( \\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( A , K ) )$ follows immediately by considering a lifting of $\\hat { \\boldsymbol { \\beta } }$ along the trivial fibration $A \\oplus C ( K ) \\to$ $A \\oplus K [ 1 ]$ . □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,256,760,298]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"97","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 Deformations of Diagrams via Projective Cofibrant Resolutions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,321,881,394]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"97","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Throughout this section we shall denote by $\\mathcal { D }$ a fixed small category. For every category $\\mathbf { M }$ we shall denote by $\\mathrm { F u n } ( \\mathcal { D } , \\mathbf { M } )$ the category of diagrams $\\mathcal { D } \\mathbf { M }$ . For every local Artin $\\mathbb { K }$ -algebra $A$ with residue field $\\mathbb { K }$ we shall denote by $\\mathbf { A } \\mathbf { g } _ { A }$ the category of unitary commutative $A$ -algebras. For simplicity of notation, if ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[312,408,683,429]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0341/bae1f2833d697a579eacca58072060af5c1972b6c120104c4423857db3dbbd04.jpg","list_items":null,"page_idx":"97","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nP _ { \\bullet } \\in \\operatorname { F u n } ( { \\mathcal { D } } , \\mathbf { A l g } _ { A } ) , \\qquad { \\mathcal { D } } \\ni a \\mapsto P _ { a } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,441,876,478]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"97","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is a diagram of $A$ -algebras and $A B$ is a morphism of algebras, we shall denote $P , \\otimes _ { A } B$ the diagram $( P _ { \\bullet } \\otimes _ { A } B ) _ { a } = P _ { a } \\otimes _ { A } B , a \\in { \\mathcal { D } }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,478,874,515]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"97","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Here we are interested in studying the deformation theory of a diagram $S _ { \\bullet } \\colon { \\mathcal { D } } $ $\\mathbf { A l g } _ { \\mathbb { K } }$ of unitary commutative algebras. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,522,881,578]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"97","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 12 A deformation over $A \\in \\mathbf { A r t } _ { \\mathbb { K } }$ of a diagram $S _ { \\bullet } \\colon { \\mathcal { D } } \\to \\mathbf { A l g } _ { \\mathbb { K } }$ is the data of a diagram $S _ { \\bullet A } : { \\mathcal { D } } \\to \\mathbf { A l g } _ { A }$ of flat $A$ -algebras and a morphism of diagrams of algebras $\\phi \\colon S _ { \\bullet A } S _ { \\bullet }$ inducing an isomorphism $S _ { \\bullet A } \\otimes _ { A } \\mathbb { K } \\simeq S _ { \\bullet }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,578,874,615]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"97","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Two deformations $\\phi \\colon S _ { \\bullet A } S _ { \\bullet }$ and $\\psi \\colon S _ { \\bullet A } ^ { \\prime } \\to S _ { \\bullet }$ are isomorphic if there exists an isomorphism of diagrams of $A$ -algebras $\\eta \\colon S _ { \\bullet A } \\to S _ { \\bullet A } ^ { \\prime }$ such that $\\phi = \\psi \\eta$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,621,881,770]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"97","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is possible to prove, see e.g. [10, A.2], that for every small category $\\mathcal { D }$ there exist model structures on $\\mathrm { F u n } ( \\mathcal { D } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } )$ and $\\mathrm { F u n } ( { \\mathcal { D } } , \\mathbf { C D G A } _ { \\mathbb { K } } )$ , called projective model structures, such that a morphism of diagrams $F G$ is a weak equivalence (resp.: fibration) if and only if $F _ { a } \\ \\to \\ G _ { a }$ is a weak equivalence (resp.: fibration) for every $a \\in \\mathcal { D }$ . The same argument used in Remark 3 shows that a morphism $f$ in $\\mathrm { F u n } ( \\mathcal { D } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } )$ ) is a weak equivalence, cofibration, trivial fibration in $\\mathrm { F u n } ( \\mathcal { D } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } )$ if and only if it is a weak equivalence, cofibration, trivial fibration in $\\mathrm { F u n } ( { \\mathcal { D } } , \\mathbf { C D G A } _ { \\mathbb { K } } )$ , respectively. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,770,867,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"97","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The notions of module and derivation extend naturally to the context of diagrams For every diagram $R _ { \\bullet } \\colon \\mathcal { D } { \\bf C D G A } _ { \\mathbb { K } } ^ { \\le 0 }$ the DG-Lie algebra of derivations is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[150,818,847,872]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0526/93baa3aed7de1749f3a1b61593cdb6abf295499cdbc75827450bbc5107868af6.jpg","list_items":null,"page_idx":"97","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { \\bullet } , R _ { \\bullet } ) = \\{ \\{ \\alpha _ { a } \\} \\in \\prod _ { a \\in \\mathcal { D } } \\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { a } , R _ { a } ) \\mid \\alpha _ { b } R _ { f } = R _ { f } \\alpha _ { a } , \\forall a \\stackrel { f } { } b \\} .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[111,882,822,905]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"97","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is plain that $\\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { \\bullet } , R _ { \\bullet } )$ is a DG-Lie subalgebra of $\\begin{array} { r } { \\prod _ { a \\in \\mathcal { D } } \\operatorname { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { a } , R _ { a } ) } \\end{array}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[685,52,879,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"97","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Lepri and M. Manetti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"97","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"92 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,204]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"98","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An $R _ { \\bullet }$ -module $M$ is a diagram of differential graded vector spaces over $\\mathcal { D }$ such that $M _ { a }$ is an $R _ { a }$ -module for every $a \\in \\mathcal { D }$ and, for every arrow $a \\ { \\overset { f } { \\to } } \\ b$ in $\\mathcal { D }$ , the map $M _ { f } \\colon M _ { a } \\to M _ { b }$ is a morphism of $R _ { a }$ -modules, where $M _ { b }$ is considered as a $R _ { a }$ -module via the morphism of DG-algebras $R _ { f } \\colon R _ { a } \\ \\ R _ { b }$ . A morphism $g \\colon M _ { \\bullet } \\to N _ { \\bullet }$ of $R _ { \\bullet }$ -modules is a morphism of diagrams of DG-vector spaces such\u0005 that $g _ { a } \\colon M _ { a } \\to N _ { a }$ is a morphism of $R _ { a }$ -modules for every $a \\in \\mathcal { D }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,204,628,222]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"98","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The differential graded vector space of derivations is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[143,234,851,288]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0192/82f41a0d5dc9969a95898df7ee3531e1921578179bee6e59bbf6d8e235505136.jpg","list_items":null,"page_idx":"98","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { \\bullet } , M _ { \\bullet } ) = \\{ \\{ \\alpha _ { a } \\} \\in \\prod _ { a \\in \\mathcal { D } } \\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { a } , M _ { a } ) \\mid \\alpha _ { b } R _ { f } = M _ { f } \\alpha _ { a } , \\forall a \\stackrel { f } { } b \\} .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,303,876,339]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"98","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The same argument used in the proof of Lemma 6 works, mutatis mutandis, also for diagrams and gives the following result. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,346,881,402]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"98","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 7 Let $R _ { \\bullet } ~ \\in ~ \\operatorname { F u n } ( { \\mathcal { D } } , \\mathbf { C D G A _ { \\mathbb { K } } } )$ be a projective cofibrant diagram and $f \\colon M _ { \\bullet } \\to N _ { \\bullet }$ a morphism of $R _ { \\bullet }$ -modules such that $f _ { a } \\colon M _ { a } \\to N _ { a }$ is a surjective quasi-isomorphism for every $a \\in \\mathcal { D }$ . Then the map ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[252,418,742,439]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0704/e66ed0b2c253743269f6ea03bab27303fa61b4b27fc5795815f53208ef115a78.jpg","list_items":null,"page_idx":"98","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nf _ { * } \\colon \\operatorname { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { \\bullet } , M _ { \\bullet } ) \\to \\operatorname { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { \\bullet } , N _ { \\bullet } ) , \\qquad \\alpha \\mapsto f \\alpha ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,456,432,474]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"98","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is a surjective quasi-isomorphism. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,481,726,501]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"98","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The main goal of this section is to prove the following theorem. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,509,881,582]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"98","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 4 Let $\\mathcal { D }$ be a small category and $S _ { \\bullet } \\colon { \\mathcal { D } } \\ \\to \\ \\mathbf { A l g } _ { \\mathbb { K } } \\ a$ diagram of unitary commutative algebras. Let $R _ { \\bullet } S _ { \\bullet }$ be a cofibrant replacement in $\\begin{array} { r } { \\mathrm { F u n } ( \\mathcal { D } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } , } \\end{array}$ • → • ) with respect to the projective model structure. Then the DG-Lie algebra $\\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { \\bullet } , R _ { \\bullet } )$ controls the deformations of $S _ { \\bullet }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,590,881,717]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"98","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In other words, the functor of isomorphism classes of deformations of $S _ { \\bullet }$ is isomorphic to the functor of Maurer-Cartan solutions in $\\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { \\bullet } , R _ { \\bullet } )$ modulus gauge equivalence. We shall prove Theorem 4 after a certain number of preliminary results. Unless otherwise specified we always equip the categories $\\mathrm { F u n } ( \\mathcal { D } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } )$ ) and $\\mathrm { F u n } ( { \\mathcal { D } } , \\mathbf { C D G A } _ { \\mathbb { K } } )$ with the projective model structure. Therefore $R _ { \\bullet } ~ ~ S _ { \\bullet }$ is a cofibrant resolution also in the model category $\\mathrm { F u n } ( { \\mathcal { D } } , \\mathbf { C D G A } _ { \\mathbb { K } } )$ and we can apply Lemma 7 to the diagram $R _ { \\bullet }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,725,658,744]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"98","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 8 Consider a commutative square of solid arrows ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[432,764,560,852]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0843/8ccbca62ea9b3d6b7b1928e0910aa2da2748785568fb890d2cde88b2b72e8bc0.jpg","list_items":null,"page_idx":"98","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { l } { P _ { \\bullet } \\xrightarrow { g } E _ { \\bullet } } \\\\ { i \\Bigg \\downarrow _ { \\begin{array} { c } { \\diagdown ^ { \\prime } } \\\\ { ⨏ _ { \\ell } ^ { \\prime } } \\end{array} } ^ { \\ell } \\Bigg \\downarrow ^ { \\prime } } \\\\ { C _ { \\bullet } \\xrightarrow [ f ^ { \\prime } ] { g } D _ { \\bullet } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,52,562,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"98","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[854,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"98","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"93 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,881,147]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"99","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"in $\\begin{array} { r } { \\mathrm { F u n } ( \\mathcal { D } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } , } \\end{array}$ ). If $i$ is a cofibration and $p _ { a } \\colon E _ { a } \\to D _ { a }$ is surjective for every $a \\in \\mathcal { D }$ , then there exists a lifting $\\gamma \\colon C _ { \\bullet } \\to E .$ in the category of diagrams of graded algebras. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,156,881,232]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"99","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Consider the contractible polynomial algebra $\\mathbb { K } [ d ^ { - 1 } ] \\ \\in \\ \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\le 0 }$ , where $\\overline { { d ^ { - 1 } } } = - 1$ and $d ( d ^ { - 1 } ) = 1$ , and notice that the natural inclusion $\\alpha \\colon \\mathbb { K } \\to \\mathbb { K } [ d ^ { - 1 } ]$ is a morphism of DG-algebras, while the natural projection $\\beta \\colon \\mathbb { K } [ d ^ { - 1 } ] \\mathbb { K }$ is a morphism of graded algebras; moreover $\\beta \\alpha$ is the identity on $\\mathbb { K }$ . Now, the morphism ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[328,247,674,276]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0194/df85b0f16bcc4974284012cb122694d8f0640576ea7aab89bc1e15fd0858ac55.jpg","list_items":null,"page_idx":"99","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nE _ { \\bullet } \\otimes _ { \\mathbb { K } } { \\mathbb { K } } [ d ^ { - 1 } ] \\xrightarrow { p \\otimes \\mathrm { I d } } D _ { \\bullet } \\otimes _ { \\mathbb { K } } { \\mathbb { K } } [ d ^ { - 1 } ]\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[116,292,678,312]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"99","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is a trivial fibration, and so there exists a commutative square ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[343,325,656,423]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0695/e5c2b70951e3ab59e732f5f5bc599ede97748dd8fd95ca5c98b4b3d4715b3f62.jpg","list_items":null,"page_idx":"99","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { P _ { \\bullet } \\xrightarrow { ( \\mathrm { I d } \\otimes \\alpha ) g } \\quad \\ J _ { \\bullet } \\otimes _ { \\mathbb K } \\mathbb K [ d ^ { - 1 } ] } \\\\ & { i \\underbrace { \\quad \\quad \\quad \\varphi \\quad \\quad } _ { ( \\mathrm { I d } \\otimes \\alpha ) f } \\quad \\quad \\ J \\otimes \\mathrm { I d } } \\\\ & { C _ { \\bullet } \\xrightarrow { ( \\mathrm { I d } \\otimes \\alpha ) g } \\quad \\ J _ { \\bullet } \\otimes _ { \\mathbb K } \\mathbb K [ d ^ { - 1 } ] } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,444,708,466]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"99","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"in $\\mathrm { F u n } ( \\mathcal { D } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } )$ ). It is now sufficient to take $\\gamma = ( \\operatorname { I d } \\otimes \\beta ) \\varphi$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,474,883,572]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"99","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 9 Let $\\begin{array} { r } { A \\in \\mathrm { ~ \\bf ~ A r t { } ~ } _ { \\mathbb { K } } } \\end{array}$ and let $N _ { \\bullet A } \\colon \\mathcal { D } \\ \\to \\ \\mathbf { C D G A } _ { A } ^ { \\leq 0 }$ be a diagram of flat $A$ -algebras. Then every cofibrant replacement $f \\colon P _ { \\bullet } \\ \\quad N _ { \\bullet A } \\otimes _ { A } \\mathbb { K }$ in $\\begin{array} { r } { \\mathrm { F u n } ( \\mathcal { D } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } ) } \\end{array}$ ) lifts to an A-linear differential on $P _ { \\bullet } \\otimes A$ and to a trivial fibration $P _ { \\bullet } \\otimes A N _ { \\bullet A }$ in the category $\\mathrm { F u n } ( \\mathcal { D } , { \\bf C D G A } _ { A } ^ { \\le 0 } )$ ). The above lifting is unique up to $A$ -linear algebra isomorphisms of $P \\cdot \\otimes A$ lifting the identity on $P _ { \\bullet }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,578,881,651]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"99","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof (Existence) We proceed by induction on the length of the Artin ring. Since for $A = \\mathbb { K }$ there is nothing to prove, we may assume $A \\in \\mathbf { A r t } _ { \\mathbb { K } }$ of length $l ( A ) > 1$ and then there exists a non-trivial element $t \\in A$ annihilated by the maximal ideal ${ \\mathfrak { m } } _ { A }$ , giving a small extension ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[268,666,728,692]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0144/57c3846e58c1fa472dd0c3c56b4c4baca4ffbdb473ba3b295208230332865ad1.jpg","list_items":null,"page_idx":"99","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n0 \\to \\mathbb { K } { \\stackrel { \\cdot t } { \\longrightarrow } } A \\to B \\to 0 , \\qquad l ( B ) = l ( A ) - 1 .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,710,879,747]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"99","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By induction there exist a $B$ -linear differential on $P _ { \\bullet } \\otimes _ { \\mathbb { K } } B$ and a commutative square in $\\mathrm { F u n } ( \\mathcal { D } , { \\bf C D G A } _ { B } ^ { \\le 0 } )$ ): ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[371,762,624,846]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0522/0bfd901b0855df75b9f62a82db725a26cf9ed54f3b8092eddffb76dfc68318a0.jpg","list_items":null,"page_idx":"99","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { c c } { P _ { \\bullet } \\otimes _ { \\mathbb { K } } B \\xrightarrow { q } \\sim N _ { \\bullet A } \\otimes _ { A } B } \\\\ { \\downarrow } \\\\ { P _ { \\bullet } \\xrightarrow { \\qquad f \\qquad \\ } N _ { \\bullet A } \\otimes _ { A } \\mathbb { K } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[687,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"99","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Lepri and M. Manetti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"99","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"94 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[858,448,879,463]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"99","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"□ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,883,181]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"100","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In view of the embedding $\\mathbb { K } \\subset B$ , the $B$ -linear morphism of diagrams $q$ is uniquely determined by its restriction $q _ { \\vert } P _ { \\bullet } \\colon P _ { \\bullet } \\to N _ { \\bullet A } \\otimes _ { A } B$ , which is a morphism in $\\begin{array} { r } { \\mathrm { F u n } ( \\mathcal { D } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } , } \\end{array}$ ). By Lemma 8 we can lift $q _ { \\lvert P _ { \\bullet } }$ to a morphism of diagrams of graded algebras $P _ { \\bullet } N _ { \\bullet A }$ and then we get a commutative diagram with (pointwise) exact rows ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[218,190,781,276]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0769/4b2e11a02a0188918b700d8fb3d62d9915bc01592cf292840fd88c7ed37289db.jpg","list_items":null,"page_idx":"100","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } { 0 \\xrightarrow { \\quad \\quad } } & { { } \\quad \\quad \\quad \\quad \\quad \\quad t \\xrightarrow { t } { } P _ { \\bullet } \\otimes _ { \\mathbb { K } } A \\xrightarrow { \\quad \\pi \\quad \\quad } { } P _ { \\bullet } \\otimes _ { \\mathbb { K } } B \\xrightarrow { \\quad \\quad \\quad } { } 0 } \\\\ { \\quad \\quad } & { { } \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ { 0 \\xrightarrow { \\quad \\quad \\quad } } & { { } \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ { \\quad \\quad \\quad } & { { } \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ { 0 \\xrightarrow { \\quad \\quad \\quad \\quad } } & { { } \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,294,881,420]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"100","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since $f$ and $q$ are trivial fibrations, to conclude the proof it is sufficient to show that there exists a lifting of the differential of $P _ { \\bullet } \\otimes _ { \\mathbb { K } } B$ to an $A$ -linear differential of $P _ { \\bullet } \\otimes _ { \\mathbb { K } } A$ making $p$ a morphism of diagrams of DG-algebras. Let $d = \\{ d _ { a } \\colon P _ { a } $ $P _ { a } \\mid a \\in { \\mathcal { D } } \\}$ be the differential of $P _ { \\bullet }$ , since the $A$ -linear derivations of $P _ { \\bullet } \\otimes _ { \\mathbb { K } } A$ of degree 1 lifting $d$ are of type $d + \\eta$ with $\\eta \\in \\mathrm { D e r } _ { \\mathbb { K } } ^ { 1 } ( P _ { \\bullet } , P _ { \\bullet } ) \\otimes \\mathfrak { m } _ { A }$ , we can lift the differential of $P _ { \\bullet } \\otimes _ { \\mathbb { K } } B$ to an $A$ -linear derivation $\\delta \\colon P _ { \\bullet } \\otimes _ { \\mathbb { K } } A \\to P _ { \\bullet } \\otimes _ { \\mathbb { K } } A$ of degree 1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,421,876,456]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"100","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now it is sufficient to prove that there exists a derivation in $\\xi \\in \\mathrm { D e r } _ { \\mathbb { K } } ^ { 1 } ( P _ { \\bullet } , P _ { \\bullet } )$ such that: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,463,330,503]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"100","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. $( \\delta + t \\xi ) ^ { 2 } = 0$ , \n2. $p ( \\delta + t \\xi ) = d _ { N _ { \\bullet } } p$ , ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,510,881,638]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"100","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and consider $\\delta + t \\xi$ as the differential of $P _ { \\bullet } \\otimes _ { \\mathbb { K } } A$ . Since $t$ is annihilated by the maximal ideal ${ \\mathfrak { m } } _ { A }$ , the condition $( \\delta + t \\xi ) ^ { 2 } = 0$ is equivalent to $d \\xi + \\xi d = 0$ , i.e., the above condition (1) holds if and only if $\\xi \\in Z ^ { 1 } ( \\operatorname { D e r } _ { \\mathbb { K } } ^ { * } ( P _ { \\bullet } , P _ { \\bullet } ) )$ . The map $\\psi = d _ { N _ { \\bullet } } p - p \\delta$ is $A$ -linear and its image is contained in $t ( N _ { \\bullet A } \\otimes _ { A } \\mathbb { K } )$ , hence it factors to a derivation $\\phi \\in Z ^ { 1 } ( \\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( P _ { \\bullet } , N _ { \\bullet A } \\otimes _ { A } \\mathbb { K } ; f ) )$ and the above condition (2) is equivalent to $\\phi = f \\xi$ . It is now sufficient to observe that since $f$ is a trivial fibration, by Lemma 7 the morphism ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[287,653,712,675]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0950/3ab1506876d8bd758d4875ec02fefd31194026deb41f79711cefc9efab65b56c.jpg","list_items":null,"page_idx":"100","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nf \\colon \\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( P _ { \\bullet } , P _ { \\bullet } ) \\to \\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( P _ { \\bullet } , N _ { \\bullet A } \\otimes _ { A } \\mathbb { K } ; f )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,690,555,708]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"100","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is a surjective quasi-isomorphism and therefore ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[246,723,751,747]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0010/83e23c77eb95ca6ebe196f35b47e5b0943e7fdcf177caa841f1219dfc577905d.jpg","list_items":null,"page_idx":"100","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nf \\colon Z ^ { 1 } ( \\mathrm { D e r } _ { \\mathbb K } ^ { * } ( P _ { \\bullet } , P _ { \\bullet } ) ) \\to Z ^ { 1 } ( \\mathrm { D e r } _ { \\mathbb K } ^ { * } ( P _ { \\bullet } , N _ { \\bullet A } \\otimes _ { A } \\mathbb K ; f ) )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,764,298,780]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"100","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is a surjective map. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,789,879,825]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"100","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Unicity Let $\\delta , \\delta ^ { \\prime }$ be two $A$ linear differentials on $P _ { \\bullet } \\otimes _ { \\mathbb { K } } A$ lifting the differential $d$ on $P _ { \\bullet }$ and let ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[252,842,742,863]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0801/eb1e7e17492d1008937dd190b9c0898fae65cb2e2371603d32452e72cb6b7142.jpg","list_items":null,"page_idx":"100","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\np \\colon ( P _ { \\bullet } \\otimes _ { \\mathbb { K } } A , \\delta ) \\to N , \\qquad q \\colon ( P _ { \\bullet } \\otimes _ { \\mathbb { K } } A , \\delta ^ { \\prime } ) \\to N ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,51,562,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"100","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[854,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"100","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"95 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,879,148]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"101","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"be two morphisms in $\\mathrm { F u n } ( \\mathcal { D } , \\mathbf { C D G A } _ { A } ^ { \\leq 0 } )$ ) lifting the trivial fibration $f \\colon P . \\ \\to$ $N _ { \\bullet A } \\otimes _ { A } \\mathbb { K }$ . We need to prove that there exists an isomorphism of diagrams of differential graded $A$ -algebras $\\phi \\colon ( P _ { \\bullet } \\otimes _ { \\mathbb { K } } A , \\delta ) ( P _ { \\bullet } \\otimes _ { \\mathbb { K } } A , \\delta ^ { \\prime } )$ such that $p = q \\phi$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,157,881,373]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"101","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By induction on the length we can assume that there exists an isomorphism of diagrams of differential graded $B$ -algebras $\\phi ^ { \\prime } \\colon ( P _ { \\bullet } \\otimes _ { \\mathbb { K } } B , \\delta ) \\ \\ ( P _ { \\bullet } \\otimes _ { \\mathbb { K } } B , \\delta ^ { \\prime } )$ such that $p = q \\phi ^ { \\prime }$ . By Lemma 8 we can lift $\\phi ^ { \\prime }$ to an isomorphism of diagrams of graded $A$ -algebras $\\psi ^ { \\prime } \\colon ( P _ { \\bullet } \\otimes _ { \\mathbb { K } } A , \\delta ) \\to ( P _ { \\bullet } \\otimes _ { \\mathbb { K } } A , \\delta ^ { \\prime } )$ ; therefore, replacing $\\delta ^ { \\prime }$ with $( \\psi ^ { \\prime } ) ^ { - 1 } \\delta ^ { \\prime } ( \\psi )$ and $q$ with $q \\psi ^ { \\prime }$ if necessary, it is not restrictive to assume $\\phi ^ { \\prime }$ equal to the identity. The derivation $p - q \\colon P _ { \\bullet } \\to t ( N _ { \\bullet A } \\otimes _ { A } \\mathbb { K } )$ can be lifted to a derivation $\\alpha \\in \\mathrm { D e r } _ { \\mathbb { K } } ^ { 0 } ( P _ { \\bullet } , P _ { \\bullet } )$ and then, replacing $q$ with $q ( \\mathrm { I d } + t \\alpha )$ and $\\delta ^ { \\prime }$ with $( \\mathrm { I d } - t \\alpha ) \\delta ^ { \\prime } ( \\mathrm { I d } + t \\alpha )$ if necessary, it is not restrictive to assume $p = q$ . This implies in particular that $\\delta ^ { \\prime } = \\delta + t \\xi$ , for some $\\xi \\in Z ^ { 1 } ( \\operatorname { D e r } _ { \\mathbb { K } } ^ { * } ( P _ { \\bullet } , P _ { \\bullet } ) )$ . Since the kernel of $f \\colon \\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( P _ { \\bullet } , P _ { \\bullet } ) \\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( P _ { \\bullet } , N _ { \\bullet A } \\otimes _ { A } \\mathbb { K } ; f ) )$ is acyclic and $p \\delta = p \\delta ^ { \\prime }$ , we have $f \\xi = 0$ and therefore $\\xi = [ d , \\alpha ]$ for some $\\alpha \\in \\mathrm { D e r } _ { \\mathbb { K } } ^ { 0 } ( P _ { \\bullet } , P _ { \\bullet } )$ such that $f \\alpha = 0$ . Now $e ^ { t \\alpha }$ is the required isomorphism. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,382,879,436]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"101","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof of Theorem 4 We assume that the reader has a certain familiarity with the theory of deformation functors associated to DG-Lie algebras; the basic facts exposed in [12, 14] are sufficient for our needs. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,438,879,512]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"101","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $\\mathcal { D }$ be a small category and $S _ { \\bullet } \\colon { \\mathcal { D } } \\to \\mathbf { A l g } _ { \\mathbb { K } }$ a diagram of unitary commutative algebras. Let $R _ { \\bullet } ~ ~ S _ { \\bullet }$ be a cofibrant replacement in $\\mathrm { F u n } ( \\mathcal { D } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } )$ with respect to the projective model structure and consider the DG-Lie algebra $L \\ =$ $\\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { \\bullet } , R _ { \\bullet } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,512,876,548]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"101","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Denoting by $ { \\mathrm { D e f } _ { S _ { \\bullet } } } \\colon { \\mathbf { A r t } } _ { \\mathbb { K } } \\ \\to$ Set the functor of isomorphism classes of deformations we want to describe an isomorphism ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[400,564,594,585]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0543/0b441880be78bf300cdbc5843f54a3e9b2a725e65ad3ff008d2db90f582eaa24.jpg","list_items":null,"page_idx":"101","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\phi \\colon \\mathrm { D e f } _ { L } \\mathrm { D e f } _ { S _ { \\bullet } } \\ .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,600,881,636]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"101","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Denoting by $d \\in \\mathrm { D e r } _ { \\mathbb { K } } ^ { 1 } ( R _ { \\bullet } , R _ { \\bullet } )$ the differential of $R _ { \\bullet }$ , for every $A \\in \\mathbf { A r t } _ { \\mathbb { K } }$ with maximal ideal ${ \\mathfrak { m } } _ { A }$ , a Maurer-Cartan element ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[246,650,751,693]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0286/e7b685f52c76054c18e2b20e81c92a50401c26139fe2f3d10930cd8d11b1362e.jpg","list_items":null,"page_idx":"101","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\xi \\in \\mathbf { M C } _ { L } ( A ) = \\left\\{ x \\in L ^ { 1 } \\otimes \\mathfrak { m } _ { A } \\ \\bigg | \\ d _ { L } x + \\frac { 1 } { 2 } [ x , x ] = 0 \\right\\}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,708,881,786]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"101","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is exactly a derivation $\\xi \\in \\mathrm { D e r } _ { \\mathbb { K } } ^ { 1 } ( R _ { \\bullet } , R _ { \\bullet } \\otimes \\mathfrak { m } _ { A } )$ such that $( R _ { \\bullet } \\otimes A , d + \\xi )$ is a flat diagram in $\\mathrm { F u n } ( \\mathcal { D } , { \\bf C D G A } _ { A } ^ { \\le 0 } )$ ). Moreover $\\boldsymbol { \\xi } , \\eta \\in \\mathbf { M C } _ { L } ( A )$ are gauge equivalent if and only if there exists an isomorphism of diagrams of DG-algebras $( R _ { \\bullet } \\otimes A , d +$ $\\xi ) \\simeq ( R _ { \\bullet } \\otimes A , d + \\eta )$ , lifting the identity over $R _ { \\bullet }$ . According to Lemma 2 the map ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[243,798,749,822]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0681/e6392e53308d2b51c69da4e5698e9e866394a81fc1dce8e6183233b899abfcc5.jpg","list_items":null,"page_idx":"101","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n{ \\bf M C } _ { L } ( A ) \\mathrm { D e f } _ { S _ { \\bullet } } ( A ) , \\qquad \\xi \\mapsto H ^ { 0 } ( R _ { \\bullet } \\otimes A , d + \\xi ) ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,839,879,875]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"101","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is properly defined and factors to a natural transformation $\\phi \\colon \\mathrm { D e f } _ { L } \\ \\ \\mathrm { D e f } _ { S _ { \\bullet } }$ . Finally Lemma 9 implies immediately that $\\phi$ is an isomorphism. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[687,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"101","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Lepri and M. Manetti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,54,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"101","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,232]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"102","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the situation of Theorem 4, according to Lemma 7, the natural map $L \\ =$ $\\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { \\bullet } , R _ { \\bullet } ) \\ \\to \\ \\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { \\bullet } , S _ { \\bullet } )$ is a quasi-isomorphism of complexes, hence $H ^ { i } ( L ) ~ = ~ 0$ for every $\\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ c \\ c \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F \\ F$ . The $S _ { \\bullet }$ -module $\\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { \\bullet } , S _ { \\bullet } )$ , defined up to quasi-isomorphism, is called the tangent complex of $S _ { \\bullet }$ . Its cohomology groups are denoted by $T ^ { i } ( S _ { \\bullet } )$ . According to [12, 14], an immediate consequence of Theorem 4 is that the space of first order deformations of the diagram $S _ { \\bullet }$ is $H ^ { 1 } ( \\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { \\bullet } , R _ { \\bullet } ) ) \\ : = \\ : T ^ { \\bar { 1 } } ( S _ { \\bullet } )$ , and obstructions to deformations are contained in the space $H ^ { 2 } ( \\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { \\bullet } , R _ { \\bullet } ) ) = T ^ { 2 } ( S _ { \\bullet } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,234,881,324]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"102","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Although in principle Theorem 4 gives a complete answer to our initial problem, for diagrams over a general small category $\\mathcal { D }$ it may be very difficult to concretely describe a cofibrant replacement, since projective cofibrations are described either as maps satisfying the left lifting property with respect to trivial fibrations, or as transfinite compositions of certain elementary cofibrations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,324,876,358]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"102","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A possible strategy to overcome this difficulty is to give an explicit functor of small categories $\\epsilon : N \\to \\mathcal { D }$ such that: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,367,879,426]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"102","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. for every diagram $S _ { \\bullet } \\in \\operatorname { F u n } ( \\mathcal { D } , \\mathbf { A l g } _ { \\mathbb { K } } )$ , the deformation theory of $S _ { \\bullet }$ is the same as the deformation theory of $S _ { \\bullet } \\circ \\epsilon \\in \\mathrm { F u n } ( N , \\mathbf { A l g _ { \\mathbb { K } } } )$ ; 2. cofibrations in $\\mathrm { F u n } ( N , { \\bf C D G A } _ { \\mathbb { K } } ^ { \\le 0 } )$ ) admit a constructive description. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,433,881,522]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"102","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the next sections we follow this strategy by setting as $\\epsilon$ a simplified version of the Bousfield-Kan approximation [3]. In our construction the category $N$ will be in particular a Reedy category (see Sect. 8) and the projective model structure in $\\mathrm { F u n } ( N , { \\bf C D G A } _ { \\mathbb { K } } ^ { \\le 0 } ,$ ) will be the same as the Reedy model structure, hence with a simpler description of cofibrations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,569,387,590]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"102","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7 Simplex Categories ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,612,881,686]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"102","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $\\pmb { \\Delta }$ be the category with objects the finite ordinals $[ n ] ~ = ~ \\{ 0 , 1 , \\cdots , n \\}$ and morphisms non-decreasing maps, also known as the simplex category. We denote by $\\delta _ { k } \\colon [ n - 1 ] \\to [ n ]$ , and by $\\sigma _ { k } \\colon [ n + 1 ] \\to [ n ]$ , $k = 0 , \\ldots , n$ , the usual face and degeneracy maps: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[159,704,833,756]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0061/1d65b4169e0d05893f821f21446670af8202efb12989f5d5244b93f8cfe4aaf1.jpg","list_items":null,"page_idx":"102","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{c} \\delta _ { k } : [ n - 1 ] \\to [ n ] , \\qquad \\delta _ { k } ( p ) = \\left\\{ p \\qquad { \\mathrm { i f ~ } } p < k \\\\ { p + 1 \\quad { \\mathrm { i f ~ } } p \\geq k } \\end{array} , \\right. \\qquad k = 0 , \\ldots , n ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[159,770,835,821]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0318/5342bde39db3c8245a0b5b3575eab11ba8ca77227c6f882bce6dd680c1c143c9.jpg","list_items":null,"page_idx":"102","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\sigma _ { k } \\colon [ n + 1 ] \\to [ n ] , \\qquad \\sigma _ { k } ( p ) = { \\left\\{ \\begin{array} { l l } { p } & { { \\mathrm { i f ~ } } p \\leq k } \\\\ { p - 1 } & { { \\mathrm { i f ~ } } p > k } \\end{array} \\right. } , \\qquad k = 0 , \\ldots , n ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,52,562,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"102","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[856,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"102","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,478,108]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"103","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"They satisfy the cosimplicial identities: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[337,123,658,256]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0686/0b60f318b4a31ab6f4af699cd3713dceb86d6bd1d4afc23dc177891e54d5c027.jpg","list_items":null,"page_idx":"103","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\sigma _ { i } \\sigma _ { j } = \\sigma _ { j } \\sigma _ { i + 1 } \\quad \\mathrm { ~ f o r ~ } i \\geq j } \\\\ & { \\delta _ { i } \\delta _ { j } = \\delta _ { j + 1 } \\delta _ { i } \\quad \\mathrm { ~ f o r ~ } i \\leq j } \\\\ & { \\sigma _ { i } \\delta _ { j } = \\left\\{ \\begin{array} { l l } { \\delta _ { j - 1 } \\sigma _ { i } , } & { \\mathrm { ~ i f ~ } j > i + 1 } \\\\ { \\mathrm { I d } , } & { \\mathrm { ~ i f ~ } j = i , i + 1 } \\\\ { \\delta _ { j } \\sigma _ { i - 1 } , } & { \\mathrm { ~ i f ~ } j < i . } \\end{array} \\right. } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,270,881,324]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"103","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We recall that a cosimplicial group is a functor $G \\colon \\Delta \\to \\mathbf { G r p } , [ n ] \\mapsto G _ { n }$ ; in the sequel we shall need the following proposition, which is an easy generalisation of a well known result about cosimplicial groups, cf. [2, Prop. X.4.9]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,331,881,387]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"103","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 6 Let $G$ be a cosimplicial group, let $n \\geq 1$ , and $I \\subseteq [ n ]$ . Assume there are given elements $x _ { i } ~ \\in ~ G _ { n }$ , $i \\in I$ , such that $\\sigma _ { i - 1 } x _ { j } = \\sigma _ { j } x _ { i }$ for all $i \\ > \\ j$ and $i , j \\in I$ . Then there exists $x \\in G _ { n + 1 }$ such that $\\sigma _ { i } x = x _ { i }$ for all $i \\in I$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,394,876,432]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"103","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Writing $I = \\{ i _ { 0 } < i _ { 1 } < \\cdots < i _ { k } \\} \\subseteq [ n ]$ , consider the sequence $z _ { i _ { 0 } } , \\ldots , z _ { i _ { k } }$ defined recursively by the formula: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[191,445,806,471]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0959/099cdb066c0b7fd39c9749fc2d0ed69b5d9be8c280024267331425b11e1619b9.jpg","list_items":null,"page_idx":"103","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nz _ { i _ { k } } = \\delta _ { i _ { k } } x _ { i _ { k } } , \\qquad z _ { i _ { p } } = z _ { i _ { p + 1 } } \\cdot ( \\delta _ { i _ { p } } \\sigma _ { i _ { p } } z _ { i _ { p + 1 } } ) ^ { - 1 } \\cdot ( \\delta _ { i _ { p } } x _ { i _ { p } } ) , \\quad p < k .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[116,486,879,522]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"103","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For later use we point out that in the construction of this sequence we have only used the group homomorphisms ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[257,539,740,558]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0868/9fe23cb1471409321f71bec21e3d374efe2ef8daa2b5318de8c57a3bd1c914b4.jpg","list_items":null,"page_idx":"103","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\sigma _ { i } \\colon G _ { n + 1 } \\to G _ { n } , \\qquad \\delta _ { i } \\colon G _ { n } \\to G _ { n + 1 } , \\qquad i \\in I .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,576,881,612]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"103","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We claim that $x = z _ { i _ { 0 } }$ is the required element: we show by induction on $k - p$ that that $\\sigma _ { i _ { m } } z _ { i _ { p } } = x _ { i _ { m } }$ for all $m \\geq p$ . For $m > p$ , ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[252,629,742,747]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0830/d37b4277c3e515d20fb104993cd5d1609ad26e0eee86b67558e917d07c3ea463.jpg","list_items":null,"page_idx":"103","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\sigma _ { i _ { m } } z _ { i _ { p } } = ( \\sigma _ { i _ { m } } z _ { i _ { p + 1 } } ) \\cdot ( \\sigma _ { i _ { m } } \\delta _ { i _ { p } } \\sigma _ { i _ { p } } z _ { i _ { p + 1 } } ) ^ { - 1 } \\cdot ( \\sigma _ { i _ { m } } \\delta _ { i _ { p } } x _ { i _ { p } } ) } \\\\ & { \\qquad = x _ { i _ { m } } \\cdot ( \\delta _ { i _ { p } } \\sigma _ { i _ { m } - 1 } \\sigma _ { i _ { p } } z _ { i _ { p + 1 } } ) ^ { - 1 } \\cdot ( \\delta _ { i _ { p } } \\sigma _ { i _ { m } - 1 } x _ { i _ { p } } ) } \\\\ & { \\qquad = x _ { i _ { m } } \\cdot ( \\delta _ { i _ { p } } \\sigma _ { i _ { p } } \\sigma _ { i _ { m } } z _ { i _ { p + 1 } } ) ^ { - 1 } \\cdot ( \\delta _ { i _ { p } } \\sigma _ { i _ { p } } x _ { i _ { m } } ) } \\\\ & { \\qquad = x _ { i _ { m } } \\cdot ( \\delta _ { i _ { p } } \\sigma _ { i _ { p } } x _ { i _ { m } } ) ^ { - 1 } \\cdot ( \\delta _ { i _ { p } } \\sigma _ { i _ { p } } x _ { i _ { m } } ) = x _ { i _ { m } } , } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,759,257,777]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"103","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and for $m = p$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[259,792,740,849]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"1004/458608a5d23cc5cc299102e0b2710e55cd7b6347c533fab143a918a69ee5ae75.jpg","list_items":null,"page_idx":"103","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\sigma _ { i _ { p } } z _ { i _ { p } } = ( \\sigma _ { i _ { p } } z _ { i _ { p + 1 } } ) \\cdot ( \\sigma _ { i _ { p } } \\delta _ { i _ { p } } \\sigma _ { i _ { p } } z _ { i _ { p + 1 } } ) ^ { - 1 } \\cdot ( \\sigma _ { i _ { p } } \\delta _ { i _ { p } } x _ { i _ { p } } ) } \\\\ & { \\qquad = ( \\sigma _ { i _ { p } } z _ { i _ { p + 1 } } ) \\cdot ( \\sigma _ { i _ { p } } z _ { i _ { p + 1 } } ) ^ { - 1 } \\cdot ( x _ { i _ { p } } ) = x _ { i _ { p } } . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[685,52,879,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"103","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Lepri and M. Manetti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,52,143,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"103","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"98 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[858,861,881,876]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"103","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"□ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,144]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"104","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The simplex category $\\pmb { \\Delta }$ admits the following useful generalisation. Let $\\mathcal { B }$ be a small category and consider, for every $n \\geq 0$ , the set $N ( { \\mathcal { B } } ) _ { n }$ of $n$ -simplexes of the nerve of $\\mathcal { B }$ : every element of $N ( { \\mathcal { B } } ) _ { n }$ is a string ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[350,159,649,183]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0482/6edc782ee05a8c1e3f782b9df3316333fb5551bef05cf5441fa9fc6485ef1c1b.jpg","list_items":null,"page_idx":"104","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nx = [ x _ { 0 } { \\overset { \\alpha _ { 1 } } { \\longrightarrow } } x _ { 1 } \\cdots x _ { n - 1 } { \\overset { \\alpha _ { n } } { \\longrightarrow } } x _ { n } ]\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,199,881,237]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"104","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"of $n$ morphisms of $\\mathcal { B }$ . The simplex category $N ( { \\mathcal { B } } )$ of $\\mathcal { B }$ is defined in the following way: the set of objects is the disjoint union of $N ( { \\mathcal { B } } ) _ { n } , n \\geq 0$ . Given two objects ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[168,252,826,280]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0063/75ee1a58b7b816b4807c0cbd2410ec4ddc9e5ec79827603f60086b4a0f6794bd.jpg","list_items":null,"page_idx":"104","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { x = [ x _ { 0 } \\xrightarrow { \\alpha _ { 1 } } x _ { 1 } \\cdots x _ { n - 1 } \\xrightarrow { \\alpha _ { n } } x _ { n } ] , \\qquad y = [ y _ { 0 } \\xrightarrow { \\beta _ { 1 } } y _ { 1 } \\cdots y _ { m - 1 } \\xrightarrow { \\beta _ { m } } y _ { m } ] , } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,297,881,352]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"104","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"a morphism $f \\colon x y$ is a monotone map $f \\colon [ n ] [ m ]$ such that $y _ { f ( i ) } = x _ { i }$ for every $i \\in [ n ]$ , and for every $0 \\leq i \\leq n$ the morphism $\\alpha _ { i }$ is the composition of $\\beta _ { j }$ , for $f ( i - 1 ) < j \\leq f ( i )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[193,366,804,397]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0423/11c905fd89535b4557a74443aaf364cfac55fdc53901f3591c4135109c691e4c.jpg","list_items":null,"page_idx":"104","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\alpha _ { i } \\colon x _ { i - 1 } = y _ { f ( i - 1 ) } { \\xrightarrow { { \\beta _ { f ( i - 1 ) + 1 } } } } \\cdots { \\xrightarrow { } } y _ { f ( i ) - 1 } { \\xrightarrow { { \\beta _ { f ( i ) } } } } y _ { f ( i ) } = x _ { i } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,411,881,472]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"104","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Notice that the equality $f ( i - 1 ) = f ( i )$ implies $x _ { i } ~ = ~ x _ { i - 1 }$ and $\\alpha _ { i } ~ = \\mathrm { I d }$ . For example, if $[ x \\stackrel { \\alpha } { } y \\stackrel { \\beta } { } z ] \\in N ( \\mathcal { B } ) _ { 2 }$ , then we have in the category $N ( { \\mathcal { B } } )$ the following morphisms: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[223,487,772,654]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0220/100a6031f004d7260c979251644d59c6ab3ffc315433c030e78696a7fbf0fc65.jpg","list_items":null,"page_idx":"104","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { [ z ] \\xrightarrow { \\delta _ { 0 } } \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\downarrow \\quad \\delta _ { 0 } \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\delta _ { 1 } } \\\\ & { [ y \\xrightarrow { \\beta } z ] \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\sigma _ { 0 } ^ { ( 1 ) } \\Bigg | } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\sigma _ { 1 } \\xrightarrow { \\beta } \\ y \\ ] { \\beta } \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\sigma _ { 1 } \\xrightarrow { \\alpha } \\ y ] } \\\\ & { [ y \\xrightarrow { \\mathrm { I d } } \\ y \\xrightarrow { \\beta } z ] { \\beta } \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\sigma _ { 1 } } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\sigma _ { 2 } ^ { ( 1 ) } \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\sigma _ { 1 } } \\\\ & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[143,671,792,690]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"104","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Notice that the simplex category of the singleton $\\mathcal { B } = \\{ * \\}$ is exactly $\\pmb { \\Delta }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,698,564,717]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"104","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 13 A morphism $f \\colon x y$ in $N ( { \\mathcal { B } } )$ : ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[287,734,708,753]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0657/abd35a29b1c17f0faf13b66ac93de985ac7d76c6d5b66f682397ae2d9adda9ae.jpg","list_items":null,"page_idx":"104","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { x \\in N ( \\mathcal { B } ) _ { n } , \\quad y \\in N ( \\mathcal { B } ) _ { m } , \\quad f \\colon [ n ] \\to [ m ] , } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,770,421,788]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"104","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is called an anchor if $f ( n ) = m$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,797,881,852]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"104","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 14 For every $k ~ \\in ~ \\mathbb { N } \\cup \\{ + \\infty \\}$ we shall denote by $N ( { \\mathcal { B } } ) _ { \\leq k }$ the full subcategory of $N ( { \\mathcal { B } } )$ with objects the (disjoint) union of $N ( { \\mathcal { B } } ) _ { i }$ for $i ~ \\leq k$ , and by ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[309,866,687,890]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0550/c31e396d5b813b9322bc4cd2c835c21dd3386ed3b30fd623f633871aa9f42882.jpg","list_items":null,"page_idx":"104","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathrm { F u n } ^ { \\star } ( N ( \\mathcal { B } ) _ { \\leq k } , \\mathbf { M } ) \\subseteq \\mathrm { F u n } ( N ( \\mathcal { B } ) _ { \\leq k } , \\mathbf { M } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,51,562,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"104","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[854,52,879,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"104","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"99 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"the full subcategory of diagrams $F \\colon N ( \\mathcal { B } ) _ { \\leq k } \\ \\ \\textbf { M }$ such that $F ( f )$ is an isomorphism for every anchor map $f$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,133,781,153]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 15 The forgetful functor $\\epsilon \\colon N ( \\mathcal { B } ) \\mathcal { B }$ is defined by setting ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[328,168,671,192]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0787/9e4b160aff8141a849d62264f34b4a584d1683261f0388fc25c6571c49f645f0.jpg","list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\epsilon ( [ x _ { 0 } \\xrightarrow { \\alpha _ { 1 } } x _ { 1 } \\cdots x _ { n - 1 } \\xrightarrow { \\alpha _ { n } } x _ { n } ] ) = x _ { n }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,210,430,228]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"on the objects. For any morphism ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[198,243,797,271]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0841/0d4fe98c248f9592bddf57c2dcf8fda0f192b45ae38477457930c27220cc1323.jpg","list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nf \\colon [ x _ { 0 } \\ { \\xrightarrow { \\alpha _ { 1 } } } \\ x _ { 1 } \\ \\cdots \\ x _ { n - 1 } \\ { \\xrightarrow { \\alpha _ { n } } } \\ x _ { n } ] \\to [ y _ { 0 } \\ { \\xrightarrow { \\beta _ { 1 } } } \\ y _ { 1 } \\ \\cdots \\ y _ { m - 1 } \\ { \\xrightarrow { \\beta _ { m } } } \\ y _ { m } ] ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,288,198,304]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[277,324,722,343]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0786/e4499e59c286d0c79795f4be903afe477698ffc4a3a083778574db53f5e5f708.jpg","list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\epsilon ( f ) = \\beta _ { m } \\circ \\cdot \\cdot \\cdot \\circ \\beta _ { f ( n ) + 1 } \\colon x _ { n } = y _ { f ( n ) } y _ { m } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[116,358,879,396]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In particular, $\\epsilon ( f ) = \\operatorname { I d }$ for any anchor $f$ . It is clear that the composition with the functor $\\epsilon$ gives, for every $k$ , a natural transformation: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[157,409,790,433]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0477/b0023dffab1bec40a4b61e84883d446794f49102a2505854399e9a12905ea44f.jpg","list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\epsilon ^ { * } \\colon \\mathrm { F u n } ( \\mathcal { B } , \\mathbf { M } ) \\to \\mathrm { F u n } ^ { \\star } ( N ( \\mathcal { B } ) _ { \\leq k } , \\mathbf { M } ) , \\qquad \\epsilon ^ { * } ( F ) = F \\circ \\epsilon _ { | N ( \\mathcal { B } ) _ { \\leq k } } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,448,571,468]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $k \\geq 2$ we also have a natural transformation ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,481,678,504]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"1005/b5d227f309b35887422b8feeab1051dad334146a3b0c989ab48d742610473118.jpg","list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\boldsymbol { \\tau } : \\mathrm { F u n } ^ { \\star } ( N ( \\mathcal { B } ) _ { \\leq k } , \\mathbf { M } ) \\to \\mathrm { F u n } ( \\mathcal { B } , \\mathbf { M } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,521,881,557]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"defined in the following way: given $G \\in \\mathrm { F u n } ^ { \\star } ( N ( \\mathcal { B } ) _ { \\leq k } , \\mathbf { M } )$ and an object $x \\in { \\mathcal { B } }$ we set ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[405,575,594,594]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0561/62340f08abc59f80ea1bb56a5124ac1588d2d65ea5b9e7e109f5435dc4bc68ca.jpg","list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\tau ( G ) ( x ) = G ( [ x ] ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,611,514,632]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Given a morphism $x { \\stackrel { \\alpha } { \\to } } y$ in $\\mathcal { B }$ we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[307,647,692,675]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0814/69cd02caef053897930fbaad011c26c477df83383f234541aa56e0e4edabfa3b.jpg","list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nG ( [ x ] ) \\xrightarrow { G ( \\delta _ { 1 } ) } G ( [ x \\xrightarrow { \\alpha } y ] ) \\xrightarrow { G ( \\delta _ { 0 } ) } G ( [ y ] )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,692,578,710]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and, since $G ( \\delta _ { 0 } )$ is an isomorphism we can define ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[280,723,719,746]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0406/97fd8332613682491530deb5bfe319fec9ea1c8380e24a8f1083b4e8c238d7be.jpg","list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\tau ( G ) ( \\alpha ) = G ( \\delta _ { 0 } ) ^ { - 1 } G ( \\delta _ { 1 } ) \\colon G ( [ x ] ) \\to G ( [ y ] ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[143,762,876,783]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We need to prove that $\\tau ( G )$ is a functor: applying $G$ to the commutative diagram ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[357,798,640,888]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0265/b558999db3772922efecc42e24d402697d11c1676e776b0e6a9ff9e2a5e65c53.jpg","list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n[ x ] \\xrightarrow [ \\mathrm { I d } ] { \\delta _ { \\begin{array} { l } { \\delta _ { 1 } } \\\\ { \\qquad \\mathrm { I d } } \\end{array} } [ x \\xrightarrow [ \\mathrm { I d } ] { \\mathrm { I d } } x ] \\mathord { \\left. \\right. } \\mathord { \\left[ x \\right] } } [ x ]\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[685,51,879,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Lepri and M. Manetti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,152,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"105","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"100 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,93,881,135]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"106","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"we prove that $\\tau ( G )$ preserves the identities. Given $[ x \\stackrel { \\alpha } { } y \\stackrel { \\beta } { } z ] \\in N ( { \\mathcal { B } } ) _ { 2 }$ , applying $G$ to the commutative diagram ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[289,151,710,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0501/db22096c72c8ad6537c2d5e51723845ab7f00f27446046f12c252991c80414d0.jpg","list_items":null,"page_idx":"106","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\left[ z \\right] \\xrightarrow { \\delta _ { 0 } } \\left[ x \\xrightarrow { \\beta \\alpha } z \\right] { - \\delta _ { 1 } } \\left[ x \\right] } \\\\ & { \\left[ \\begin{array} { l } { \\delta _ { 0 } } \\\\ { \\delta _ { 0 } } \\end{array} \\right] \\xrightarrow [ { \\delta _ { 1 } } ] { \\left[ \\begin{array} { l l l } { \\delta _ { 1 } } & & { \\left\\lfloor x \\right\\rfloor } \\\\ { \\delta _ { 2 } } \\end{array} \\right] { - \\delta _ { 2 } } } \\left[ x \\xrightarrow [ { \\delta _ { 2 } ] { \\delta _ { 2 } } } y \\right] } \\\\ & { \\left[ y \\xrightarrow [ { \\delta _ { 1 } } ] { \\delta _ { 1 } } \\bigoplus _ { \\substack { \\left[ \\underline { { y } } \\right] } } \\right] { - \\delta _ { 1 } } \\left[ x \\xrightarrow [ { \\delta _ { 2 } ] { \\delta _ { 3 } } } z \\right] { - \\delta _ { 2 } } \\left[ x \\xrightarrow [ { \\delta _ { 3 } ] { \\alpha } } y \\right] } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,328,881,366]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"106","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"we obtain $\\tau ( G ) ( \\beta \\alpha ) = \\tau ( G ) ( \\beta ) \\circ \\tau ( G ) ( \\alpha )$ . Therefore $\\tau$ is properly defined and its functoriality is clear. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[113,373,808,393]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"106","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 7 The above functors $\\epsilon ^ { * }$ and $\\tau$ are equivalences of categories. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,400,881,438]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"106","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof It is immediate from the definition that $\\tau \\circ \\epsilon ^ { * }$ is the identity. On the other hand every anchor map of type ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[375,454,621,474]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0706/b4980cdffe8b54c63d1b86823091e2fb0f794397306514883d998451130d6170.jpg","list_items":null,"page_idx":"106","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n[ x _ { n } ] \\to [ x _ { 0 } \\to \\cdot \\cdot \\cdot \\to x _ { n } ]\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,490,756,510]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"106","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"induces, for every $G \\in \\mathrm { F u n } ^ { \\star } ( N ( \\mathcal { B } ) _ { \\leq k } , \\mathbf { M } )$ , a canonical isomorphism ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[134,525,863,551]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0070/91076d9e75ee0fb7bbfc41fd7633dce6c9bc62a9b461bc4ac0bfab006c828d46.jpg","list_items":null,"page_idx":"106","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\epsilon ^ { * } \\tau ( G ) ( [ x _ { 0 } \\cdot \\cdot \\cdot x _ { n } ] ) = \\tau ( G ) ( x _ { n } ) = G ( [ x _ { n } ] ) \\xrightarrow { \\simeq } G ( [ x _ { 0 } \\cdot \\cdot \\cdot x _ { n } ] ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,594,881,650]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"106","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 8 Let $S _ { \\bullet } \\in \\mathrm { F u n } ( \\mathcal { B } , \\mathbf { A l g } _ { \\mathbb { K } } )$ be a diagram of commutative algebras and $k \\geq 2$ . Then the isomorphism classes of deformations of S are the same as the isomorphism classes of deformations of $\\epsilon ^ { * } S _ { \\bullet } \\in \\mathrm { F u n } ( N ( \\mathcal { B } ) _ { \\leq k } , \\mathbf { A l g } _ { \\mathbb { K } } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,657,876,693]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"106","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof It is immediate from the definition that for every $A \\in \\mathbf { A r t } _ { \\mathbb { K } }$ the equivalences of categories ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[291,710,706,744]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0628/4eaeff36674d51e43561662ffdee785e58da5551ffe9b72b143fe587d51f3f40.jpg","list_items":null,"page_idx":"106","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathrm { F u n } ( \\mathbf { \\mathcal { B } } , \\mathbf { A } \\mathbf { l g } _ { A } ) \\xrightarrow [ \\tau ] { \\epsilon ^ { * } } \\mathrm { F u n } ^ { \\star } ( N ( \\mathbf { \\mathcal { B } } ) _ { \\leq k } , \\mathbf { A } \\mathbf { l g } _ { A } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,765,876,801]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"106","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"preserve flatness. Therefore, for every deformation $\\phi \\colon S _ { A , \\bullet } \\to S .$ of $S _ { \\bullet }$ the map $\\epsilon ^ { * } \\phi \\colon \\epsilon ^ { * } S _ { A , \\bullet } \\to \\epsilon ^ { * } S _ { \\bullet }$ is a deformation of the diagram $\\epsilon ^ { * } S _ { \\bullet }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,803,879,858]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"106","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In order to conclude the proof we only need to show that, if $R _ { \\bullet } \\colon N ( \\mathcal { B } ) _ { \\leq k } \\ $ $\\mathbf { A } \\mathbf { g } _ { A }$ is a deformation of $\\epsilon ^ { * } S _ { \\bullet }$ , and $f \\colon \\alpha \\beta$ is an anchor in $N ( { \\mathcal { B } } ) _ { \\leq k }$ , then $R _ { f }$ is an isomorphism. This implies that $R _ { \\bullet } \\in \\mathrm { F u n } ^ { \\star } ( N ( \\mathcal { B } ) _ { \\leq k } , \\mathbf { A l g } _ { A } )$ . We have by definition ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,51,564,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"106","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"106","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"101 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[861,570,879,584]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"106","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"□ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,876,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"107","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"that $\\phi \\colon R _ { \\bullet } \\to \\epsilon ^ { * } S _ { \\bullet }$ induces an isomorphism $R _ { \\bullet } \\otimes _ { A } \\mathbb { K } \\to \\epsilon ^ { * } S _ { \\bullet }$ and, since $\\epsilon ^ { * } S _ { \\alpha } $ $\\epsilon ^ { * } S _ { \\beta }$ is the identity, by the commutativity of the diagram ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[371,142,628,223]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0929/602f0df01959acdab795da4e6e46a0ead9e360d5384c89e581e82657a7e8763c.jpg","list_items":null,"page_idx":"107","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l r } { { R _ { \\alpha } \\otimes _ { A } \\mathbb { K } \\longrightarrow R _ { \\beta } \\otimes _ { A } \\mathbb { K } } } \\\\ & { } & { \\cong \\downarrow \\phi } \\\\ & { } & { \\epsilon ^ { * } S _ { \\alpha } \\longrightarrow \\epsilon ^ { * } S _ { \\beta } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,244,881,282]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"107","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"we obtain that $R _ { \\alpha } \\otimes _ { A } \\mathbb { K } \\to R _ { \\beta } \\otimes _ { A } \\mathbb { K }$ is also an isomorphism. By Lemma 2, $R _ { \\alpha } $ $R _ { \\beta }$ is an isomorphism too. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,327,446,348]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"107","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"8 Reedy Model Structures ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,372,881,426]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"107","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We briefly recall the notion of Reedy category, for more details see [6]. We do so in view of Theorem 5, which yields a model structure on the category $\\mathrm { F u n } ( \\mathcal { D } , \\mathbf { M } )$ of diagrams on a model category $\\mathbf { M }$ indexed by a Reedy category $\\mathcal { D }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,433,876,472]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"107","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 16 A Reedy category is a small category $\\mathcal { D }$ together with two subcategories $\\vec { \\mathcal { D } } , \\overleftarrow { \\mathcal { D } }$ , such that: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,480,881,584]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"107","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. $\\mathrm { O b } ( \\mathcal { D } ) = \\mathrm { O b } ( \\mathcal { \\vec { D } } ) = \\mathrm { O b } ( \\mathcal { \\overleftarrow { D } } )$ \n2. Every morphism $f$ in $\\mathcal { D }$ has a unique factorisation $f = g h$ , where $g$ is in $\\vec { \\mathcal { D } }$ and $h$ is in . \n3. There exists a function deg $\\mathrm { O b } ( \\mathcal { D } ) \\mathbb { N }$ such that every non-identity morphism in $\\vec { \\mathcal { D } }$ raises degree and every non-identity morphism in lowers degree. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,591,881,669]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"107","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is easy to see that in a Reedy category every isomorphism is an identity: if $f$ is an isomorphism and $f = g _ { 1 } h _ { 1 }$ $g _ { 1 } h _ { 1 } , f ^ { - 1 } = g _ { 2 } h _ { 2 } , h _ { 1 } g _ { 2 } = g _ { 3 } h _ { 3 }$ are factorisations as in (2), then $g _ { 1 } g _ { 3 }$ must be the identity. A Reedy category is called direct if $\\mathcal { D } = \\overline { { \\mathcal { D } } }$ , or equivalently if contains only the identities. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,671,758,689]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"107","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For instance, we have the following examples of Reedy categories: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,696,883,855]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"107","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. A category whose only morphisms are the identities is called discrete. Every discrete category is trivially a Reedy category. \n2. Let $I$ be a finite poset such that there exists a function deg $I \\mathbb { N }$ such that $\\deg ( x ) > \\deg ( y )$ for every $x > y$ . Then $I$ is a direct Reedy category. \n3. The simplex category $\\pmb { \\Delta }$ is a Reedy category, with de ${ \\mathfrak { g } } ( [ n ] ) = n$ , $\\vec { \\Delta }$ the injective maps and $\\overleftarrow { \\Delta }$ the surjective maps. \n4. If $c$ and $\\mathcal { D }$ are Reedy categories then so is the product $\\boldsymbol { C } \\times \\boldsymbol { \\mathcal { D } }$ , with ${ \\overrightarrow { C \\times { \\mathcal { D } } } } =$ −→ × −→, ←−−− × = ←− × ←− and deg(c, d) = deg(c) + deg(d). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,861,881,899]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"107","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $a$ is an object of a category $\\mathcal { D }$ we denote by $a \\downarrow \\mathcal { D }$ the undercategory of maps $a b$ in $\\mathcal { D }$ and by $\\mathcal { D } \\downarrow a$ the overcategory of maps $b \\to a$ in $\\mathcal { D }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[687,52,879,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"107","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Lepri and M. Manetti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,152,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"107","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"102 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,706,108]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"108","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 17 Let $\\mathcal { D }$ be a Reedy category and $a$ an object in $\\mathcal { D }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,117,881,193]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"108","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. The matching category $M _ { a } \\mathcal { D }$ of $\\mathcal { D }$ at $a$ is the full subcategory of $a \\downarrow \\overleftarrow { \\mathcal { D } }$ containing all objects except the identity map of $a$ . \n2. The latching category $L _ { a } \\mathcal { D }$ of $\\mathcal { D }$ at $a$ is the full subcategory of $\\vec { \\mathcal { D } } \\downarrow a$ containing all objects except the identity map of $a$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,202,881,276]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"108","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 18 Let $\\mathcal { D }$ be a Reedy category, let $\\mathbf { M }$ be a complete and cocomplete category, let $X$ be a $\\mathcal { D }$ -diagram in $\\mathbf { M }$ , and $a$ be an object in $\\mathcal { D }$ . For notational simplicity $X$ also denotes the induced $M _ { a } \\mathcal { D }$ -diagram, with $X _ { a b } = X _ { b }$ , and the induced $L _ { a } \\mathcal { D }$ -diagram, with $X _ { b a } = X _ { b }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,283,630,321]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"108","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. The matching object of $X$ at $a$ is $M _ { a } X = \\operatorname* { l i m } _ { M _ { a } \\mathcal { D } } X$ . \n2. The latching object of $X$ at $a$ is $L _ { a } X = \\operatorname { c o l i m } _ { L _ { a } \\mathcal { D } } X$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,328,699,348]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"108","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There are natural morphisms $L _ { a } X \\longrightarrow X$ and $X \\longrightarrow M _ { a } X$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,355,881,409]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"108","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The main use of Reedy categories originates from the following theorem: the category of diagrams in a model category indexed by a Reedy category has a model category structure. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,417,881,459]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"108","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 5 (Reedy-Kan) Let $\\mathcal { D } = ( \\overline { { \\mathcal { D } } } , \\overleftarrow { \\mathcal { D } } )$ be a Reedy category, and M a model category. There is a model structure on \" $\\mathrm { F u n } ( \\mathcal { D } , \\mathbf { M } )$ where a map $f \\colon X \\longrightarrow Y$ is: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,466,788,522]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"108","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. a weak equivalence iff $X _ { i } \\longrightarrow Y _ { i }$ is a weak equivalence for all $i \\in \\mathcal { D }$ ; \n2. a fibration iff $X _ { i } \\longrightarrow M _ { i } X \\times _ { { M _ { i } } Y } Y _ { i }$ is a fibration for all $i \\in \\mathcal { D }$ ; \n3. a cofibration iff $X _ { i } \\coprod _ { L _ { i } X } L _ { i } Y \\longrightarrow Y _ { i }$ is a cofibration for all $i \\in \\mathcal { D }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,530,382,548]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"108","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For a proof, see [6, 15.3]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,549,881,620]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"108","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We call Reedy weak equivalences, Reedy fibrations and Reedy cofibrations the weak equivalences, fibrations and cofibrations of this model structure, to avoid confusion with other model structures on the same category. For example, the commutative square ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[441,638,558,719]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0009/b119f35d10a0556fdd3323de4232f8f55865a94f142599f82b6e560942972d12.jpg","list_items":null,"page_idx":"108","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\left. \\begin{array} { c } { { A \\longrightarrow B } } \\\\ { { g \\left\\downarrow \\begin{array} { c } { { } } \\\\ { { } } \\end{array} \\right\\downarrow } } \\\\ { { C \\longrightarrow D } } \\end{array} \\right.\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,740,881,812]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"108","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"may be considered as a diagram over the Reedy poset of subsets of $\\{ 0 , 1 \\}$ . Then it is a Reedy fibrant diagram if and only if $A , B , C , D$ are fibrant objects; it is a Reedy cofibrant diagram if and only if $A$ is a cofibrant object and the three maps $f , g$ and $B \\ U _ { A } \\ C \\to D$ are cofibrations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,813,881,866]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"108","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We say that a map $f \\colon X \\to Y$ in $\\mathrm { F u n } ( \\mathcal { D } , \\mathbf { M } )$ is a pointwise weak equivalence (cofibration, fibration) if $f _ { i } : X _ { i } Y _ { i }$ is a weak equivalence (cofibration, fibration) for all $i \\in \\mathcal { D }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,562,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"108","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"108","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"103 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,879,142]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"109","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 10 Let $\\mathcal { D }$ be a Reedy category. Then the Reedy model structure on $\\begin{array} { r } { \\mathrm { F u n } ( \\mathcal { D } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } ) } \\end{array}$ ) coincides with the projective model structure if and only if every object is Reedy fibrant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,151,881,223]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"109","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof By definition the two model structures have the same weak equivalences. Let $\\varphi \\colon X Y$ be a morphism in $\\begin{array} { r } { \\mathrm { F u n } ( \\mathcal { D } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } ) } \\end{array}$ ). If $\\varphi$ is a Reedy fibration, then it is not difficult to prove it is a pointwise fibration ([6], 15.3.11); therefore if every pointwise fibration is a Reedy fibration then the two model structures coincide. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,225,881,372]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"109","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Assume that every object is Reedy fibrant and that $\\varphi _ { i } : X _ { i } \\longrightarrow Y _ { i }$ is a fibration in $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ for all $i \\in \\mathcal { D }$ ; we want to prove that $\\varphi$ is a Reedy fibration. We denote by $c ( \\mathbb { K } ) \\colon \\mathcal { D } \\to \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\le 0 }$ the constant diagram $i \\mapsto \\mathbb { K }$ , and by $K = c ( \\mathbb { K } ) \\times _ { Y } X$ the fibre product of $\\varphi$ and the initial morphism $c ( \\mathbb { K } ) \\to Y$ . Note that $M _ { i } c ( \\mathbb { K } )$ is concentrated in degree 0. Since the fibre product and the matching objects are both limits, they commute by Fubini’s theorem [9, Prop. 6.2.8], and we have $M _ { i } ( K ) =$ $M _ { i } ( c ( \\mathbb { K } ) \\times _ { Y } X ) \\cong M _ { i } c ( \\mathbb { K } ) \\times _ { M _ { i } Y } M _ { i } X$ . Thus we have a morphism of cartesian squares: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[339,388,656,561]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0967/0a233222d76221dd8756a48c257e25d3403127fe7ac6992cf0fefb4dc4c9581a.jpg","list_items":null,"page_idx":"109","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[118,584,881,711]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"109","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The map $\\varphi _ { i } \\colon X _ { i } \\to Y _ { i }$ induces $\\widehat { \\varphi _ { i } } \\colon M _ { i } X \\to M _ { i } Y$ ; let $Y _ { i } \\times _ { M _ { i } Y } M _ { i } X$ be the fibre product of $\\widehat { \\varphi _ { i } }$ and the natural map $Y _ { i } M _ { i } Y$ . We have to show that the map $X _ { i } $ $Y _ { i } \\times _ { M _ { i } Y } M _ { i } X$ is surjective in strictly negative degree. Let $( \\alpha , \\beta ) \\in Y _ { i } \\times _ { M _ { i } Y } M _ { i } X$ , with $\\overline { { \\alpha } } = \\overline { { \\beta } } < 0$ . Without loss of generality, because of the surjectivity of $\\varphi _ { i }$ , we can assume $\\alpha = 0$ , so $\\widehat { \\varphi _ { i } } ( \\beta ) = 0$ , which means $\\beta$ lifts to $( 0 , \\beta ) \\in M _ { i } K$ , and then to $K _ { i }$ , since the map $K _ { i } M _ { i } K$ is by hypothesis a fibration. By the commutativity of the above diagram, we have the thesis. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,711,881,767]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"109","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conversely, if the Reedy and projective model structures coincide, an object is Reedy fibrant if and only if it is point-wisely fibrant, and that is clearly true in CDGA≤0K . □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,774,410,792]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"109","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The following result is clear. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,801,879,858]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"109","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 11 The simplex category $N ( { \\mathcal { B } } )$ is a Reedy category, where the direct subcategory $\\overrightarrow { N ( \\mathcal { B } ) }$ is defined by injective maps and the inverse subcategory $\\overleftarrow { \\Lambda } ( \\mathcal { B } )$ by surjective maps. The same applies to its subcategories $N ( { \\mathcal { B } } ) _ { \\leq k }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[687,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"109","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Lepri and M. Manetti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,54,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"109","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"110","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In particular, when $\\mathcal { B } = \\{ * \\}$ we recover the usual Reedy structure on $\\pmb { \\Delta }$ . By Theorem 5 we have the Reedy model structure on the category $\\mathrm { F u n } ( N ( \\mathcal { B } ) , { \\bf C D G A } _ { \\mathbb { K } } ^ { \\le 0 } )$ ; we show that the Reedy and projective model structures on this category coincide, using Lemma 10. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,168,876,207]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"110","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 6 Every object in $\\mathrm { F u n } ( N ( \\mathbf { \\mathcal { B } } ) _ { \\leq k } , \\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 } )$ is Reedy fibrant, and so the Reedy model structure coincides with the projective model structure. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,213,881,270]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"110","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof In view of the definition of fibrations in $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ , it is sufficient to show that for every $X \\in \\mathrm { F u n } ( N ( \\mathcal { B } ) _ { \\leq k } , \\mathbf { G r p } )$ the map $X _ { \\alpha } M _ { \\alpha } X$ is surjective for all $\\alpha \\in N ( \\mathcal { B } ) _ { \\leq k }$ . Fix $n \\leq k$ and let $\\alpha ^ { ( m ) } \\in N ( { \\mathcal { B } } ) _ { n }$ , ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[337,286,658,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0420/1c84b992339e1bc40f24b741ee8ffaa8593f7d33ddbac5c5466a2b84f6dd713f.jpg","list_items":null,"page_idx":"110","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\alpha ^ { ( m ) } = [ x _ { 0 } \\xrightarrow { h _ { 1 } } x _ { 1 } \\xrightarrow { h _ { 2 } } \\cdot \\cdot \\cdot \\xrightarrow { h _ { n } } x _ { n } ] ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,328,881,382]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"110","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $m$ is the number of morphisms $h _ { i }$ equal to the identity, $0 \\leq m \\leq n$ . If $m = 0$ there is nothing to prove, because the matching category of $\\alpha ^ { ( 0 ) }$ is empty, so every $X _ { \\alpha ^ { ( 0 ) } }$ is automatically fibrant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,385,410,400]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"110","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In case $n = m = 1$ , we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[432,418,562,490]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0643/67dc1dfa0a85b73e7a58f8d8de6305dd1ff338e06d88bc40573a9078fa320524.jpg","list_items":null,"page_idx":"110","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { c } { { [ a a ] } } \\\\ { { \\nonumber } } \\\\ { { \\delta _ { 0 } } } \\end{array} \\overset { \\triangledown } { ( \\begin{array} { l } { { \\overline { { { \\sigma _ { 0 } } } } } } \\\\ { { \\overline { { { a } } } } } \\end{array} ) } _ { \\delta _ { 1 } }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,510,876,546]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"110","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"with $\\sigma _ { 0 } \\delta _ { 0 } = \\sigma _ { 0 } \\delta _ { 1 } = \\mathrm { I d }$ by the cosimplicial identities, so ${ \\cal X } ( \\sigma _ { 0 } ) \\colon X _ { \\alpha ^ { ( 1 ) } } \\to M _ { \\alpha ^ { ( 1 ) } } X$ has a section and hence is surjective. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,546,881,647]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"110","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In general, assume $n \\geq 2$ and $0 < m \\le n$ ; let ${ \\cal I } = \\{ i \\in \\{ 0 , \\cdot \\cdot \\cdot n - 1 \\} \\mid h _ { i + 1 } = \\mathrm { I d } \\}$ , $| I | = m$ . For every $i \\in I$ we have a degeneracy map $\\sigma _ { i } \\colon \\alpha ^ { ( m ) } \\alpha _ { i } ^ { ( m - 1 ) }$ , where the α(m−1)i a re suitable objects in $N ( { \\mathcal { B } } ) _ { n - 1 }$ . From each $\\alpha _ { i } ^ { ( m - 1 ) }$ there are $m - 1$ degeneracy maps to other objects $\\alpha _ { i , j } ^ { ( m - 2 ) } \\in N ( \\mathcal { B } ) _ { n - 2 }$ , and so on. For example, for $n = 3 , m = 2 , I = \\{ 0 , 2 \\}$ : ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[236,662,763,789]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0199/cf2a827d638e4153098586550bf73920b82f0d8c38ac90862bee8e089819ddfd.jpg","list_items":null,"page_idx":"110","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n[ a b \\underbrace { b } _ { \\sigma _ { 1 } } ^ { [ a a b b ] } \\underbrace { \\vphantom { \\int } _ { [ a b ] } ^ { \\sigma _ { 2 } } } _ { [ a b ] } \\underbrace { \\vphantom { \\int } _ { \\sigma _ { 1 } } ^ { \\sigma _ { 2 } } } _ { [ a b ] } \\nonumber\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,801,881,863]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"110","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For every map $\\sigma _ { i } \\colon \\alpha ^ { ( m ) } \\to \\alpha _ { i } ^ { ( m - 1 ) }$ we also have two sections $\\delta _ { i } , \\delta _ { i + 1 } \\colon \\alpha _ { i } ^ { ( m - 1 ) } \\to$ $\\alpha ^ { ( m ) }$ , so using an identical computation to Lemma 6, we have that $X _ { \\alpha ^ { ( m ) } }$ maps surjectively into ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[177,878,815,902]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0628/df863ab338a6fcdb179f9473a35b8b64dd69cd6787cbbea4b8a5fa34db5f3cb0.jpg","list_items":null,"page_idx":"110","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nV : = \\{ x _ { l } \\in \\alpha _ { l } ^ { ( m - 1 ) } , \\ l \\in I \\ | \\ \\sigma _ { i - 1 } x _ { j } = \\sigma _ { j } x _ { i } \\ \\mathrm { ~ f o r ~ e v e r y } \\ i , j \\in I , \\ i > j \\} .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,51,562,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"110","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"110","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"105 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,881,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"111","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is clear that the map $X _ { \\alpha ^ { ( m ) } } V$ factors through the inclusion $M _ { \\alpha ^ { ( m ) } } X V$ , so $X _ { \\alpha ^ { ( m ) } }$ also maps surjectively into $M _ { \\alpha ^ { ( m ) } } X$ , and we have the thesis. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[136,135,876,153]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"111","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finally, the following corollary is an immediate consequence of the above results. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,160,881,252]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"111","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 1 Let $\\mathcal { B }$ be a small category and $S _ { \\bullet } \\colon { \\mathcal { B } } \\to \\mathbf { A l g } _ { \\mathbb { K } }$ a diagram of unitary commutative algebras. Let $\\epsilon \\colon N ( \\mathcal { B } ) _ { \\leq k } \\to \\mathcal { B }$ be the functor defined in Definition 15 for some $k \\_ \\geq \\ 2$ and let $R _ { \\bullet } ~ ~ S _ { \\bullet } ~ \\circ ~ \\epsilon$ be a Reedy cofibrant replacement in $\\mathrm { F u n } ( N ( { \\mathcal { B } } ) _ { \\leq k }$ , $\\mathbf { C D G A } _ { \\mathbb { K } } ^ { \\leq 0 }$ ). Then the DG-Lie algebra $\\mathrm { D e r } _ { \\mathbb { K } } ^ { * } ( R _ { \\bullet } , R _ { \\bullet } )$ controls the deformations of $S _ { \\bullet }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,259,881,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"111","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An example of deformation problem which is naturally encoded by a diagram over a non-Reedy category is the case of deformations of pairs (algebra, idempotent), cf. [16, Example 5.1]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,315,876,351]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"111","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $e \\colon R R$ be an idempotent morphism of an algebra $R \\in \\mathbf { A l g } _ { \\mathbb { K } }$ , then the deformations of the pair $( R , e )$ can be interpreted as the deformations of the diagram ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[428,366,569,397]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0508/412cb5667aaef80639436b46a455bd54d1bc901e121c5b903b143351828e7a63.jpg","list_items":null,"page_idx":"111","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nR _ { \\bullet } \\colon \\textit { \\textbf { R } } ^ { \\sim } \\bigcup \\mathfrak { e }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,415,881,507]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"111","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"over the (non-Reedy) category $\\mathcal { B }$ that has one object $\\bullet$ and two morphisms $\\{ \\mathrm { I d } , \\alpha \\}$ , with $\\alpha ^ { 2 } = \\alpha$ . By Proposition 8, the diagrams $R$ and $\\epsilon ^ { * } R _ { \\bullet } \\in \\mathrm { F u n } ( N ( \\mathcal { B } ) _ { \\leq 2 }$ , $\\mathbf { A l g } _ { \\mathbb { K } } )$ have the same deformation theory. Moreover, since $\\epsilon ^ { * } R _ { \\bullet } \\in \\mathrm { F u n } ^ { \\star } ( N ( \\mathcal { B } ) _ { \\leq 2 } ^ { - } , \\mathbf { A l g _ { \\mathbb { K } } } )$ , it is easy to see that the diagram $\\epsilon ^ { * } R$ has the same deformation theory of the diagram ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[225,525,772,612]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0807/4f2d4fbcea69a6ce67589dd1f7d079212a7edd9f9a1a96e6cf4f2bee8a48dd26.jpg","list_items":null,"page_idx":"111","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[118,635,760,653]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"111","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"of algebras over the following (direct Reedy) subcategory of $N ( { \\mathcal { B } } ) _ { \\leq 2 }$ : ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[164,669,831,762]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0225/78887379985eb22ec2d9948ac4d858029d19c6b94468e8e1e5e59faa08576c95.jpg","list_items":null,"page_idx":"111","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { \\overbrace { \\phantom { \\left( \\frac { \\partial \\theta } { \\partial \\theta } \\right) ^ { 2 } } \\left[ \\bullet \\xrightarrow { \\alpha } \\bullet \\right] \\xrightarrow [ \\delta _ { 0 } ] { \\delta _ { 1 } } \\left[ \\bullet \\xrightarrow [ \\delta ] { \\delta _ { 2 } } \\bullet \\xrightarrow [ \\delta ] { \\alpha } \\bullet \\right] \\delta _ { 0 } ^ { 2 } \\left[ \\delta _ { 1 } = \\delta _ { 2 } \\delta _ { 0 } \\right] , } ^ { \\delta _ { 1 } } \\qquad \\delta _ { 0 } ^ { 2 } = \\delta _ { 1 } \\delta _ { 0 } , } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,785,879,831]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"111","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acknowledgements Both authors thank Francesco Meazzini for useful discussions about the topics of this paper, and the anonymous referee for useful comments and remarks. M.M. is partially supported by Italian MIUR under PRIN project 2017YRA3LK “Moduli and Lie theory”. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[687,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"111","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E. Lepri and M. Manetti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,52,152,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"111","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"106 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,248,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"112","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,121,879,737]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"112","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Bousfield, A., Gugenheim, V.: On PL De Rham Theory and Rational Homotopy Type. Memoirs of the American Mathematical Society, vol. 8, no. 179. American Mathematical Society, Providence (1976) \n2. Bousfield, A.K., Kan, D.M.: Homotopy Limits, Completions and Localizations. Springer Lecture Notes in Mathematics, vol. 304. Springer, New York (1972) \n3. Chachólski, W., Scherer, J.: Homotopy Theory of Diagrams. Memoirs of the American Mathematical Society, no. 736. American Mathematical Society, Providence (2002) \n4. Gelfand, S.I., Manin, Y.I.: Methods of Homological Algebra. Springer Monographs in Mathematics. Springer, Berlin (2003) \n5. Hinich, V.: Deformations of homotopy algebras. Commun. Algebra 32(2), 473–494 (2004) \n6. Hirschhorn, P.S.: Model Categories and their Localizations. Mathematical Surveys and Monographs, vol. 99. American Mathematical Society, Providence (2003) \n7. Hovey, M.: Model Categories. Mathematical Surveys and Monographs, vol. 63. American Mathematical Society, Providence (1999) \n8. Jardine, J.F.: A closed model structure for differential graded algebras. In: Cyclic Cohomology and Noncommutative Geometry (Waterloo, ON, 1995). Fields Institute Communications, vol. 17, pp. 55–58. American Mathematical Society, Providence (1997) \n9. Leinster, T.: Basic Category Theory. Cambridge Studies in Advanced Mathematics, vol. 143. Cambridge University Press, Cambridge (2014) \n10. Lurie, J.: Higher Topos Theory. Annals of Mathematical Studies, vol. 170. Princeton University Press, Princeton (2009) \n11. Lurie, J.: Higher Algebra. Available on J. Lurie’s web page (2014). Preprint \n12. Manetti, M.: Deformation theory via differential graded Lie algebras. In: Seminari di Geometria Algebrica 1998–1999. Scuola Normale Superiore, Pisa (1999). arXiv: math.AG/0507284 \n13. Manetti, M.: Deformation of singularities via differential graded Lie algebras. In: Seminar Notes, Roma (2001). www1.mat.uniroma1.it/people/manetti/dispense/defosing.pdf \n14. Manetti, M.: Differential graded Lie algebras and formal deformation theory. In: Algebraic Geometry: Seattle 2005. Proceedings of Symposia in Pure Mathematics, vol. 80, pp. 785–810 (2009) \n15. Manetti, M., Meazzini, F.: Formal deformation theory in left-proper model categories. New York J. Math. 25, 1259–1311 (2019) \n16. Manetti, M., Meazzini, F.: Deformations of algebraic schemes via Reedy-Palamodov cofibrant resolutions. Indag. Math. (N.S.) (2019). https://doi.org/10.1016/j.indag.2019.08.007 \n17. Palamodov, V.P.: Deformations of complex spaces. Usp. Mat. Nauk 31(3), 129–194 (1976). English translation: Russian Math. Surveys 31, 129–197 (1976) \n18. Quillen, D.: Homotopical Algebra. Lecture Notes in Mathematics, vol. 43. Springer, New York (1967) \n19. Sernesi, E.: Deformations of Algebraic Schemes. Grundlehren der mathematischen Wissenschaften, vol. 334. Springer, New York (2006) \n20. Tate, J.: Homology of Noetherian rings and local rings. Illinois J. Math. 1, 14–27 (1957) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,562,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"112","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On Deformations of Diagrams of Commutative Algebras ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"112","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,694,141]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"113","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,175,537,192]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"113","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"César Lozano Huerta and Alex Massarenti ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,286,881,412]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"113","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract Inspired by the Weak Lefschetz Principle, we study when a smooth projective variety fully determines the birational geometry of some of its subvarieties. In particular, we consider the natural embedding of the space of complete quadrics into the space of complete collineations and we observe that their birational geometry, from the point of view of Mori theory, fully determines each other. When two varieties are related in this way, we call them birational twins. We explore this notion and its various flavors for other embeddings between Mori dream spaces. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,432,822,466]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"113","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords Mori dream spaces $\\cdot$ Cox rings $\\cdot$ Spherical varieties $\\cdot$ Complete collineations and quadrics ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,486,840,521]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"113","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2010 Mathematics Subject Classification Primary 14E30; Secondary 14J45, 14N05, 14E07, 14M27 ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[123,558,309,579]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"113","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,603,881,710]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"113","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The celebrated Lefschetz hyperplane theorem asserts that if there is a smooth complex projective variety $Y$ with the inclusion of a smooth ample hypersurface i $X Y$ , then the ambient variety $Y$ often fully determines many of the topological invariants of $X$ . Such invariants include homology, cohomology (compatible with Hodge structures) and homotopy groups among others. This phenomenon leads to the question: what are some algebro-geometric invariants to which a result of this type applies? In other words, we are asking if the Weak Lefschetz Principle holds beyond topological invariants. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,897,692,941]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"113","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_7 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[804,91,876,139]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"113","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,899,876,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"113","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[116,743,867,879]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"113","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. L. Huerta Universidad Nacional Autónoma de México, Instituto de Matemáticas, Oaxaca, Mexico e-mail: lozano@im.unam.mx A. Massarenti $( \\boxtimes )$ ) Dipartimento di Matematica e Informatica, Università di Ferrara, Ferrara, Italy Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Rio de Janeiro, Brazil e-mail: alex.massarenti@unife.it; alexmassarenti@id.uff.br ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,88,876,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"114","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,126,879,214]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"114","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The purpose of this paper is to investigate the Weak Lefschetz Principle in the context of birational geometry. In other words, if we consider two varieties $X$ , Y as well as an embedding between them $i : X \\to Y$ , then we ask about the birational invariants $X$ which are determined by those of $Y$ . Such invariants may include effective and ample cones, or finer information as we now explain. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,216,881,466]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"114","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let us recall that a normal $\\mathbb { Q }$ -factorial projective variety $X$ with finitely generated Picard group is a Mori dream space if and only if its Cox ring is finitely generated [11, Proposition 2.9]. The birational geometry of these varieties is encoded in this ring and we may ask whether the Weak Lefschetz Principle applies to it. This question has recently been studied in the case of hypersurfaces [10, 12, 20, 22], and for Mori dream spaces of Picard rank two [13]. In addition to considering the Cox ring, let us recall that the pseudo-effective cone ${ \\overline { { \\operatorname { E f f } } } } ( X )$ of a projective variety $X$ with irregularity zero, such as a Mori dream space, can be decomposed into chambers depending on the stable base locus of the corresponding linear series. This decomposition is called stable base locus decomposition and in general it is coarser than the so-called Mori chamber decomposition; which encodes the isomorphism type of the birational models of $X$ . Mori dream spaces and these decompositions have been widely studied by many authors recently, for instance see [1, 7, 17, 18] and the references therein. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,468,879,540]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"114","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We may also ask whether the previous two decompositions satisfy the Weak Lefschetz Principle. We then summarize the previous discussion in the following definition. Let us denote by SBLD $( X )$ the stable base locus decomposition of ${ \\overline { { \\operatorname { E f f } } } } ( X )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,549,881,602]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"114","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 1.1 Let $X , Y$ be $\\mathbb { Q }$ -factorial projective varieties and $i : X \\to Y$ be an embedding. These varieties are said Lefschetz divisorially equivalent if the pull-back $i ^ { * } : \\operatorname { P i c } ( Y ) \\to \\operatorname { P i c } ( X )$ induces an isomorphism such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[173,618,822,639]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0048/dfcee5d0e3152c3baf93195dbf27a5663215d2ba2d002dc66aa66ab67070daf3.jpg","list_items":null,"page_idx":"114","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\ni ^ { * } \\operatorname { E f f } ( Y ) = \\operatorname { E f f } ( X ) , \\quad i ^ { * } \\operatorname { M o v } ( Y ) = \\operatorname { M o v } ( X ) , \\quad i ^ { * } \\operatorname { N e f } ( Y ) = \\operatorname { N e f } ( X )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,656,879,711]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"114","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Furthermore, $X$ and $Y$ are said strongly Lefschetz divisorially equivalent if they are Lefschetz divisorially equivalent and in addition $i ^ { * } \\operatorname { S B L D } ( Y ) \\ : = \\ : \\operatorname { S B L D } ( X )$ , with $h ^ { 1 } ( X , { \\mathcal { O } } _ { X } ) = h ^ { 1 } ( Y , { \\mathcal { O } } _ { Y } ) { = } 0 .$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,719,881,845]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"114","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is an immediate consequence of the Lefschetz hyperplane theorem that a smooth variety $Y$ , of Picard rank one and dimension at least four, is Lefschetz divisorially equivalent to any of its smooth ample divisors. If the assumption that the Picard rank is one is removed in this case, B. Hassett et al. [10] exhibited an example of a variety and an ample divisor $D \\subset Y$ whose Picard groups are isomorphic by the Lefschetz hyperplane theorem, but such that the nef cone is not preserved; hence $D$ and $Y$ are not Lefschetz divisorially equivalent. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,846,876,881]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"114","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The previous definition is not restricted to subvarieties of codimension one. In fact, the results in this paper involve classical spaces and subvarieties of high codimension, such that the birational geometry of both objects can now be linked using Definition 1.1. Let us recall those spaces so we can state our results precisely. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[633,54,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"114","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. L. Huerta and A. Massarenti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,152,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"114","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"110 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,90,876,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"115","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,127,881,484]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"115","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $V$ , W be two $K$ -vector spaces of dimensions $n + 1$ and $m + 1$ , respectively, with $n ~ \\leqslant m$ . The field $K$ is algebraically closed of characteristic zero. We will denote by ${ \\mathcal { X } } ( n , m )$ the space of complete collineations $V W$ and by $\\boldsymbol { \\mathcal { Q } } ( \\boldsymbol { n } )$ the space of complete $( n - 1 )$ -dimensional quadrics of $V$ . In [23, 24], I. Vainsencher showed that these spaces can be understood as sequences of blow-ups along the subvariety parametrizing rank one matrices and the strict transforms of its secant varieties. Such blowups are central characters of this paper, so let us be more precise about them. Given an irreducible and reduced non-degenerate variety $X \\subset \\mathbb { P } ^ { N }$ , and a positive integer $h \\leqslant N$ , the $h$ -secant variety $\\mathbb { S } e c _ { h } ( X )$ of $X$ is the subvariety of $\\mathbb { P } ^ { N }$ obtained as the closure of the union of all $( h - 1 )$ -planes spanned by $h$ general points of $X$ . Spaces of matrices and symmetric matrices admit a natural stratification dictated by the rank and observe that a general point of the $h$ -secant variety of a Segre, or a Veronese, corresponds to a matrix of rank $h$ . More precisely, let $\\mathbb { P } ^ { N }$ be the projective space parametrizing $( n + 1 ) \\times ( n + 1 )$ matrices modulo scalars, $\\mathbb { P } ^ { N _ { + } }$ the subspace of symmetric matrices, $\\mathcal { S } \\subset \\mathbb { P } ^ { N }$ the Segre variety, and $\\mathcal { V } \\subset \\mathbb { P } ^ { N _ { + } }$ the Veronese variety. Set $\\mathcal { X } ( n ) : = \\mathcal { X } ( n , n )$ and denote by $\\mathcal { X } ( n ) _ { i } , \\mathcal { Q } ( n ) _ { i }$ respectively the varieties obtained at the step $i$ in Vainsencher’s constructions 3.4; which roughly says that $\\mathcal { X } ( n ) _ { i }$ , ${ \\mathcal { Q } } ( n ) _ { i }$ are the blow-ups respectively of $\\mathbb P ^ { N }$ and $\\mathbb { P } ^ { N _ { + } }$ along all the $j$ -secant varieties of $s$ and $\\nu$ for $j \\ \\leq \\ i$ . Since $\\mathbb { S } e c _ { h } ( \\mathcal { V } ) = \\mathbb { S } e c _ { h } ( \\mathcal { S } ) \\cap \\mathbb { P } ^ { N _ { + } }$ , the natural inclusion $\\mathbb { P } ^ { N _ { + } } \\hookrightarrow \\mathbb { P } ^ { N }$ lifts to an embedding ${ \\mathcal { Q } } ( n ) _ { i } \\hookrightarrow { \\mathcal { X } } ( n ) _ { i }$ for any $i$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,486,879,557]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"115","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"These families of spaces $\\mathcal { X } ( n ) _ { i }$ and $\\mathcal { Q } ( n ) _ { i }$ are very different in may aspects, for instance their dimensions are different. However, when it comes to certain birational invariants they are indistinguishable. Indeed, by Proposition 3.12 and Lemma 3.18, we have our first result. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,567,879,638]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"115","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem A The varieties $\\mathcal { X } ( n ) _ { i }$ and ${ \\mathcal { Q } } ( n ) _ { i }$ are Lefschetz divisorially equivalent via the usual embedding for any i. In particular, the spaces of complete collineations ${ \\mathcal { X } } ( n )$ and quadrics $\\mathcal { Q } ( n )$ are Lefschetz divisorially equivalent via the usual embedding. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,648,881,773]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"115","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A more thorough version of the Weak Lefschetz Principle in our context would be the following: a small modification of the ambient variety $Y$ restricts to a small modification, possibly an isomorphism, of the subvariety $X$ , and all the small modifications of $X$ can be obtained in this way. This is stronger than determining invariants and it is the content of the following definition. We will test this definition on some Mori dream spaces. The Mori chamber decomposition of Eff $( X )$ , of a Mori dream space $X$ , will be denoted by $\\operatorname { M C D } ( X )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,783,879,855]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"115","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 1.2 Let $X , Y$ be as in the previous definition, we say that they are birational twins if they are Lefschetz divisorially equivalent Mori dream spaces and in addition $i ^ { * } \\mathsf { M C D } ( Y ) \\ : = \\ : \\mathbf { M C D } ( X )$ . Also, they are said to be strong birational twins if they are birational twins and in addition $i ^ { * } \\operatorname { S B L D } ( Y ) = \\operatorname { S B L D } ( X )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,694,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"115","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,874,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"115","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,879,142]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"116","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our next result shows that the relationship between the birational geometry of complete collineations and complete quadrics, in some cases, goes deeper than Theorem A. The following result follows from Theorem 3.20 and Corollary 3.22. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,153,874,187]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"116","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem B The varieties $\\mathcal { X } ( n ) _ { 3 }$ and $\\mathcal { Q } ( n ) _ { 3 }$ are birational twins for any $n \\geqslant 1$ Furthermore, $\\mathcal { X } ( 3 )$ and $\\mathcal { Q } ( 3 )$ are strong birational twins. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,198,881,286]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"116","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In general, the spaces $\\mathcal { X } ( n ) _ { i } , \\mathcal { Q } ( n ) _ { i }$ are Mori dream spaces for any $i$ ; we can actually count the number of chambers in the Mori chamber decomposition in various cases. Furthermore, the ample, movable and effective cones are preserved by Theorem A. However, we do not know if the structure of the Mori chambers is also preserved. In other words, if Theorem B holds in general. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,297,829,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"116","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Question 1.3 Are $\\mathcal { X } ( n ) _ { i }$ and ${ \\mathcal { Q } } ( n ) _ { i }$ birational twins for $n \\geqslant 4$ and any $i \\geq 4 2$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,324,881,484]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"116","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We organize this paper as follows. In Sect. 2, we discuss Definitions 1.1 and 1.2 and the relations among the various flavors of these definitions. In particular, Theorem 2.6 provides examples of Mori dream spaces that are strongly Lefschetz divisorially equivalent but not birational twins, and of Mori dream spaces that are birational twins but not strong birational twins. In Sect. 3, we introduce the spaces of complete collineations and quadrics, and we study their birational geometry from a Mori theoretic viewpoint. Finally, in Sect. 4 we recall the modular description of the unique flip of $\\mathcal { Q } ( 3 )$ given in [14, Section 5.2], and based on it, we give a conjectural description of the unique flip of $\\mathcal { X } ( 3 )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,530,658,572]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"116","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 Lefschetz Divisorially Equivalent Varieties and Birational Twins ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,596,879,650]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"116","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This section contains preliminaries and explores various aspects of Definitions 1.1 and 1.2. In particular, Theorem 2.6 and Proposition 2.9 are the main results of this section. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,651,881,794]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"116","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $X$ be a normal projective variety over an algebraically closed field of characteristic zero. We denote by $N ^ { 1 } ( X )$ the real vector space of $\\mathbb { R }$ -Cartier divisors modulo numerical equivalence. The nef cone of $X$ is the closed convex cone $\\mathsf { N e f } ( X ) \\subset N ^ { 1 } ( X )$ generated by classes of nef divisors. The movable cone of $X$ is the convex cone $\\mathsf { M o v } ( X ) \\subset N ^ { 1 } ( X )$ generated by classes of movable divisors; these are Cartier divisors whose stable base locus has codimension at least two in $X$ . The effective cone of $X$ is the convex cone $\\operatorname { E f f } ( X ) \\subset N ^ { 1 } ( X )$ generated by classes of effective divisors. We have inclusions $\\operatorname { N e f } ( X ) \\subset { \\overline { { \\operatorname { M o v } ( X ) } } } \\subset { \\overline { { \\operatorname { E f f } ( X ) } } }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,795,881,848]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"116","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We will denote by $N _ { 1 } ( X )$ the real vector space of numerical equivalence classes of 1-cycles on $X$ . The closure of the cone in $N _ { 1 } ( X )$ generated by the classes of irreducible curves in $X$ is called the Mori cone of $X$ , we will denote it by $\\operatorname { N E } ( X )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,849,879,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"116","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A class $[ C ] \\in N _ { 1 } ( X )$ is called moving if the curves in $X$ of class $[ C ]$ cover a dense open subset of $X$ . The closure of the cone in $N _ { 1 } ( X )$ generated by classes of ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[633,54,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"116","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. L. Huerta and A. Massarenti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,54,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"116","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"112 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,88,879,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"117","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"moving curves in $X$ is called the moving cone of $X$ and we will denote it by mov $( X )$ . \nWe refer to [9, Chapter 1] for a comprehensive treatment of these topics. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,126,881,252]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"117","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We say that a birational map $f : X \\to X ^ { \\prime }$ to a normal projective variety $X ^ { \\prime }$ is a birational contraction if its inverse does not contract any divisor. We say that it is a small $\\mathbb { Q }$ -factorial modification if $X ^ { \\prime }$ is $\\mathbb { Q }$ -factorial and $f$ is an isomorphism in codimension one. If $f \\ : \\ X \\quad - \\quad X ^ { \\prime }$ is a small $\\mathbb { Q }$ -factorial modification, then the natural pull-back map $f ^ { * } ~ : ~ N ^ { 1 } ( X ^ { \\prime } ) ~ ~ N ^ { 1 } ( X )$ sends $\\operatorname { M o v } ( X ^ { \\prime } )$ and $\\operatorname { E f f } ( X ^ { \\prime } )$ isomorphically onto $\\operatorname { M o v } ( X )$ and $\\operatorname { E f f } ( X )$ , respectively. In particular, we have $f ^ { * } ( \\operatorname { N e f } ( { \\bar { X ^ { \\prime } } } ) ) \\subset { \\bar { \\operatorname { M o v } } } ( X )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,259,879,297]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"117","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 2.1 A normal projective $\\mathbb { Q }$ -factorial variety $X$ is called a Mori dream space if the following conditions hold: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,304,881,396]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"117","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"– Pic# $( X )$ is finitely generated, or equivalently $h ^ { 1 } ( X , { \\mathcal { O } } _ { X } ) = 0$ , $- \\mathrm { ~ \\ N e f ~ } ( X )$ is generated by the classes of finitely many semi-ample divisors, there is a finite collection of small $\\mathbb { Q }$ -factorial modifications $f _ { i } : X \\to X _ { i }$ , such that each $X _ { i }$ satisfies the second condition above, and Mov $\\begin{array} { r l } { ( X ) } & { { } = } \\end{array}$ $\\textstyle \\bigcup _ { i } \\ f _ { i } ^ { * } ( { \\mathrm { N e f } } ( X _ { i } ) )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,405,881,493]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"117","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By [3, Corollary 1.3.2] smooth Fano varieties are Mori dream spaces. In fact, there is a larger class of varieties called log Fano varieties which are Mori dream spaces as well. By work of M. Brion [5] we have that $\\mathbb { Q }$ -factorial spherical varieties are Mori dream spaces. An alternative proof of this result can be found in [21, Section 4]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,495,881,603]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"117","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The collection of all faces of all cones $f _ { i } ^ { * } ( \\mathrm { N e f } \\left( X _ { i } \\right) )$ in Definition 2.1 forms a fan which is supported on $\\operatorname { M o v } ( X )$ . If two maximal cones of this fan, say $f _ { i } ^ { * } ( \\mathrm { N e f } \\left( X _ { i } \\right) )$ and $f _ { j } ^ { * } ( \\mathrm { N e f } ( X _ { j } ) )$ , meet along a facet, then there exist a normal projective variety $Y$ , a small modification $\\varphi : X _ { i } \\ \\to \\ Y _ { j }$ , and $h _ { i } : X _ { i } \\to Y$ and $h _ { j } : X _ { j } \\to Y$ small birational morphism of relative Picard number one such that $h _ { j } \\circ \\varphi = h _ { i }$ . The fan structure on $\\operatorname { M o v } ( X )$ can be extended to a fan supported on $\\operatorname { E f f } ( X )$ as follows. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,611,881,755]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"117","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 2.2 Let $X$ be a Mori dream space. We describe a fan structure on the effective cone $\\operatorname { E f f } ( X )$ , called the Mori chamber decomposition. We refer to [11, Proposition 1.11] and [19, Section 2.2] for details. There are finitely many birational\\$ \u0003 \u0004 % contractions from $X$ to Mori dream spaces, denoted by $g _ { i } \\ : \\ X \\to \\ Y _ { i }$ . The set $\\operatorname { E x c } ( g _ { i } )$ of exceptional prime divisors of $g _ { i }$ has cardinality $\\rho ( X / Y _ { i } ) = \\rho ( X ) -$ $\\rho ( Y _ { i } )$ . The maximal cones $\\mathcal { C }$ of the Mori chamber decomposition of $\\operatorname { E f f } ( X )$ are of the form: $\\begin{array} { r l } { \\mathcal { C } _ { i } } & { = \\Big \\langle g _ { i } ^ { * } \\big ( \\mathrm { N e f } ( Y _ { i } ) \\big ) , \\mathrm { E x c } ( g _ { i } ) \\Big \\rangle } \\end{array}$ . We call $\\mathcal { C } _ { i }$ or its interior $\\boldsymbol { \\mathcal { C } } _ { i } ^ { \\circ }$ a maximal chamber of $\\operatorname { E f f } ( X )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,764,881,819]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"117","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 2.3 Let $X$ be a normal projective variety with finitely generated divisor class group $\\mathrm { C l } ( X ) : = \\mathrm { W D i v } ( X ) / \\mathrm { P D i v } ( X )$ , in particular $h ^ { 1 } ( X , { \\mathcal { O } } _ { X } ) = 0$ The $C o x$ sheaf and Cox ring of $X$ are defined as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[248,833,749,878]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0798/561115f984aaf982d26fe361a71e991727dcf985a59c5484724f14e8b5197554.jpg","list_items":null,"page_idx":"117","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n{ \\mathcal { R } } : = \\bigoplus _ { [ D ] \\in \\operatorname { C l } ( X ) } { \\mathcal { O } } _ { X } ( D ) \\qquad \\operatorname { C o x } ( X ) : = \\Gamma ( X , { \\mathcal { R } } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,52,694,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"117","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"117","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"113 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,234]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"118","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recall that $\\mathcal { R }$ is a sheaf of $\\operatorname { C l } ( X )$ -graded ${ \\mathcal { O } } _ { X }$ -algebras, whose multiplication maps are discussed in [2, Section 1.4]. In case the divisor class group is torsionfree, one can just take the direct sum over a subgroup of $\\mathrm { W D i v } ( X )$ , isomorphic to $\\operatorname { C l } ( X )$ via the quotient map, getting immediately a sheaf of ${ \\mathcal { O } } _ { X }$ -algebras. Denote by $\\widehat { X }$ the relative spectrum of $\\mathcal { R }$ and by $\\overline { { X } }$ the spectrum of $\\operatorname { C o x } ( X )$ . The $\\operatorname { C l } ( X )$ - grading induces an action of the quasitorus $H _ { X } : = \\operatorname { S p e c } \\mathbb { C } [ \\operatorname { C l } ( X ) ]$ on both spaces. The inclusion $\\mathcal { O } _ { X } \\mathcal { R }$ induces a good quotient $p _ { X } \\colon { \\widehat { X } } \\to X$ with respect to this action. Summarizing, we have the following diagram ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[448,247,548,321]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0180/d08fd5d27f24c3b014cdebeae8dd36ac6afc3495c5ff101038110b7bd951d57a.jpg","list_items":null,"page_idx":"118","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n{ \\underset { X } { \\widehat { X } } } \\subseteq { \\overline { { X } } }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,340,881,414]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"118","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and we will refer to it as the Cox construction of $X$ . In case $\\operatorname { C o x } ( X )$ is a finitely generated algebra, the complement of $\\widehat { X }$ in the affine variety $\\overline { { X } }$ has codimension $\\geqslant 2$ . This subvariety is the irrelevant locus and its defining ideal is the irrelevant ideal $\\mathcal { I } _ { \\mathrm { i r r } } ( X ) \\subseteq \\mathbf { C o x } ( X )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,414,881,486]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"118","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $X$ be a normal $\\mathbb { Q }$ -factorial projective variety, and let $D$ be an effective $\\mathbb { Q }$ - divisor on $X$ . The stable base locus $\\mathbf { B } ( D )$ of $D$ is the set-theoretic intersection of the base loci of the complete linear systems $| s D |$ for all positive integers $s$ such that $s D$ is integral. In other words, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[405,501,589,542]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0976/f94fd44579a0426f2054ac1c966a7ebfecad80fe9d65401c0a01e96343769600.jpg","list_items":null,"page_idx":"118","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf { B } ( D ) = \\bigcap _ { s > 0 } B ( s D )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,555,881,609]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"118","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since stable base loci do not behave well with respect to numerical equivalence, we will assume that $h ^ { 1 } ( X , { \\mathcal { O } } _ { X } ) = 0$ . This implies that linear and numerical equivalence of $\\mathbb { Q }$ -divisors coincide. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,611,881,681]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"118","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since numerically equivalent $\\mathbb { Q }$ -divisors on $X$ have the same stable base locus, then ${ \\overline { { \\operatorname { E f f } } } } ( X )$ the pseudo-effective cone of $X$ can be decomposed into chambers depending on the stable base locus of the corresponding linear series. This decomposition is called stable base locus decomposition. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,683,881,737]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"118","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $X$ is a Mori dream space, satisfying then the condition $h ^ { 1 } ( X , { \\mathcal { O } } _ { X } ) = 0$ , determining the stable base locus decomposition of $\\operatorname { E f f } ( X )$ is a first step in order to compute its Mori chamber decomposition. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,744,881,782]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"118","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 2.4 Recall that two divisors $D _ { 1 } , D _ { 2 }$ are said to be Mori equivalent if $\\mathbf { B } ( D _ { 1 } ) = \\mathbf { B } ( D _ { 2 } )$ and the following diagram of rational maps is commutative ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[364,798,635,869]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0579/0a89dc884b4dce3b83c7b11c8b4ab4d5d76749b8a41f89bc32b4c04e772940b8.jpg","list_items":null,"page_idx":"118","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { c } { { \\phi _ { D _ { 1 } } \\_ { \\stackrel { \\displaystyle \\bigwedge } \\gamma } , \\ ~ \\stackrel { \\displaystyle } { \\bigwedge } \\ _ { { { { \\mathrm { \\Large ~ \\ddots ~ } } \\atop { { \\mathrm { \\Large ~ \\ddots ~ } } \\atop { { \\mathrm { \\Large ~ \\ddots ~ } } \\atop { { \\mathrm { \\Large ~ \\ddots ~ } } \\atop { { \\mathrm { \\Large ~ \\ddots ~ } } \\atop { { \\mathrm { \\Large ~ \\ddots ~ } } \\atop { { \\mathrm { \\scriptsize ~ \\ddots ~ } } \\atop { { \\mathrm { \\scriptsize ~ \\ddots ~ } } \\scriptstyle { { \\mathrm { \\scriptsize ~ \\ � } } } \\left( D _ { 2 } \\right) } } } } } } } } } } } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[633,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"118","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. L. Huerta and A. Massarenti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"118","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,88,879,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"119","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where the horizontal arrow is an isomorphism. Therefore, the Mori chamber decomposition is a refinement of the stable base locus decomposition. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,133,876,171]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"119","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $X$ be a Mori dream space with Cox ring $\\operatorname { C o x } ( X )$ and grading matrix $Q$ . The matrix $Q$ defines a surjection ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[423,187,576,207]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0836/743d7d472d5fc4df516f9b22d11de9b2a65858daae45c1a0c3ee39ee68bca007.jpg","list_items":null,"page_idx":"119","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nQ \\colon E \\to \\mathbf { C l } ( X )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,223,881,403]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"119","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"from a free, finitely generated, abelian group $E$ to the divisor class group of $X$ . Denote by $\\gamma$ the positive quadrant of $E _ { \\mathbb { Q } } : = E \\otimes _ { \\mathbb { Z } } \\mathbb { Q }$ . Let $e _ { 1 } , \\ldots , e _ { r }$ be the canonical basis of $E _ { \\mathbb { Q } }$ . Given a face $\\gamma _ { 0 } \\preceq \\gamma$ , we say that $i \\in \\{ 1 , \\ldots , r \\}$ is a cone index of $\\gamma _ { 0 }$ if $e _ { i } \\in \\gamma _ { 0 }$ . The face $\\gamma _ { 0 }$ is an $\\mathfrak { F }$ -face if there exists a point of ${ \\overline { { X } } } = \\operatorname { S p e c } ( \\operatorname { C o x } ( X ) )$ whose $i$ -th coordinate is non-zero exactly when $i$ is a cone index of $\\gamma _ { 0 }$ [2, Construction 3.3.1.1]. The set of these points is denoted by $\\overline { { X } } ( \\gamma _ { 0 } )$ . Given the Cox construction of $X$ we denote by $X ( \\gamma _ { 0 } ) \\subseteq X$ the image of $\\overline { { X } } ( \\gamma _ { 0 } )$ , and given an $\\mathfrak { F }$ -face $\\gamma _ { 0 }$ its image $Q ( \\gamma _ { 0 } ) \\subseteq \\mathbf { C l } ( X ) _ { \\mathbb { Q } }$ is an orbit cone of $X$ . The set of all orbit cones of& $X$ is denoted by $\\Omega$ . Accordingly to [2, Definition 3.1.2.6] an effective class $w \\in \\operatorname { C l } ( X )$ defines the GIT chamber ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[396,417,601,460]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0187/a449d6d9146d9be7f65dd9ab9d76ae7e5898f59df4fb5b09a8ae252441241ba9.jpg","list_items":null,"page_idx":"119","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\lambda ( w ) : = \\bigcap _ { \\{ \\omega \\in \\Omega : w \\in \\omega \\} } \\omega\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,474,881,545]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"119","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $w$ is an ample class of $X$ , then the corresponding GIT chamber is the semi-ample cone of $X$ . The variety $X$ can be reconstructed from the pair $( \\operatorname { C o x } ( X ) , \\Phi )$ formed by the Cox ring together with a bunch of cones, consisting of certain subsets of the orbit cones [2, Definition 3.1.3.2]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,548,881,618]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"119","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We now explore different aspects of Definitions 1.1 and 1.2. We exhibit two varieties which are birational twins but fail to be strong birational twins. We do this inspired by [13, Theorem 1.1.]. We also exhibit two Mori dream spaces which are strongly Lefschetz divisorially equivalent but fail to be birational twins. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,627,879,663]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"119","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2.6 Let $Z$ be the toric variety with Cox ring $K [ T _ { 1 } , \\dots , T _ { 1 1 } ]$ whose grading matrix and irrelevant ideal are the following ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[177,678,820,722]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0217/92f05396ad6a75b1cc07ef615de96342d6636f4ad754e27010c8e0749ecd5a88.jpg","list_items":null,"page_idx":"119","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nQ = \\left[ \\begin{array} { c c } { 1 \\mathrm { ~ 1 ~ 2 ~ 2 ~ 2 ~ 2 ~ 1 ~ 0 ~ 0 ~ 0 ~ 0 ~ } } \\\\ { 0 \\mathrm { ~ 0 ~ 1 ~ 1 ~ 1 ~ 2 ~ 1 ~ 1 ~ 1 ~ 1 ~ } } \\end{array} \\right] \\qquad \\mathcal { J } _ { \\mathrm { i r r } } ( Z ) = \\langle T _ { 1 } , T _ { 2 } \\rangle \\cap \\langle T _ { 3 } , \\ldots , T _ { 1 1 } \\rangle\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,735,881,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"119","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and let $F _ { \\mathrm { ~ } } = T _ { 3 } T _ { 8 } + T _ { 4 } T _ { 9 } + T _ { 1 } T _ { 7 } , G _ { \\mathrm { ~ } } = T _ { 5 } T _ { 9 } + T _ { 6 } T _ { 1 1 } + T _ { 2 } T _ { 7 }$ . Then the ring K[T1,...,T11](F,G) is the Cox ring of a projective normal Q-factorial Mori dream space $X \\subset Z$ of Picard rank two. The varieties $X$ and $Z$ are birational twins but fail to be strong birational twins. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,806,881,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"119","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Furthermore, the ring $\\frac { K [ T _ { 1 } , . . . , T _ { 1 1 } ] } { ( F , G , T _ { 7 } + T _ { 1 } T _ { 1 1 } ^ { 2 } + T _ { 2 } T _ { 8 } T _ { 1 1 } + T _ { 2 } T _ { 1 0 } ^ { 2 } + T _ { 2 } T _ { 8 } T _ { 9 } ) }$ is the Cox ring of a projective normal $\\mathbb { Q }$ -factorial Mori dream space $Y ~ \\subset ~ X$ of Picard rank two. The varieties $Y$ and $X$ are strongly Lefschetz divisorially equivalent but fail to be birational twins. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,51,697,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"119","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"119","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"115 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,879,147]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"120","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Let us denote by $w _ { i } \\in \\mathbb { Z } ^ { 2 }$ the degree of $T _ { i }$ . We will denote by $V ( T _ { i _ { 1 } } , \\dots , T _ { i _ { r } } ) : = \\{ T _ { i _ { 1 } } = \\dots = T _ { i _ { r } } = 0 \\}$ . The following picture represents the degrees of the generators of the Cox ring ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[312,166,683,286]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0649/8bcbdabe131d1e8122aab3314e3bb2f0ee59012581777e5ff9b54aae9ea3369c.jpg","list_items":null,"page_idx":"120","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[118,307,881,363]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"120","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\operatorname { E f f } ( Z ) = \\operatorname { M o v } ( Z ) = \\lambda _ { A } \\cup \\lambda \\cup \\lambda ^ { \\prime }$ and $\\mathbf { N e f } ( Z ) = \\lambda _ { A }$ . Let ${ \\overline { { Z } } } = K ^ { 1 1 }$ be the spectrum of the Cox ring of $Z$ and let $\\overline { { X } }$ be the affine subvariety defined by $\\overline { { X } } = \\overset { \\cdot } { \\{ F = G = 0 \\} } \\subset \\overline { { Z } }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,363,881,456]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"120","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A standard computation shows that $\\overline { { X } }$ is irreducible and $\\mathrm { c o d i m } _ { \\overline { { X } } } ( \\mathrm { S i n g } ( \\overline { { X } } ) ) \\geqslant 2$ , and then Serre’s criterion on normality yields that $\\overline { { X } }$ is normal. Let $p _ { Z } \\colon { \\widehat { Z } } \\to Z$ be the characteristic space morphism of $Z$ , and let ${ \\widehat { X } } : = { \\overline { { X } } } \\cap { \\widehat { Z } }$ . Note that the irrelevant locus with respect to the chamber $\\lambda _ { A }$ has two components given by $\\Gamma _ { 1 } = \\{ T _ { 1 } =$ $T _ { 2 } = 0 \\}$ and $\\Gamma _ { 2 } = \\{ T _ { 3 } = \\cdot \\cdot \\cdot = T _ { 1 1 } = 0 \\}$ . Hence $\\widehat { Z } = K ^ { 1 1 } \\setminus \\overline { { \\{ \\Gamma _ { 1 } \\cup \\bar { \\Gamma _ { 2 } } \\} } }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,456,879,509]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"120","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The image of $\\widehat { X }$ via $p _ { Z }$ is a subvariety $X$ of $Z$ , let $i : X \\to Z$ be the inclusion. Since $\\overline { { X } }$ is irreducible and normal, and $X$ is a GIT quotient of $\\overline { { X } }$ by a reductive group [6, Theorem 1.24 (vi)] yields that $X$ is irreducible and normal as well. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,510,879,584]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"120","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $Z ^ { \\prime }$ be the smooth locus of $Z$ . Note that $Z ^ { \\prime }$ contains the open subset $Z ^ { \\prime \\prime }$ of $Z$ obtained by removing the union of all the toric subvarieties of the form $p z ( V ( T _ { i } , T _ { j } ) )$ . Since the Zariski closure of $p _ { Z } ^ { - 1 } ( X \\cap Z ^ { \\prime \\prime } )$ in $\\widehat { Z }$ is $\\widehat { X }$ we conclude that $\\widehat { X }$ is the Zariski closure of $p _ { Z } ^ { - 1 } ( X \\cap Z ^ { \\prime } )$ in $\\widehat { Z }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[113,585,876,623]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"120","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note that, according to the grading matrix $Q$ , the $K ^ { * } \\times K ^ { * }$ action on $\\overline { { X } }$ is given by ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,636,847,680]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0810/703f8ccb4d358800d43439bac6b635b57fc31006625183e2df148fb6a441780e.jpg","list_items":null,"page_idx":"120","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { c c } { { ( K ^ { * } \\times K ^ { * } ) \\times \\overline { { { X } } } } } & { { \\longrightarrow } } \\\\ { { ( ( \\lambda , \\mu ) , ( T _ { 1 } , \\dots , T _ { 1 1 } ) ) \\longmapsto ( \\lambda T _ { 1 } , \\lambda T _ { 2 } , \\lambda ^ { 2 } \\mu T _ { 3 } , \\dots , \\lambda \\mu ^ { 2 } T _ { 7 } , \\mu T _ { 8 } , \\dots , \\mu T _ { 1 1 } ) } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,695,881,894]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"120","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Therefore, if $T _ { 2 } ~ \\ne ~ 0 , T _ { 8 } ~ \\ne ~ 0$ we may set $\\lambda ~ = ~ \\mu ~ = ~ 1$ , and write $\\begin{array} { r l } { T _ { 3 } } & { { } = } \\end{array}$ \u0012=T4T9+T1T7 , T $\\begin{array} { r } { T _ { 7 } ~ { \\stackrel { - } { = } } ~ - \\frac { T _ { 5 } T _ { 9 } + T _ { 6 } T _ { 1 1 } } { T _ { 2 } } } \\end{array}$ − T5T9+T6T11 . Therefore, if we remove the images of V (T2) ∪ $V ( T _ { 8 } )$ 8 from $X$ 2 , the resulting variety is isomorphic to an affine space. Therefore, $\\operatorname { C l } ( X )$ is generated by the classes of the images of the two irreducible divisors $V ( T _ { i } ) \\cap \\overline { { X } }$ , with $i \\in \\{ 2 , 8 \\}$ , and $\\rho ( X ) \\quad \\leqslant \\quad 2$ . Moreover, crossing the wall corresponding to $w _ { 1 } , w _ { 2 }$ we get a morphism $f : Z \\to \\mathbb { P } ^ { 1 }$ . Furthermore, $X \\subset Z$ is not contained in any fiber of $f$ , and hence $f$ restricts to a surjective morphism $f _ { | X } : X \\to \\mathbb { P } ^ { 1 }$ . This forces $\\rho ( X ) \\ \\geqslant \\ 2$ . Finally, we conclude that the images of $V ( T _ { 2 } ) , V ( T _ { 8 } )$ form a basis of $\\operatorname { C l } ( X )$ . So $\\rho ( X ) = 2$ and the pull-back $i ^ { * } : \\mathbf { C l } ( Z ) \\to$ $\\operatorname { C l } ( X )$ , induced by the inclusion, is an isomorphism. Now, note that $\\Gamma _ { 2 } \\subset { \\overline { { X } } }$ has codimension greater that two in $\\overline { { X } }$ , and $\\Gamma _ { 1 } \\cap { \\overline { { X } } } = \\{ T _ { 1 } = T _ { 2 } = T _ { 3 } T _ { 8 } + T _ { 4 } T _ { 9 } =$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[633,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"120","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. L. Huerta and A. Massarenti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,152,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"120","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"116 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,876,129]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"121","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$T _ { 5 } T _ { 9 } + T _ { 6 } T _ { 1 1 } = 0 \\}$ has codimension two in $\\overline { { X } }$ . Hence codi $\\operatorname { r } _ { \\overline { { X } } } ( { \\overline { { X } } } \\setminus { \\widehat { X } } ) \\geqslant 2$ and [2, Corollary 4.1.1.5] yields that the Cox ring of $X$ is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[371,142,628,184]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0070/4a3d9c9156c39e25ebc454a4d31fb0f75f8605bd0dc4242e6f5602b808974357.jpg","list_items":null,"page_idx":"121","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname { C o x } ( X ) = { \\frac { K [ T _ { 1 } , \\dots , T _ { 1 1 } ] } { I ( { \\overline { { X } } } ) } }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,196,881,270]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"121","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In general, if $X$ is a Mori dream space such that $\\operatorname { C l } ( X )$ has rank two we can fix a total order on the classes in the effective cone $w \\leqslant w ^ { \\prime }$ if $w$ is on the left of $w ^ { \\prime }$ . Given two convex cones $\\lambda , \\lambda ^ { \\prime }$ contained in the effective cone we will write- - $\\lambda \\leqslant \\lambda ^ { \\prime }$ if $w \\leqslant w ^ { \\prime }$ for any $w \\in \\lambda$ and $w ^ { \\prime } \\in \\lambda$ . Note that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[284,285,712,307]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0793/045e62935d08cc53fcbee6ca860aec11434092bb954919b0d3407400b1ed4dc6.jpg","list_items":null,"page_idx":"121","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nV ( f _ { i } : w _ { i } \\geqslant \\lambda _ { A } ) = \\{ T _ { 1 } = T _ { 2 } = \\widetilde { F } = \\widetilde { G } = 0 \\}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,327,660,348]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"121","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\widetilde { F } = T _ { 3 } T _ { 8 } + T _ { 4 } T _ { 9 }$ , $\\widetilde { G } = T _ { 5 } T _ { 9 } + T _ { 6 } T _ { 1 1 }$ . Furthermore ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,364,874,408]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0273/8817a17b7160d9c7377aabe65bb9cee2854ee8258e0b2046ad6cb9f0a6c81702.jpg","list_items":null,"page_idx":"121","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { V ( f _ { i } : w _ { i } \\leqslant \\lambda ) = \\{ T _ { 7 } = T _ { 8 } = T _ { 9 } = T _ { 1 0 } = T _ { 1 1 } = 0 \\} ; } \\\\ & { V ( f _ { i } : w _ { i } \\leqslant \\lambda ^ { \\prime } ) = \\{ T _ { 1 } = T _ { 2 } = T _ { 8 } = T _ { 9 } = T _ { 1 0 } = T _ { 1 1 } = 0 \\} \\cup \\{ T _ { 7 } = T _ { 8 } = T _ { 9 } = T _ { 1 0 } = T _ { 1 1 } = 0 \\} . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,421,881,495]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"121","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"So $V ( f _ { i } ~ : ~ w _ { i } ~ \\geqslant ~ \\lambda _ { A } ) ~ \\supset ~ \\{ T _ { 1 } ~ = ~ T _ { 2 } ~ = ~ T _ { 8 } ~ = ~ T _ { 9 } ~ = { \\cal T } _ { 1 0 } ~ = ~ T _ { 1 1 } ~ = ~ 0 \\}$ contains a component of the set $V ( f _ { i } : w _ { i } \\leqslant \\lambda ^ { \\prime } )$ . Finally, [13, Lemmas 4.3, 4.4, 4.5] yields that $\\mathsf { M C D } ( X ) \\ = \\ i ^ { * } \\mathbf { M C D } ( Z )$ , and that $\\operatorname { S B L D } ( X )$ has just two chambers while $\\mathrm { S B L D } ( Z ) = \\mathbf { M C D } ( Z )$ has three chambers. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,495,576,513]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"121","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let ${ \\overline { { Y } } } \\subset { \\overline { { Z } } } = K ^ { 1 1 }$ be the subvariety cut out by ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[205,528,792,551]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0935/8edfc9a44eda826f29d2410d875be0f9c24b278345517be0c7c58da59962be93.jpg","list_items":null,"page_idx":"121","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n{ \\overline { { Y } } } = \\{ F = G = T _ { 7 } + T _ { 1 } T _ { 1 1 } ^ { 2 } + T _ { 2 } T _ { 8 } T _ { 1 1 } + T _ { 2 } T _ { 1 0 } ^ { 2 } + T _ { 2 } T _ { 8 } T _ { 9 } = 0 \\}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,564,881,621]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"121","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We have that $\\overline { { Y } }$ is irreducible of dimension eight and $\\mathrm { c o d i m } _ { \\overline { { { Y } } } } ( \\mathrm { S i n g } ( { \\overline { { { Y } } } } ) ) ~ \\geqslant ~ 2$ , so Serre’s criterion on normality yields that $\\overline { { Y } }$ is normal, and [6, Theorem 1.24 (vi)] yields that the GIT quotient $Y$ is irreducible and normal as well. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,623,881,831]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"121","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $T _ { 2 } \\neq 0 , T _ { 8 } \\neq 0$ we may write $\\begin{array} { r } { T _ { 3 } ~ = ~ - \\frac { T _ { 4 } T _ { 9 } + T _ { 1 } T _ { 7 } } { T _ { 8 } } } \\end{array}$ T4T9+T1T7 , T $\\begin{array} { r } { T _ { 7 } ~ = ~ - \\frac { T _ { 5 } T _ { 9 } + T _ { 6 } T _ { 1 1 } } { T _ { 2 } } } \\end{array}$ and − T2T 210+T2T8T11+T1T 211+T7 . Therefore, Y \\ (V (T2) ∪ V (T8)) is isomorphic to an affine space and $\\operatorname { \\mathbb { C } l } ( Y )$ is generated by the classes of the images of the two irreducible divisors $V ( T _ { i } ) \\cap \\overline { { Y } }$ , with $i \\in \\{ 2 , 8 \\}$ , and $\\rho ( Y ) \\ \\leqslant \\ 2$ . Again crossing the wall corresponding to $w _ { 1 }$ , $w _ { 2 }$ we get a morphism $f : Z \\to \\mathbb { P } ^ { 1 }$ and since $Y \\subset Z$ is not contained in any fiber of $f$ this morphism restricts to a surjective morphism $f _ { | Y } : Y \\to \\mathbb { P } ^ { 1 }$ . This forces $\\rho ( Y ) \\geqslant 2$ . We conclude that the images of $V ( T _ { 2 } ) , V ( T _ { 8 } )$ form a basis of $\\operatorname { C l } ( Y )$ . So $\\rho ( Y ) = 2$ and the pull-back $j ^ { * } : \\operatorname { C l } ( X ) \\to \\operatorname { C l } ( Y )$ , induced by the inclusion $j : Y X$ , is an isomorphism. As before $\\Gamma _ { 2 } \\subset \\overline { { Y } }$ has codimension greater that two in $\\overline { { Y } }$ , and $\\Gamma _ { 1 } \\cap \\overline { { Y } }$ has codimension two in $\\overline { { Y } }$ . Hence [2, Corollary 4.1.1.5] yields that the Cox ring of $Y$ is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[371,845,626,887]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0068/0c8f52042474e60817ce67d007517fa902f491a5393b838c4ea372edbe5dee27.jpg","list_items":null,"page_idx":"121","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf { C o x } ( Y ) = \\frac { K [ T _ { 1 } , \\hdots , T _ { 1 1 } ] } { I ( \\overline { { Y } } ) }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,51,697,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"121","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"121","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"117 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,507,108]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"122","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now, for the Mori dream space $Y$ we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,124,874,166]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0043/df8f48cb0e01dc454535fb538d1584d2444f3b76af7da34ab6c5ad166d4c2197.jpg","list_items":null,"page_idx":"122","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { V ( f _ { i } : w _ { i } \\leqslant \\lambda ) = V ( f _ { i } : w _ { i } \\leqslant \\lambda ^ { \\prime } ) = \\{ T _ { 7 } = T _ { 8 } = T _ { 9 } = T _ { 1 0 } = T _ { 1 1 } = 0 \\} ; } \\\\ & { V ( f _ { i } : w _ { i } \\geqslant \\lambda ) = V ( f _ { i } : w _ { i } \\geqslant \\lambda ^ { \\prime } ) = \\{ T _ { 1 } = T _ { 2 } = T _ { 3 } = T _ { 4 } = T _ { 5 } = T _ { 6 } = T _ { 7 } = 0 \\} . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,180,876,216]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"122","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hence [13, Lemma 4.3] yields that $\\operatorname { M C D } ( Y )$ has two chambers while $\\mathbf { M C D } ( X ) =$ $i ^ { * } \\mathrm { M C D } ( Z )$ has three chambers. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,225,881,315]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"122","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 2.7 Note that any pair of varieties among the Mori dream spaces $Y \\subset X \\subset$ $Z$ in Theorem 2.6 gives a pair of Lefschetz divisorially equivalent varieties. On the other hand, $i ^ { * } \\operatorname { S B L D } ( Z ) \\neq \\operatorname { S B L D } ( X )$ and $( i \\circ j ) ^ { * } \\mathsf { M C D } ( Z ) \\ne \\mathsf { M C D } ( Y )$ . Therefore, Lefschetz divisorially equivalent does not imply strongly Lefschetz divisorially equivalent nor birational twins. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,325,881,360]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"122","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"However, when a Mori dream space inherits the stable base locus decomposition from an ambient toric variety something more can be said. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,369,881,496]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"122","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 2.8 Given a Mori dream space $X$ , then there is an embedding $i : X \\to \\mathcal { T } _ { X }$ into a simplicial projective toric variety $\\mathcal { T } _ { X }$ such that $i ^ { * } : \\operatorname { P i c } ( \\mathcal T _ { X } ) \\to \\operatorname { P i c } ( X )$ is an isomorphism which induces an isomorphism $\\begin{array} { l l l } { \\operatorname { E f f } ( { \\mathcal { T } } _ { X } ) } & { \\to } & { \\operatorname { E f f } ( X ) } \\end{array}$ , [11, Proposition 2.11]. Furthermore, the Mori chamber decomposition of $\\operatorname { E f f } ( { \\mathcal { T } } _ { X } )$ is a refinement of the Mori chamber decomposition of $\\operatorname { E f f } ( X )$ . Indeed, if $\\operatorname { C o x } ( X ) \\ \\cong$ $\\textstyle \\frac { K [ T _ { 1 } , \\ldots , T _ { s } ] } { I }$ where the $T _ { i }$ = are homogeneous generators with non-trivial effective $\\operatorname { P i c } ( X )$ -degrees then $\\operatorname { C o x } ( \\mathcal { T } _ { X } ) \\cong K [ T _ { 1 } , \\dots , T _ { s } ]$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,504,879,558]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"122","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 2.9 Let $X \\mathcal { T } _ { X }$ be a Mori dream space embedded in its canonical toric embedding. Assume that both $X$ and $\\mathcal { T } _ { X }$ have Picard rank two. If $X$ and $\\mathcal { T } _ { X }$ are strongly Lefschetz divisorially equivalent then they are strong birational twins. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,567,881,677]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"122","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof By Remark $2 . 8 \\mathrm { M C D } ( \\mathcal { T } _ { X } )$ refines $\\operatorname { M C D } ( X )$ , and moreover $\\operatorname { M C D } ( X )$ refines $\\mathrm { S B L D } ( X )$ . By our hypothesis $\\mathrm { S B L D } ( X ) = i ^ { * } \\mathrm { S B L D } ( \\mathcal { T } _ { X } )$ , where $i : X \\to \\mathcal { T } _ { X }$ is the inclusion. Now, since $\\mathcal { T } _ { X }$ is a toric projective variety of Picard rank two [13, Theorem 1.2] yields $\\mathrm { S B L D } ( \\mathcal { T } _ { X } ) = \\mathbf { M C D } ( \\mathcal { T } _ { X } )$ . Summing up, $\\mathrm { M C D } ( \\mathcal { T } _ { X } ) = \\mathrm { S B L D } ( \\mathcal { T } _ { X } )$ refines $\\operatorname { M C D } ( X )$ which in turn refines $\\Re \\mathrm { L D } ( X ) = \\Im \\mathrm { B L D } ( \\mathcal { T } _ { X } ) = \\Re \\mathrm { C D } ( \\mathcal { T } _ { X } )$ , and this forces $\\mathsf { M C D } ( X ) = i ^ { * } \\mathsf { M C D } ( \\mathcal { T } _ { X } )$ . □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,720,744,764]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"122","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 Birational Twin Varieties: Complete Collineations and Quadrics ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,786,876,840]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"122","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This section is the core of the paper. It contains the definitions and crucial properties of the spaces of complete collineations and complete quadrics. It also contains the details of the proofs of Theorems A and B listed in the introduction. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,842,881,912]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"122","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $V$ and $W$ be $K$ -vector spaces of dimension $n + 1$ and $m + 1$ , respectively with $n \\leqslant m$ . Let $\\mathbb { P } ^ { N }$ denote the projective space whose points parametrize non-zero linear maps $W V$ up to a scalar multiple. We recall that a point in this spaces is called a collineation from $W$ to $V$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[633,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"122","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. L. Huerta and A. Massarenti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"122","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"118 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,144]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"123","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Observe that a full-rank collineation $p \\in \\mathsf { H o m } ( W , V )$ induces another collineation ${ \\textstyle \\bigwedge } ^ { k } W \\ \\to \\ { \\textstyle \\bigwedge } ^ { k } V$ , for any $\\textit { k } \\leq n + 1$ , by taking the wedge product. Consider the rational map ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,157,879,202]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0262/c7dd9085129764287dbb18d31dc5faff338da4a3453dfee8b32fb007048557a2.jpg","list_items":null,"page_idx":"123","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\phi : \\mathbb { P } ( \\mathbf { H o m } ( W , V ) ) \\ \\xrightarrow { } \\ \\mathbb { P } ( \\mathbf { H o m } ( { \\textstyle \\bigwedge } ^ { 2 } W , { \\textstyle \\bigwedge } ^ { 2 } V ) ) \\times \\cdots \\times \\mathbb { P } ( \\mathbf { H o m } ( { \\textstyle \\bigwedge } ^ { n + 1 } W , { \\textstyle \\bigwedge } ^ { n + 1 } V ) )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,235,879,273]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"123","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3.2 The space of complete collineations ${ \\mathcal { X } } ( n , m )$ from $W$ to $V$ is defined as the closure of the graph of the rational map (3.1). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,282,876,316]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"123","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In order to understand the closure of the image of $\\phi$ , we now recall an alternative description of the space ${ \\mathcal { X } } ( n , m )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,318,881,477]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"123","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Given an irreducible, reduced and non-degenerate variety $X \\subset \\mathbb { P } ^ { N }$ , for any $h <$ $N$ , we can consider the $h$ -secant variety of $X$ , which is defined as the closure of the union of all $( h - 1 )$ -planes spanned by $h$ general points in $X$ . We denote this variety by $\\mathbb { S } e c _ { h } ( X )$ . Observe that any point $p \\in \\mathbb { P } ^ { N } = \\mathbb { P } ( \\operatorname { H o m } ( W , V ) )$ can be represented by a $( n + 1 ) { \\times } ( m + 1 )$ matrix $Z$ , and that the locus of matrices of rank one is the Segre variety $\\mathcal { S } \\subset \\mathbb { P } ^ { N }$ . More generally, any $p \\in \\mathbb { S } e c _ { h } ( S )$ can be represented by a matrix $Z$ which is a linear combination of $h$ matrices of rank one, and conversely. In other words, a point $p$ is contained in $\\mathbb { S } e c _ { h } ( S )$ if and only if the rank of its representing matrix is at most $h$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,480,876,515]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"123","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Setting up notation, we will think of the point $p = [ z _ { 0 , 0 } : \\cdot \\cdot \\cdot : z _ { n , m } ] \\in \\mathbb { P } ^ { N }$ as the matrix ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[391,528,605,599]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0376/050266ccd7dbe7273bb1fe29a7753fbf5dbc30a66cefdd91b8ea1e6d9fb4dcdf.jpg","list_items":null,"page_idx":"123","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nZ = \\left( \\begin{array} { c } { { z _ { 0 , 0 } \\ldots z _ { 0 , m } } } \\\\ { { \\vdots \\ddots \\vdots } } \\\\ { { z _ { n , 0 } \\ldots z _ { n , m } } } \\end{array} \\right)\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[136,614,863,632]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"123","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We can now describe the space ${ \\mathcal { X } } ( n , m )$ as a sequence of blow-ups as follows. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,641,774,659]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"123","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Construction 3.4 Let us consider the following sequence of blow-ups: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,666,856,795]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"123","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\begin{array} { r l } { - } & { { } \\mathcal { X } ( n , m ) _ { 1 } } \\end{array}$ is the blow-up of $\\mathcal { X } ( n , m ) _ { 0 } : = \\mathbb { P } ^ { N }$ along the Segre variety $s$ ; \n– $\\mathbf { \\chi } - \\mathcal { X } ( n , m ) _ { 2 }$ is the blow-up of $\\mathcal { X } ( n , m ) _ { 1 }$ along the strict transform of $\\mathbb { S } e c _ { 2 } ( S )$ ; . \n$\\mathbf { \\chi } - \\mathbf { \\chi } \\mathcal { X } ( n , m ) _ { i }$ is the blow-up of $\\mathcal { X } ( n , m ) _ { i - 1 }$ along the strict transform of $\\mathbb { S } e c _ { i } ( \\boldsymbol { S } )$ ; .. \n$\\vdash \\mathcal { X } ( n , m ) _ { n }$ is the blow-up of $\\mathcal { X } ( n , m ) _ { n - 1 }$ along the strict transform of $\\mathbb { S } e c _ { n } ( S )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,803,881,894]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"123","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It follows from [24] that the variety $\\mathcal { X } ( n , m ) _ { n }$ is isomorphic to ${ \\mathcal { X } } ( n , m )$ , the space of complete collineations from $W$ to $V$ . Furthermore, we have that for any $i = 1 , \\ldots , n$ the variety $\\mathcal { X } ( n , m ) _ { i }$ is smooth, the strict transform of $\\mathbb { S } e c _ { i } ( \\boldsymbol { S } )$ in $\\mathcal { X } ( n , m ) _ { i - 1 }$ is smooth, and the divisor $E _ { 1 } \\cup E _ { 2 } \\cup \\dots \\cup E _ { i - 1 }$ in $\\mathcal { X } ( n , m ) _ { i - 1 }$ is simple normal crossing. When $n = m$ we will write ${ \\mathcal { X } } ( n )$ for $\\mathcal { X } ( n , n )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,51,697,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"123","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"123","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"119 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,881,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"124","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Setting up notation, let $f _ { i } : \\mathcal { X } ( n , m ) _ { i } \\to \\mathcal { X } ( n , m ) _ { i - 1 }$ be the blow-up morphism and $E _ { i } = \\operatorname { E x c } ( f _ { i } )$ its exceptional divisor. By abusing notation, we will also denote by $E _ { i }$ the strict transform in the subsequent blow-ups. Let $H$ denote the pull-back to $\\mathcal { X } ( n , m ) _ { i }$ of the hyperplane section of $\\mathbb { P } ^ { N }$ . We will denote by $f : \\mathcal { X } ( n , m ) \\to \\mathbb { P } ^ { N }$ the composition of the $f _ { i }$ ’s. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,180,881,325]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"124","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $n = m$ , then the last blow-up in Construction 3.4 is the blow-up of a Cartier divisor and hence ${ \\mathcal { X } } ( n ) _ { n } \\cong { \\mathcal { X } } ( n ) _ { n - 1 }$ . Furthermore, we have a distinguished linear subspace in $\\mathbb { P } ^ { N } = \\mathbb { P } ( \\mathrm { H o m } ( V , V ) )$ . Indeed, the space of symmetric matrices is defined by $\\mathbb { P } ^ { N _ { + } } : = \\mathbb { P } ( \\operatorname { S y m } ^ { 2 } V ) = \\{ z _ { i , j } = z _ { j , i } | i \\neq j \\}$ , where $N _ { + } = { \\binom { n + 2 } { 2 } } - 1$ . Furthermore, the space $\\mathbb { P } ^ { N _ { + } }$ cuts out scheme-theoretically on $s$ the Veronese variety $\\nu \\subseteq \\mathbb { P } ^ { N _ { + } }$ ; which parametrizes $( n + 1 ) { \\times } ( n + 1 )$ symmetric matrices of rank one. More generally, the space $\\mathbb { P } ^ { N _ { + } }$ cuts out scheme-theoretically on $\\mathbb { S } e c _ { h } ( \\cal S )$ the $h$ - secant variety $\\mathbb { S } e c _ { h } ( \\mathbb { \\nu } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,325,879,379]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"124","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Observe that a full-rank symmetric matrix ) $q \\in \\mathsf { S y m } ^ { 2 } ( V )$ induces another symmetric matrix ${ \\textstyle \\bigwedge } ^ { k } q$ , for any $k \\ \\leq \\ n + 1$ , by taking the wedge product. There is a rational map ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[186,393,744,438]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0154/9dba7ddf42fa877cdf813a9e7d7ce96f37e046475ec165ef2a393f19f707c473.jpg","list_items":null,"page_idx":"124","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { \\rho : \\mathbb { P } ( \\mathbf { S y m } ^ { 2 } V ) \\xrightarrow { } \\mathbb { P } ( \\mathbf { S y m } ^ { 2 } \\bigwedge ^ { 2 } V ) \\times \\dots \\times \\mathbb { P } ( \\mathbf { S y m } ^ { 2 } \\bigwedge ^ { n } V ) } \\\\ { q \\longmapsto \\qquad ( \\bigwedge ^ { 2 } q , \\dots , \\bigwedge ^ { n } q ) } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,453,876,489]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"124","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3.6 The space of complete quadrics in $\\mathbb P ^ { n }$ is defined as the closure of the graph of the map (3.5). We denote this space by $\\boldsymbol { \\mathcal { Q } } ( \\boldsymbol { n } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,498,881,554]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"124","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"One can apply Construction 3.4 in the symmetric setting to the spaces $\\mathbb { P } ^ { N _ { + } }$ . In doing so, we obtain the following blow-up construction for the space of complete quadrics [23, Theorem 6.3]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,561,772,579]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"124","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Construction 3.7 Let us consider the following sequence of blow-ups: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,587,797,716]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"124","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\begin{array} { r l } { - } & { { } \\mathcal { Q } ( n ) _ { 1 } } \\end{array}$ is the blow-up of $\\mathcal { Q } ( n ) _ { 0 } : = \\mathbb { P } ^ { N _ { + } }$ along the Veronese variety $\\nu$ ; \n– $\\mathcal { Q } ( n ) _ { 2 }$ is the blow-up of $\\mathcal { Q } ( n ) _ { 1 }$ along the strict transform of $\\mathbb { S } e c _ { 2 } ( \\mathcal { V } )$ ; . \n$\\ c - \\mathcal { Q } ( n ) _ { i }$ is the blow-up of $\\mathcal { Q } ( n ) _ { i - 1 }$ along the strict transform of $\\mathbb { S } e c _ { i } ( \\gamma )$ ; . \n$\\begin{array} { r l } { - } & { { } \\mathcal { Q } ( n ) _ { n } } \\end{array}$ is the blow-up of $\\scriptstyle { \\mathcal { Q } } ( n ) _ { n - 1 }$ along the strict transform of $\\mathbb { S } e c _ { n } ( \\mathbb { \\nu } )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,723,881,798]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"124","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It follows from [23] that the variety ${ \\mathcal { Q } } ( n ) _ { n - 1 } \\cong { \\mathcal { Q } } ( n ) _ { n }$ is isomorphic to the space of complete $( n - 1 )$ -dimensional quadrics $\\boldsymbol { \\mathcal { Q } } ( \\boldsymbol { n } )$ . Furthermore, the variety ${ \\mathcal { Q } } ( n ) _ { i }$ is smooth, the strict transform of $\\mathbb { S } e c _ { i } ( \\gamma )$ in $\\mathcal { Q } ( n ) _ { i - 1 }$ is smooth, and the divisor $E _ { 1 } ^ { + } \\cup E _ { 2 } ^ { + } \\cup \\cdots \\cup E _ { i - 1 } ^ { + }$ in $\\mathcal { Q } ( n ) _ { i - 1 }$ − is simple normal crossing for any $i = 1 , \\ldots , n$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,798,881,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"124","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Setting up notation, let $f _ { i } ^ { + } : \\mathcal { Q } ( n ) _ { i } \\ \\to \\ \\mathcal { Q } ( n ) _ { i - 1 }$ be the blow-up morphism and $E _ { i } ^ { + } = \\mathrm { E x c } ( f _ { i } ^ { + } )$ its exceptional divisor. We will also denote by $E _ { i } ^ { + }$ the strict transforms of this divisor in the subsequent blow-ups, if confusion does not arise. Let $H ^ { + }$ stand for the pull-back to ${ \\mathcal { Q } } ( n ) _ { i }$ of the hyperplane section of $\\mathbb { P } ^ { N _ { + } }$ . We will denote by $f ^ { + } : \\mathcal { Q } ( n ) \\to \\mathbb { P } ^ { N _ { + } }$ the composition of the $f _ { i } ^ { + }$ ’s. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[633,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"124","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. L. Huerta and A. Massarenti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,152,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"124","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"120 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,722,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"125","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.1 Birational Geometry of the Intermediate Spaces ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,132,879,187]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"125","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section we will investigate the birational geometry of the intermediate blowups $\\mathcal { X } ( n ) _ { i }$ , ${ \\mathcal { Q } } ( n ) _ { i }$ appearing in Constructions 3.4 and 3.7. We begin by recalling the notion of spherical variety. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,195,881,250]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"125","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3.8 A spherical variety is a normal variety $X$ together with an action of a connected reductive affine algebraic group $\\mathcal { G }$ , a Borel subgroup $\\mathcal { B } \\subset \\mathcal { G }$ , and a base point $x _ { 0 } \\in X$ such that the ${ \\mathcal { B } }$ -orbit of $x _ { 0 }$ in $X$ is a dense open subset of $X$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,250,881,322]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"125","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $( X , { \\mathcal { G } } , { \\mathcal { B } } , x _ { 0 } )$ be a spherical variety. We distinguish two types of $\\mathcal { B }$ -invariant prime divisors: a boundary divisor of $X$ is a $\\mathcal { G }$ -invariant prime divisor on $X$ , a color of $X$ is a $\\mathcal { B }$ -invariant prime divisor that is not $\\mathcal { G }$ -invariant. We will denote by $B ( X )$ and ${ \\mathcal { C } } ( X )$ respectively the set of boundary divisors and colors of $X$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,330,876,367]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"125","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Notation 3.9 For any $i = 1 , \\ldots , n + 1$ let us denote by $D _ { i }$ the strict transform of the divisor in $\\mathbb { P } ^ { N }$ given by⎜⎝ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[312,384,687,454]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0067/c92405c831209ae328b2b506d66793060dfcb9b000a0b1a1aa3814d0ee4ce96b.jpg","list_items":null,"page_idx":"125","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname* { d e t } \\left( \\begin{array} { c c c } { { z _ { n - i + 1 , m - i + 1 } } } & { { \\ldots } } & { { z _ { n - i + 1 , m } } } \\\\ { { \\vdots } } & { { \\ddots } } & { { \\vdots } } \\\\ { { z _ { n , m - i + 1 } } } & { { \\ldots } } & { { z _ { n , m } } } \\end{array} \\right) = 0\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,468,879,504]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"125","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"in the intermediate spaces of complete collineations. Similarly, we will denote by $D _ { i } ^ { + }$ the analogous divisors in the intermediate spaces of complete quadrics. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,513,879,569]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"125","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 3.11 Let $\\mathcal { X } ( n , m ) _ { i }$ be the intermediate space appearing at the step $1 \\leqslant$ $i ~ \\leqslant ~ n - 1$ of Construction 3.4. Then \u0005 $\\mathcal { X } ( n , m ) _ { i }$ is spherical. Furthermore,\u0005 $\\operatorname { P i c } ( \\mathcal X ( n , m ) _ { i } ) = \\mathbb Z [ H , E _ { 1 } , \\dots , E _ { i } ]$ and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,582,876,626]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0711/e8e4c53c2e6ade0c62a15dd43ea8d49cbd058ccd7f7c176d5803a0732b38f1d7.jpg","list_items":null,"page_idx":"125","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nB ( \\mathcal { X } ( n , m ) _ { i } ) = \\left\\{ \\begin{array} { l l } { \\{ E _ { 1 } , \\dots , E _ { i } \\} } & { i f n < m } \\\\ { \\{ E _ { 1 } , \\dots , E _ { i } , D _ { n + 1 } \\} } & { i f n = m } \\end{array} \\right. \\quad \\mathcal { C } ( \\mathcal { X } ( n , m ) _ { i } ) = \\left\\{ \\begin{array} { l l } { \\{ D _ { 1 } , \\dots , D _ { n + 1 } \\} } & { i f n < m } \\\\ { \\{ D _ { 1 } , \\dots , D _ { n } \\} } & { i f n = m } \\end{array} \\right.\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,639,769,659]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"125","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finally, for the intermediate spaces $\\mathcal { Q } ( n ) _ { i }$ in Construction 3.7 we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[191,672,806,698]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0970/480167ce8f1e09357959a43dda9923e1c38c81b644b6809ab5a03658b07a1e32.jpg","list_items":null,"page_idx":"125","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { \\mathcal { B } ( \\mathcal { Q } ( n ) _ { i } ) = \\{ E _ { 1 } ^ { + } , \\ldots , E _ { i } ^ { + } , D _ { n + 1 } ^ { + } \\} , \\quad \\mathcal { C } ( \\mathcal { Q } ( n ) _ { i } ) = \\{ D _ { 1 } ^ { + } , \\ldots , D _ { n } ^ { + } \\} . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,711,881,857]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"125","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Note that $\\mathcal { X } ( n , m ) _ { i }$ is spherical, even though it is not wonderful, with respect to the action of $\\begin{array} { l } { \\mathcal { G } ~ = ~ \\varsigma L ( n + 1 ) \\times S L ( m + 1 ) } \\end{array}$ , with the Borel subgroup $\\mathcal { B }$ given by pair of upper triangular matrices. Indeed, by [24, Theorem 1] ${ \\mathcal { X } } ( n , m )$ is a wonderful variety and $\\mathcal { B }$ acts on it with a dense open orbit, and since the actions of $\\mathcal { B }$ on ${ \\mathcal { X } } ( n , m )$ and $\\mathcal { X } ( n , m ) _ { i }$ are compatible with the blow-up maps in Construction 3.4 we conclude that $\\mathcal { B }$ acts on $\\mathcal { X } ( n , m ) _ { i }$ with a dense open orbit as well. For the definition and the basic properties of wonderful varieties we refer to [2, Section 4.5.5]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,51,697,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"125","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"125","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"121 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,198]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"126","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since the actions of $\\mathcal { G }$ on ${ \\mathcal { X } } ( n , m )$ and $\\mathcal { X } ( n , m ) _ { i }$ are compatible with the blow-up morphism ${ \\mathcal { X } } ( n , m ) \\ \\to \\ { \\mathcal { X } } ( n , m ) _ { i }$ , the intermediate space $\\mathcal { X } ( n , m ) _ { i }$ has at most the same number of boundary divisors and colors as ${ \\mathcal { X } } ( n , m )$ . Now, to compute $B ( \\mathcal { X } ( n , m ) _ { i } )$ and $\\mathcal { C } ( \\mathcal { X } ( n , m ) _ { i } )$ it is enough to note that the ones listed in the statement are clearly boundary divisors and colors, and to apply [15, Proposition 3.6]. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,205,574,225]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"126","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 3.12 Let $n < m$ . Then $D _ { 1 } \\sim H$ and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[373,238,626,292]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0519/d50ebee67001ab2b1a20f06917d57af0de5b4fba8e321ae610dd40f585842ae1.jpg","list_items":null,"page_idx":"126","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nD _ { k } \\sim k H - \\sum _ { h = 1 } ^ { k - 1 } ( k - h ) E _ { h }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[116,306,879,342]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"126","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for $k = 2 , \\ldots , n + 1$ . Furthermore, $\\operatorname { E f f } ( \\mathcal { X } ( n , m ) _ { i } ) \\ = \\ \\langle E _ { 1 } , \\dots , E _ { i } , D _ { n + 1 } \\rangle$ and $\\mathrm { N e f } ( \\mathcal { X } ( n , m ) _ { i } ) = \\langle D _ { 1 } , . . . , D _ { i + 1 } \\rangle$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,343,546,361]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"126","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $n = m$ we have $D _ { 1 } \\sim H$ , $D _ { 1 } ^ { + } \\sim H ^ { + }$ and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[179,378,740,432]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0644/7b7e7b795997405e56a3aecb00770d3ce551b215369ea9605fb0135b7e931d35.jpg","list_items":null,"page_idx":"126","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nD _ { k } \\sim k H - \\sum _ { h = 1 } ^ { k - 1 } ( k - h ) E _ { h } , ~ D _ { k } ^ { + } \\sim k H ^ { + } - \\sum _ { h = 1 } ^ { k - 1 } ( k - h ) E _ { h } ^ { + }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[113,447,744,466]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"126","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for $k = 2 , \\ldots , n$ , and for the boundary divisors $E _ { n }$ and $E _ { n } ^ { + }$ we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,481,874,536]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0695/13ebbf8adb88cf947056f952313c8509f72c6dffa34e0bdb9146bb7fc5633cac.jpg","list_items":null,"page_idx":"126","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nE _ { n } \\sim ( n + 1 ) H - \\sum _ { h = 1 } ^ { n - 1 } ( n - h + 1 ) E _ { h } , \\quad E _ { n } ^ { + } \\sim ( n + 1 ) H ^ { + } - \\sum _ { h = 1 } ^ { n - 1 } ( n - h + 1 ) E _ { h } ^ { + } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,567,881,623]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"126","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Furthermore, $\\begin{array} { r } { \\mathrm { E f f } ( \\mathcal { X } ( n ) _ { i } ) = \\langle E _ { 1 } , \\ldots , E _ { i } , D _ { n + 1 } \\rangle , \\mathrm { N e f } ( \\mathcal { X } ( n ) _ { i } ) = \\langle D _ { 1 } , \\ldots , D _ { i + 1 } \\rangle } \\end{array}$ and similarly $\\mathrm { E f f } ( \\boldsymbol { \\mathcal { Q } } ( n ) _ { i } ) = \\left. E _ { 1 } ^ { + } , \\ldots , E _ { i } ^ { + } , D _ { n + 1 } ^ { + } \\right. , \\mathrm { N e f } ( \\boldsymbol { \\mathcal { Q } } ( n ) _ { i } ) = \\left. D _ { 1 } ^ { + } , \\ldots , D _ { i + 1 } ^ { + } \\right.$ for $i = 1 , \\ldots , n - 1$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,632,881,669]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"126","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Consider the case $n < m$ , a similar argument works for the case $n = m$ . Let $f : \\mathcal { X } ( n , m ) \\to \\mathbb { P } ^ { N }$ be the blow-up morphism in Construction 3.4. We have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[275,683,722,737]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0327/7bf2f410e40b5942d4e388ee73965afbf16d47d03719156415f286db45742042.jpg","list_items":null,"page_idx":"126","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nD _ { k } = \\deg ( f _ { * } D _ { k } ) H - \\sum _ { h = 1 } ^ { i } \\operatorname { m u l t s } _ { e c _ { h } ( \\mathcal { S } ) } ( f _ { * } D _ { k } ) E _ { h }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,750,881,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"126","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now, since $f _ { * } D _ { k }$ is the zero locus of a $k \\times k$ minor of the matrix $Z$ then $\\deg ( f _ { * } D _ { k } ) = k$ . Furthermore, [15, Lemma 3.10] yields $\\mathrm { m u l t } _ { \\mathbb { S } e c _ { h } ( S ) } ( f _ { * } D _ { k } ) = k - h$ if $h = 1 , \\ldots k - 1$ and $\\mathrm { m u l t } _ { \\mathbb { S } e c _ { h } ( { \\cal S } ) } ( f _ { * } D _ { k } ) = 0$ if $h \\geqslant k$ , and hence we get (3.13). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[633,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"126","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. L. Huerta and A. Massarenti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"126","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"122 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,881,142]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"127","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By [2, Proposition 4.5.4.4] and [4, Section 2.6] the generators of the effective and the nef cones are respectively generated by the boundary divisors and the colors. Lemma 3.11 then concludes the proof. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,178,478,198]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"127","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.1.1 Generators of $\\mathbf { C o x } ( \\mathcal { X } ( \\pmb { n } , \\pmb { m } ) _ { i } )$ ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,214,881,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"127","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $I = \\{ i _ { 0 } , \\ldots , i _ { k } \\}$ , $J = \\{ j _ { 0 } , \\ldots , j _ { k } \\}$ be two ordered sets of indexes with $0 \\leqslant$ $i _ { 0 } \\leqslant \\cdots \\leqslant i _ { k } \\leqslant n$ and $0 \\ \\leqslant \\ j _ { 0 } \\ \\leqslant \\ \\cdots \\ \\leqslant \\ j _ { k } \\ \\leqslant \\ m$ , and denote by $Z _ { I , J }$ the $( k + 1 ) \\times ( k + 1 )$ minor of $Z$ built with the rows indexed by $I$ and the columns indexed by $J$ . Similarly, let $Z _ { I , J } ^ { + }$ be the $( k + 1 ) \\times ( k + 1 )$ minor of the symmetrization of $Z$ built with the rows indexed by $I$ and the columns indexed by $J$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,315,879,387]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"127","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 3.16 Let $T _ { I , J }$ be the canonical section associated to the strict transform of the hypersurface $\\{ \\operatorname* { d e t } ( Z _ { I , J } ) = 0 \\} \\subset \\mathbb { P } ^ { N }$ , and let $S _ { j }$ be the canonical section associated to the exceptional divisor $E _ { j }$ in Construction 3.4. Then $\\mathbf { C o x } ( \\mathcal { X } ( n , m ) ) _ { i }$ is generated by the $T _ { I , J }$ for $1 \\leqslant | I | , | J | \\leqslant n + 1$ and the $S _ { j }$ for $j = 1 , \\dotsc , i$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,387,879,487]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"127","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now, consider $\\mathcal { Q } ( n ) _ { i }$ . Let $T _ { I , J } ^ { + }$ be the canonical section associated to the strict transform of the hypersurface $\\{ \\mathsf { d e t } ( Z _ { I , J } ^ { + } ) ~ = ~ 0 \\} ~ \\subset ~ \\mathbb { P } ^ { N _ { + } }$ , and let $S _ { j } ^ { + }$ be the canonical section associated to the exceptional divisor $E _ { j } ^ { + }$ in Construction 3.7. Then $\\mathrm { C o x } ( \\mathcal { Q } ( n ) _ { i } )$ is generated by the $T _ { I , J } ^ { + }$ for $1 \\leqslant | I | , | J | \\leqslant n + 1$ and the $S _ { j } ^ { + }$ for $j = 1 , \\dots , i$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,495,881,585]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"127","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof By [2, Theorem 4.5.4.6] if $\\mathcal { G }$ is a semi-simple and simply connected algebraic group and $( X , { \\mathcal { G } } , { \\mathcal { B } } , x _ { 0 } )$ is a spherical variety with boundary divisors $E _ { 1 } , \\ldots , E _ { r }$ and colors $D _ { 1 } , \\ldots , D _ { s }$ , then $\\operatorname { C o x } ( X )$ is generated as a $K$ -algebra by the canonical sections of the $E _ { i }$ ’s and the finite dimensional vector subspaces $\\operatorname* { l i n } _ { K } ( { \\mathcal { G } } \\cdot D _ { i } ) \\subseteq \\mathbf { C o x } ( X )$ for $1 \\leqslant i \\leqslant s$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,587,881,677]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"127","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Consider the case $n < m$ . For $\\mathcal { X } ( n ) _ { i }$ and $\\mathcal { Q } ( n ) _ { i }$ an analogous argument works. By Lemma 3.11 we have that $B ( \\mathcal { X } ( n , m ) _ { i } ) = \\{ E _ { 1 } , . . . , E _ { i } \\}$ and $\\mathcal { C } ( \\mathcal { X } ( n , m ) _ { i } ) =$ $\\{ D _ { 1 } , \\ldots , D _ { n + 1 } \\}$ . Recall that for any $k = 0 , \\ldots , n$ the divisor $D _ { k + 1 }$ is the strict transform of the hypersurface in $\\mathbb { P } ^ { N }$ defined by the determinant of the $( k + 1 ) \\times$ $( k + 1 )$ most right down minor of the matrix $Z$ . Let us denote by $Z _ { I , J } ^ { r d }$ such minor. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,678,881,758]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"127","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note that $\\operatorname* { d e t } ( Z _ { I , J } ^ { r d } ) \\ \\in \\ I ( \\mathbb { S } e c _ { k } ( { \\cal S } ) )$ . Therefore, considering the action of $\\mathcal { G } =$ $S L ( n + 1 ) \\times S L ( m + 1 )$ we have that $g \\cdot \\operatorname* { d e t } ( Z _ { I , J } ^ { r d } ) \\in I ( \\mathbb { S } e c _ { k } ( { \\cal S } ) )$ for any $g \\in { \\mathcal { G } }$ , and hence $\\begin{array} { r } { \\operatorname* { l i n } _ { K } ( \\mathscr { G } \\cdot \\mathrm { d e t } ( Z _ { I , J } ^ { r d } ) ) \\subseteq I ( \\mathbb { S } e c _ { k } ( S ) ) } \\end{array}$ . To conclude it is enough to recall that $I ( \\mathbb { S } e c _ { k } ( \\mathcal { S } ) )$ is generated by the $( k + 1 ) \\times ( k + 1 )$ minors of $Z$ . □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,694,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"127","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"127","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"123 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[861,129,879,141]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"127","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"□ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,799,130]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"128","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.2 The Mori Chamber Decomposition of the Intermediate Space of Picard Rank Three ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,153,876,207]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"128","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section, using the techniques introduced in [15, 16], we will study the Mori chamber and stable base locus decomposition for the spaces \\$ $\\mathcal { X } ( n ) _ { 3 }$ in Construction 3.4.\\$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,217,879,273]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"128","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Notation 3.17 We will denote by $\\langle v _ { 1 } , \\ldots , v _ { s } \\rangle$ the cone in $\\mathbb { R } ^ { n }$ generated by the vectors $v _ { 1 } , \\dots , v _ { s } \\in \\mathbb { R } ^ { n }$ . Given two vectors $v _ { i } , v _ { j }$ we set $( v _ { i } , v _ { j } ] : = \\left. v _ { i } , v _ { j } \\right. \\backslash \\mathbb { R } _ { > 0 } v _ { i }$ and $( v _ { i } , v _ { j } ) : = \\big \\langle v _ { i } , v _ { j } \\big \\rangle \\setminus ( \\mathbb { R } _ { > 0 } v _ { i } \\cup \\mathbb { R } _ { > 0 } v _ { j } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,280,876,316]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"128","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 3.18 Consider the natural inclusion $i \\quad : \\quad \\mathcal { Q } ( n ) _ { i } \\quad \\quad \\mathcal { X } ( n ) _ { i }$ . Then $i ^ { * } \\operatorname { M o v } ( \\mathcal { X } ( n ) _ { i } ) = \\operatorname { M o v } ( \\mathcal { Q } ( n ) _ { i } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,325,879,361]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"128","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof By Proposition $3 . 1 6 \\mathrm { C o x } ( \\mathcal { Q } ( n ) _ { i } )$ is generated by the pull-backs of the generators of $\\mathrm { C o x } ( \\mathcal { X } ( n ) _ { i } )$ . Hence the statement follows from [2, Proposition 3.3.2.3]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,369,881,423]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"128","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 3.19 The movable cone of $\\mathcal { X } ( n ) _ { 3 }$ is generated by $D _ { 1 } \\sim H , D _ { 2 } \\sim$ $2 H - E _ { 1 }$ , $D _ { n } \\sim n H + ( 1 - n ) E _ { 1 } + ( 2 - n ) E _ { 2 }$ , $P _ { n } \\sim n ( n - 1 ) H + n ( 2 - n ) E _ { 1 } +$ $( n - 1 ) ( 2 - n ) E _ { 2 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,424,876,462]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"128","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Similarly, $\\mathbf { M o v } ( \\mathcal { Q } ( n ) _ { 3 } )$ is generated by $D _ { 1 } ^ { + } \\sim H ^ { + }$ , $D _ { 2 } ^ { + } \\sim 2 H ^ { + } - E _ { 1 } ^ { + }$ , $D _ { n } ^ { + } \\sim$ $n H ^ { + } + ( 1 - n ) E _ { 1 } ^ { + } + ( 2 - n ) E _ { 2 } ^ { + }$ , $P _ { n } ^ { + } \\sim n ( n - 1 ) H ^ { + } + n ( 2 - \\bar { n } ) E _ { 1 } ^ { + } + ( n - 1 ) ( \\bar { 2 } - n ) E _ { 2 } ^ { + } ;$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,468,881,558]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"128","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof We use [2, Proposition 3.3.2.3] which gives a way of computing the movable cone starting from the generators of the Cox ring. More specifically, in order to compute the movable cone it is enough to intersect all the cones generated by the sets of vectors obtained by removing just one element from the set of generators of the Cox ring. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,560,881,666]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"128","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In our specific case, using the generators of $\\mathrm { C o x } ( \\mathcal { X } ( n ) _ { 3 } )$ in Proposition 3.16, it is immediate to see that this procedure give us the vectors $( 1 , 0 , 0 )$ , $( 2 , - 1 , 0 )$ , $( n , 1 -$ $n , 2 - n )$ and the vector $( n ( n - 1 ) , n ( 2 - n )$ , $( n - 1 ) ( 2 - n ) )$ that is the intersection of the plane generated by $( 1 , 0 , 0 )$ , $( n + 1 , - n , - n + 1 )$ and the plane generated by $( 0 , 1 , 0 )$ , $( n , 1 - n , 2 - n )$ . Finally, the statement on $\\mathbf { M o v } ( \\mathcal { Q } ( n ) _ { 3 } )$ follows from Lemma 3.18. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,677,879,731]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"128","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3.20 Let $f ( n )$ be the number of chambers in the Mori chamber decomposition of $\\mathcal { X } ( n ) _ { 3 }$ . Then $f ( n + 1 ) = f ( n ) + n + 1$ . Furthermore, the decomposition can be described by the following 2-dimensional cross-section of $\\mathrm { E f f } ( \\mathcal { X } ( n ) _ { 3 } )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[633,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"128","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. L. Huerta and A. Massarenti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"128","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"124 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,93,872,450]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0970/a12e8f6c7295dee86d37b566ab60c2902b398dd1eba16ab1a8ef755ce988c8ba.jpg","list_items":null,"page_idx":"129","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[118,471,881,525]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"129","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Furthermore, the same statements hold, by replacing the relevant divisors with their pull-backs via the embedding $i : \\mathcal { Q } ( n ) _ { 3 } \\to \\mathcal { X } ( n ) _ { 3 }$ , for the intermediate space of quadrics $\\mathcal { Q } ( n ) _ { 3 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,534,881,587]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"129","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof By [15, Theorem 6.11] the statement holds for $n = 3$ . Consider the case $n = 4$ . Then the Cox ring of $\\mathcal { X } ( 4 ) _ { 3 }$ has additional generators belonging to the class of $D _ { 5 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,588,879,642]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"129","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now, let $\\tau _ { \\mathcal { X } ( 4 ) _ { 3 } }$ be a projective toric variety as in Remark 2.8. Then there is an embedding $i : \\mathcal { X } ( 4 ) _ { 3 } \\mathcal { T } _ { \\mathcal { X } ( 4 ) _ { 3 } }$ such that $i ^ { * } : \\mathrm { P i c } ( \\mathcal { T } _ { \\mathcal { X } ( 4 ) _ { 3 } } ) \\mathrm { P i c } ( \\mathcal { X } ( 4 ) _ { 3 } )$ is an isomorphism inducing an isomorphism $\\mathrm { E f f } ( \\mathcal T _ { \\mathcal X ( 4 ) _ { 3 } } ) \\to \\mathrm { E f f } ( \\mathcal X ( 4 ) _ { 3 } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,644,881,839]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"129","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since $\\tau _ { \\mathcal { X } ( 4 ) _ { 3 } }$ is toric, the Mori chamber decomposition of $\\mathrm { E f f } ( \\mathcal T _ { \\mathcal { X } ( 4 ) _ { 3 } } )$ can be computed by means of the Gelfand-Kapranov-Zelevinsky decomposition [2, Section 2.2.2]. Roughly speaking such decomposition is given by all the convex cones we can construct using all the generators of $\\cos ( \\mathcal { T } _ { \\mathcal { X } ( 4 ) _ { 3 } } )$ . Starting from the decomposition for $\\tau _ { \\mathcal { X } ( 4 ) _ { 3 } }$ we are allowed to add just two more segments, namely $[ E _ { 2 } , D _ { 5 } ] , [ E _ { 1 } , D _ { 5 } ]$ which introduce a new chamber each, and $[ D _ { 1 } , D _ { 5 } ]$ which gives rise two three new chambers by dividing $\\left. D _ { 1 } , D _ { 3 } , E _ { 1 } \\right. , \\left. E _ { 1 } , D _ { 3 } , D _ { 4 } \\right. , \\left. E _ { 1 } , D _ { 4 } , D _ { 5 } \\right.$ . Note that we get exactly $\\begin{array} { r l } { f ( 4 ) } & { { } = } \\end{array}$ $f ( 3 ) + 4 + 1 = 9 + 4 + 1 = 1 4$ chambers. On the other hand, a priori this is just a refinement of the Mori chamber decomposition of Ef $\\yen ( \\mathcal { X } (4 ) _ { 3 } )$ . We will prove that it is indeed the Mori chamber decomposition. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,840,879,894]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"129","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note that by considering a general $\\bar { \\mathbb { P } } ^ { 3 } \\times \\mathbb { P } ^ { 3 } \\subset \\mathbb { P } ^ { 4 } \\times \\mathbb { P } ^ { 4 }$ we get a natural embedding $j : \\mathcal { X } ( 3 ) _ { 3 } \\ \\ \\mathcal { X } ( 4 ) _ { 3 }$ . Furthermore, this embedding preserves the generators of the Picard groups in Lemma 3.11 and the generators of the Cox rings in Proposition 3.16. Therefore, all the chambers of $\\mathbf { M C D } ( \\mathcal { X } ( 3 ) _ { 3 } )$ appear in $\\mathbf { M C D } ( \\mathcal { X } ( 4 ) _ { 3 } )$ . Furthermore, the segments $[ E _ { 2 } , D _ { 5 } ] , [ E _ { 1 } , D _ { 5 } ]$ must appear in $\\mathbf { M C D } ( \\mathcal { X } ( 4 ) _ { 3 } )$ simply because by Proposition 3.12 they define two external walls of $\\operatorname { E f f } ( \\mathcal { X } ( 4 ) _ { 3 } )$ , and the segment $[ D _ { 1 } , D _ { 5 } ]$ must appear as well since it is needed to intercept the vector $P _ { 4 }$ which by Proposition 3.19 is an extremal ray of $\\mathbf { M o v } ( \\mathcal { X } ( 4 ) _ { 3 } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,694,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"129","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"129","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"130","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,180,881,304]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"130","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now, assuming that the statement holds for $\\mathcal { X } ( n \\mathrm { ~ - ~ } 1 ) _ { 3 }$ we can prove it for $\\mathcal { X } ( n ) _ { 3 }$ arguing exactly as we did in order to pass from $\\mathcal { X } ( 3 ) _ { 3 }$ to $\\mathcal { X } ( 4 ) _ { 3 }$ . Indeed, arguing exactly as in the previous part of the proof we see that all the chambers of $\\mathrm { M C D } ( \\mathcal { X } ( n \\mathrm { ~ - ~ } 1 ) _ { 3 } )$ must appear in $\\mathbf { M C D } ( \\mathcal { X } ( n ) _ { 3 } )$ . Moreover, we have to add the segments $[ E _ { 1 } , D _ { n + 1 } ] , [ E _ { 2 } , D _ { n + 1 } ]$ which by Proposition 3.12 give external walls of $\\mathrm { E f f } ( \\mathcal { X } ( n ) _ { 3 } )$ , and $[ D _ { 1 } , D _ { n + 1 } ]$ needed to intercept the vectors $P _ { n }$ which by Proposition 3.19 is an extremal ray of $\\mathbf { M o v } ( \\mathcal { X } ( n ) _ { 3 } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,306,879,394]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"130","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finally, note that $n - 2$ of the chambers in $\\mathcal { X } ( n - 1 ) _ { 3 }$ are subdivided in two chambers each by $[ D _ { 1 } , D _ { n + 1 } ]$ , the new cone $\\langle E _ { 1 } , D _ { n } , D _ { n + 1 } \\rangle$ also is subdivided in two chambers each by $[ D _ { 1 } , D _ { n + 1 } ]$ , and we must add also the chamber $\\langle E _ { 2 } , D _ { n } , D _ { n + 1 } \\rangle$ . Hence the number of chambers of $\\mathrm { E f f } ( \\mathcal { X } ( n ) _ { 3 } )$ is given by $f ( n ) = f ( n - 1 ) + n -$ $2 + 2 + 1 = f ( n - 1 ) + n + 1$ . □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,403,879,495]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"130","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark $3 . 2 I$ By [15, Theorem 6.11] $\\mathcal { X } ( 3 ) _ { 3 }$ has nine Mori chambers but just eight stable base locus chambers. Indeed, within the non-convex chamber determined by the divisors $D _ { 1 } , D _ { 2 } , D _ { 3 } , E _ { 2 }$ the stable base locus is $E _ { 2 }$ . Note that this last fact holds true more generally for the space $\\mathcal { X } ( n ) _ { 3 }$ for $n \\geqslant 3$ . Similarly, this holds also for the spaces of quadrics $\\mathcal { Q } ( n ) _ { 3 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,503,879,558]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"130","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 3.22 The varieties $\\mathcal { X } ( n ) _ { i }$ and ${ \\mathcal { Q } } ( n ) _ { i }$ are Lefschetz divisorially equivalent, and moreover $\\mathcal { X } ( n ) _ { 3 }$ and $\\mathcal { Q } ( n ) _ { 3 }$ are birational twins for any $n \\geqslant 1$ . Furthermore, $\\mathcal { X } ( 3 ) _ { 3 }$ and $\\mathcal { Q } ( 3 ) _ { 3 }$ are strong birational twins. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,567,879,620]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"130","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof The first statement follows from Proposition 3.12 and Lemma 3.18. The second statement follows then from Theorem 3.20. Finally, the third statement is a consequence of Remark 3.21. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,665,781,687]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"130","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 The Flip of the Spaces of Complete Quadric Surfaces ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,710,879,818]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"130","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3.20 and Corollary 3.22 imply that each of the spaces $\\mathcal { X } ( 3 ) , \\mathcal { Q } ( 3 )$ of complete collineations of $\\mathbb { P } ^ { 3 }$ and complete quadric surfaces, admits a single flip. Moreover, such flips are strongly related to each other. Indeed, by computing the Mori chamber decomposition in both cases, we observe that $i ^ { * } \\mathbf { M C D } ( \\mathcal { X } ( 3 ) ) \\ =$ $\\mathbf { M C D } ( \\mathcal { Q } ( 3 ) )$ , thus these two spaces are birational twins and the unique flip on $\\mathcal { X } ( 3 )$ induces that of $\\mathcal { Q } ( 3 )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,819,881,890]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"130","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section we exhibit the flip of $\\mathcal { Q } ( 3 )$ , denoted by $\\mathcal { Q } ( 3 ) ^ { + }$ , the space of complete quadric surfaces following [14]. Towards the end of the section, we conjecture the geometry of the flip of $\\mathcal { X } ( 3 )$ . We construct $\\mathcal { Q } ( 3 ) ^ { + }$ by analyzing a $\\mathbb { Z } / 2$ -action on the following Hilbert scheme. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[633,54,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"130","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. L. Huerta and A. Massarenti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,54,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"130","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,881,147]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"131","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 4.1 Let Hilb $\\begin{array} { r l } { = } & { { } \\mathbf { H i l b } ^ { 2 x + 1 } ( \\mathbb { G } ( 1 , 3 ) ) } \\end{array}$ denote the Hilbert scheme parametrizing subschemes of $\\mathbb { G } ( 1 , 3 ) \\subset \\mathbb { P } ^ { 5 }$ whose Hilbert polynomial is $P ( x ) = 2 x + 1$ . This space is isomorphic to the following blow-up ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[396,163,599,183]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0111/7ae5ee027e68f225dfca7c9fc77a21c85b467ea2382c02dedcd92e292b7a3a06.jpg","list_items":null,"page_idx":"131","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbf { H i l b } \\cong B l _ { \\mathbb { O G } } \\mathbb { G } ( 2 , 5 )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,199,879,235]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"131","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\mathbb { O } \\mathbb { G } \\subset \\mathbb { G } ( 2 , 5 )$ denotes the orthogonal Grassmannian inside the Grassmannian of 2-planes in $\\mathbb { P } ^ { 5 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,244,881,336]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"131","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Observe that a generic smooth curve with Hilbert polynomial $P ( x ) = 2 x + 1$ in $\\mathbb { P } ^ { 5 }$ is a plane conic $C$ . Thus, its ideal $I _ { C } \\subset k [ p _ { 0 } , \\dotsc , p _ { 5 } ]$ is generated by a quadric $F$ and three independent linear forms $L _ { 1 } , L _ { 2 } , L _ { 3 }$ . Since $C \\subset \\mathbb { G } = \\mathbb { G } ( 1 , 3 )$ , the equation $F$ is the quadric corresponding to the Grassmannian $\\mathbb { G } \\subset \\mathbb { P } ^ { 5 }$ under the Plücker embedding. This description gives rise to a rational map ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[400,352,599,373]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"1015/a0627f4570e7965d7811b967972613faacfe7fc059537cb6f027e5a6ede654af.jpg","list_items":null,"page_idx":"131","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nf : \\mathbb { G } ( 2 , 5 ) \\xrightarrow { } \\xrightarrow { \\smile } \\mathbb { H i l b }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,388,881,498]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"131","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"by assigning the 2-plane $P$ defined by the independent linear forms $( L _ { 1 } , L _ { 2 } , L _ { 3 } )$ to the ideal $\\langle L _ { 1 } , L _ { 2 } , L _ { 3 } \\rangle + \\langle F \\rangle \\subset k [ p _ { 0 } , \\ldots , p _ { 5 } ] .$ . Observe that the exceptional locus of $f$ consists of planes in $\\mathbb { P } ^ { 5 }$ such that there is a containment of ideals $\\langle F \\rangle \\subset$ $\\langle L _ { 1 } , L _ { 2 } , L _ { 3 } \\rangle$ , that is planes $P$ which are contained in the quadric $\\mathbb { G } \\subset \\mathbb { P } ^ { 5 }$ . The locus parametrizing such planes is exactly the orthogonal Grassmannian $\\mathbb { O G }$ . Now, we resolve the rational map $f$ , ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[366,513,633,596]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0297/4acb7cfd4891dff40f397d8f6494e3b322dcda96129e3fa837bb2eb6bb856805.jpg","list_items":null,"page_idx":"131","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { l } { { \\displaystyle B l _ { \\mathbb { O G } } \\mathbb { G } ( 2 , 5 ) } } \\\\ { { \\displaystyle \\qquad \\pi \\int _ { { \\bf \\Phi } } \\widetilde { { \\bf \\Phi } } } } \\\\ { { \\displaystyle \\mathbb { G } ( 2 , 5 ) \\mathrm { ~ . ~ . ~ . ~ } \\ f \\longrightarrow \\pmb { \\mathrm { H i l b } } } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,615,881,765]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"131","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The morphism $\\tilde { f }$ is an isomorphism. Indeed, the rational map $f$ is birational as it has an inverse morphism $g : \\mathbf { H i l b } \\mathbb { G } ( 2 , 5 )$ defined as follows: let $[ C ] \\in \\mathbf { H i l b }$ be a point, then the ideal $I ( C ) = ( F ) + ( \\mathrm { p l a n e } ) \\stackrel { \\xi } { \\mapsto }$ (plane) $\\in \\mathbb { G } ( 2 , 5 )$ . It is clear that $f \\circ g = I d$ , hence $f$ , and consequently $\\tilde { f }$ , is birational. Furthermore, $\\tilde { f }$ is a bijection. Indeed, since the exceptional divisor $E \\subset B l _ { \\mathbb { O G } } \\mathbb { G } ( 2 , 5 )$ is a $\\mathbb { P } ^ { 5 }$ -bundle over $\\mathbb { O G }$ , whose points can be thought of as pairs $( P , C )$ , where $P \\subset \\mathbb { P } ^ { 5 }$ is a 2- plane and $C \\subset P$ is a plane conic. Thus, Zariski’s Main Theorem implies that $\\tilde { f }$ is an isomorphism. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,773,879,833]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"131","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 4.2 Let Hilb be as above, then $\\mathrm { P i c } ( \\mathbf { H i l b } ) \\cong \\langle H ^ { + } , E _ { 2 } ^ { + } , E _ { 1 , 1 } ^ { + } \\rangle$ where $H ^ { + }$ is the pullback of the unique generator of the group $A ^ { 1 } ( \\mathbb { G } ( 2 , 5 ) )$ and the $E ^ { + } \\ '$ are the exceptional divisors of the blow-up. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,51,697,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"131","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"131","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"127 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"132","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof The orthogonal Grassmannian $\\mathbb { O G }$ has two components, hence the result follows. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,135,879,189]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"132","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If the base field $K$ is algebraically closed, then for a given smooth quadric $Q \\subset$ $\\mathbb { P } ^ { 3 }$ , the Fano variety of lines $F _ { 1 } ( Q ) \\subset \\mathbb { G } ( 1 , 3 )$ consists of two smooth conics. We get a $\\mathbb { Z } / 2$ -action on $\\mathbf { H i l b } ^ { 2 x + 1 } ( \\mathbb { G } ( 1 , 3 ) )$ by exchanging such conics. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,196,874,232]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"132","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 4.3 There is a nontrivial globally defined $\\mathbb { Z } / 2$ -action on $\\mathbf { H } \\mathbf { i } \\mathbf { l } \\mathbf { b } ^ { 2 x + 1 }$ $( \\mathbb { G } ( 1 , 3 ) )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,241,876,279]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"132","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Let $Q \\subset \\mathbb { P } ^ { 3 }$ be a smooth quadric hypersurface. The Fano variety of lines $F _ { 1 } ( Q )$ is the zero locus of a section of the following bundle, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[444,294,553,376]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0425/d8d0ac3738467b31146619af5a014c0bc6fa244e5889b2b20da7cb4ddd17bac3.jpg","list_items":null,"page_idx":"132","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { c } { { \\mathrm { S y m } ^ { 2 } ( S ^ { * } ) } } \\\\ { { Q _ { \\mid L } ~ { \\Bigg \\{ } ~ { \\Bigg \\{ } ~ \\pi ~ } } \\\\ { { \\mathbb { G } ( 1 , 3 ) } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,399,879,453]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"132","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $S ^ { * }$ is the dual of the tautological bundle $s$ over $\\mathbb { G } ( 1 , 3 )$ . A smooth conic in $\\mathbb { P } ^ { 5 }$ determines uniquely a 2-plane, thus in the Plücker embedding $\\mathbb { G } ( 1 , 3 ) \\subset \\mathbb { P } ^ { 5 }$ , we have that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,462,708,500]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"132","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"– $F _ { 1 } ( Q )$ determines two 2-planes if $\\operatorname { r a n k } ( Q )$ is either 2 or 4, – $F _ { 1 } ( Q )$ determines a single 2-plane if $\\operatorname { r a n k } ( Q )$ is either 1 or 3. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,507,881,581]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"132","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Exchanging such planes gives rise to a $\\mathbb { Z } / 2$ -action on $\\mathbb { G } ( 2 , 5 )$ , the Grassmannian of 2-planes in $\\mathbb { P } ^ { 5 }$ . The second condition above, says that such a $\\mathbb { Z } / 2$ -action on $\\mathbb { G } ( 2 , 5 )$ preserves the orthogonal Grassmannian $\\mathbb { O G }$ , hence inducing a $\\mathbb { Z } / 2$ -action on the blow-up $\\mathbf { H i l b } ^ { 2 x + 1 } ( \\bar { \\mathbb { G } } ( 1 , 3 ) )$ . □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,588,881,683]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"132","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Observe that there is an $S L _ { 4 } ( \\mathbb { C } )$ -action on Hilb induced by the action of $S L _ { 4 }$ on $\\mathbb { P } ^ { 3 }$ . This action stratifies Hilb into $S L _ { 4 }$ -orbits compatible with the exceptional divisors $E _ { 2 } ^ { + }$ , $E _ { 1 , 1 } ^ { + }$ . Notice that $\\mathbb { Z } / 2$ acts trivially on $S L _ { 4 }$ -orbits of codimension 2. In codimension 1, we have that $\\mathbb { Z } / 2$ acts as the identity on the exceptional divisors $E _ { 2 } ^ { + }$ and $E _ { 1 , 1 } ^ { + }$ . Consider the quotient ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[391,699,605,722]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0764/069044543fb0195871dd520d50d2e03948ae92d6b477220fddc265a5895852e1.jpg","list_items":null,"page_idx":"132","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathcal { Q } ( 3 ) ^ { + } : = \\mathbf { H i l b } / ( \\mathbb { Z } / 2 )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,738,881,774]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"132","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where the $\\mathbb { Z } / 2$ -action is defined in the previous Lemma 4.3.The main result of this section is the following. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,783,879,839]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"132","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 4.4 There is a flip $f ~ : ~ \\mathcal { Q } ( 3 ) \\quad \\xrightarrow { } - ~ \\mathcal { Q } ( 3 ) ^ { + }$ over the Chow variety $\\mathbf { C h o w } ^ { 2 x + 1 } ( \\mathbb { G } ( 1 , 3 ) )$ , and the flipping locus in $\\mathcal { Q } ( 3 )$ is the intersection the divisors $E _ { 1 } \\cap E _ { 3 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[633,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"132","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. L. Huerta and A. Massarenti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"132","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"128 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[145,88,710,108]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"133","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The previous result can summarized in the following diagram where $\\phi _ { 1 }$ and $\\phi _ { 2 }$ are small contractions and the other maps, except for the flip, are all divisorial contractions described in previous sections. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,123,792,208]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0869/2f483848b3007c149d4bee2b4e98581b57335906a62b3de92124e15eeea07209.jpg","list_items":null,"page_idx":"133","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[120,237,876,271]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"133","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,273,881,363]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"133","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In order to show that $\\mathcal { Q } ( 3 )$ and $\\mathcal { Q } ( 3 ) ^ { + }$ are related through a flip, we lift the maps so we avoid the action of $\\mathbb { Z } / 2$ . In this scenario, we need to describe the flip for the Hilbert scheme $\\mathbf { H i l b } ^ { 2 x + 1 } ( \\mathbb { G } ( 1 , 3 ) )$ . In doing so, we are lead to consider the Kontsevich moduli space $\\overline { { \\mathcal { M } } } _ { 0 , 0 } ( \\mathbb { G } , 2 )$ of degree two stable maps into the Grassmannian $\\mathbb { G } = \\mathbb { G } ( 1 , 3 )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,372,876,408]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"133","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 4.5 There is a nontrivial globally defined $\\mathbb { Z } / 2$ -action on the Kontsevich space $\\overline { { \\mathcal { M } } } _ { 0 , 0 } ( \\mathbb { G } ( 1 , 3 ) , 2 )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,415,881,469]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"133","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof We have a generic 2 to 1 morphism from the Kontsevich moduli space\u0016 \u0017 $\\mathcal { M } \\ = \\ \\overline { { \\mathcal { M } } } _ { 0 , 0 } ( \\mathbb { G } ( 1 , 3 ) , 2 ) \\ = \\ \\{ ( C , C ^ { * } ) \\}$ to the space , $\\mathcal { Q } ( 3 )$ of complete quadric surfaces defined as follows ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[378,481,621,534]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0693/6f7a4ed64ed12ae1620a0654b4d401c04ca0462ec8c873cb72d00ee467758add.jpg","list_items":null,"page_idx":"133","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n( C , C ^ { * } ) \\mapsto \\left( \\bigcup _ { L \\in C } L , C ^ { * } \\right)\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,548,881,675]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"133","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where the notation $( S , C ^ { * } )$ means a surface $s$ , and a curve $C ^ { * }$ as its marking. This map is 2 to 1 over the open subset parametrizing smooth quadric surfaces as well as over the divisor of complete quadrics of rank two. Indeed, if $\\cup _ { L \\in C } L$ sweeps out a smooth quadric $s$ , then $L$ is a ruling of $s$ . The other ruling induces another conic $C ^ { \\prime }$ which gets mapped to $s$ . The situation is similar over the locus of complete quadrics of rank two. Note that this map is 1 to 1 outside two such regions. We now define the $\\mathbb { Z } / 2$ -action on $\\overline { { \\mathcal { M } } }$ by identifying the two curves $C$ and $C ^ { \\prime }$ . □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,683,876,720]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"133","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corollary 4.6 The quotient of $\\overline { { \\mathcal { M } } } _ { 0 , 0 } ( \\mathbb { G } , 2 )$ by the $\\mathbb { Z } / 2$ -action is isomorphic to $\\mathcal { Q } ( 3 )$ . In particular, the quotient is smooth. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,728,881,783]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"133","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Let $Z$ denote the quotient of $\\overline { { \\mathcal { M } } }$ by the $\\mathbb { Z } / 2$ -action defined above. Observe that $X _ { 3 }$ and $Z$ are birational and there is a bijection between them. Zariski’s Main Theorem implies now the corollary. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,789,881,864]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"133","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It follows from [8] that $\\overline { { \\mathcal { M } } } _ { 0 , 0 } ( \\mathbb { G } , 2 )$ and $\\mathbf { H i l b } ^ { 2 x + 1 } ( \\mathbb { G } ( 1 , 3 ) )$ are related through a flip over the Chow variety. Furthermore, the maps in such a flip diagram between these two spaces are $\\mathbb { Z } / 2$ -equivariant [14, Section 5.2]. The result in Theorem 4.4 now follows from the previous lemmas. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,864,879,900]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"133","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Following the construction of the space $\\mathcal { Q } ( 3 ) ^ { + }$ above, we conjecture a modular interpretation for the flip of the space of complete collineations $\\mathcal { X } ( 3 )$ . Let $A \\in \\mathcal { X } ( 3 )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,51,697,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"133","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,879,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"133","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"129 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,91,881,199]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"134","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"be a generic complete collineation of $\\mathbb { P } ^ { 3 }$ . Then, we associate to it a hypersurface of $X = \\mathbf { \\bar { P } } ^ { 3 } \\times \\mathbf { \\mathbb { P } } ^ { 3 }$ of bidegree $( 1 , 1 )$ , denoted by $Y _ { A }$ ; this is an element of the complete linear system $| \\mathcal { O } _ { X } ( 1 , 1 ) | \\cong \\mathbb { P } ^ { 1 5 }$ . The singular locus of $Y _ { A }$ , when non-empty, is a product of projective spaces. Hence, the space $\\mathcal { X } ( 3 )$ parametrizes hypersurfaces $Y _ { A }$ with a marking: complete collineation of the type $Y _ { Q }$ , where $Q \\in \\mathcal { X } ( i )$ , for some $i < 3$ . The hypersurface $Y _ { A }$ is smooth for generic $A \\in \\mathcal { X } ( 3 )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,201,881,253]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"134","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On another hand, let $F ( Y _ { A } ) \\subset \\mathbb { G } ( 1 , 3 ) \\times \\mathbb { G } ( 1 , 3 )$ be the Fano scheme of ruled surfaces, isomorphic to $\\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 }$ , embedded in $Y _ { A }$ . This is the zero locus of a global section induced by $Y _ { A }$ , ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[412,270,587,351]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0422/300742c1e1a8269ba5e0516e410b805eee0e0696c4537766353891c1f4bdce21.jpg","list_items":null,"page_idx":"134","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { c } { S ^ { * } \\times S ^ { * } } \\\\ { { } } \\\\ { Y _ { A } \\left( \\begin{array} { l } { { } } \\\\ { { } } \\\\ { { \\pi } } \\end{array} \\right) \\pi } \\\\ { \\mathbb { G } ( 1 , 3 ) \\times \\mathbb { G } ( 1 , 3 ) } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,369,669,387]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"134","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $S ^ { * }$ is the dual of the tautological bundle over $\\mathbb { G } ( 1 , 3 )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,388,881,477]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"134","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We claim that the subscheme $F ( Y _ { A } )$ fully determines the hypersurface $Y _ { A }$ . Hence, the Hilbert scheme $\\mathbf { H i l b } _ { 1 , 1 }$ parametrizing subschemes $F ( Y _ { A } )$ inside the product $\\mathbb { G } ( 1 , 3 ) \\times \\mathbb { G } ( 1 , 3 )$ is birational to $\\mathcal { X } ( 3 )$ . In other words, $\\mathbf { H i l b } _ { 1 , 1 } \\cong \\mathcal { X } ( 3 )$ . We conjecture that this rational map is a flip over the Chow variety that parametrizes the cycle classes of $F ( Y _ { A } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,496,881,633]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"134","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acknowledgements The first named author is currently a CONACYT-Research Fellow and during the preparation of this article he was partly supported by the CONACYT Grant CB-2015/253061. The second named author is a member of the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni of the Istituto Nazionale di Alta Matematica “F. Severi” (GNSAGA-INDAM). We thank the organizers of the conferences Birational Geometry and Moduli Spaces, held in Rome, Italy, and Moduli Spaces in Algebraic Geometry and Applications, held in Campinas, Brazil, where part of this work was carried out. We have benefited from discussion with G. Reyes-Ahumada. We thank the referee for the comments that helped us to improve the paper. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,680,248,699]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"134","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[127,722,881,887]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"134","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Araujo, C., Massarenti, A.: Explicit log Fano structures on blow-ups of projective spaces. Proc. Lond. Math. Soc. (3) 113(4), 445–473 (2016). MR 3556488 \n2. Arzhantsev, I., Derenthal, U., Hausen, J., Laface, A.: Cox rings. Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge (2015). MR 3307753 \n3. Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010). MR 2601039 \n4. Brion, M.: Groupe de Picard et nombres caractéristiques des variétés sphériques. Duke Math. J. 58(2), 397–424 (1989). MR 1016427 \n5. Brion, M.: Variétés sphériques et théorie de Mori. Duke Math. J. 72(2), 369–404 (1993). MR 1248677 \n6. Brion, M.: Introduction to actions of algebraic groups. Les cours du CIRM, 1 no. 1: Hamiltonian Actions: invariants et classification (2010) \n7. Castravet, A.-M., Tevelev, J.: Hilbert’s 14th problem and Cox rings. Compos. Math. 142(6), 1479–1498 (2006). MR 2278756 8. Chen, D., Coskun, I.: Stable base locus decompositions of Kontsevich moduli spaces. Mich. Math. J. 59(2), 435–466 (2010). MR 2677631 \n9. Debarre, O.: Higher-Dimensional Algebraic Geometry. Universitext. Springer, New York (2001). MR 1841091 \n10. Hassett, B., Lin, H.W., Wang, C.L.: The weak Lefschetz principle is false for ample cones. Asian J. Math. 6(1), 95–100 (2002) \n11. Hu, Y., Keel, S.: Mori dream spaces and GIT. Mich. Math. J. 48, 331–348 (2000). Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786494 \n12. Jow, S.-Y.: A Lefschetz hyperplane theorem for Mori dream spaces. Math. Z. 268(1–2), 197– 209 (2011). MR 2805430 \n13. Laface, A., Massarenti, A., Rischter, R.: On Mori chamber and stable base locus decompositions. Trans. Am. Math. Soc. 373(3), 1667–1700 (2020) \n14. Lozano Huerta, C.: Birational geometry of the space of complete quadrics. Int. Math. Res. Not. IMRN 2015(23), 12563–12589 (2015). MR 3431630 \n15. Massarenti, A.: On the birational geometry of spaces of complete forms I: collineations and quadrics (2018). arXiv:1803.09161 \n16. Massarenti, A.: On the birational geometry of spaces of complete forms II: skew-forms. J Algebra 546, 178–200 (2020) \n17. Massarenti, A., Rischter, R.: Spherical blow-ups of Grassmannians and Mori dream spaces. J. Algebra 494, 188–219 (2018). MR 3723177 \n18. Mukai, S.: Counterexample to hilbert’s fourteenth problem for the 3-dimensional additive group. Technical report, Kyoto University, Research Institute for Mathematical Sciences (2001) \n19. Okawa, S.: On images of Mori dream spaces. Math. Ann. 364(3–4), 1315–1342 (2016). MR 3466868 \n20. Ottem, J.C.: Birational geometry of hypersurfaces in products of projective spaces. Math. Z. 280(1–2), 135–148 (2015). MR 3343900 \n21. Perrin, N.: On the geometry of spherical varieties. Transform. Groups 19(1), 171–223 (2014). MR 3177371 \n22. Szendroi, B.: On the ample cone of an ample hypersurface. Asian J. Math. 7(1), 1–5 (2003). MR 2015238 \n23. Vainsencher, I.: Schubert calculus for complete quadrics. In: Enumerative Geometry and Classical Algebraic Geometry (Nice, 1981). Progress in Mathematics, vol. 24, pp. 199–235. Birkhäuser, Boston (1982). MR 685770 \n24. Vainsencher, I.: Complete collineations and blowing up determinantal ideals. Math. Ann. 267(3), 417–432 (1984). MR 738261 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[633,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"134","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. L. Huerta and A. Massarenti ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,52,152,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"134","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"130 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[116,90,881,680]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"135","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,694,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"135","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"135","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"131 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,649,166]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"136","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,202,277,220]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"136","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Luigi Lunardon ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,288,879,376]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"136","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract In this survey, we discuss the state of art about the monodromy property for Calabi-Yau varieties. We explain what is the monodromy property for Calabi-Yau varieties and then discuss some examples of Calabi-Yau varieties that satisfy this property. After this recap, we discuss a possible approach to future research in this area. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,396,874,430]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"136","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords Degenerations of Calabi-Yau varieties $\\cdot$ Motivic integration $\\cdot$ Motivic monodromy conjecture $\\cdot$ Motivic zeta function $\\cdot$ Triple-points-free models ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,453,861,486]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"136","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2010 Mathematics Subject Classification Primary: 14E18; Secondary: 14G10, \n14D06 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,525,309,545]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"136","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,569,881,677]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"136","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our aim is the study of the monodromy of degenerations of Calabi-Yau varieties, and we are interested, in particular, in the so-called motivic monodromy conjecture. While degenerations and monodromy are intuitive concepts in complex geometry, translating these ideas in the setting of algebraic geometry is not trivial. In this section we try to give an intuitive description of our topic; hopefully, this picture will guide the reader through the understanding of this paper. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,678,876,713]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"136","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Denote by $D \\subset \\mathbb { C }$ the set $\\{ x \\ \\in \\mathbb { C } \\colon | x | \\ < \\ 1 \\}$ . For the moment, a model for a degeneration of a Calabi-Yau is the datum of: a Calabi-Yau variety $X$ over $\\mathbb { C }$ , a smooth complex variety $\\mathcal { X }$ , endowed with a proper holomorphic map $\\pi : \\mathcal { X } D$ which is smooth on $D ^ { * } \\ = \\ D \\ \\backslash \\ \\{ 0 \\}$ and satisfying the following additional properties: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,897,692,941]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"136","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_8 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,834,546,879]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"136","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"L. Lunardon $( \\boxtimes )$ Office 671, Kensington, London e-mail: l.lunardon16@imperial.ac.uk; ll3916@ic.ac.uk ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[804,91,876,139]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"136","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,899,876,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"136","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"133 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,881,142]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"137","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,151,879,205]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"137","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(1) there exists $z _ { 0 } \\in D , \\ z _ { 0 } \\not = 0$ such that $\\pi ^ { - 1 } ( z _ { 0 } ) = \\mathcal { X } _ { z _ { 0 } } \\cong X$ ; (2) $X _ { 0 }$ is a strict normal crossing divisor on $\\mathcal { X }$ —by this we mean that the irreducible components of $( X _ { 0 } ) _ { r e d }$ are smooth and with transverse intersection. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,216,881,268]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"137","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The divisor $\\begin{array} { r c l } { X _ { 0 } } & { = } & { \\sum _ { i \\in I } N _ { i } E _ { i } } \\end{array}$ contains some of the geometrical data of the degeneration, but it is not enough to determine the motivic zeta function. As we see in Sect. 3 some additional information is required. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,270,881,324]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"137","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Up to homotopy, we have a natural action of $\\mathbb { Z } = \\pi _ { 1 } ( D ^ { * } )$ on the underlying topological space of $X$ . This action induces an action on the cohomology groups $H ^ { i } ( X , \\mathbb { Q } )$ . We can think of this as the cohomological datum of the degeneration. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,325,881,521]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"137","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A natural question is how the geometrical and the cohomological data of a degeneration are related. The motivic monodromy conjecture suggests a possible answer to this question; this conjecture states that poles of the motivic zeta function of a degeneration of Calabi-Yau varieties are linked to monodromy eigenvalues. The motivic zeta function is a power series which depends on $\\mathcal { X } ^ { \\ast }$ —the restriction of the model $\\mathcal { X }$ over $D ^ { * }$ —and on a relative volume form $\\omega$ on $\\mathcal { X } ^ { \\ast }$ . The motivic zeta function encodes all these data and some more information about the geometry of the special fiber. A degeneration of $X$ satisfies the monodromy property if any pole of the motivic zeta function determines an eigenvalue in monodromy. The motivic monodromy conjecture states that Calabi-Yau varieties satisfy the monodromy property. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,522,881,701]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"137","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The motivic monodromy conjecture is the more recent of a series of similar conjectures in different settings. The first conjecture of this series is known as the $p$ -adic monodromy conjecture and it was suggested by Igusa. Given a polynomial $f \\in \\mathbb { Z } [ x _ { 1 } , \\ldots , x _ { n } ]$ , this conjecture suggests a connection between the associated $p$ -adic zeta function and the local monodromy eigenvalues of $f \\colon \\mathbb { C } ^ { n } \\to \\mathbb { C }$ . After Kontsevich’s work on motivic integration, Denef and Loeser proposed an upgrade of this conjecture in [6]. Given a polynomial $f \\in \\mathbb { C } [ x _ { 1 } , \\ldots , x _ { n } ]$ , they introduced the associated motivic zeta function and formulated the motivic monodromy conjecture for hypersurface singularities, which statement is analogue to the one of the $p$ -adic monodromy conjecture. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,704,881,900]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"137","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The main aim of this survey is to present this topic in a rigorous way. In Sect. 2 we give the definition of a degeneration of a Calabi-Yau varieties and we explain what a model is. In Sect. 3 we recall some basic definitions in motivic integration and state the monodromy property for Calabi-Yau varieties. The main technical issue is that the motivic zeta function is a power series on a certain localization of an equivariant Grothendieck ring of varieties, so we have to be careful when we talk about poles of the function. For the definition of the monodromy zeta function and the A’Campo formula we refer to [16]. The equivariant version of the Grothendieck ring of varieties was introduced in [10]. The definition of the motivic zeta function and the Denef and Loeser’s formula are the ones given in [9]. The definitions of poles of a power series with coefficients in the Grothendieck ring of varieties were given in [17]. In Sect. 4 we talk about abelian varieties, Calabi-Yau varieties which admit an equivariant Kulikov model and Kummer surfaces (we refer to [9] for the first two cases and to [18] for the last one). In Sect. 5 we discuss triple-points-free models of K3 surfaces (we refer to [11, 12] and [15]). Finally, in Sect. 6 we briefly discuss some directions of the research in this topic. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[779,54,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"137","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"L. Lunardon ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,54,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"137","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,180]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"138","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,225,624,246]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"138","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 Models for Calabi-Yau Varieties over $\\pmb { K }$ ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,268,881,340]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"138","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the introduction we talked about degenerations of Calabi-Yau varieties, the picture we described was intuitive and geometric, however, it was not rigorous. In this section we define rigorously what we mean by degenerations of Calabi-Yau varieties and what a model is. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,342,881,394]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"138","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We fix the following notations $k ~ = ~ \\mathbb { C }$ , $K \\ = \\ \\mathbb { C } ( ( t ) )$ and $R ~ = ~ \\mathbb { C } [ [ t ] ]$ . For every positive integer $d$ , we have ${ \\cal K } ( d ) = \\mathbb { C } ( ( \\sqrt [ d ] { t } ) ) , { \\cal R } ( d ) = \\mathbb { C } [ [ \\sqrt [ d ] { t } ] ]$ . We set $\\overline { { K } } = \\bar { \\mathbb { C } } \\{ \\{ t \\} \\}$ , the field of Puiseux series: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[403,409,589,460]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0953/17364bcff87f24ece907057e436793fab65fbd7d7db37ec1d78219be67e5ea89.jpg","list_items":null,"page_idx":"138","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathbb { C } \\{ \\{ t \\} \\} = \\bigcup _ { d = 1 } ^ { \\infty } K ( d ) ;\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,474,881,533]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"138","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"this is an algebraic closure of $K$ . The group $\\operatorname { G a l } ( K ( d ) / K )$ is canonically isomorphic to $\\mu _ { d }$ , the group of $d$ -th roots of unity in $k$ . The group $\\widehat { \\mu }$ is the profinite group of roots of unity in $k$ , it is obtained as $\\varprojlim \\mu _ { d }$ . We have that ${ \\widehat { \\mu } } \\cong \\operatorname { G a l } ( { \\overline { { K } } } / K )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,539,879,575]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"138","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 2.1 (Calabi-Yau Varieties) A Calabi-Yau variety $X$ over $K$ is a smooth, proper, geometrically connected variety with trivial canonical sheaf. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,584,876,620]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"138","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 2.2 (Abelian Variety) An abelian variety $A$ over $K$ is a smooth, proper, geometrically connected commutative group-scheme over $K$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,627,879,683]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"138","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 2.3 All abelian varieties are Calabi-Yau varieties. In fact, since they are group varieties, their tangent bundle is trivial. Thus it follows that the top exterior power of the cotangent bundle is the trivial line bundle. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,692,876,728]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"138","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 2.4 (K3 Surfaces) A K3 surface over $K$ is a 2-dimensional Calabi-Yau variety $X$ over $K$ with $H ^ { 1 } ( X , { \\mathcal { O } } _ { X } ) = 0$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,735,881,791]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"138","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 2.5 If we restrict out attention to surfaces over $\\overline { { K } }$ , it follows by the Enriques-Kodaira classification, that a 2 dimensional Calabi-Yau variety is either an abelian or a K3 surface. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,798,879,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"138","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 2.6 (Model) Let $X$ be a proper and smooth $K$ -scheme; a model for $X$ over $R$ is a flat $R$ -algebraic space $\\mathcal { X }$ endowed with an isomorphism of $K$ -schemes: $\\mathcal { X } _ { K } = \\mathcal { X } \\times _ { R } K \\to X$ . We say that $\\mathcal { X }$ is a strict normal crossing model (snc model) for $X$ if it is regular and proper over $R$ , and $\\mathcal { X } _ { k } = \\mathcal { X } \\times _ { R } k$ is a strict normal crossing divisor on $\\mathcal { X }$ . The variety $\\mathcal { X } _ { k }$ is called special fiber, while $\\mathcal { X } _ { K }$ is called generic fiber. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,719,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"138","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"138","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,88,876,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"139","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The special fiber, under this definition, need not to be reduced. If it is reduced, then the model is called semistable. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,135,881,225]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"139","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 2.7 In this remark we compare this definition of a model to the one that we gave in the introduction. The space SpecR corresponds to the disk $D$ ; the generic fiber corresponds to the degeneration over the punctured disk, while the special fiber corresponds to the fiber over 0. The generic fiber over $\\overline { { K } }$ corresponds to the universal fiber of the degeneration, i.e. the base change to a universal covering space of $D ^ { * }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,232,881,340]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"139","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 2.8 Snc models always exist by Hironaka’s resolution of singularities, while semistable models do not, in general. However, by the semi-stable reduction Theorem [13], given any proper model $\\mathcal { X }$ of $X$ there exist a positive integer $d$ and a semistable model $\\mathcal { X } _ { d }$ of $X \\times _ { K } K ( d )$ that dominates $\\mathcal { X } \\times _ { R } R ( d )$ . If, moreover, $X$ is projective, then also $\\mathcal { X } _ { d }$ may be taken projective. It is important to notice that \b $\\mathcal { X } _ { d }$ is not a model for $X$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,349,881,459]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"139","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Notation 2.9 Choose an snc model $\\mathcal { X }$ for the Calabi-Yau variety $X$ . Express the special fiber as the divisor # $\\mathcal { X } _ { k } ~ = ~ \\sum _ { i \\in I } N _ { i } E _ { i }$ . For the rest of the paper we fix the following notation: for any $\\textit { \\textbf { J } } \\subset \\textit { \\textbf { I } }$ , we define $E _ { J } ~ = ~ \\bigcap _ { j \\in J } E _ { j }$ and $E _ { J } ^ { \\circ } \\ =$ $E _ { J } \\setminus { \\left( \\bigcup _ { i \\not \\in J } { E _ { i } } \\right) } .$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,466,744,486]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"139","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 2.10 Let $X$ be the K3 surface in $\\mathbb { P } _ { K } ^ { 3 }$ given by the equation: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[261,501,733,525]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0962/9dac773462ac603338b6bf42432f91c5c29795294965d4bf06895f35c139a8ed.jpg","list_items":null,"page_idx":"139","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nx ^ { 2 } w ^ { 2 } + y ^ { 2 } w ^ { 2 } + z ^ { 2 } w ^ { 2 } + x ^ { 4 } + y ^ { 4 } + z ^ { 4 } + t w ^ { 4 } = 0 .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,540,881,612]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"139","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $\\mathcal { X }$ be the closed subscheme of $\\mathbb { P } _ { R } ^ { 3 }$ given by the same equation. The scheme $\\mathcal { X }$ is regular, but the special fiber $\\mathcal { X } _ { k }$ is a singular surface. The only singular point is $P = [ 0 , 0 , 0 , 1 ]$ ; the singularity at that point is a canonical singularity of type $A _ { 1 }$ . If we blow up $\\mathcal { X }$ at $P$ , we obtain an snc model for $X$ , call it $\\mathcal { X } ^ { \\prime }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,614,879,687]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"139","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The special fiber of this model is non-reduced, so in particular it is not a semistable model. As a divisor the special fiber of this model is $\\mathcal { X } _ { k } ^ { \\prime } = D + 2 E$ , where $D$ is the proper transform of $\\mathcal { X } _ { k }$ , which is a smooth K3 surface, and $E \\cong \\mathbb { P } _ { k } ^ { 2 }$ . The intersection of $D$ and $E$ is transverse and it is a smooth conic in $E$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,732,708,753]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"139","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 Monodromy Property for Calabi-Yau Varieties ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,777,881,831]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"139","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section we explain what the motivic monodromy property for Calabi-Yau varieties is. First we explain what is the monodromy action; then we introduce the motivic zeta function and explain how they are, conjecturally, related. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[779,54,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"139","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"L. Lunardon ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"139","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,453,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"140","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.1 The Monodromy Action ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,130,881,199]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"140","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recall that we fixed ${ \\overline { { K } } } = \\mathbb { C } \\{ \\{ t \\} \\}$ , the field of Puiseux series; let $\\sigma$ be the canonical topological described as is groupand is cal $\\operatorname { G a l } ( { \\overline { { K } } } / K )$ . The generator odromy operator $\\sigma$ can be $\\begin{array} { r } { \\sigma = \\left( \\exp \\left( \\frac { 2 \\pi i } { d } \\right) \\right) _ { d > 0 } } \\end{array}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,204,881,279]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"140","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3.1 (Monodromy Eigenvalue) If $X$ is a smooth, proper variety over $K$ , then, for all $i$ , the monodromy operator $\\sigma$ acts on the $l$ -adic étale cohomology group $H _ { \\mathit { e t } } ^ { i } ( X \\times _ { K } { \\overline { { K } } } , \\mathbb { Q } _ { l } )$ . We say that $\\lambda$ is a monodromy eigenvalue if there exists an index $i$ such that $\\lambda$ is an eigenvalue for the action of $\\sigma$ on $H _ { { \\acute { e } } t } ^ { i } ( X \\times _ { K } { \\overline { { K } } } , \\mathbb { Q } _ { l } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,286,879,322]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"140","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3.2 (Monodromy Zeta Function) The monodromy zeta function of $X$ is defined as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[209,334,788,384]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0267/b720da0428fab0941a53e24cff7a8bfc6a2bd055ab1dcc3fee4af4badc30eb91.jpg","list_items":null,"page_idx":"140","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\zeta _ { X } ( T ) = \\prod _ { i > 0 } \\left( { \\operatorname * { d e t } \\left( { T \\cdot \\operatorname { I d } - \\sigma | _ { H _ { \\mathit { e t } } ^ { i } ( X \\times _ { K } \\overline { { K } } , \\mathbb { Q } _ { l } ) } } \\right) } \\right) ^ { ( - 1 ) ^ { i + 1 } } \\in \\mathbb { Q } _ { l } ( T ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,396,881,543]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"140","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 3.3 The monodromy zeta function does not encode all the information about the monodromy eigenvalues, in fact, some cancelations may occur. Moreover, there is an even more natural function that encodes all the monodromy eigenvalue— namely, the product of all the characteristic polynomials. One of the main reason why the monodromy zeta function is such a useful tool to study the monodromy eigenvalues is that Theorem 3.5 gives an alternative, and often easier, way to compute this function, while computing all the characteristic polynomials of the monodromy action is usually more complicated. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,549,881,605]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"140","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 3.4 If $X$ is a K3 surface over $K$ , then $H _ { { \\acute { e } } t } ^ { i } ( X \\times _ { K } { \\overline { { K } } } \\mathbb { Q } _ { l } )$ is trivial in odd degrees; as a consequence, the monodromy zeta function of a K3 surface has all the\b monodromy eigenvalues as poles. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,612,881,669]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"140","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3.5 (A’Campo’s Formula, [16] Theorem 2.6.2) Let \u0006 \u0007 $X$ be a smooth and proper $K$ -variety. Fix an snc model $\\mathcal { X }$ for $X$ with special fiber $\\begin{array} { r } { \\mathcal { X } _ { k } = \\sum _ { i \\in I } N _ { i } E _ { i } } \\end{array}$ . Then the monodromy zeta function is given by ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[339,681,656,729]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0894/3ac903e14405c12b72bcd605cbfbc3910afa03758192da86f3f8d0cd23cb05ad.jpg","list_items":null,"page_idx":"140","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\zeta _ { X } ( T ) = \\prod _ { i \\in I } \\left( T ^ { N _ { i } } - 1 \\right) ^ { - \\chi _ { t o p } ( E _ { i } ^ { \\circ } ) } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[111,744,808,764]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"140","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where χt op is the topological Euler characteristic and $E _ { i } ^ { \\circ } = E _ { i } \\setminus \\bigcup _ { j \\neq i } E _ { j }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[113,770,820,791]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"140","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 3.6 If follows from A’Campo formula that $\\zeta _ { X } ( T ) \\in \\mathbb { Q } ( T ) \\subset \\mathbb { Q } _ { l } ( T )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,51,722,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"140","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"140","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"137 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,487,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"141","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.2 The Motivic Zeta Function ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,132,881,205]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"141","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Next we discuss the second main ingredient to understand the monodromy property: the motivic zeta function. First we need some background in motivic integration. Since we want to keep track of the Galois action of $\\operatorname { \\bar { G } a l } ( { \\overline { { K } } } / K )$ , instead of working with the usual Grothendieck ring of varieties, we use an equivariant version of it. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,213,879,250]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"141","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3.7 (Category of $G$ -schemes Over $k$ ) Given any group $G$ , we define the category of $G$ -schemes over $k$ , denote by $\\mathbf { S c h } _ { k , G }$ , as the category that has: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,259,760,295]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"141","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(1) as objects separated $k$ -schemes of finite type with good $G$ -action; \n(2) as morphisms $G$ -equivariant morphisms of $k$ -schemes. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,304,874,340]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"141","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We say that the action of the group $G$ on the scheme $X$ is good if $X$ has a finite partition in $G$ -stable affine subschemes. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,348,881,439]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"141","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 3.8 The definition of good action of the group $G$ on the scheme $X$ that we use is the one given in [9, (2.2.1)], and it is weaker than the one that is commonly used, i.e. that $X$ can be covered by $G$ -stable affine open subschemes. The reason we preferred our definition is that it can be generalized to algebraic spaces, and it gives rise to the same Grothendieck ring as the usual one. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,447,881,539]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"141","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3.9 (Equivariant Grothendieck Ring, [10] Definition 4.1) Fix a finite group $G$ . The equivariant Grothendieck ring of $G$ -varieties over $k$ is the ring generated as an abelian group by the isomorphism classes of objects $X \\in \\mathbf { S c h } _ { k , G }$ , with the ring structure given by the fiber product over $k$ , with the additional relations: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,548,838,566]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"141","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(1) (Scissor relation) Given a $G$ -scheme $X$ and a closed $G$ -subscheme $Y$ , then ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[414,582,621,603]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0453/b0f88a161f5db8c0f178c137f2c49ee0e969d17b3c9f9e08467171cc3b6adb8b.jpg","list_items":null,"page_idx":"141","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n[ X ] = [ Y ] + [ X \\setminus Y ] .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,618,879,656]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"141","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(2) Given $A _ { 1 }$ and $A _ { 2 }$ two $G$ -equivariant affine bundles of rank $d$ over a scheme $S \\in \\mathbf { S c h } _ { k , G }$ we have $[ A _ { 1 } ] = [ A _ { 2 } ]$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,663,380,684]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"141","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$K _ { 0 , k } ^ { G }$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,690,879,764]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"141","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 3.10 An algebraic space $X$ with good $G$ -action defines a class in the equivariant Grothendieck ring. In fact, since $X$ admits a partition in $G$ -stable affine schemes, it is possible to use the scissor relation to construct the class in the equivariant Grothendieck ring. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,771,881,830]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"141","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For our purposes, the ring $K _ { 0 , k } ^ { G }$ is not enough, we have to invert an element, namely the class $\\mathbb { L } = [ \\mathbb { A } ^ { 1 } ]$ , with trivial action of the group $G$ . Thus we define the ring $\\mathcal { M } _ { k } ^ { G }$ to be the ring ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[410,846,589,872]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0401/0bf5cb21ddebb82f06532c6f58f0c26118e2e6aa19e7b9e13e62f580146e0804.jpg","list_items":null,"page_idx":"141","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r } { \\mathcal { M } _ { k } ^ { G } = K _ { 0 , k } ^ { G } [ \\mathbb { L } ^ { - 1 } ] . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[779,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"141","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"L. Lunardon ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"141","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"138 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,879,127]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"142","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If we have a profinite group ${ \\widehat { G } } = \\varprojlim G _ { i }$ , with all the groups $G _ { i }$ finite, then we define: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[416,138,578,169]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0860/77a561fbda99e45b89a3813aa345f30075390715ded431d3957fa2f25702e29a.jpg","list_items":null,"page_idx":"142","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathcal { M } _ { k } ^ { \\widehat { G } } = \\varinjlim \\mathcal { M } _ { k } ^ { G _ { i } }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,180,881,291]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"142","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3.11 (Equivariant Weak Néron Model) Let $X$ be a smooth proper $K$ - scheme. For every $d > 0$ , set $X ( d ) = X \\times _ { K } K ( d )$ . There is an action of $\\mu _ { d }$ on $X ( d )$ . An equivariant weak Néron model for $X ( d )$ is a separated and smooth $R ( d )$ -scheme $\\mathcal { X }$ , with a good action of $\\mu _ { d }$ . Moreover, we require that there exists an isomorphism of $K ( d )$ -schemes ${ \\mathcal { X } } _ { K ( d ) } \\to X ( d )$ which is $\\mu _ { d }$ -equivariant and such that the natural map ${ \\mathcal { X } } ( R ( d ) ) \\to X ( K ( d ) )$ is a bijection. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,298,881,388]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"142","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 3.12 An equivariant weak Néron model always exists, as explained in [9, 2.2.3]. If we have $\\mathcal { X }$ an snc model for a Calabi-Yau variety $X$ , then it is possible to construct an equivariant weak Néron model for $X ( d )$ . We have just to normalize the pullback of $\\mathcal { X }$ on the new base, apply a $\\mu _ { d }$ -equivariant resolution of singularities and then restrict to the $R ( d )$ -smooth locus. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,397,881,543]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"142","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now we are ready to talk about motivic integration. We fix our variety $X$ over $K$ with trivial canonical bundle, and choose a volume form $\\omega$ on $X$ . Denote by $\\omega _ { d }$ the pullback of $\\omega$ to $X ( d )$ . Choose a weak equivariant Néron model $\\mathcal { X }$ for $X ( d )$ . For every connected component $C$ of the special fiber $\\mathcal { X } _ { k }$ , the order of $\\omega _ { d }$ along $C$ is the unique integer $n$ such that $t ^ { - n / d } \\omega _ { d }$ is a generator for the sheaf $\\omega _ { \\mathcal { X } / R ( d ) }$ locally at the generic point of $C$ . For every integer $i$ , let $C ( i )$ the union of the connected components of $\\mathcal { X } _ { k }$ of order $i$ . It is important to remark that $C ( i )$ is stable under the action of $\\mu _ { d }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,551,881,587]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"142","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3.13 (Motivic Integral) With the above notation the motivic integral of $\\omega _ { d }$ on $X ( d )$ is defined as: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[334,600,658,653]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0197/7a42795e2a595061233637fee611dce32691c4e9ea4b7a09bb5d009b65e43d0e.jpg","list_items":null,"page_idx":"142","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\int _ { X ( d ) } | \\omega _ { d } | = \\sum _ { i \\in \\mathbb { Z } } [ C ( i ) ] \\mathbb { L } ^ { - i } \\in \\mathcal { M } _ { k } ^ { \\widehat { \\mu } } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,668,879,704]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"142","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 3.14 ([9], Proposition 2.2.5) The motivic integral of $\\omega _ { d }$ on $X ( d )$ is independent of the choice of the weak Néron model $\\mathcal { X }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,713,881,767]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"142","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3.15 (Motivic Zeta Function) Let ⎛ ⎞ $X$ be a Calabi-Yau variety over $K$ , choose $\\omega$ a volume form on $X$ . The motivic zeta function of\t - $X$ with respect to the volume form $\\omega$ is: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[289,777,710,848]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0978/eeeaaa24e623d7d60ad896ece68843d802e93e9601512227f5a02f4e40e0057b.jpg","list_items":null,"page_idx":"142","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nZ _ { X , \\omega } ( T ) = \\sum _ { d > 0 } \\left( \\intop _ { X \\left( d \\right) } \\left| \\omega _ { d } \\right| \\right) T ^ { d } \\in \\mathcal { M } _ { k } ^ { \\widehat { \\mu } } [ [ T ] ] .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,861,881,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"142","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As in the case of the monodromy zeta function, once we fix an snc model, there exists an alternative way to compute the motivic zeta function. Assume that $\\mathcal { X }$ is an snc model for $X$ , and let $\\begin{array} { r } { \\mathcal { X } _ { k } = \\sum _ { i \\in I } N _ { i } E _ { i } } \\end{array}$ be the special fiber, the reduced special fiber is $\\begin{array} { r } { \\mathcal { X } _ { k , r e d } = \\sum _ { i \\in I } E _ { i } } \\end{array}$ ∈. The volume form $\\omega$ on $X$ defines a rational section of the line bundle $\\omega \\chi _ { / R } ( \\mathcal { X } _ { k , r e d } - \\mathcal { X } _ { k } )$ on $\\mathcal { X }$ . To this section we can associate a divisor supported on the special fiber. This divisor is of the form . $\\textstyle \\sum _ { i \\in I } \\nu _ { i } E _ { i }$ . By the numerical data of $E _ { i }$ we mean the couple $( N _ { i } , \\nu _ { i } )$ . For any subset $J \\subset I$ consider the varieties $E _ { J }$ and $E _ { J } ^ { \\circ }$ , defined as in Notation 2.9, and set $N _ { J } = \\operatorname* { g c d } \\{ N _ { j } | j \\in J \\}$ . Let $\\mathcal { X } ( N _ { J } )$ be the normalization of $\\mathcal { X } \\times _ { R } R ( N _ { J } )$ . Then $\\widetilde { E _ { J } ^ { \\circ } } = \\mathcal { X } ( N _ { J } ) \\times _ { \\mathcal { X } } E _ { J } ^ { \\circ }$ is a Galois cover of $E _ { J } ^ { \\circ }$ (an explicit description of this cover can be found in Section 2.3 of [16]). Now we are ready to give an alternative description of the motivic zeta function. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,51,722,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"142","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"142","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"139 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,271]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"143","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,280,876,316]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"143","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3.16 (Denef and Loeser’s Formula, [2], Corollary 4.3.2) In the above\t . setting we have: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[189,331,810,382]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0693/f2ca1d1ac6c5c4456609b7779ce60a276572d358912f27a431234b3216c06867.jpg","list_items":null,"page_idx":"143","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nZ _ { X , \\omega } ( T ) = \\sum _ { \\theta \\neq J \\subset I } [ \\widetilde { E _ { J } ^ { \\circ } } ] ( \\mathbb { L } - 1 ) ^ { | J | - 1 } \\prod _ { j \\in J } \\frac { \\mathbb { L } ^ { - \\nu _ { j } } T ^ { N _ { j } } } { 1 - \\mathbb { L } ^ { - \\nu _ { j } } T ^ { N _ { j } } } \\in \\mathcal { M } _ { k } ^ { \\widehat { \\mu } } [ [ T ] ]\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,396,881,453]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"143","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is clear from Definition 3.15 that the motivic zeta function is a power series with coefficients in the ring $\\mathcal { M } _ { k } ^ { \\widehat { \\mu } }$ . However, the definition of poles of this function is not immediate. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,456,881,546]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"143","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3.17 (Poles, [17]) Let $\\begin{array} { r } { Z ( T ) \\ \\in \\ \\mathcal { M } _ { k } ^ { \\widehat { \\mu } } \\left[ T , \\frac { 1 } { 1 - \\mathbb { L } ^ { a } T ^ { b } } \\right] _ { ( a , b ) \\in S } } \\end{array}$ be a rational function over $\\mathcal { M } _ { k } ^ { \\widehat { \\mu } }$ , and choose $q \\in \\mathbb { Q }$ . We say that the rational number $q$ is a pole of order at most $m \\geq 1$ for the function $Z ( T )$ if there exists a set $\\mathcal { P }$ whose elements are multisets contained in $\\mathbb { Z } \\times \\mathbb { Z } _ { > 0 }$ such that: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,554,881,612]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"143","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(1) each multiset $P \\in \\mathcal { P }$ contains at most $m$ elements $( a , b )$ such that ${ \\frac { a } { b } } = q$ and (2) $Z ( T )$ is an element in the $\\mathcal { M } _ { k } ^ { \\widehat { \\mu } } [ T ]$ -submodule of $\\mathcal { M } _ { k } ^ { \\widehat { \\mu } } [ [ T ] ]$ generated by the elements of the form ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[419,624,612,671]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0856/a435a40866eca6781a068f25d802297fe31dc21f2821b509773c1f1e1a8c3aad.jpg","list_items":null,"page_idx":"143","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\prod _ { ( a , b ) \\in P } { \\frac { 1 } { ( 1 - \\mathbb { L } ^ { a } T ^ { b } ) } } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[157,686,378,704]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"143","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for all multisets $P \\in \\mathcal { P }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,713,881,749]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"143","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The order of a pole at $q$ is the minimal $m$ such that $Z ( T )$ has a pole at $q$ of order at most $m$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,758,881,890]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"143","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 3.18 The reason why the definition of a pole is so involved is that the ring $\\mathcal { M } _ { k } ^ { \\widehat { \\mu } }$ is not a domain. Actually in [19] it is proven that, if the base field has characteristic zero, not even the usual Grothendieck ring of varieties $K _ { 0 , k }$ is a domain. Moreover, if $k = \\mathbb { C }$ it was proven in [1] that $\\mathbb { L }$ is a zero-divisor in $K _ { 0 , k }$ . It was proven in [7] that $\\mathcal { M } _ { k } = K _ { 0 , k } [ \\mathbb { L } ^ { - 1 } ]$ is not a domain, the background to this result is presented in Appendix A of [3]; since $\\mathcal { M } _ { k }$ injects into $\\mathcal { M } _ { k } ^ { \\hat { \\mu } }$ (as varieties with trivial $\\widehat { \\mu }$ action) it follows that the latter is not a domain either. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[779,54,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"143","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"L. Lunardon ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,152,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"143","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"140 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,876,124]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"144","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"At this point we have all the background needed to explain what the monodromy property for Calabi-Yau varieties is. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,133,879,207]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"144","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3.19 ([9], Definition 2.3.5) Given a Calabi-Yau variety $X$ over $K$ and a volume form $\\omega$ on $X$ we say that the couple ' ( $( X , \\omega )$ satisfies the monodromy property if there exists a finite set $S \\subset \\mathbb { Z } \\times \\mathbb { Z } _ { > 0 }$ such that $Z _ { X , \\omega } ( T )$ is an element of the sub-ring ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[307,220,690,265]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0821/580c61c42f1413625ed4f22c559d5f120a1a903d4bad7992b33d27b7693117cf.jpg","list_items":null,"page_idx":"144","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\mathcal { M } _ { k } ^ { \\widehat { \\mu } } \\left[ T , \\frac { 1 } { 1 - \\mathbb { L } ^ { a } T ^ { b } } \\right] _ { ( a , b ) \\in S } \\subset \\mathcal { M } _ { k } ^ { \\widehat { \\mu } } [ [ T ] ] ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,280,881,337]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"144","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and, for any couple $( a , b ) \\ \\in \\ S$ , we have that $\\textstyle \\exp { \\bigl ( } 2 \\pi { \\sqrt { - 1 } } { \\frac { a } { b } } { \\bigr ) }$ is a monodromy eigenvalue as in definition 3.1; which means that it is an eigenvalue of the action of $\\operatorname { G a l } ( { \\overline { { K } } } / K )$ on $H _ { e t } ^ { i } ( X \\times _ { K } { \\overline { { K } } } , \\mathbb { Q } _ { l } )$ for some $i \\geq 0$ and every embedding of $\\mathbb { Q } _ { l }$ in $\\mathbb { C }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,381,505,402]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"144","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 Some Well-Understood Cases ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,426,881,642]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"144","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We are far from a complete understanding of whether or not Calabi-Yau varieties satisfy the monodromy property. It was proven for certain families, under some more restrictive hypothesis on the type of Calabi-Yau variety or on the type of degeneration. The conjecture was first proven for abelian surfaces in [8], this proof uses the theory of Néron models. A generalization of this result was obtained in [9]; in this paper it was proven that degenerations of Calabi-Yau varieties which admit equivariant Kulikov models satisfy the monodromy property and that abelian varieties admit such models. Using the aforementioned result of [9], it was also possible to prove the monodromy property for Kummer surfaces: it was showed in [18] that they admit equivariant Kulikov models. All the examples we mentioned have in common that the motivic zeta function has a unique pole. This is not by\b chance, in fact, we have the following, more general, result. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,651,879,746]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"144","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 4.1 ([9, Theorem 3.3.3]) Let $X$ be a Calabi-Yau variety of dimension $n$ over $K$ , with a volume form ω. Choose an snc model for $X$ , let $\\begin{array} { r } { \\mathcal { X } _ { k } = \\sum _ { i } N _ { i } E _ { i } } \\end{array}$ be the special fiber of this model and denote with $\\nu _ { i }$ the vanishing order of $\\omega$ on $E _ { i }$ . $\\begin{array} { r } { L e t \\operatorname* { m i n } ( \\omega ) = \\operatorname* { m i n } _ { i } \\frac { \\nu _ { i } } { N _ { i } } } \\end{array}$ . Then exp $\\left( - 2 \\pi i \\operatorname* { m i n } ( \\omega ) \\right)$ ) is an eigenvalue of the action of $\\sigma$ on $H _ { e t } ^ { n } ( X \\times _ { K } { \\overline { { K } } } , \\mathbb { C } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,755,881,881]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"144","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For any Calabi-Yau varieties $X$ and any volume form $\\omega$ on $X$ , we have that $1 - \\operatorname* { m i n } ( \\omega )$ is always a pole of $Z _ { X , \\omega }$ . Indeed, in all the cases we mentioned at the beginning of this section, it was proven that $1 - \\operatorname* { m i n } ( \\omega )$ is the unique pole, and since $\\exp ( - 2 \\pi i \\operatorname* { m i n } ( \\omega ) )$ is a monodromy eigenvalue, then the monodromy property holds. In Sect. 5 we present a family of degenerations of K3 surfaces that satisfies the monodromy property, but whose motivic zeta functions may have more than one pole, most of these results are the work of [11] and [12]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,719,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"144","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"144","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"141 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"145","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 4.2 ([9, Theorem 4.2.2] ) Fix an abelian variety A with a volume form $\\omega$ . Choose an snc model $\\mathcal { A }$ , and let' $\\begin{array} { r } { \\mathcal { A } _ { k } = \\sum _ { i } N _ { i } E _ { i } } \\end{array}$ be the special fiber. We denote with $\\nu _ { i }$ the vanishing order of $\\omega$ on $E _ { i }$ . The motivic zeta function $Z _ { A , \\omega } ( T )$ has $a$ unique pole at $q = 1 - \\operatorname* { m i n } ( \\omega )$ . More precisely: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[225,175,774,220]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0099/b7cc313c7eae395075086f47f25823a22aee0254a341e58377c2128f7d0485a2.jpg","list_items":null,"page_idx":"145","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nZ _ { A , \\omega } ( T ) \\in \\mathcal { M } _ { k } ^ { \\widehat { \\mu } } \\left[ T , \\frac { 1 } { 1 - \\mathbb { L } ^ { a } T ^ { b } } \\right] _ { ( a , b ) \\in S ; a / b = q } \\subset \\mathcal { M } _ { k } ^ { \\widehat { \\mu } } [ [ T ] ] .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[123,234,876,270]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"145","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It follows that the monodromy property holds for abelian varieties from Theorems 4.2 and 4.1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,270,876,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"145","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now we are ready to introduce the definition of equivariant Kulikov models for Calabi-Yau varieties. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,315,881,369]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"145","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 4.3 (Equivariant Kulikov Model) Let $X$ be a Calabi-Yau variety over $K$ , and fix a positive integer $d$ . A Kulikov model for $X$ over $R ( d )$ is a regular, proper and flat algebraic space $\\mathcal { X }$ over $R ( d )$ such that: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,378,551,396]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"145","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(1) there is an isomorphism of $K ( d )$ -schemes: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[318,414,710,500]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0929/31030d739e4572b61c1e78e13af6a98e2480be0c67aff328a11121589078d6d8.jpg","list_items":null,"page_idx":"145","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l r } { \\mathcal { X } _ { K ( d ) } \\xrightarrow { \\sim \\sim \\sim X \\times _ { K } K ( d ) } \\xrightarrow { \\sim X } X } & { { } \\mathrm { ~ , ~ } } & { } \\\\ { \\bigoplus _ { \\substack { \\operatorname * { s p e c } ( K ( d ) ) \\xrightarrow [ ] { } \\sim \\operatorname { S p e c } K } } \\mathcal { Y } } & { { } \\quad } & { } \\\\ { \\operatorname { S p e c } ( K ( d ) ) \\xrightarrow [ ] { } \\operatorname { S p e c } K } & { { } \\quad } & { } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,519,826,557]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"145","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(2) the special fiber $\\mathcal { X } _ { k }$ is a divisor with normal crossing (3) the logarithmic relative canonical bundle $\\omega \\chi _ { / R ( d ) } ( \\mathcal { X } _ { k , r e d } - \\mathcal { X } _ { k } )$ is trivial. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,564,876,600]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"145","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We say that the Kulikov model $\\mathcal { X }$ is equivariant if the Galois action of $\\mu _ { d }$ on $X \\times _ { K } K ( d )$ extends to $\\mathcal { X }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,609,881,719]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"145","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 4.4 Not all snc models are Kulikov models; for instance, the model we constructed in Example 2.10 is not a Kulikov model over $K$ . It is easy to check that the K3 surface of Example 2.10 admits a Kulikov model over $K ( 2 )$ , however, we will show in Example 5.8 that this model is not equivariant. Indeed, in Example 5.8 we show something stronger, i.e. that the K3 surface of Example 2.10 does not admit an equivariant Kulikov model for any $d$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,725,881,801]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"145","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 4.5 ([9, Theorem 5.3.2]) Let $X$ be a Calabi-Yau variety, with a volume form ω. Assume that $X$ admits an equivariant Kulikov model over ' ( $R ( d )$ for some positive $d$ . Then the motivic zeta function of $( X , \\omega )$ has a unique pole at $q = 1 -$ $\\begin{array} { r } { \\operatorname* { m i n } _ { i } \\frac { \\nu _ { i } } { N _ { i } } } \\end{array}$ . More precisely: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[227,816,772,863]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0213/6a45d45e52ebde4dbcd369da15989464c4df1d6802e33f12f15e3606ee7675cc.jpg","list_items":null,"page_idx":"145","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nZ _ { X , \\omega } ( T ) \\in \\mathcal { M } _ { k } ^ { \\widehat { \\mu } } \\left[ T , \\frac { 1 } { 1 - \\mathbb { L } ^ { a } T ^ { b } } \\right] _ { ( a , b ) \\in S , a / b = q } \\subset \\mathcal { M } _ { k } ^ { \\widehat { \\mu } } [ [ T ] ] .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[779,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"145","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"L. Lunardon ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,152,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"145","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"142 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,879,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"146","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 4.6 As in the previous case, the monodromy property for Calabi-Yau varieties admitting an equivariant Kulikov model follows from Theorems 4.5 and 4.1. This result generalize Theorem 4.2, in fact it was proven that abelian varieties admit an equivariant Kulikov model in [9, Theorem 5.1.6]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,171,876,205]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"146","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Another consequence of Theorem 4.5 is that the monodromy property holds for-Kummer surfaces, this was proven in [18].- ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,216,879,304]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"146","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 4.7 (Kummer Surface) Let $A$ be an abelian surface over $K$ , consider an involution $\\iota$ , and call $A _ { \\iota }$ the fixed point scheme of $\\iota$ . Let $\\widetilde { A }$ be the blow of $A$ at $A _ { \\iota }$ ; the involution $\\iota$ acts regularly on $\\dot { A }$ . Call $X$ the quotient of $\\widetilde { A }$ by the action of $\\iota$ ; $X$ is a smooth K3 surface over $K$ . Any K3 surface that can be obtained in this way is called a Kummer surface. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,315,879,385]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"146","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 4.8 ([18, Theorem 6.2]) Let $X$ be a Kummer surface, then there exists a minimal $d _ { 0 } > 0$ such that $X ( d _ { 0 } )$ has an equivariant Kulikov model, moreover, if $d \\ > \\ 0$ is such that $X ( d )$ admits an equivariant Kulikov model, then $d _ { 0 } | d$ . In particular, Kummer surfaces satisfy the monodromy property. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,396,879,466]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"146","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 4.1 was also used in [9] to prove the monodromy property for some Calabi-Yau varieties which were not in any of these families. As far as we know, the cases we listed in this section are the only ones where Theorem 4.1 was used to prove the monodromy property. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,513,708,555]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"146","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 Triple-Points-Free Models of K3 and Why they are Interesting ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,578,881,668]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"146","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section we discuss triple-points-free models of K3 surfaces. If a K3 surface admits a triple-points-free model, it satisfies the monodromy property, however, the motivic zeta function of this K3 surface may have more then a single pole. In particular K3 surfaces admitting a triple-points-free model may not admit an equivariant Kulikov model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,669,881,812]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"146","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In [4] Crauder and Morrison described special the fiber of relatively minimal, triple-points-free snc models of surfaces with trivial pluricanonical bundles. Many additional results to describe the combinatorics of the special fiber were given by Jaspers in [12]. The motivic zeta function and the monodromy property for these surfaces were studied in [11] and [12]. In Corollary 4.2.4 of [12] it was shown that poles of the motivic zeta function may be recovered from the combinatorial data of the model, in particular it was shown that besides $1 - \\operatorname* { m i n } ( \\omega )$ , there are additional poles as soon as the triple-points-free model has a so-called conic flower. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,813,879,884]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"146","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Regarding the monodromy property, it was proven that under some additional conditions on the special fiber, K3 surfaces admitting a triple-points-free model satisfy the monodromy property. In Appendix B of [12] there is a classification of the possible special fibers of triple-points-free models that do not satisfy the monodromy property; such surfaces are known as combinatorial countercandidates. In [15] we proved that these combinatorial countercandidates do not exist. As a consequence, the monodromy property holds for K3 admitting triple-points-free models. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,719,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"146","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"146","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"147","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,169,879,225]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"147","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 5.1 (Triple-Points-Free Model) Given a K3 surface $X$ , a triple-pointsfree model $\\mathcal { X }$ of $X$ is a relatively minimal snc model such that given any three distinct irreducible components $E _ { i } , E _ { j }$ , $E _ { k }$ of the special fiber $\\mathcal { X } _ { k }$ , ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[407,241,589,262]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0483/8bad81a2d70f0cf9a04a0b0cef8bba63bbe72af939d972dc760b82e8eb3fc7a0.jpg","list_items":null,"page_idx":"147","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nE _ { i } \\cap E _ { j } \\cap E _ { k } = \\emptyset .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,277,881,315]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"147","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since there are not triple intersections, the dual complex of the special fiber of this model is a graph, we call it $\\Gamma$ ; to each vertex of $\\Gamma$ we associate the weight ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[432,328,562,364]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0891/5e00c990411023c2c8b51938b5130f59197b466dc53c4d91a404705464386534.jpg","list_items":null,"page_idx":"147","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\rho _ { i } = \\frac { \\nu _ { i } } { N _ { i } } + 1 ;\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,378,779,397]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"147","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"we denote by $\\Gamma _ { \\mathrm { m i n } }$ the subgraph of $\\Gamma$ of components of minimal weight. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,405,876,441]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"147","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There is a very explicit description of possible special fibers of triple-points-free models for K3 surfaces. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,450,879,506]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"147","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 5.2 (Crauder-Morrison Classification for K3 Surfaces, [4] and [12]) Let X be a smooth, proper $K 3$ surface over $K$ , let $\\mathcal { X }$ be a relatively minimal triplepoints-free model of $X$ , then $\\mathcal { X }$ has the following properties: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,513,760,533]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"147","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(1) $\\Gamma _ { \\mathrm { m i n } }$ is a connected subgraph of \b. It is either a vertex or a chain. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,533,879,639]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"147","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(2) Each connected component of $\\Gamma \\backslash \\Gamma _ { \\mathrm { m i n } }$ is a chain (called flower) $F _ { 0 } - F _ { 1 } - \\cdot \\cdot \\cdot -$ $F _ { l }$ where only $F _ { l }$ meets $\\Gamma _ { \\mathrm { m i n } }$ . The weights strictly decrease along these flowers, $F _ { 0 }$ being the one with maximal weight. The surface $F _ { 0 }$ is either minimal ruled or isomorphic to $\\mathbb { P } ^ { 2 }$ . If it is isomorphic to $\\mathbb { P } ^ { 2 }$ then $F _ { 0 } \\cap F _ { 1 }$ is either a line or a conic. The other components are minimal ruled surfaces, and $F _ { i } \\cap F _ { i + 1 }$ and $F _ { i } \\cap F _ { i - 1 }$ are both sections of the ruling. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,641,881,711]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"147","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(3) If $\\Gamma _ { \\mathrm { m i n } }$ is a single vertex, there are three possible cases for the corresponding surface. It is either isomorphic to $\\mathbb { P } ^ { 2 }$ or it is a ruled surface or the canonical divisor is numerically trivial. If $\\Gamma _ { \\mathrm { m i n } }$ is a point, we call the model a flowerpot degeneration. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,713,881,801]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"147","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(4) If $\\Gamma _ { \\mathrm { m i n } }$ is a chain $V _ { 0 } - V _ { 1 } - \\cdot \\cdot \\cdot - V _ { k + 1 }$ , then we can describe the components of the chain. I $f i \\neq 0 , k + 1 ,$ , $V _ { i }$ is an elliptic ruled surface, and $V _ { i - 1 } \\cap V _ { i }$ and $V _ { i } \\cap V _ { i + 1 }$ are both sections of the ruling; $i f i = 0 , k + 1$ , then $V _ { i }$ is isomorphic to $\\mathbb { P } ^ { 2 }$ , or it is $a$ , rational or elliptic, ruled surface. If $\\Gamma _ { \\mathrm { m i n } }$ is a chain we call the model a chain degeneration. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,810,881,902]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"147","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 5.3 In [4] there is an even stronger result. Indeed, they classified the special fiber of triple-points-free models with generic fiber $\\mathcal { X } _ { K }$ with trivial pluricanonical bundle. In [4] there is also a complete classification of the possible flowers, divided in 21 combinatorial classes. If $\\mathcal { X } _ { K }$ is a K3 surface, this classification was refined in Chapter 3 of [12]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[779,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"147","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"L. Lunardon ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"147","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"144 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"148","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"One of the main results of [12] is the description of the motivic zeta functions of these models. This description is extremely explicit and using it, it was possible to study the poles. It turned out that poles of the motivic zeta function are closely related to the presence of conic flowers (i.e. $F _ { 0 }$ is isomorphic to $\\mathbb { P } ^ { 2 }$ and $F _ { 0 } \\cap F _ { 1 }$ is a conic). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,187,881,261]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"148","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 5.4 ([12, Theorem 4.3.8]) Let $X$ be a $K 3$ surface over $K$ with a volume form ω. Choose a triple-points-free model $\\mathcal { X }$ , whose special fiber is $\\begin{array} { r } { \\mathcal { X } _ { k } = \\sum _ { i } N _ { i } E _ { i } } \\end{array}$ , with numerical data $( N _ { i } , \\nu _ { i } )$ . Then $q \\in \\mathbb { Q }$ is a pole of $Z _ { X , \\omega } ( T )$ if and only if there exists an $i$ such that the numerical data of $E _ { i }$ satisfy $q = - \\nu _ { i } / N _ { i }$ and such that: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,270,462,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"148","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(1) either $\\rho _ { i }$ is minimal (2) or $E _ { i }$ is the top of a conic flower. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,315,879,367]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"148","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moreover, while in the second case the pole is always of order 1, in the first one it is of order $I$ if $\\mathcal { X }$ is a flowerpot degeneration and of order 2 if it is a chain degeneration. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,378,874,412]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"148","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 5.5 In [12, Appendix A] there is a Python code that describes the contribution of each flower to the motivic zeta function. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,421,881,531]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"148","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 5.6 In [12] it was also discussed whether or not a K3 surface $X$ admitting a triple-points-free model $\\mathcal { X }$ satisfies the monodromy property. In [12, Theorem 5.2.1] it was proven that flowerpot degenerations satisfy the monodromy property. In [12, Theorem 5.3.1] it was proven that chain degenerations with some extra assumptions satisfy the monodromy property. However, it was not clear if those assumptions were enough to prove the monodromy property for any chain degeneration. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,531,881,710]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"148","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The strategy adopted in [12] was to try to prove the conjecture by contradiction. The assumption was that there exists a chain degeneration that does not satisfy the monodromy property, this mean that some poles of the motivic zeta function of this degeneration do not correspond to monodromy eigenvalues. The first step was to use A’Campo’s formula and Denef and Loeser’s formula to deduce some information on the geometry of the special fiber of this triple-points-free model. The last step would have been to prove that these surfaces—called combinatorial countercandidates— do not appear as the special fiber of any triple-points-free model. This would have proven the monodromy property for all K3 surfaces admitting a triple-points-free model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,711,879,782]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"148","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In [12, Section 6] there is a description of some topological properties of these combinatorial countercandidates, and this description become even more explicit in [12, Appendix B], where all the possible countercandidates are listed. The kind of information we have about the numerical countercandidates is: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[113,791,815,810]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"148","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(1) the central fiber of the degeneration is a chain degeneration, the chain is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[412,827,626,846]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0472/c843f2cca4a856936b94dc62f811a97a6f576689a25d7648053dbcd47be230ae.jpg","list_items":null,"page_idx":"148","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nV _ { 0 } - V _ { 1 } - \\cdot \\cdot \\cdot - V _ { k + 1 } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,52,719,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"148","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"148","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,605,108]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"149","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(2) The surface $V _ { 0 }$ satisfies the following properties: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[157,115,879,225]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"149","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(a) it contains a smooth elliptic curve $D$ such that $D = - K _ { V _ { 0 } }$ ; \n(b) it is a smooth rational ruled non-minimal surface obtained from an Hirzebruch surface by $l _ { 0 }$ blow-ups whose centers lies in the image of $D$ ; \n(c) it contains at least $h$ smooth rational curves of self intersection $^ { - 2 }$ , we denote them by $C _ { i }$ . The curves $C _ { i }$ are all disjoint, moreover from the adjunction formula it follows that they do not intersect $D$ ; ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,232,881,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"149","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(3) the other internal components of the chain are elliptic or rational ruled surfaces, and they are obtained from minimal surface by some blow-up along the intersection of two components. Moreover, we know that there is a certain number of $^ { - 2 }$ rational curves, disjoint from the sections $V _ { i } \\cap V _ { i + 1 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,315,881,369]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"149","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Whether or not any combinatorial countercandidate exists was left as an open problem in [12]. It was proven in [15] that they do not exist, all the information required to prove this result are the one regarding the surface $V _ { 0 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,378,876,414]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"149","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 5.7 ([12, 15]) Let $X$ be a $K 3$ surface that admits a triple-points-free model $\\mathcal { X }$ . Then the monodromy property holds for $X$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,421,838,441]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"149","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To conclude this section we come back to the quartic surface of Example 2.10. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,448,867,468]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"149","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 5.8 ([9, 11, 12]) Let $\\mathrm { X }$ be the K3 surfaces in $\\mathbb { P } _ { K } ^ { 3 }$ , given by the equation: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[261,483,733,507]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0361/e220eee700cd14e56fa8e0fb09f7c28c4efc15e23bad1b6f12bf7d53d446b1e2.jpg","list_items":null,"page_idx":"149","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nx ^ { 2 } w ^ { 2 } + y ^ { 2 } w ^ { 2 } + z ^ { 2 } w ^ { 2 } + x ^ { 4 } + y ^ { 4 } + z ^ { 4 } + t w ^ { 4 } = 0 .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,522,881,650]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"149","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From Example 2.10 we already know that there exists a model $\\mathcal { X } ^ { \\prime }$ such that the special fiber is $\\mathcal { X } _ { \\mathbb { C } } ^ { \\prime } = D + 2 E$ , where $D$ is a K3 surface and $E \\cong \\mathbb { P } _ { \\mathbb { C } } ^ { 2 }$ . The curve $C = D \\cap E$ is a smooth conic in $E$ ; thus $E$ is a conic flower of type 2B. We can now choose a volume form on $\\mathrm { X }$ (for instance the natural volume form induced by the embedding in $\\mathbb { P } ^ { 3 }$ ). This form extends to a relative volume form on $\\mathcal { X } ^ { \\prime }$ . We can now compute the numerical data of this relative volume form and we obtain- - - $\\nu _ { D } = 0$ and $\\nu _ { E } = 1$ . The motivic zeta function of this K3 is: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[125,663,870,705]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0069/380d890f0d550b67df66138a84ad1e10a8e694721526e9bcd2faed6840083136.jpg","list_items":null,"page_idx":"149","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nZ _ { X , \\omega } ( T ) = [ \\widetilde { D } ^ { \\circ } ] \\frac { T } { 1 - T } + [ \\widetilde { E } ^ { \\circ } ] \\frac { \\mathbb { L } ^ { - 1 } T ^ { 2 } } { 1 - \\mathbb { L } ^ { - 1 } T ^ { 2 } } + [ \\widetilde { C } ] ( \\mathbb { L } - 1 ) \\frac { \\mathbb { L } ^ { - 1 } T ^ { 3 } } { ( 1 - T ) ( 1 - \\mathbb { L } ^ { - 1 } T ^ { 2 } ) } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,723,740,743]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"149","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The motivic zeta function of this K3 has two poles, namely 0 and $\\frac { 1 } { 2 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,750,881,804]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"149","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 5.9 The quartic surface of Example 5.8 does not admit any equivariant Kulikov model. This follows from Theorem 4.5, since its motivic zeta function has two poles. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[779,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"149","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"L. Lunardon ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,152,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"149","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"146 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,307,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"150","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 What’s Next ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,133,879,205]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"150","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this section we describe a possible approach to future research in this area. Our idea is to construct further examples of K3 surfaces whose motivic zeta function has more than one pole. Once we have done this, the natural question is whether or not these K3 surfaces satisfy the monodromy property. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,205,881,331]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"150","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From the results of Theorem 5.4 it is clear that, in the case of triple-pointsfree models, additional poles come only from conic flowers. However, it is not immediately clear what distinguishes these flowers geometrically. To obtain some understanding of this, we tried to study what happens to the model if we contract some flowers. The idea of contracting flowers already appears in [5], in this paper there is also a list of the singularities caused by the contraction of some family of flowers. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,333,881,456]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"150","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From the computations of Example 5.8, we can see that the contraction of the flower of type 2B gives a regular model with a special fiber which is irreducible but with a singular point. The situation with non-conic flowers is different, L. Halle showed me that, up to a finite base change, it is possible to contract them smoothly— by this we mean that the resulting model is an snc model. This suggested to us that there might be a relation between poles of the motivic zeta function and singularities of the models. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,459,881,636]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"150","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Assume now that $X$ is a K3 surface over $K$ , $\\omega$ a volume form on $X$ and let $\\mathcal { X }$ be a model for $X$ . To $X$ we can associate a motivic zeta function $Z _ { X , \\omega }$ ; however, it may not be possible to use Denef and Loeser’s formula with the model $\\mathcal { X }$ . In general it is not regular, and even if it were, the irreducible components of the special fiber $\\mathcal { X } _ { k }$ might be singular. Of course, by Hironaka’s resolution of singularities it is possible to construct an snc model $\\mathcal { V }$ which dominates $\\mathcal { X }$ , but what we would like to understand is whether or not it is possible to deduce the presence of additional poles of $Z _ { X , \\omega }$ from the singularities of $\\mathcal { X }$ . This problem, however, is very generic, and thus pretty tough to approach. So we had to restrict the class of singularities we would consider. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,647,879,737]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"150","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 6.1 (Rational Double Points) Given $X$ a normal surface over $\\mathbb { C }$ , we say that a point $x \\in X$ is a rational double point (or ADE or Du Val singularity) if it is a canonical singularity. Given a rational double point singularity, we have that étale locally around the point the surface is isomorphic to the closed subset of $\\mathbb { A } ^ { 3 }$ given by one of the following equations: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,744,719,836]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"150","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(1) $x ^ { 2 } + y ^ { 2 } + z ^ { n + 1 } = 0$ ; these are $A _ { n }$ singularities; \n(2) $x ^ { 2 } + y ^ { 2 } z + z ^ { n - 1 } = 0$ , with $n > 3$ ; these are $D _ { n }$ singularities; \n(3) $x ^ { 2 } + y ^ { 3 } + z ^ { 4 } = 0$ ; this is the $E _ { 6 }$ singularity; \n(4) $x ^ { 2 } + y ^ { 3 } + z ^ { 3 } y = 0$ ; this is the $E _ { 7 }$ singularity; \n(5) $x ^ { 2 } + y ^ { 3 } + z ^ { 5 } = 0$ ; this is the $E _ { 8 }$ singularity. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,845,881,899]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"150","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Fix a K3 surface $X$ which admits a regular model $\\mathcal { X }$ whose special fiber $\\mathcal { X } _ { k }$ has some rational double points (for instance, the K3 surface of Example 5.8). Assume, furthermore, that none of the rational double points lie the intersection of some irreducible components of the special fiber and call the set of rational double points S. Then we can construct an snc model $\\mathcal { V }$ for $X$ , with a morphism $\\pi \\colon \\mathcal { V } \\mathcal { X }$ blowing-up $\\mathcal { X }$ at some smooth points and at some smooth curves contained in $\\mathcal { X } _ { k }$ . Once we have constructed such an snc model, it is possible to compute the motivic zeta function using Denef and Loeser’s formula. We can split the motivic zeta function $Z _ { X , \\omega }$ in two parts, the one which depends on the strata contained in $\\pi ^ { - 1 } ( { \\mathcal { X } } _ { k } \\setminus S )$ and the one which depends on the strata contained in $\\pi ^ { - 1 } ( S )$ . We denote the second part by $Z _ { X , \\omega , s i n g }$ , and we refer to it as the contribution of the singularities to $Z _ { X , \\omega }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,719,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"150","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"150","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,252]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"151","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,259,876,279]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"151","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 6.2 For instance, in Example 5.8, the contribution of the $A _ { 1 }$ singularity is: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[191,292,801,336]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0745/a38e94cadbcd7cd175077178521a38010bf49c943b546ac7a0c61edb0d330fd8.jpg","list_items":null,"page_idx":"151","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nZ _ { X , \\omega , s i n g } = [ \\widetilde { E } ^ { \\circ } ] \\frac { \\mathbb { L } ^ { - 1 } T ^ { 2 } } { 1 - \\mathbb { L } ^ { - 1 } T ^ { 2 } } + [ \\widetilde { C } ] ( \\mathbb { L } - 1 ) \\frac { \\mathbb { L } ^ { - 1 } T ^ { 3 } } { ( 1 - T ) ( 1 - \\mathbb { L } ^ { - 1 } T ^ { 2 } ) } ;\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,349,879,402]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"151","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"since the motivic zeta function in this example has more than one pole, then $X$ can’t have good reduction over $K$ ; in particular we have that the action of $\\operatorname { G a l } ( { \\overline { { K } } } / K )$ is not trivial. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,412,881,521]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"151","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We are studying the contributions of ADE singularities to the motivic zeta function. In the last part of this paper we briefly discuss some of the questions we would like to answer. Questions 6.3, 6.4, and 6.5 may be considered short term goals, while Question 6.7 is much wilder, and should be considered a long term goal. Question 6.9 was for us a further motivation to study this topic, and a natural follow-up question to Question 6.7. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,530,879,584]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"151","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Question 6.3 Assume that $X$ is a K3 surface that admits a regular model whose special fiber has rational double points, is it true that the motivic zeta function of $X$ has at least two poles? If not, for which singularities is this true? ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,584,876,620]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"151","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A positive answer to this question would provide more examples of Calabi-Yau varieties whose motivic zeta function has multiple poles. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,629,876,665]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"151","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Question 6.4 In the above setting, can we describe explicitly the contributions of the various singularities? ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,674,879,728]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"151","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Having an explicit description of the motivic zeta function would be not only interesting by itself, but it is a necessary step in a better understanding of the monodromy property for K3 surfaces. This brings us to the next question. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,737,876,773]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"151","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Question 6.5 In the above setting, does $X$ satisfy the monodromy property? If not, which obstructions are encountered? ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,782,879,834]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"151","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Requiring that the model $\\mathcal { X }$ is regular is a strong assumption; we would like to relax this hypothesis. The natural class of threefold singularities that extends Du Val singularities is known as compound Du Val singularities. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,845,879,899]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"151","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 6.6 (Compound Du Val Singularities) Given $X$ a normal threefold over $\\mathbb { C }$ , we say that a point $x \\in X$ is a compound Du Val singularity if for some general hyperplane section $H$ through $x$ we have that $x \\in H$ is a Du Val singularity. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[779,54,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"151","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"L. Lunardon ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,54,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"151","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,879,142]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"152","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Question 6.7 Assume that $X$ is a K3 surface that admits a model $\\mathcal { X }$ whose singularities are compound Du Val singularities. What can we say of the analogs of Questions 6.3, 6.4, 6.5 under these less restrictive hypothesis? ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,151,772,169]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"152","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There exists a criterion to show that a K3 surface $X$ has good reduction. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,178,876,216]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"152","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 6.8 ([14], Theorem 6.1) Let $X$ be a $K 3$ surface over $K$ , then $X$ has good reduction if and only if the action of $\\operatorname { \\overrightarrow { { r a l } } } ( \\overline { { K } } / K )$ on $H _ { \\mathit { e t } } ^ { 2 } ( X _ { \\overline { { K } } } , \\mathbb { Q } _ { l } )$ is trivial. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,223,876,259]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"152","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An interesting problem is to find sufficient conditions to prove that a K3 surface does not have good reduction. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,270,881,358]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"152","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Question 6.9 Assume that $X$ is a K3 surface that admits a model $\\mathcal { X }$ whose singularities are compound Du Val singularities or a regular model whose special fiber has rational double points. Is that enough to prove that $X$ does not have good reduction? If $X$ satisfies the monodromy property, can this obstruction be detected from the motivic zeta function? ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,379,879,439]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"152","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acknowledgements I would like to thank Prof. J. Nicaise for his help during the preparation of this paper. This work was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1]. The EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,486,248,506]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"152","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[125,530,881,887]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"152","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Borisov, L.A.: The class of the affine line is a zero divisor in the Grothendieck ring. J. Algebraic Geom. 27(2), 203–209 (2018) \n2. Bultot, E., Nicaise, J.: Computing zeta functions on log smooth models. C. R. Math. Acad. Sci. Paris 353(3), 261–264 (2015) \n3. Cauwbergs, T.: Splicing motivic zeta functions. Rev. Mat. Complut. 29(2), 455–483 (2016) \n4. Crauder, B., Morrison, D.R.: Triple-point-free degenerations of surfaces with Kodaira number zero. In: The Birational Geometry of Degenerations (Cambridge, Mass, 1981). Progress in Mathematics, vol. 29, pp. 353–386. Birkhäuser, Boston (1983) \n5. Crauder, B., Morrison, D.R.: Minimal models and degenerations of surfaces with Kodaira number zero. Trans. Am. Math. Soc. 343(2), 525–558 (1994) \n6. Denef, J., Loeser, F.: Motivic Igusa zeta functions. J. Algebraic Geom. 7(3), 505–537 (1998) \n7. Ekedahl, T.: The Grothendieck group of algebraic stacks. arXiv:0903.3143 [math.AG] (2009) \n8. Halvard Halle, L., Nicaise, J.: Motivic zeta functions of abelian varieties, and the monodromy conjecture. Adv. Math. 227(1), 610–653 (2011) \n9. Halvard Halle, L., Nicaise, J.: Motivic zeta functions of degenerating Calabi-Yau varieties. Math. Ann. 370(3–4), 1277–1320 (2018) \n10. Hartmann, A.: The quotient map on the equivariant Grothendieck ring of varieties. Manuscripta Math. 151(3–4), 419–451 (2016) \n11. Jaspers, A.: The global monodromy property for $K 3$ surfaces allowing a triple-point-free model. C. R. Math. Acad. Sci. Paris 355(2), 200–204 (2017) \n12. Jaspers, A.: The Monodromy Property for K3 Surfaces Allowing a Triple-Point-Free Model (2017). Ph. D. Thesis, KU Leuven. arXiv:1706.07086 \n13. Kempf, G. Knudsen, F.F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings. I. Lecture Notes in Mathematics, vol. 339. Springer, Berlin (1973) \n14. Liedtke, C., Matsumoto, Y.: Good reduction of K3 surfaces. Compos. Math. 154(1), 1–35 (2018) \n15. Lunardon, L.: The monodromy property for certain families of K3 surfaces and higher dimesional Calabi-Yau varieties, Ph.D. Thesis, Imperial College London (2020) \n16. Nicaise, J.: Geometric criteria for tame ramification. Math. Z. 273(3–4), 839–868 (2013) \n17. Nicaise, J., Xu, C.: Poles of maximal order of motivic zeta functions. Duke Math. J. 165(2), 217–243 (2016) \n18. Overkamp, O.: Degeneration of Kummer surfaces. arXiv:1806.10105 [math.AG] (2018) \n19. Poonen, B.: The Grothendieck ring of varieties is not a domain. Math. Res. Lett. 9(4), 493–497 (2002) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,719,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"152","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"152","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,91,881,240]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"153","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[779,54,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"153","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"L. Lunardon ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,52,152,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"153","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"150 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,676,139]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"154","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Examples of Irreducible Symplectic Varieties ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,175,250,193]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"154","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Arvid Perego ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,286,881,412]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"154","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract Irreducible symplectic manifolds are one of the three building blocks of compact Kähler manifolds with numerically trivial canonical bundle by the Beauville-Bogomolov decomposition theorem. There are several singular analogues of irreducible symplectic manifolds, in particular in the context of compact Kähler orbifolds, and in the context of normal projective varieties with canonical singularities. In this paper we will collect their definitions, analyze their mutual relations and provide a list of known examples. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,432,824,465]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"154","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords Irreducible symplectic varieties $\\cdot$ Moduli spaces of sheaves $\\cdot$ K3 surfaces ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,504,309,525]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"154","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,549,881,638]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"154","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A central problem in complex geometry is the classification of Ricci-flat compact Kähler manifolds. By Yau’s theorem [51], these are exactly the compact Kähler manifolds whose first Chern class is zero in $H ^ { 2 } ( X , \\mathbb { R } )$ or, equivalently, whose canonical bundle is numerically trivial. This implies that the Kodaira dimension is zero. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,639,879,710]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"154","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In dimension 1, compact Kähler manifolds of Kodaira dimension zero are exactly elliptic curves. The birational classification of compact complex surfaces shows that compact Kähler surfaces of Kodaira dimension zero are K3 surfaces, 2-dimensional complex tori, Enriques surfaces and bielliptic surfaces. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,711,879,764]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"154","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Among compact Kähler manifolds of Kodaira dimension zero, a very special role is played by those manifolds whose canonical bundle is trivial (which are sometimes called Calabi-Yau manifolds). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,834,630,878]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"154","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. Perego $( \\boxtimes )$ Dipartimento di Matematica, Università di Genova, Genova, Italy e-mail: perego@dima.unige.it ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,897,692,941]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"154","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_9 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[804,91,876,139]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"154","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,899,876,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"154","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,196]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"155","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The first family of examples is given by complex tori, i.e. quotients of a complex vector space $V$ of dimension $n$ by a rank $2 n$ lattice $\\Gamma$ in $V$ . A projective complex torus is called Abelian variety: in dimension 1 all complex tori are projective, and they are exactly the elliptic curves. In higher dimension there are complex tori which are not projective. In any case, if $X$ is a complex torus of dimension $n$ , then $\\pi _ { 1 } ( X ) \\simeq$ $\\mathbb { Z } ^ { 2 n }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,198,881,304]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"155","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A second family of examples is given by special unitary manifolds, i.e. compact Kähler manifolds with trivial canonical bundle, whose every finite étale covering has no non-trivial holomorphic $p$ -form for $0 ~ < ~ p ~ < ~ n$ (where $n$ is the complex dimension). They are projective for $n \\geq 3$ , and have finite fundamental group, which is trivial if $n$ is even (see Proposition 2 of [2]). Simply connected special unitary manifolds are called irreducible Calabi-Yau manifolds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,306,881,412]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"155","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A third family of examples is given by irreducible symplectic manifolds, that will be our main interest. We recall that if $X$ is a complex manifold, a holomorphic symplectic form on $X$ is a closed, holomorphic 2-form $\\sigma$ on $X$ which is everywhere non-degenerate. A holomorphic symplectic manifold is a complex manifold admitting a holomorphic symplectic form. Among holomorphic symplectic manifolds we find all even-dimensional complex tori. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,414,881,504]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"155","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $X$ is a holomorphic symplectic manifold, then its complex dimension is even. If $2 n$ is the complex dimension of $X$ , and $\\sigma$ is a holomorphic symplectic form on $X$ , then $\\sigma ^ { n }$ is a nowhere vanishing section of $K _ { X }$ : it follows that a holomorphic symplectic manifold has trivial canonical bundle. Moreover, as $\\sigma$ is closed, then it defines a nontrivial cohomology class in $H ^ { 0 } ( X , \\Omega _ { X } ^ { 2 } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,513,879,549]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"155","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 1 An irreducible symplectic manifold is a compact, Kähler, holomorphic symplectic manifold $X$ which is simply connected and such that $h ^ { 2 , 0 } ( X ) = 1$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,557,876,594]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"155","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark $I$ As shown in Propositions 3 and 4 of [2], for an irreducible symplectic manifold $X$ of dimension $n$ and for every $0 \\leq p \\leq n$ we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[369,609,626,651]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0546/8c02461fecb4973f964bf2160918d6d2edc690c5f490c3aa91167064ec0948ad.jpg","list_items":null,"page_idx":"155","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nh ^ { p , 0 } ( X ) = { \\left\\{ \\begin{array} { l l } { 0 , { \\mathrm { i f ~ } } p { \\mathrm { ~ i s ~ o d d } } } \\\\ { 1 , { \\mathrm { o t h e r w i s e } } } \\end{array} \\right. }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,666,879,720]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"155","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conversely, by Proposition A.1 of [21] every compact, Kähler, holomorphic symplectic manifold $X$ of complex dimension $n$ such that $h ^ { p , 0 } ( X )$ is as in (1) is simply connected, and hence an irreducible symplectic manifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,729,879,801]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"155","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 2 As shown in Proposition 4 of [2], a compact Kähler manifold of complex dimension $2 n$ is a holomorphic symplectic manifold if and only if its holonomy group is contained in the symplectic group $S p ( r )$ . The holonomy group is precisely $S p ( r )$ if and only if $X$ is an irreducible symplectic manifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,810,879,864]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"155","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As already recalled, Ricci-flat compact Kähler surfaces are K3 surfaces, complex tori of dimension 2, Enriques surfaces and bielliptic surfaces. Among them, K3 surfaces are both irreducible Calabi-Yau and irreducible symplectic. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,866,876,902]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"155","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An Enriques surface is a finite quotient of a K3 surface by a fixed point free involution, while a bielliptic surface is quotient of an Abelian surface (product of two elliptic curves) by the free action of a finite Abelian group. It follows that a Ricci-flat compact Kähler surface has a finite étale covering which is either an irreducible symplectic (Calabi-Yau) surface or a complex torus of dimension 2. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"155","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"152 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[799,54,879,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"155","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,879,142]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"156","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,144,881,252]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"156","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The same phenomenon occurs for higher dimensional Ricci-flat compact Kähler manifolds, giving to complex tori, irreducible Calabi-Yau manifolds and irreducible symplectic manifolds a special role in the classification. This is the content of the following, which goes under the name of Beauville-Bogomolov Decomposition Theorem, and whose proof (based on several results of differential geometry) is contained in [6] and [2]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,261,879,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"156","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 1 Let $X$ be a Ricci-flat compact Kähler manifold. Then $X$ has a finite étale covering $f : Y \\longrightarrow X$ , where $Y$ is a product of complex tori, irreducible Calabi-Yau manifolds and irreducible symplectic manifolds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,324,881,378]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"156","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"While it is not difficult to provide examples of complex tori and irreducible Calabi-Yau manifolds, it is a hard problem to give examples of irreducible symplectic manifolds. The list of all the known deformation classes is very short: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,387,881,584]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"156","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Irreducible symplectic surfaces are exactly K3 surfaces. \n2. For $n \\geq 2$ , the Hilbert scheme $H i l b ^ { n } ( S )$ of $n$ points on a K3 surface $s$ is an irreducible symplectic manifold of dimension $2 n$ (see Théorème 3 of [2]). \n3. For $n \\geq 2$ , let $T$ be a complex torus of dimension 2 and $s : H i l b ^ { n + 1 } ( T ) \\longrightarrow T$ the sum morphism. Then $K u m ^ { n } ( T ) : = s ^ { - 1 } ( 0 _ { S } )$ , called generalized Kummer variety, is an irreducible symplectic manifold of dimension $2 n$ (see Théorème 4 of [2]). \n4. Two more deformation classes, $O G _ { 6 }$ of dimension 6 and $O G _ { 1 0 }$ of dimension 10, were constructed by O’Grady (in [44] and [43] respectively) as a symplectic resolution of some singular moduli spaces of semistable sheaves on a projective K3 or on an Abelian surface. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,594,879,647]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"156","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The previous examples form different deformation classes because even if they can have the same dimension, they have different second Betti number (which is, following the previous ordering, 22, 23, 7, 8 and 24, see [2, 43, 44] and [48]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,656,881,782]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"156","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 3 Different constructions of examples of irreducible symplectic manifolds were presented in [4, 9, 22, 45] (deformation equivalent to $H i l b ^ { 2 } ( K 3 ) )$ and [28] (deformation equivalent to $H i l b ^ { 4 } ( K 3 ) )$ . Moduli spaces of stable sheaves or of Bridgeland stable complexes on projective K3 surfaces or on Abelian surfaces give rise to examples of irreducible symplectic manifolds which are deformation equivalent to Hilbert schemes of points on K3 surfaces or to generalized Kummer varieties on Abelian surfaces. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,792,881,899]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"156","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 1 is stated only for compact Kähler manifolds. Anyway, if $X$ is a complex projective manifold with Kodaira dimension zero, the minimal model $Y$ of $X$ (whose conjectural existence is predicted by the Minimal Model Program) is a projective variety which is birational to $X$ , and has terminal singularities and nef canonical divisor. Assuming the abundance conjecture, it follows that a multiple of the canonical divisor $K _ { Y }$ of $Y$ is trivial. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"156","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,471,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"156","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Examples of Irreducible Symplectic Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,881,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"157","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For the classification of projective varieties whose Kodaira dimension is 0 it is then central to extend Theorem 1 to normal projective varieties having terminal singularities and torsion (i.e. numerically trivial by Theorem 8.2 of [26]) canonical divisor. This implies the need for a definition of singular analogues of irreducible Calabi-Yau and symplectic manifolds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,180,881,252]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"157","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Various definitions have been proposed and studied over the years, and the main purpose of this survey is to present a list of definitions of irreducible Calabi-Yau and symplectic varieties which can be found in the literature, together with a list of known example. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,297,537,318]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"157","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 Irreducible Symplectic Varieties ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,340,881,412]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"157","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There are two main settings we will consider: orbifolds and varieties with canonical singularities. The second one is more natural for the purposes of the Minimal Model Program, while the first one has the advantage to be more similar to the smooth case (and it was the first one to be considered). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,459,287,478]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"157","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.1 Orbifolds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,503,879,557]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"157","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A first generalization of the decomposition theorem was obtained by Campana in [8], in the setting of orbifolds. We refer the reader to [10] or to [15] (where orbifolds are called $V .$ -manifolds) for precise definitions and results on these analytic spaces. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,558,879,665]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"157","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An orbifold is a connected, para-compact analytic space $X$ such that for every $x \\in X$ there is an open neighborhood $U$ of $x$ in $X$ , an open subset $V$ of $\\mathbb { C } ^ { n }$ (with respect to the Euclidean topology) and a finite group $G$ of automorphisms of $V$ such that there is an isomorphism $\\phi : V / G \\longrightarrow U$ . The composition $\\pi : V \\longrightarrow U$ of the projection $V \\longrightarrow V / G$ with $\\phi$ is called uniformization map, and the triple $( V , G , \\pi )$ a uniformizing system for $U$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,666,874,701]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"157","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It follows from Proposition 1.3 of [5] and by Proposition 5.15 of [27] that an orbifold is normal, $\\mathbb { Q }$ -factorial, Cohen-Macaulay and has rational singularities. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,702,881,810]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"157","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On an orbifold $X$ of dimension $n$ one can then always find a uniformizing open cover $\\{ U _ { i } \\} _ { i \\in I }$ , i.e. for each $i \\in I$ there is a uniformizing system $( V _ { i } , G _ { i } , \\pi _ { i } )$ for $U _ { i }$ . The sheaf of differential $p$ -forms on $X$ is denoted $\\mathcal { A } _ { X } ^ { p }$ , and by definition it is the sheaf which restricted to $U _ { i }$ is $\\pi _ { i * } ( \\mathcal { A } _ { V _ { i } } ^ { p } ) ^ { G _ { i } }$ , i.e. the push-forward under $\\pi _ { i }$ of the $G _ { i }$ -invariant part of the sheaf of differential $p$ -forms on $V _ { i }$ . Similarly one defines the sheaf of holomorphic $p$ -forms on $X$ , denoted $\\Omega _ { X } ^ { p }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,810,879,864]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"157","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A differential 2-form $\\omega \\in \\mathcal { A } _ { X } ^ { 2 } ( X )$ is a Kähler form for $X$ if for every $x \\in X$ and every uniformizing system $( V , G , \\pi )$ of an open neighborhood $U$ of $x$ , we have that $\\omega _ { | U } \\in ( \\mathcal { A } _ { V } ^ { 2 } ) ^ { G } ( V )$ is a Kähler form on $V$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,866,876,900]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"157","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moreover, one can still define the notion of first Chern class for orbifolds: the canonical sheaf $K _ { X }$ (which, restricted to every uniformized open subset $U$ with uniformizing system $( V , G , \\pi )$ , is just $\\pi _ { * } K _ { V }$ ) is not in general locally free, but $K _ { X } ^ { m }$ is if $m$ is a multiple of $| G |$ . It follows that if $X$ is compact, then there is $m \\gg 0$ such that $K _ { X } ^ { m }$ is locally free, and one defines $\\begin{array} { r } { c _ { 1 } ( X ) : = - \\frac { \\bar { 1 } } { m } c _ { 1 } ( K _ { X } ^ { m } ) } \\end{array}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,54,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"157","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[799,54,879,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"157","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,879,144]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"158","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,145,879,196]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"158","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As shown by Theorem 1.1 of [8], if $X$ is a compact, connected Kähler orbifold such that $c _ { 1 } ( X ) = 0$ in $H ^ { 2 } ( X , \\mathbb { R } )$ , every Kähler class is represented by a unique Ricci-flat Kähler metric on $X$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,232,605,250]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"158","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.1.1 The Decomposition Theorem for Orbifolds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[123,270,879,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"158","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As in the smooth case, one has two important kinds of Ricci-flat compact Kähler orbifolds which arise. The first one generalizes irreducible Calabi-Yau manifolds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,313,881,367]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"158","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 2 An irreducible Calabi-Yau orbifold is a compact, connected Kähler orbifold $X$ having simply connected smooth locus, $K _ { X } \\simeq O _ { X }$ and $H ^ { 0 } ( X , \\Omega _ { X } ^ { p } ) = 0$ for every $0 < p < \\dim ( X )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,378,881,448]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"158","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By Proposition 6.6 of [8] an irreducible Calabi-Yau orbifold of dimension $n$ is equivalently a compact, connected Kähler orbifold whose smooth locus is simply connected, and which has a Ricci-flat Kähler metric whose holonomy group is $S U ( n )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,450,881,557]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"158","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The second kind of Ricci-flat compact Kähler orbifolds is a generalization of irreducible symplectic manifolds. A closed holomorphic 2-form on $X$ , i.e. a global section of $\\Omega _ { X } ^ { 2 }$ , which is everywhere non-degenerate is called symplectic form. An orbifold admitting a holomorphic symplectic form is called holomorphic symplectic orbifold. As for holomorphic symplectic manifolds, a holomorphic symplectic orbifold has even complex dimension and trivial canonical sheaf. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,567,881,621]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"158","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3 An irreducible symplectic orbifold is a compact, connected Kähler orbifold which is holomorphic symplectic, has simply connected smooth locus and $h ^ { 0 } ( X , \\Omega _ { X } ^ { p } ) = 1$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,630,879,702]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"158","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By Proposition 6.6 of [8] an irreducible symplectic orbifold of dimension $2 n$ is equivalently a compact, connected Kähler orbifold whose smooth locus is simply connected, and which has a Ricci-flat Kähler metric whose holonomy group is $S p ( n )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,702,874,738]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"158","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The decomposition theorem for Ricci-flat compact Kähler orbifolds is the following (see Théorème 6.4 of [8]): ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,746,879,801]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"158","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2 Let $X$ be a Ricci-flat compact Kähler orbifold. Then $X$ has a finite quasi-étale covering $f : Y \\longrightarrow X$ , where $Y$ is a product of complex tori, irreducible Calabi-Yau orbifolds and irreducible symplectic orbifolds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,810,879,864]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"158","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A finite quasi-étale morphism is a finite morphism which is étale in codimension 1. The statement of Théorème 6.4 of [8] is more precise about the finite quasi-étale covering: it is indeed a finite orbifold covering (see Définition 5.1 of [8]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,864,876,900]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"158","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The proof of Theorem 2 relies first on a generalization to orbifolds of the de Rham decomposition theorem (Proposition 5.4 of [8]), which provides a decomposition of the universal orbifold covering $\\widetilde { X }$ of $X$ as a product $M _ { 0 } \\times M _ { 1 } \\times \\cdots \\times M _ { k }$ , where $M _ { 0 }$ is a Euclidean space, and $M _ { 1 } , \\dots , M _ { k }$ have all irreducible holonomy representation. The orbifold version of the Cheeger-Gromoll theorem is provided by Borzellino and Zhu (see [7]), and implies that $M _ { 1 } , \\dots , M _ { k }$ are all compact and their Ricci curvature is zero. The remaining part of the proof is similar to the one for compact manifolds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,471,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"158","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Examples of Irreducible Symplectic Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"158","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,881,199]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"159","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,235,487,253]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"159","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.1.2 Related Notions and Examples ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,271,849,289]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"159","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There are other definitions which are related to irreducible symplectic orbifolds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,298,879,369]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"159","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 4 A primitively symplectic V-manifold is a holomorphic symplectic orbifold $X$ such that $h ^ { 2 , 0 } ( X ) \\ = \\ 1$ . An irreducible symplectic V-manifold is a compact, connected Kähler holomorphic symplectic orbifold $X$ which is simply connected and such that $h ^ { 2 , 0 } ( X ) = 1$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,379,879,468]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"159","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Primitively symplectic V-manifolds are introduced in (section 2.1, [15]), while irreducible symplectic V-manifolds are introduced in (Definition 1.3.(iv), [30]). Clearly an irreducible symplectic V-manifold is a compact, connected Kähler primitively symplectic V-manifold which is simply connected. Moreover, an irreducible symplectic orbifold is an irreducible symplectic V-manifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,471,881,542]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"159","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We will discuss in the following several examples of irreducible symplectic V-manifolds/orbifolds of dimension 4 and 6 which appear in the literature. In particular, we will see that irreducible symplectic V-manifolds are not always irreducible symplectic orbifolds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,578,448,594]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"159","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Symmetric Products of K3 Surfaces ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,614,881,722]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"159","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $s$ is a K3 surface and $m \\ \\geq \\ 2$ , the symmetric product $X \\ : = \\ S y m ^ { m } ( S )$ is an irreducible symplectic $\\mathrm { V } .$ -manifold. Indeed, it is a compact, connected Kähler orbifold having $Y \\ : = \\ H i l b ^ { m } ( S )$ as a resolution of the singularities. As $Y$ is an irreducible symplectic manifold, it is simply connected and has $h ^ { 2 , 0 } ( Y ) \\ =$ 1: it follows that $X$ is simply connected and $h ^ { 2 , 0 } ( X ) \\ = 1$ (see as instance Proposition 2.13 of [15]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,723,876,756]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"159","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We notice that the smooth locus $X ^ { s }$ of $X$ is not simply connected, since $\\pi _ { 1 } ( X ^ { s } ) \\simeq$ $\\Sigma _ { m }$ . It follows that $X$ is not an irreducible symplectic orbifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,794,287,812]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"159","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Fujiki’s Examples ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,830,879,884]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"159","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Tables I and $\\mathrm { I I }$ in section 13 of [15] present a list of 18 examples of irreducible symplectic V-manifolds of dimension 4 constructed as follows: let $s$ be a K3 surface or a 2-dimensional complex torus, $H$ a finite group acting symplectically on $s$ (i.e. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,54,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"159","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[799,54,879,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"159","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"160","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"each element $h \\in H$ preserves the holomorphic symplectic form of $s$ ) and $\\tau$ an automorphism of $H$ of order 2. Then $H$ acts on $S \\times S$ via the action mapping $( h , \\left( s , t \\right) )$ to $( h ( s ) , \\tau ( h ) ( t ) )$ . Let $G ( H )$ be the subgroup of $A u t ( S \\times S )$ generated by $H$ and by the involution $\\iota$ mapping $( s , t )$ to $( t , s )$ , and consider $Y : = S \\times S / G ( H )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,162,881,250]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"160","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The fixed locus of the action of $G ( H )$ may have 2-dimensional components, and apart from that it has only isolated points: the singular locus of $Y$ is then given by a (possibly empty) 2-dimensional locus $\\Sigma$ and by a finite number of points. Blowing up $\\Sigma$ , one gets a $\\mathrm { V } .$ -manifold $X$ whose singular locus is given by a finite number of points. Theorem 13.1 of [15] shows that $X$ is an irreducible symplectic V-manifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,252,881,363]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"160","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We reproduce here Tables I and $\\mathrm { I I }$ for the convenience of the reader: here $b _ { 2 }$ is the second Betti number of the variety, and $a _ { k }$ is the number of singular points of type $\\widehat { A } _ { k } = ( \\mathbb { C } ^ { 4 } / g _ { k } , 0 )$ , where $g _ { k } = ( \\zeta _ { k } , \\zeta _ { k } , \\zeta _ { k } ^ { - 1 } , \\zeta _ { k } ^ { - 1 } )$ and $\\zeta _ { k } : = e ^ { \\frac { 2 \\pi i } { k } }$ (the sum $a _ { 2 } + a _ { 3 } + a _ { 4 } + a _ { 6 }$ is the number of singular points of the variety). The varieties $X _ { p }$ are obtained starting from a K3 surface $s$ , while the varieties $Y _ { n }$ are obtained starting from a 2-dimensional complex torus. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[259,409,738,756]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0612/03d34fa0df96f0bb4d725c2cb0f41ddd22363529baaaebb51379b19812144989.jpg","list_items":null,"page_idx":"160","sub_type":null,"table_body":"<table><tr><td rowspan=1 colspan=1>Symbol</td><td rowspan=1 colspan=1>H</td><td rowspan=1 colspan=1>b</td><td rowspan=1 colspan=1>a2</td><td rowspan=1 colspan=1>a3</td><td rowspan=1 colspan=1>a4</td><td rowspan=1 colspan=1>a6</td></tr><tr><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>ids</td><td rowspan=1 colspan=1>23</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>L</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>Z/2Z</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>D</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>Z/2Z)2</td><td rowspan=1 colspan=1>14</td><td rowspan=1 colspan=1>36</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>D</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X4</td><td rowspan=1 colspan=1>Z/2Z)3</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X5</td><td rowspan=1 colspan=1>Z/3Z</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>15</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X6</td><td rowspan=1 colspan=1>（Z/2Zz)2</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X7</td><td rowspan=1 colspan=1>Z/4Z</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X8</td><td rowspan=1 colspan=1>(Z/4Z)2</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X9</td><td rowspan=1 colspan=1>Z/6Z</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>X10</td><td rowspan=1 colspan=1>(Z/2Z)(Z/4Z)</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X11</td><td rowspan=1 colspan=1>(Z/2Z)(Z/6Z)</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Y1</td><td rowspan=1 colspan=1>Z/3Z</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>36</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Y</td><td rowspan=1 colspan=1>(Z/3Z)2</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>27</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Y</td><td rowspan=1 colspan=1>Z/3Z)3</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Y4</td><td rowspan=1 colspan=1>Z/4Z</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>54</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Y5</td><td rowspan=1 colspan=1>(Z/4Z)(Z/2Z)</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>52</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Y</td><td rowspan=1 colspan=1>(Z/4Z) (Z/2Z)2</td><td rowspan=1 colspan=1>14</td><td rowspan=1 colspan=1>48</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Y7</td><td rowspan=1 colspan=1>Z/6Z</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>35</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr></table>","table_caption":"[\"Table 1 Fujiki’s examples \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[120,776,881,848]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"160","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In all the previous examples, the singular locus of $S \\times S / G ( H )$ has always at least one irreducible component of dimension 2: we have 1 component for $X _ { 1 } , X _ { 5 }$ , $X _ { 6 } , Y _ { 1 }$ , $Y _ { 2 }$ and $Y _ { 3 }$ ; 2 components for $X _ { 2 }$ , $X _ { 7 }$ , $X _ { 9 }$ , $Y _ { 4 }$ and $Y _ { 7 }$ ; 4 components for $X _ { 3 }$ , $X _ { 8 }$ , $X _ { 1 0 }$ , $X _ { 1 1 }$ and $Y _ { 5 }$ ; 8 components for $X _ { 4 }$ and $Y _ { 6 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,849,876,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"160","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The varieties $X _ { 1 }$ and $Y _ { 3 }$ are the only smooth examples we get, and they are both irreducible symplectic manifolds: $X _ { 1 }$ is the Hilbert scheme of two points on $s$ , while ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,471,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"160","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Examples of Irreducible Symplectic Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"160","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,91,876,127]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"161","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$Y _ { 3 }$ is deformation equivalent to $K u m ^ { 2 } ( T )$ (see Remark 13.2.3 and Proposition 14.3 of [15]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,129,881,235]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"161","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $p \\neq 1$ and $n \\neq 3$ , then $X _ { p }$ and $Y _ { n }$ are singular symplectic varieties whose singular locus has codimension 4: by Corollary 1 of [39] they all have terminal singularities, and by the Main Theorem of [41] their deformations are locally trivial. It follows that if $p \\neq q$ and $( p , q ) \\neq ( 2 , 4 )$ then $X _ { p }$ and $X _ { q }$ are not deformation equivalent; if $n \\neq m$ then $Y _ { n }$ is not deformation equivalent to $Y _ { m }$ , and for every $p , n$ we have that $X _ { p }$ is not deformation equivalent to $Y _ { n }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,237,881,325]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"161","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is not known if $X _ { 2 }$ and $X _ { 4 }$ give different deformation classes. Anyway, the Fujiki examples provide at least 17 different deformation classes of irreducible symplectic V-manifolds in dimension 4, 15 of which are singular. The values of $b _ { 2 }$ of these examples are 7, 8, 10, 11, 14, 16 and 23. It is still an open problem to determine if $X _ { p }$ and $Y _ { n }$ are irreducible symplectic orbifolds if $p \\neq 1$ and $n \\neq 3$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,361,348,379]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"161","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Quotients of $H i l b ^ { n } ( K 3 )$ ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,397,881,506]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"161","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We now consider quotients of $H i l b ^ { n } ( S )$ where $s$ is a projective K3 surface. We start with the case $n = 2$ , so that the fixed locus $F i x ( G )$ of $G$ is a union of a finite number of points (depending on the order of $G$ ) and a (possibly empty) holomorphic symplectic surface, and the same holds for the singular locus of $H i l b ^ { 2 } ( S ) / G$ . By blowing up the singular surface one then gets an irreducible symplectic V-manifold $M _ { G }$ (see [15]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,507,881,596]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"161","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A particular case is when $G = \\langle \\phi \\rangle$ , where $\\phi$ is a symplectic automorphism of prime order. By Corollary 2.13 of [36], the order $p$ of $\\phi$ can only be 2, 3, 5, 7 or 11. The case of $p = 2$ , i.e. $\\phi$ is a symplectic involution, is studied in [30] and [32]. The case $p = 3$ is studied in [32]. The case $p = 1 1$ is studied in [33]. The cases $p = 5 , 7$ are treated in [35]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,633,851,723]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"161","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 1 (Quotient by a Symplectic Involution) If $G ~ = ~ \\langle \\phi \\rangle$ where $\\phi$ is a symplectic involution, then $M _ { 2 } \\ : = \\ M _ { G }$ is an irreducible symplectic Vmanifold with 28 isolated singular points, and by Theorem 2.5 of [32] we have $b _ { 2 } ( M _ { 2 } ) = 1 6$ . By Corollary 3.23 of [34] and Proposition 5.1 of [30] we have that the topological Euler number of $M _ { 2 }$ is 268. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,725,851,795]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"161","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It follows that $M _ { 2 }$ is not deformation equivalent to any of the Fujiki examples above: the only examples in Table 1 having 28 singular points are $X _ { 2 }$ and $X _ { 4 }$ (for both of which $b _ { 2 } = 1 6$ ), whose topological Euler number is 226 (see Remark 13.2.4 of [15]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,54,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"161","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[799,54,879,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"161","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[145,109,854,180]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"162","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 2 (Quotient by a Natural Symplectic Automorphism of Order 3) If ${ \\cal G } ~ = ~ \\langle \\phi \\rangle$ where $\\phi$ is a natural symplectic automorphism of order 3, then $F i x ( G )$ is given by 27 isolated points. We get that $M _ { 3 } : = M _ { G }$ is an irreducible symplectic V-manifold with 27 isolated singular points. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,183,851,252]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"162","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This example is then not deformation equivalent to $M _ { 2 }$ , and neither to any of the Fujiki examples in Table 1: the only one of that list having 27 singular points is $Y _ { 2 }$ , whose $b _ { 2 } = 7 .$ . But by Theorem 1.3 of [33] we have $b _ { 2 } ( M _ { 3 } ) = 1 1$ , so $M _ { 3 }$ is a new deformation class. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,309,854,399]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"162","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example 3 (Quotient by a Symplectic Automorphism of Order $1 l$ ) Examples 4.5.1 and 4.5.2 of [36] provide two K3 surfaces $S _ { 1 }$ and $S _ { 2 }$ such that $H i l b ^ { 2 } ( S _ { i } )$ has an automorphism $\\sigma _ { i }$ of order 11. Let $G _ { i } : = \\langle \\sigma _ { i } \\rangle$ : in both cases $F i x ( G _ { i } )$ is given by 5 isolated points. Then $M _ { 1 1 } ^ { i } : = M _ { G _ { i } }$ is an irreducible symplectic V-manifold with 5 singular points. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,399,851,435]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"162","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By [33] we have $b _ { 2 } ( M _ { 1 1 } ^ { i } ) = 3$ and that $M _ { 1 1 } ^ { 1 }$ and $M _ { 1 1 } ^ { 2 }$ are not deformation equivalent, so they provide two more deformation classes. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,472,881,633]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"162","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $G = \\phi$ where $\\phi$ has order 5, then $M _ { 5 } : = M _ { G }$ is an irreducible symplectic V-manifold with 14 singular points and $b _ { 2 } ( M _ { 5 } ) = 7$ . If $G = \\phi$ where $\\phi$ has order 7, then $M _ { 7 } : = M _ { G }$ is an irreducible symplectic V-manifold with 9 singular points and $b _ { 2 } ( M _ { 7 } ) = 5$ (see Theorems 1.2 and 1.3 of [35] and Examples 4.3.1 and 4.4.1 of [36]). We then get 6 more deformation classes of singular irreducible symplectic V-manifolds of dimension 4 to add to the 15 classes presented by Fujiki. If we now consider a K3 surface $s$ and $H i l b ^ { n } ( S )$ for $n \\geq 3$ , let $\\phi$ be a symplectic involution on $H i l b ^ { n } ( S )$ . The quotient $H _ { n }$ of $H i l b ^ { n } ( S )$ by the action of $\\phi$ is again an irreducible symplectic $V$ -manifold (see [15], or Lemma 1 below). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,635,876,671]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"162","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We are in the position to describe which of these examples are irreducible symplectic orbifolds, as the following shows: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,680,876,716]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"162","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 1 Let $s$ be a projective $K 3$ surface and $G$ a finite group of automorphisms of $H i l b ^ { n } ( S )$ acting symplectically. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,723,879,759]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"162","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. If the codimension of $F i x ( G )$ is at least 4, then $M _ { G } : = H i l b ^ { 2 } ( S ) / G$ is not an irreducible symplectic orbifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,761,881,833]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"162","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. If $n = 2$ and $G$ is generated by a symplectic involution, then the singular locus of $H i l b ^ { 2 } ( S ) / G$ is given by 28 isolated points and a K3 surface $\\Sigma$ . Then the partial resolution $M _ { G }$ of $H i l b ^ { 2 } ( S ) / G$ obtained by blowing-up $\\Sigma$ is an irreducible symplectic orbifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,842,879,896]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"162","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In particular $M _ { 2 }$ is an irreducible symplectic orbifold, while $M _ { 3 }$ , $M _ { 5 }$ , $M _ { 7 }$ , $M _ { 1 1 } ^ { 1 }$ , $M _ { 1 1 } ^ { 2 }$ and $H _ { n }$ are irreducible symplectic $V$ -manifolds which are not irreducible symplectic orbifolds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,471,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"162","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Examples of Irreducible Symplectic Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"162","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,879,196]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"163","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Suppose first that $F i x ( G )$ has codimension at least 4 in $H i l b ^ { n } ( S )$ , so the singular locus of $M _ { G }$ has codimension at least 4. We know that $M _ { G }$ is an irreducible symplectic V-manifold: we let $U$ be its smooth locus and $f : H i l b ^ { n } ( S ) \\longrightarrow M _ { G }$ the quotient morphism. Hence $f : f ^ { - 1 } ( U ) \\longrightarrow U$ is a finite étale covering whose degree is the order of $G$ : hence $U$ is not simply connected, and $M _ { G }$ is not an irreducible symplectic orbifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,196,881,286]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"163","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We notice that for $M _ { 3 } , M _ { 5 } , M _ { 7 } , M _ { 1 1 } ^ { 1 }$ and $M _ { 1 1 } ^ { 2 }$ we have that $F i x ( G )$ is given by isolated points (see Examples 2 and 3 above, and Examples 4.3.1 and 4.4.1 of [36]), so they are not irreducible symplectic orbifolds. The fixed locus $F i x ( G )$ in the case of $H _ { n }$ has codimension at least 4 by Theorem 1.1 of [24], so that $H _ { n }$ is not an irreducible symplectic orbifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,288,879,360]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"163","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The case of $n = 2$ and $G$ generated by a symplectic involution is due to Menet (see Remark 3.22 of [34]). We need to show that the smooth locus $U$ of $M _ { G }$ is simply connected. Let $D$ be the exceptional divisor of $M _ { G }$ coming from the blowup $b : M _ { G } \\longrightarrow X _ { G } : = H i l b ^ { 2 } ( S ) / G$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,360,881,484]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"163","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now, let $\\Sigma ^ { \\prime }$ be the 2-dimensional component of the singular locus of $X _ { G }$ , and $U ^ { \\prime }$ the smooth locus of $X _ { G }$ . We notice that the blow-up morphism $^ b$ gives an isomorphism between $U \\backslash D$ and $U ^ { \\prime }$ . Moreover the quotient morphism $f :$ $H i l b ^ { 2 } ( S ) \\longrightarrow X _ { G }$ gives a double étale covering $H i l b ^ { 2 } ( S ) \\setminus F i x ( G ) \\longrightarrow U ^ { \\prime }$ . As $F i x ( G )$ has codimension 2 in $H i l b ^ { 2 } ( S )$ , and $H i l b ^ { 2 } ( S )$ is smooth, it follows that $\\pi _ { 1 } ( H i l b ^ { 2 } ( S ) ) \\simeq \\pi _ { 1 } ( H i l b ^ { 2 } ( S ) \\setminus F i x ( G ) )$ , so that $H i l b ^ { 2 } ( S ) \\setminus F i x ( G )$ is simply connected. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,486,879,557]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"163","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As a consequence, we see that $\\pi _ { 1 } ( U \\setminus D ) \\simeq \\pi _ { 1 } ( U ^ { \\prime } ) \\simeq \\mathbb { Z } / 2 \\mathbb { Z } .$ . Hence, there is an étale covering $\\pi ^ { \\prime } : Y ^ { \\prime } \\longrightarrow U \\setminus D$ of degree 2, which extends to a finite covering-$\\pi : Y \\longrightarrow U$ branched along $D$ . Notice that $Y ^ { \\prime }$ is simply connected, hence it follows that $Y$ is simply connected as well. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,558,881,648]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"163","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let now $x _ { 0 } \\in D$ and consider a loop - $\\gamma$ in $U$ pointed at $x _ { 0 }$ . Let $\\widetilde { \\gamma }$ be a lift of $\\gamma$ to $Y$ : notice that as $x _ { 0 }$ is a branching point, the fiber of $\\pi$ over $x _ { 0 }$ is given by a unique point $y _ { 0 }$ , so $\\widetilde { \\gamma }$ is a loop in $Y$ pointed at $y _ { 0 }$ . Since $Y$ is simply connected, it follows that the homotopy class of $\\widetilde { \\gamma }$ is zero in $\\pi _ { 1 } ( Y , y _ { 0 } )$ , so the homotopy class of $\\gamma$ is zero as well in $\\pi _ { 1 } ( U , x _ { 0 } )$ , and we are done. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,671,330,689]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"163","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Quotients of $K u m ^ { 2 } ( S )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,707,879,761]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"163","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Similar considerations are done in [25] for quotients of $K u m ^ { 2 } ( S )$ for an Abelian surface S. If $\\sigma$ is a symplectic involution on $K u m ^ { 2 } ( S )$ , by Theorem 7.5 of [25] the fixed locus of $\\sigma$ is given by 36 points together with a K3 surface $\\Sigma$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,762,881,854]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"163","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The quotient $K u m ^ { 2 } ( S ) / \\sigma$ has then a singular locus given by 36 points and a K3 surface. Blowing up the image of $\\Sigma$ in $K u m ^ { 2 } ( S ) / \\sigma$ one gets a $V$ -manifold $K$ which has 36 singular points. Again $K$ is an irreducible symplectic V-manifold, and $b _ { 2 } ( K ) = 8$ (see [25]), hence $K$ is not deformation equivalent to $M _ { 2 } , M _ { 3 } , M _ { 1 1 } ^ { 1 }$ and $M _ { 1 1 } ^ { 2 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,854,876,890]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"163","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moreover, the only examples in Table 1 having 36 singular points are $X _ { 3 }$ and $Y _ { 1 }$ , whose $b _ { 2 }$ is 14 and 7 respectively, so $K$ is not deformation equivalent to any of the Fujiki examples: we then have another deformation class of singular irreducible symplectic V-manifolds to add the previously mentioned 21 classes. The same proof of point 2 of Proposition 1 gives the following: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,54,152,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"163","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[799,54,879,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"163","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,879,142]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"164","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,153,731,169]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"164","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 2 The variety $K$ is an irreducible symplectic orbifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,198,487,213]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"164","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Examples of Markushevich-Tikhomirov ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,232,881,358]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"164","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Markushevich and Tikhomirov present in [30] a different construction of irreducible symplectic $\\mathrm { V } .$ -manifolds of dimension 4. Let $X$ be a del Pezzo surface which is the double cover of $\\mathbb { P } ^ { 2 }$ branched along a smooth quartic $B _ { 0 }$ with 28 bitangent lines, and let $s$ be the double cover of $X$ branched along the curve $\\Delta _ { 0 }$ such that $\\Delta _ { 0 } + i ( \\Delta _ { 0 } )$ is the inverse image of a smooth quartic curve of $\\mathbb { P } ^ { 2 }$ which is totally tangent to $B _ { 0 }$ at eight distinct points (here $i$ is the involution on $X$ induced by the double cover $X \\longrightarrow \\mathbb { P } ^ { 2 }$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,360,879,468]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"164","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The surface $s$ is a K3 surface, and let $M _ { k }$ be the moduli space of torsion sheaves on $s$ with first Chern class $H$ (the pull-back of $- K _ { X } )$ and Euler character $k -$ 2 for $k \\in 2 { \\mathbb { Z } }$ , i.e. of Mukai vector $\\ : v = ( 0 , H , k - 2 ) \\ :$ . We notice that $M _ { k }$ is a projective variety of dimension 6: as $k$ is even, $M _ { k }$ has exactly 28 singular points (see Proposition 1.12.(iii) of [30]). Moreover, the moduli space $M _ { k }$ has an involution $\\sigma$ mapping $L \\in M _ { k }$ to $\\mathcal { E } x t _ { O _ { S } } ^ { 1 } ( L , O _ { S } ( - H ) )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,468,879,539]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"164","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $\\tau$ Ois the involution on $S$ induced by the double cover $s \\longrightarrow X$ , we let $\\kappa : =$ $\\tau ^ { * } \\circ \\sigma$ : the fixed locus of $\\kappa$ has a 4-dimensional irreducible component, denoted $\\mathcal { P } ^ { k }$ . The morphism mapping $L \\in { \\mathcal { P } } ^ { k }$ to $L ( H ) \\in \\mathcal { P } ^ { k + 2 }$ is an isomorphism, hence we get at most two non-isomorphic varieties 0 and 2. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,540,879,611]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"164","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By Markushevich and Tikhomirov [30] we know that $\\mathcal { P } ^ { 0 }$ and $\\mathcal { P } ^ { 2 }$ are both irreducible symplectic V-manifolds having 28 singular points. By Lemma 5.2 and Corollary 5.7 of [30] we have that $\\mathcal { P } ^ { 0 }$ is birational to $M _ { 2 }$ via a Mukai flop, and by Corollary 3.23 of [34]: it follows that $\\mathcal { P } ^ { 0 }$ is an irreducible symplectic orbifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,612,879,647]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"164","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is not known if $\\mathcal { P } ^ { 0 }$ and $\\mathcal { P } ^ { 2 }$ are birational, isomorphic nor deformation equivalent, and it is still an open question if $\\mathcal { P } ^ { 2 }$ is an irreducible symplectic orbifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,675,309,692]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"164","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Example of Matteini ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,711,879,800]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"164","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A similar construction to that of Markushevich and Tikhomirov is presented in [31]. Take a K3 surface $s$ which is a double cover of a generic cubic surface $Y$ with involution $\\tau$ , and take $M$ to be the moduli space of semistable torsion sheaves with first Chern class $H$ (the pull-back of $- K _ { Y }$ ) and Euler character $^ { - 3 }$ : this is a singular projective variety of dimension 8. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,801,879,890]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"164","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"One still has an involution $\\sigma$ on $M$ obtained as before, and we let $\\kappa : = \\tau ^ { * } \\circ \\sigma$ . The fixed locus of $\\kappa$ has a 6-dimensional irreducible component $\\mathcal { P }$ , whose singular locus is the union of 27 singular K3 surfaces. In [31] it is shown that $\\mathcal { P }$ is an irreducible symplectic V-manifold of dimension 6. It is not known if this example is an irreducible symplectic orbifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,471,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"164","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Examples of Irreducible Symplectic Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,874,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"164","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[861,156,879,168]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"164","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"□ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,610,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"165","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2 Varieties with Canonical Singularities ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,132,881,205]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"165","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A further generalization of the Beauville-Bogomolov decomposition theorem was obtained for projective varieties with canonical singularities by the works of Druel, Greb, Guenancia, Höring, Kebekus and Peternell, in particular [11, 12, 16, 18] and [19]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,240,697,259]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"165","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2.1 The Decomposition for Singular Projective Varieties ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,276,881,331]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"165","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We introduce the following notation: if $X$ is a normal variety and \u0003 \u0004 $X _ { r e g }$ is the smooth locus of $X$ whose open embedding in $X$ is $j : X _ { r e g } \\longrightarrow X$ , for every $p \\in \\mathbb N$ such that $0 \\leq p \\leq \\dim ( X )$ we let ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[348,345,649,375]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0251/03ecdb855e65b1d597a8e39742521892ca6834ceaf7901fb6aeee798e178c074.jpg","list_items":null,"page_idx":"165","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\Omega _ { X } ^ { [ p ] } : = j _ { * } \\Omega _ { X _ { r e g } } ^ { p } = \\big ( \\wedge ^ { p } \\Omega _ { X } \\big ) ^ { * * } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,388,633,406]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"165","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"whose global sections are called reflexive- $p$ -forms on $X$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,406,881,507]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"165","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Notice that by definition of $\\Omega _ { X } ^ { [ p ] }$ we have $H ^ { 0 } ( X , \\Omega _ { X } ^ { [ p ] } ) \\ = \\ H ^ { 0 } ( X _ { r e g } , \\Omega _ { X _ { r e g } } ^ { p } )$ Theorem 1.4 of [17] shows that if -- $X$ is a quasi-projective variety with klt singularities and $\\pi : { \\widetilde { X } } \\longrightarrow X$ is a log-resolution, then for every $p \\in \\mathbb { N }$ such that $0 \\leq p \\leq \\dim ( X )$ the sheaf $\\pi _ { * } \\Omega _ { \\widetilde { X } } ^ { p }$ is reflexive. This implies in particular that $H ^ { 0 } ( X , \\Omega _ { X } ^ { [ p ] } ) \\simeq H ^ { 0 } ( \\widetilde { X } , \\Omega _ { \\widetilde { X } } ^ { p } )$ (see Observation 1.3 therein). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,509,881,603]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"165","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As shown in [17], if ${ \\hat { f } } : Y \\longrightarrow X$ is a finite, dominant morphism between two irreducible normal varieties, then there is a morphism of reflexive sheaves $f ^ { * } \\Omega _ { X } ^ { [ p ] } \\longrightarrow \\Omega _ { Y } ^ { [ p ] }$ , induced by the usual pull-back morphism of forms on the smooth loci, giving a morphism $f ^ { [ * ] } : H ^ { 0 } ( X , \\Omega _ { X } ^ { [ p ] } ) \\longrightarrow H ^ { 0 } ( Y , \\Omega _ { Y } ^ { [ p ] } )$ , called reflexive pullback morphism. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[138,603,845,621]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"165","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We recall the definitions of symplectic form and symplectic variety (see [3]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,630,487,648]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"165","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 5 Let $X$ be a normal variety. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,657,881,803]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"165","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. A symplectic form on $X$ is a closed reflexive 2-form $\\sigma$ on $X$ which is nondegenerate at each point of $X _ { r e g }$ . 2. If $\\sigma$ is a symplectic form on $X$ , the pair $( X , \\sigma )$ is a symplectic variety if for everyresolution $f : { \\widetilde { X } } \\longrightarrow X$ of the singularities of $X$ , the holomorphic symplectic form $\\sigma _ { r e g } : = \\sigma _ { | X _ { r e g } }$ extends to a holomorphic 2-form on $\\widetilde { X }$ . 3. If $( X , \\sigma )$ is a symplectic variety and $f : { \\widetilde { X } } \\ \\longrightarrow \\ X$ is a resolution of the singularities over which $\\sigma _ { r e g }$ extends to a holomorphic symplectic form, we say that $f$ is a symplectic resolution. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,812,879,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"165","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A normal variety having a symplectic form and whose singular locus has codimension at least 4 is a symplectic variety (see [14]), and a symplectic variety has terminal singularities if and only if its singular locus has codimension at least 4 (Corollary 1 of [39]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[799,54,879,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"165","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"165","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"162 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,879,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"166","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We now define irreducible Calabi-Yau and irreducible symplectic varieties following [18]. If $X$ and $Y$ are two irreducible normal projective varieties, a finite quasi-étale morphism $f ~ : ~ Y ~ \\longrightarrow ~ X$ is a finite morphism which is étale in codimension one. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,169,876,205]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"166","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 6 Let $X$ be an irreducible normal projective variety with trivial canonical divisor and canonical singularities, of dimension $d \\ge 2$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,216,881,309]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"166","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. The variety $X$ is irreducible Calabi-Yau if for every $0 < p < d$ and for every finite quasi-étale morphism $Y \\longrightarrow X$ , we have $H ^ { 0 } ( Y , \\Omega _ { Y } ^ { [ p ] } ) = 0$ . 2. The variety $X$ is irreducible symplectic if it has a symplectic form $\\sigma \\in$ $H ^ { 0 } ( X , \\Omega _ { X } ^ { [ 2 ] } )$ , and for every finite quasi-étale morphism $f : Y \\longrightarrow X$ the exterior algebra of reflexive forms on $Y$ is spanned by $f ^ { [ * ] } \\sigma$ . □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,318,881,390]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"166","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A description of irreducible Calabi-Yau and irreducible symplectic varieties in terms of holonomy is available by the work of Greb et al. [16]. More precisely, if $H$ is an ample divisor on $X$ , by Eyssidieux et al. [13] there is a singular Ricci-flat Kähler metric $\\omega _ { H }$ in $c _ { 1 } ( H )$ , inducing a Riemannian metric $g _ { H }$ on $X _ { r e g }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,391,881,480]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"166","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We let $H o l ( X _ { r e g } , g _ { H } )$ be the holonomy group of this metric. Proposition F of [18] shows that a normal projective variety of dimension $n$ with klt singularities and trivial canonical bundle is an irreducible Calabi-Yau variety if and only if $H o l ( \\boldsymbol { X _ { r e g } } , g _ { H } )$ is isomorphic to $S U ( n )$ , and it is an irreducible symplectic variety if and only if $H o l ( X _ { r e g } , g _ { H } )$ is isomorphic to $S p ( n / 2 )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[141,481,847,498]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"166","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The decomposition theorem for singular projective varieties is the following: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,507,881,596]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"166","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3 Let $X$ be a normal projective varieties with klt singularities and numerically trivial canonical bundle. Then $X$ has a finite quasi-étale covering $f : Y \\longrightarrow X$ , where $Y$ is a normal projective variety with canonical singularities which is a product of complex tori, irreducible Calabi-Yau varieties and irreducible symplectic varieties. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,606,881,750]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"166","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This is Theorem 1.15 of [19], and the proof can be found therein. It consists of three major parts: one is the holonomy decomposition obtained by Greb, Guenancia and Kebekus in [16] (namely Theorem B and Proposition D therein); a second one is an algebraic integrability theorem of Druel, which is Theorem 1.4 of [11]; the final ingredient is Theorem 1.1 of [19]. Less general versions of the Bogomolov decomposition theorem in the projective singular setting were previously obtained in [11, 18] and [12]. The complete proof can be found in section 4 of [19] (the proof of Theorem 1.5 therein). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,759,879,831]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"166","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 4 We notice that even if the statement of Theorem 1.15 of [19] gives the existence of a quasi-étale covering $f : Y \\implies X$ , this is consistent with the statement of Theorem 3 since they define quasi-étale morphisms as finite morphisms whose ramification divisor is empty. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,471,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"166","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Examples of Irreducible Symplectic Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"166","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,567,108]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"167","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2.2 Relation Between the Previous Notions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,126,879,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"167","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The first result we state is about the relation between irreducible symplectic manifolds, orbifolds and varieties. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,169,539,189]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"167","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 3 The following properties hold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,198,867,252]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"167","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Irreducible symplectic manifolds are irreducible symplectic orbifolds. \n2. Projective irreducible symplectic orbifolds are irreducible symplectic varieties. \n3. Smooth irreducible symplectic varieties are irreducible symplectic manifolds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,279,879,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"167","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof The first point is trivial, and the last is a consequence of Proposition A.1 of [21]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,315,881,421]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"167","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Suppose that $X$ is a projective irreducible symplectic orbifold. Then $X$ is an normal projective variety with rational Cohen-Macaulay singularities and trivial canonical bundle. It follows that $X$ has rational Gorenstein singularities. Moreover, it has a holomorphic symplectic form on its singular locus, so by Theorem 6 of [40] it follows that $X$ is a projective symplectic variety. In particular it has canonical singularities (see [3]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,423,881,530]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"167","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By Theorem 3 there is then a finite quasi-étale covering $f : Y \\longrightarrow X$ , where $Y$ is a product of Abelian varieties, irreducible Calabi-Yau varieties and irreducible symplectic varieties. As $X _ { r e g }$ is simply connected by definition of irreducible symplectic orbifold, it follows that $f$ is an isomorphism. As a consequence $Y$ is simply connected, so it has no factor which is an Abelian variety, and it is a symplectic variety, hence it has no factor which is an irreducible Calabi-Yau variety. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,533,879,624]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"167","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hence $Y$ is a product of irreducible symplectic varieties. But as $H ^ { 0 } ( X , \\Omega _ { X } ^ { [ 2 ] } )$ is one dimensional by definition of irreducible symplectic orbifold, the same holds for $Y$ . If $Y$ is a product of $m$ irreducible symplectic varieties, we have that $H ^ { 0 } ( Y , \\Omega _ { Y } ^ { [ 2 ] } )$ has dimension $m$ : it follows that $m = 1$ , so $Y$ is an irreducible symplectic variety. As $X$ is isomorphic to $Y$ , we are done. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,632,881,723]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"167","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Irreducible symplectic $\\mathrm { V } .$ -manifolds are not necessarily irreducible symplectic varieties. A first example of this is given by symmetric products of K3 surfaces: if $s$ is a K3 surface and $m \\in \\mathbb { N } , m \\geq 2$ , then $X : = S y m ^ { m } ( S )$ is an irreducible symplectic V-manifold. Anyway it has a finite quasi-étale covering $S ^ { m } \\longrightarrow X$ , and $h ^ { 0 } ( S ^ { m } , \\Omega _ { S ^ { m } } ^ { 2 } ) = m$ , so $X$ is not an irreducible symplectic variety. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,723,881,831]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"167","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By Theorem I of [18] we know that all irreducible symplectic varieties are simply connected, so if $X$ is a primitively symplectic V-manifold which is an irreducible symplectic variety, then $X$ is an irreducible symplectic V-manifold. Anyway, as symplectic singularities are not, in general, quotient singularities, we cannot expect that an irreducible symplectic variety is an irreducible symplectic V-manifold: an example will be given in the last section. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,833,881,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"167","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The following is a criterion to guarantee that some quotients of an irreducible symplectic manifold are irreducible symplectic varieties (and irreducible symplectic $V$ -manifolds as well). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[799,54,879,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"167","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[861,255,876,267]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"167","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"□ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"167","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"164 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,876,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"168","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 1 Let $X$ be an irreducible symplectic manifold, $G \\subseteq A u t ( X )$ a finite subgroup acting symplectically on $X$ and $Y : = X / G$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,135,881,187]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"168","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. The quotient Y is an irreducible symplectic $V$ -manifold. 2. If Y has terminal singularities and $X$ is projective, then $Y$ is an irreducible symplectic variety. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,196,881,340]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"168","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof The proof of the first part is basically contained in [15], but we present it here for the reader’s sake. The fact that $X$ is a compact, connected Kähler orbifold is Lemma 1.4 of [15], and by Lemma 2.4 of [15] we have that $Y$ is symplectic. As $X$ is simply connected, by Lemma 1.2 of [15] it follows that $Y$ is simply connected as well. Finally, we have $H ^ { 2 , 0 } ( Y ) \\simeq H ^ { 2 , 0 } ( X ) ^ { G }$ , where $H ^ { 2 , 0 } ( X ) ^ { G }$ is the space of $G$ - invariant sections. But as $G$ acts symplectically and $X$ is an irreducible symplectic manifold, it follows that $H ^ { 2 , 0 } ( Y )$ is 1-dimensional, so that $Y$ is an irreducible symplectic $V$ -manifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,342,879,448]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"168","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For the second part, as we know that $Y$ is a normal projective symplectic variety with terminal singularities, in order to show that it is an irreducible symplectic variety we just need to look at its finite quasi-étale coverings. So, let $f : Y ^ { \\prime } \\longrightarrow Y$ be a finite quasi-étale covering of Y . As the singularities of $Y$ are terminal and as $X$ is simply connected (being an irreducible symplectic manifold), it follows that $X$ is the universal covering of $Y$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,450,881,539]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"168","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In particular, it follows that we have a finite quasi-étale covering $\\pi ^ { \\prime } : X \\longrightarrow Y ^ { \\prime }$ given by the quotient by a subgroup $G ^ { \\prime }$ of $G$ , and if we let $\\pi : X \\longrightarrow Y$ , then $\\pi = f \\circ \\pi ^ { \\prime }$ . As $X$ is an irreducible symplectic manifold, its exterior algebra of holomorphic forms is spanned by the symplectic form on $X$ , which is the reflexive pull-back of the one on $Y$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,539,881,639]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"168","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moreover, we notice that $H ^ { 0 } ( Y ^ { \\prime } , \\Omega _ { Y ^ { \\prime } } ^ { [ p ] } ) \\simeq H ^ { p , 0 } ( X ) ^ { G ^ { \\prime } }$ , and as $X$ is an irreducible symplectic manifold and $G$ (and hence $G ^ { \\prime }$ ) acts symplectically we see that if $p$ is even then $H ^ { 0 } ( Y ^ { \\prime } , \\Omega _ { Y ^ { \\prime } } ^ { [ p ] } )$ is spanned by $f ^ { [ * ] } \\sigma ^ { p / 2 }$ (where $\\sigma$ is the symplectic form on $Y$ ) and if $p$ is odd then $H ^ { 0 } ( Y ^ { \\prime } , \\Omega _ { Y ^ { \\prime } } ^ { [ p ] } ) = 0 .$ . But this shows that $Y$ is an irreducible symplectic variety. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,647,881,701]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"168","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As an application of this, we see that the examples $M _ { 3 }$ , $M _ { 5 }$ , $M _ { 7 }$ , $M _ { 1 1 } ^ { 1 }$ , $M _ { 1 1 } ^ { 2 }$ and $H _ { n }$ presented in Sect. 2.1.2 are all examples of irreducible symplectic $V$ -manifolds which are irreducible symplectic varieties (but not irreducible symplectic orbifolds). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,702,881,791]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"168","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The partial resolution $M _ { 2 }$ of the quotient of $H i l b ^ { 2 } ( S )$ by a symplectic involution (where $s$ is a K3 surface), the partial resolution $K _ { 2 }$ of the quotient of $K u m ^ { 2 } ( T )$ by a symplectic involution (where $T$ is a 2-dimensional complex torus), and the example $\\mathcal { P } ^ { 0 }$ of [30] (which is deformation equivalent to $M _ { 2 }$ ) are all irreducible symplectic varieties as they are irreducible symplectic orbifolds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,792,876,827]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"168","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For all other examples in Sect. 2.1.2 (those of Fujiki, the example $\\mathcal { P } ^ { 2 }$ of [30] and the example of Matteini), it is not known if they are irreducible symplectic varieties. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,471,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"168","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Examples of Irreducible Symplectic Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"168","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,346,106]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"169","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2.3 Related Notions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,126,879,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"169","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Irreducible symplectic varieties appear in several papers under different definitions. A first one appears in [1], and it is defined as follows (the name given to these varieties in [1] is irreducible symplectic varieties). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,187,879,259]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"169","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 7 A resolvable symplectic variety is a normal, compact Kähler space $X$ whose smooth locus has a holomorphic symplectic form $\\sigma$ , and which has a symplectic resolution of the singularities which is an irreducible symplectic manifold. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,268,881,358]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"169","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A projective resolvable symplectic variety is not always an irreducible symplectic variety: if $s$ is a projective K3 surface and $m \\ge 2$ , then $S y m ^ { m } ( S )$ is a projective resolvable symplectic variety (since it is a normal projective symplectic variety having $H i l b ^ { m } ( S )$ as a symplectic resolution), but it is not an irreducible symplectic variety. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,360,879,448]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"169","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Similarly, singular Kummer surfaces (i.e. a surface $s$ obtained as quotient of an Abelian surface $A$ by the involution mapping $p \\in A$ to $- p \\in A$ ) are resolvable symplectic surfaces (they have a symplectic resolution which is a K3 surface), irreducible symplectic V-manifolds but not irreducible symplectic varieties (as the quotient map $A \\longrightarrow S$ is a finite quasi-étale covering and $h ^ { 1 , 0 } ( A ) \\neq 0 \\}$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,450,879,503]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"169","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We will see in Sect. 2.2.4 examples of irreducible symplectic varieties which are not resolvable symplectic varieties, and of resolvable symplectic varieties which are not irreducible symplectic V-manifolds. Anyway we have the following: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,513,879,566]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"169","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 4 If $X$ is an irreducible symplectic variety (resp. an irreducible symplectic $V .$ -manifold) having $a$ symplectic resolution Y , then $Y$ is an irreducible symplectic manifold. In particular, $X$ is a resolvable symplectic variety. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,575,879,666]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"169","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof If $X$ is an irreducible symplectic variety, this is Remark 1.16 of [47]. If $X$ is an irreducible symplectic $\\mathrm { v } .$ -manifold, then $X$ has canonical singularities. By Takayama [50] we get $\\pi _ { 1 } ( X ) \\ \\simeq \\ \\pi _ { 1 } ( Y )$ , so $Y$ is simply connected. Finally, by Theorem 1.4 of [17] we have $h ^ { 0 } ( Y , \\Omega _ { Y } ^ { 2 } ) = h ^ { 0 } ( X , \\Omega _ { X } ^ { [ 2 ] } )$ , which is 1 by definition, and we are done. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,677,879,750]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"169","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A further definition of irreducible symplectic variety appears in [49], where it is defined as a projective symplectic variety $X$ such that $h ^ { 1 } ( X , O _ { X } ) \\ = \\ 0$ and $h ^ { 0 } ( X , \\Omega _ { X } ^ { [ 2 ] } ) = \\bar { 1 }$ O. It is called Namikawa symplectic variety if it is moreover $\\mathbb { Q }$ factorial and its singular locus has codimension at least 4 (see Definition 1 therein). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,759,876,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"169","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 8 We will call Namikawa symplectic variety a normal, compact Kähler complex space $X$ such that $h ^ { 1 } ( X , O _ { X } ) = 0$ and $h ^ { 0 } ( X , \\Omega _ { X } ^ { [ 2 ] } ) = 1$ . □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,806,876,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"169","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Namikawa symplectic varieties are the most general kind of varieties we will deal with. Namely: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,851,881,887]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"169","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 5 Irreducible symplectic varieties, resolvable symplectic varieties and irreducible symplectic V-manifolds are all Namikawa symplectic varieties. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[799,54,879,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"169","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,152,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"169","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"166 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"170","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof The proof that irreducible symplectic and resolvable symplectic varieties are Namikawa symplectic is given in Propositions 1.9 and 1.10 of [47]. If $X$ is an irreducible symplectic V-manifold, then - $X$ has rational Gorenstein singularities and has a symplectic form on its smooth locus, hence by Theorem 6 of [40] it is a symplectic variety. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,180,881,291]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"170","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moreover, if ${ \\check { f } } : { \\widetilde { X } } \\longrightarrow X$ is a resolution of the singularities, as- $X$ has rational singularities we have an isomorphism between- $H ^ { 1 } ( X , { \\bar { O } } _ { X } )$ and $H ^ { 1 } ( \\widetilde { X } , O _ { \\widetilde { X } } )$ . As $X$ has klt singularities we have- $\\pi _ { 1 } ( X ) \\simeq \\pi _ { 1 } ( \\widetilde { X } )$ (by Takayama [50]), and since $X$ is simply connected it follows that $\\tilde { X }$ is simply connected. As a consequence of this we see that $H ^ { 1 } ( \\widetilde { X } , O _ { \\widetilde { X } } ) = 0$ , and hence $H ^ { 1 } ( X , O _ { X } ) = 0$ . As by definition of irreducible symplectic V-manifold we have that $H ^ { 0 } ( X , \\Omega _ { X } ^ { [ 2 ] } )$ is one dimensional, we are done. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,298,881,352]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"170","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"No example of Namikawa symplectic variety which is not an irreducible symplectic variety, nor an irreducible symplectic V-manifold, nor a resolvable symplectic variety is known. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,388,287,406]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"170","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2.4 Examples ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,424,879,496]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"170","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We now introduce two families of examples of irreducible symplectic varieties. In what follows $s$ will denote a projective K3 surface or an Abelian surface, and welet $\\epsilon ( S ) : = 1$ if $s$ is K3, and 0 if $s$ is Abelian. We let $\\rho ( S )$ be the rank of the Néron-Severi group $N S ( S )$ of $s$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,496,879,569]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"170","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An element $v \\in \\widetilde { H } ( S , \\mathbb { Z } ) : = H ^ { 2 * } ( S , \\mathbb { Z } )$ will be written $\\boldsymbol { v } = ( v _ { 0 } , v _ { 1 } , v _ { 2 } )$ , where $v _ { i } \\in H ^ { 2 i } ( S , \\mathbb { Z } )$ , and - $v _ { 0 } , v _ { 2 } \\ \\in \\ \\mathbb { Z }$ . It will be called Mukai vector if $v _ { 0 } \\geq 0$ , $v _ { 1 } ~ \\in$ $N S ( S )$ and if $v _ { 0 } = 0$ , then either $v _ { 1 }$ is the first Chern class of an effective divisor, or $v _ { 1 } = 0$ and $v _ { 2 } > 0$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,570,881,659]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"170","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The $\\mathbb { Z }$ -module $\\widetilde { H } ( S , \\mathbb { Z } )$ has a pure weight-two Hodge structure and a lattice structure with respect to the Mukai pairing $\\left( . , . \\right)$ (see [20], Definitions 6.1.5 and 6.1.11). We let $\\boldsymbol { v } ^ { 2 } : = ( v , v )$ for every Mukai vector $v$ , and we refer to ${ \\widetilde { H } } ( S , \\mathbb { Z } )$ as the Mukai lattice of $s$ . We will always write 1 $v \\ = \\ m w$ , where $m \\in \\mathbb { N }$ and $w$ is a primitive Mukai vector on $s$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,659,681,677]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"170","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To any coherent sheaf $\\mathcal { F }$ on $s$ we associate a Mukai vector ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[337,692,658,717]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0700/6b5f1e7818b4194258c2ae7cdc920554ee04251afd90d10daa2ab472773cb867.jpg","list_items":null,"page_idx":"170","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\boldsymbol { v } ( \\mathcal { F } ) : = c h ( \\mathcal { F } ) \\sqrt { t d ( S ) } \\in \\widetilde { H } ( S , \\mathbb { Z } ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,732,881,824]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"170","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Taking $v$ a Mukai vector on $s$ and supposing that $H$ is a $v$ -generic polarization (see as instance section 2.1 of [47] for the precise definition), we consider the moduli space $M _ { v } ( S , H )$ (resp. $M _ { v } ^ { s } ( S , H ) _ { , } ^ { \\setminus }$ ) of Gieseker $H$ -semistable (resp. $H$ - stable) sheaves on $s$ with Mukai vector $v$ . Then $M _ { v }$ is a projective variety and $M _ { v } ^ { s } \\subseteq M _ { v }$ is open (see [20]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,471,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"170","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Examples of Irreducible Symplectic Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"170","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[145,88,425,108]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"171","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The following properties hold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,117,881,169]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"171","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. If $s$ is a K3 surface and $v = m w$ for a primitive Mukai vector $w$ , then $M _ { v } \\ne \\emptyset$ if and only if $w ^ { 2 } \\geq - 2$ (see [38] and [52]). If $s$ is an Abelian surface, then $M _ { v } \\ne \\emptyset$ if and only $w ^ { 2 } \\geq 0$ (see [53]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,171,876,205]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"171","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. If $s$ is a K3 surface and $v = m w$ for a primtive Mukai vector $w$ such that $w ^ { 2 } =$ $^ { - 2 }$ , then $M _ { v }$ is a point (see [38]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,207,881,295]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"171","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3. If $s$ is a K3 surface, $v = m w$ and $w ^ { 2 } = 0$ , then $M _ { v } \\simeq S y m ^ { m } ( S ^ { \\prime } )$ for a projective K3 surface $S ^ { \\prime }$ (see [38] and [23]). If $m = 1$ we then get a projective K3 surface, while if $m \\ \\geq \\ 2$ we then get an irreducible symplectic V-manifold, which is a resolvable symplectic variety but which is neither an irreducible symplectic variety nor an irreducible symplectic orbifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,297,881,421]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"171","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4. If $s$ is an Abelian surface, $v = m w$ and $w ^ { 2 } = 0$ , then $M _ { v } \\simeq S y m ^ { m } ( T )$ for an Abelian surface $T$ (see [38] and [23]). If $m = 1$ we then get an Abelian surface, and hence a K3 surface via the Kummer construction. If $m \\ge 2$ we consider the natural sum morphism $s : M _ { v } \\longrightarrow T$ , and let $K : = s ^ { - 1 } ( 0 )$ : then $K$ is an irreducible symplectic V-manifold, which is a resolvable symplectic variety but which is neither an irreducible symplectic variety nor an irreducible symplectic orbifold (see Example 1.13 of [47]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,423,881,493]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"171","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5. If $s$ is a K3 surface or an Abelian surface and $v = m w$ for a primitive Mukai vector $w$ such that $w ^ { 2 } > 0$ , then $M _ { v }$ is a normal, irreducible projective variety of dimension $v ^ { 2 } + 2$ whose smooth locus is $M _ { v } ^ { s }$ (see [52]). By [37] $M _ { v }$ has a symplectic form. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,495,881,585]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"171","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6. If $s$ is an Abelian surface and $v \\ = \\ m w$ for a primitive Mukai vector $w$ such that $w ^ { 2 } > 0$ , by section 4.1 of [53] we have a dominant isotrivial fibration $a _ { v } :$ $M _ { v } ( S , H ) \\longrightarrow S \\times { \\widehat S }$ , where $\\widehat { S }$ is the dual of $s$ . We let $K _ { v } : = a _ { v } ^ { - 1 } ( 0 _ { S } , O _ { S } )$ , and $K _ { v } ^ { s } : = K _ { v } \\cap M _ { v } ^ { s }$ . The restriction of the symplectic form of $M _ { v }$ to $K _ { v }$ is a symplectic form (see [53]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,594,881,648]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"171","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The moduli spaces $M _ { v }$ and $K _ { v }$ described above give us examples of irreducible symplectic varieties if $v ^ { 2 } > 0$ . This is the content of the following result, which is Theorem 1.19 of [47]: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,656,879,711]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"171","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 6 Let $s$ be a projective $K 3$ surface of an Abelian surface, v a Mukai vector on $s$ such that $v = m w$ for a primitive Mukai vector $w$ with $w ^ { 2 } > 0$ , and $H$ a v-generic polarization. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,719,715,738]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"171","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. If $s$ is $K 3$ , then $M _ { v } ( S , H )$ is an irreducible symplectic variety. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[143,746,881,819]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"171","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"a. If $m = 1$ , it is an irreducible symplectic manifold. \n$b _ { \\cdot }$ . If $m = 2$ and $w ^ { 2 } = 2$ , it has a symplectic resolution which is an irreducible symplectic manifold. \nc. In all other cases it has terminal singularities. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,827,881,864]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"171","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. Suppose that $s$ is Abelian. If $m = 1$ and $w ^ { 2 } = 2$ then $K _ { v } ( S , H )$ is a point. In all other cases $K _ { v } ( S , H )$ is an irreducible symplectic variety. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,872,742,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"171","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"a. If $m = 1$ and $w ^ { 2 } > 2$ , it is an irreducible symplectic manifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[799,54,879,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"171","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"171","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"168 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[143,90,879,145]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"172","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"b. If $m = 2$ and $w ^ { 2 } = 2$ , it has a symplectic resolution which is an irreducible symplectic manifold. \nc. In all other cases it has terminal singularities. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,154,881,244]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"172","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The case $m = 1$ was proved in its final form by Yoshioka in [52] and [53] (but with important steps towards the complete proof given in [2, 37, 42]). The case $m = 2$ , $w ^ { 2 } = 2$ was studied first by O’Grady in [43] and [44] for $v = 2 ( 1 , 0 , - 1 )$ , where it was shown that $M _ { v }$ (resp. $K _ { v }$ ) has a symplectic resolution which is an irreducible symplectic manifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,246,879,316]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"172","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In [29] it is shown that the symplectic resolution exists for all $v = 2 w$ , where $w$ is primitive and $w ^ { 2 } = 2$ . In [46] it is shown that such a symplectic resolution is an irreducible symplectic manifold, deformation equivalent to $O G _ { 1 0 }$ (resp. $O G _ { 6 }$ ). The proof of the statement for all other cases is contained in [47]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,318,881,424]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"172","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The cases $m = 1$ and $m = 2$ , $w ^ { 2 } = 2$ then recover all the known deformation classes of irreducible symplectic manifolds. Moreover, the case $m = 2$ , $w ^ { 2 } = 2$ gives examples of irreducible symplectic varieties which are resolvable symplectic varieties. We notice in particular that if $s$ is Abelian, then the smooth locus of $K _ { v }$ in this case is not simply connected (see Theorem 3.7 of [47]), giving then an example of irreducible symplectic variety which is not an irreducible symplectic orbifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,426,879,478]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"172","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The remaining cases give examples of irreducible symplectic varieties having no symplectic resolution, hence they are not resolvable symplectic varieties. Moreover, we have the following result: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,487,881,576]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"172","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 7 Let $S _ { 1 }$ and $S _ { 2 }$ be two projective $K 3$ surfaces (resp. Abelian surfaces), $v _ { i } = m _ { i } w _ { i }$ a Mukai vector on $S _ { i }$ for $m _ { i } > 0$ and $w _ { i }$ primitive Mukai vector, and $H _ { i } a$ $v _ { i }$ -generic polarization on $S _ { i }$ . Then $M _ { v _ { 1 } } ( S _ { 1 } , H _ { 1 } )$ (resp. $K _ { v _ { 1 } } ( S _ { 1 } , H _ { 1 } ) ,$ ) is deformation equivalent to $M _ { v _ { 2 } } ( S _ { 2 } , H _ { 2 } )$ (resp. $K _ { v _ { 2 } } ( S _ { 2 } , H _ { 2 } ) )$ 1 if and only if $m _ { 1 } = m _ { 2 }$ and $w _ { 1 } ^ { 2 } = w _ { 2 } ^ { 2 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,587,881,696]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"172","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The sufficient condition is essentially proved by Yoshioka in [54] and [55] (see even [47]). The converse is proved in a forthcoming paper of the author and Rapagnetta. As a consequence we see that in dimension $2 n$ we find a deformation class of irreducible symplectic varieties of dimension $2 n$ for each pair $( m , k )$ such that $m ^ { 2 } k + 1 = n$ or ${ \\dot { m } } ^ { 2 } { \\dot { k } } - 1 = n$ (just take the moduli space of semistable sheaves of Mukai vector $( m , 0 , - m k ) )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,704,881,831]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"172","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remark 5 The deformation classes in dimension 4 and 6 we obtain in this way are different from those of the examples of singular irreducible symplectic V-manifolds we presented in Sect. 2.1.2. Indeed, if $s$ is K3 we have $\\dim ( M _ { v } ) \\ : = \\ : 4 , 6$ only if $v ^ { 2 } = 2 , 4$ , so $v$ must be primitive and $M _ { v }$ is then smooth. If $s$ is Abelian we have $\\mathrm { d i m } ( K _ { v } ) = 4 , 6$ only if $v ^ { 2 } = 6 , 8$ . If $\\upsilon ^ { 2 } = 6$ then $v$ is primitive and $K _ { v }$ smooth; if $\\upsilon ^ { 2 } = 8$ then either $v$ is primitive and $K _ { v }$ is smooth, or $v = 2 w$ for $w$ primitive with $w ^ { 2 }$ , and $K _ { v }$ has a symplectic resolution. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,840,881,894]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"172","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finally, we remark that the singular moduli spaces in the statement of Proposition 6 give examples of irreducible symplectic varieties which are not irreducible symplectic V-manifolds, as their singularities are not quotient singularities. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,471,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"172","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Examples of Irreducible Symplectic Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"172","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[861,130,879,144]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"172","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"□ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[861,564,876,576]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"172","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"□ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,91,876,121]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"173","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acknowledgements The author wishes to thank Chiara Camere and Baohua Fu for useful discussions, and especially Grégoire Menet and Antonio Rapagnetta for all the fruitful exchanges. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,168,248,187]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"173","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,202,881,900]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"173","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Bakker, B., Lehn, C.: A Global Torelli Theorem for Singular Symplectic Varieties (2016). Preprint arxiv:1612.07894 \n2. Beauville, A.: Variétés kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18, 755–782 (1983) \n3. Beauville, A.: Symplectic singularities. Invent. Math. 139(3), 541–549 (2000) \n4. Beauville, A., Donagi, R.: La variété des droites d’une hypersurface cubique de dimension 4. C. R. Acad. Sci. Paris Sr. I Math. 301(14), 703–706 (1985) \n5. Blache, R.: Chern classes and Hirzeburch-Riemann-Roch theorem for coherent sheaves on complex projective orbifolds with isolated singularities. Math. Z. 222, 7–57 (1996) \n6. Bogomolov, F.: On the decomposition of Kähler manifolds with trivial canonical class. Math. USSR-Sb. 22, 580–583 (1974) \n7. Borzellino, J., Zhu, S.-H.: The Splitting Theorem for Orbifolds. Ill. J. Math. 38, 679–691 (1994) \n8. Campana, F.: Orbifoldes à première classe de Chern nulle. In: The Fano Conference, pp. 339– 351, University Torino, Torino (2004) \n9. Debarre, O., Voisin, C.: Hyper-Kähler fourfolds and Grassmann Geometry. J. Reine Angew. Math. 649, 63–87 (2010) \n10. Demailly, J.-P., Kollár, J.: Semi-continuity of Complex Singularity Exponents and Kähler-Einstein Metrics on Fano Orbifolds. Ann. Sci. Ec. Norm. Sup. Série 4 34(4), 525–556 (2001) \n11. Druel, S.: A decomposition theorem for singular spaces with trivial canonical class of dimension at most five. Invent. Math. 211(1), 245–296 (2018) \n12. Druel, S., Guenancia, H.: A decomposition theorem for smoothable varieties with trivial canonical class. J. de l’École Polytechnique—Mathématiques 5, 117–147 (2018) \n13. Eyssidieux, P., Guedj, V., Zeriahi, A.: Singular Kähler-Einstein metrics. J. Am. Math. Soc. 22(3), 607–639 (2009) \n14. Flenner, H.: Extendability of differential forms on non-isolated singularities. Invent. Math. 94, 317–326 (1988) \n15. Fujiki, A.: On primitively symplectic compact Kähler V-Manifolds of dimension four. In: Classification of Algebraic and Analytic Manifolds. Progress in Mathematics 39, 71–250 (1983) \n16. Greb, D., Guenancia, H., Kebekus, S.: Klt with Trivial Canonical Class—Holonomy, Differential Forms, and Fundamental Group. Geom. Topol. 23, 2051–2124 (2019) \n17. Greb, D., Kebekus, S., Kovács, S., Peternell, T.: Differential forms on log-canonical spaces. Publ. Math. IHES 114(1), 87–169 (2011) \n18. Greb, D., Kebekus, S., Peternell, T.: Singular spaces with trivial canonical class. In: Proceedings of the conference in honor of Shigefumi Mori’s 60th birthday, minimal models and extreme rays. Advanced Studies in Pure Mathematics. Kinokuniya Publishing House, Tokyo (2011) \n19. Höring, A., Peternell, T.: Algebraic Integrability of Foliations with Numerically Trivial Canonical Bundle. Invent. Math. 216, 395–419 (2019) \n20. Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. Aspects of Mathematics E, vol. 31. Vieweg, Braunschweig (1997) \n21. Huybrechts, D., Nieper-Wisskirchen, M.: Remarks on derived equivalence of Ricci-flat manifolds. Math. Z. 267(3–4), 939–963 (2011) \n22. Iliev, A., Ranestad, K.: K3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds. Trans. Am. Math. Soc. 353, 1455–1468 (2001) \n23. Kaledin, D., Lehn, M., Sorger, C.: Singular symplectic moduli spaces. Invent. Math. 164, 591– 614 (2006) \n24. Kamenova, L., Mongardi, G., Oblomkov, A.: Symplectic involutions of $K 3 ^ { [ n ] }$ type and Kummer n type manifolds (2018). Preprint arxiv:1809.02810 \n25. Kapfer, S., Menet, G.: Integral cohomology of the generalized Kummer fourfold. Algebr. Geom. 5(5), 523–567 (2018) \n26. Kawamata, Y.: Minimal models and the Kodaira dimension of algebraic fiber spaces. J. Reine Angew. Math. 363, 1–46 (1985) \n27. Kollár, J., Mori, S.: Birational geometry of algebraic varieties. In: Cambridge Tracts in Mathematics, vol. 134. Cambridge University, Cambridge (1998) \n28. Lehn, C., Lehn, M., Sorger, C., van Straten, D.: Twisted cubics on cubic fourfolds. J. Reine Angew. Math. 731, 87–128 (2017) \n29. Lehn, M., Sorger, C.: La singularité de O’Grady. J. Algebraic Geom. 15, 756–770 (2006) \n30. Markushevich, D., Tikhomirov, A.S.: New symplectic V-manifolds of dimension four via the relative compactified Prymian. Int. J. Math. 18(10), 1187–1224 (2007) \n31. Matteini, T.: A new singular symplectic variety of dimension 6 with a Lagrangian Prym fibration. Manuscripta Math. 149(1), 131–151 (2016) \n32. Menet, G.: Beauville-Bogomolov lattice for a singular symplectic variety of dimension 4. J. Pure App. Math. 219(5), 1455–1495 (2014) \n33. Menet, G.: On the integral cohomology of quotients of manifolds by cycling groups. J. Math. Pures Appl. 119(9), 280–325 (2018) \n34. Menet, G.: A Global Torelli Theorem for irreducible symplectic orbifolds. Preprint arxiv:1805.12426 \n35. Menet, G.: Integral cohomology of quotients via toric geometry (2019). Preprint \n36. Mongardi, G.: Automorphisms of Hyperkähler Manifolds. PhD Thesis arxiv:1303.4670 \n37. Mukai, S.: Symplectic structure of the moduli space of sheaves on an Abelian or K3 surface. Invent. Math. 77, 101–116 (1984) \n38. Mukai, S.: On the moduli space of bundles on K3 surfaces I. In: Vector Bundles on Algebraic Varieties, Bombay (1984). Tata Institute of Fundamental Research Studies on Mathematics, vol. 11, , pp. 341–413. Tata Institute of Fundamental Research, Bombay (1987) \n39. Namikawa, Y.: A Note on Symplectic Singularities (2001). Preprint arxiv:0101028 \n40. Namikawa, Y.: Extension of 2-forms and symplectic varieties. J. Reine Angew. Math. 539, 123–147 (2001) \n41. Namikawa, Y.: On deformations of $\\mathbb { Q }$ -factorial symplectic varieties. J. Reine Angew. Math. 599, 97–110 (2006) \n42. O’Grady, K.: The Weight-two Hodge Structure of Moduli Spaces of Sheaves on a K3 Surface. J. Algebraic Geom. 6(4), 599–644 (1997) \n43. O’Grady, K.: Desingularized Moduli spaces of sheaves on a K3 surface. J. Reine. Angew. Math. 512, 49–117 (1999) \n44. O’Grady, K.: A new six dimensional irreducible symplectic variety. J. Algebraic Geom. 12, 435–505 (2003) \n45. O’Grady, K.: Irreducible symplectic 4-folds and Eisenbud-Popescu-Walters sextics. Duke Math. J. 134, 99–137 (2006) \n46. Perego, A., Rapagnetta, A.: Deformation of the O’Grady moduli spaces. J. Reine Angew. Math. 678, 1–34 (2013) \n47. Perego, A., Rapagnetta, A.: The moduli spaces of sheaves on K3 surfaces are irreducible symplectic varieties. Preprint arxiv:1802.01182 \n48. Rapagnetta, A.: On the Beauville form of the known irreducible symplectic varieties. Math. Ann. 340(1), 77–95 (2008) \n49. Schwald, M.: Fujiki Relations and Fibrations of Irreducible Symplectic Varities (2017). Preprint arxiv:1701.09069 \n50. Takayama, S.: Local simple connectedness of resolutions of log-terminal singularities. Int. J. Math. 14(8), 825–836 (2003) \n51. Yau, S.-T.: On the ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. Commun. Pure App. Math 31, 339–411 (1978) \n52. Yoshioka, K.: Irreducibility of Moduli Spaces of Vector Bundles on K3 Surfaces (1999). Preprint arxiv:math/9907001 \n53. Yoshioka, K.: Moduli spaces of stable sheaves on Abelian surfaces. Math. Ann. 321(4), 817– 884 (2001) \n54. Yoshioka, K.: Twisted stability and Fourier-Mukai functor I. Compos. Math. 138(3), 261–288 (2003) \n55. Yoshioka, K.: Twisted stability and Fourier-Mukai functor II. Compos. Math. 145, 112–142 (2009) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,54,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"173","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[799,54,879,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"173","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[116,99,883,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"174","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,471,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"174","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Examples of Irreducible Symplectic Varieties ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"174","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"171 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[116,91,881,240]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"175","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[799,54,879,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"175","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"175","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,644,141]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"176","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An Example of Mirror Symmetry for Fano Threefolds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[123,175,280,193]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"176","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Andrea Petracci ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,288,881,340]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"176","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract In this note we illustrate the Fanosearch programme of Coates, Corti, Galkin, Golyshev, and Kasprzyk in the example of the anticanonical cone over the smooth del Pezzo surface of degree 6. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,360,835,394]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"176","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords Mirror symmetry $\\cdot$ Fano varieties $\\cdot$ Toric varieties $\\cdot$ Deformation theory ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,412,797,432]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"176","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2010 Mathematics Subject Classification 14J45, 14M25, 14J33, 14D15 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,468,309,489]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"176","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,507,230,527]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"176","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1.1 Aim ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,552,881,731]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"176","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Fanosearchprogramme of Coates et al. [8] studies Fano varieties via Mirror Symmetry. In this context it is crucial to study toric degenerations of smooth Fano varieties, or conversely smoothings of toric Fano varieties. Toric Fano varieties are associated to certain lattice polytopes, called Fano polytopes; some combinatorial input on a Fano polytope conjecturally allows to construct a deformation of the corresponding toric Fano. This is also reflected by Mirror Symmetry, where the combinatorial input is encoded by certain special Laurent polynomials. The goal of this note is to illustrate this programme in a specific example where two different combinatorial inputs on the same polytope produce two different smoothings of the same toric Fano threefold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,834,635,878]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"176","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. Petracci $( \\boxtimes )$ Institut für Mathematik, Freie Universität Berlin, Berlin, Germany e-mail: andrea.petracci $@$ fu-berlin.de ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,897,706,941]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"176","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_10 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[804,91,876,139]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"176","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,899,876,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"176","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,332,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"177","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1.2 The Example ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,132,881,277]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"177","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The example we consider is the projective cone over the anticanonical embedding of the smooth del Pezzo surface of degree 6. This threefold, denoted by $X$ , is a toric Fano and has an isolated Gorenstein canonical non-terminal singularity at the vertex of the cone. The deformations of this singularity have been studied by Altmann [4]; we will recall Altmann’s results in Sect. 2.3. In Sect. 2.4 we will see that the base of the miniversal deformation (or equivalently the Kuranishi family) of the projective threefold $X$ has two irreducible components, which deform $X$ to two different smooth Fano threefolds, namely: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,286,669,321]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"177","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"• a general element $X _ { 2 }$ of the linear system $| O _ { \\mathbb { P } ^ { 2 } \\times \\mathbb { P } ^ { 2 } } ( 1 , 1 ) |$ , • $\\bar { X _ { 3 } } = \\bar { \\mathbb { P } } ^ { 1 } \\times \\bar { \\mathbb { P } } ^ { 1 } \\times \\bar { \\mathbb { P } } ^ { 1 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,331,881,384]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"177","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(The reason for the subscripts 2 and 3 will be evident later.) These two smooth Fanos are connected via deformation through $X$ , but cannot be connected via a deformation with smooth fibres, as their Betti numbers are different. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,385,881,530]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"177","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As $X$ is toric, by means of toric geometry, we can associate to $X$ a 3-dimensional lattice polytope $P \\subseteq \\mathbb { R } ^ { 3 }$ which is a hexagonal pyramid (see the precise definition in (2) and Fig. 1). The hexagonal facet of $P$ is denoted by $F$ (see (1) and the left part of Fig. 2). In Proposition 2.2 we will see that the two smoothings of $X$ are associated to some combinatorial additional data on the polytope $P$ . More precisely, they correspond to the two maximal Minkowski decompositions of the hexagon $F$ (see (3) and (4), and Fig. 2). We will introduce the notion of Minkowski sum and Minkowski decomposition in Sect. 2.3. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[332,561,662,744]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 1 The 3-dimensional lattice polytope $P$ associated to $X$ \"]","image_footnote":"[]","img_path":"images/0842/aea482f8093324f43b4758359351cedb8a78d6dddeb3b74abb5eb9f1254739c7.jpg","list_items":null,"page_idx":"177","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[177,785,820,869]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 2 The two maximal Minkowski decompositions of the hexagon $F$ \"]","image_footnote":"[]","img_path":"images/0925/dedcce463875c0ebf0feb24831a72d2b18f5490860875bebdd9a565458742ce4.jpg","list_items":null,"page_idx":"177","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[790,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"177","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. Petracci ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"177","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"174 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,198]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"178","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now we consider the Laurent polynomials in 3 variables which are supported on $P$ , i.e. Laurent polynomials $f ~ \\in ~ \\mathbb { C } [ x ^ { \\pm } , y ^ { \\pm } , z ^ { \\pm } ]$ such that if the monomial $x ^ { i } y ^ { j } z ^ { k }$ appears in $f$ then the point $( i , j , k ) \\in \\mathbb { Z } ^ { 3 }$ lies in $P$ . Among these Laurent polynomials, we consider those which have coefficient 1 on the vertices of2 3 $P$ and have coefficient 0 on the origin of ${ \\mathbb { R } } ^ { 3 }$ ; this gives rise to the following 1-dimensional family: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[277,211,722,253]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0858/8288fcbf3aee22daf5ba42e9b14f861ff07b45f7537e611d081ae3c98ca1a5a4.jpg","list_items":null,"page_idx":"178","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nf _ { a } = z \\left( a + x + x y + y + { \\frac { 1 } { x } } + { \\frac { 1 } { x y } } + { \\frac { 1 } { y } } \\right) + { \\frac { 1 } { z } }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,268,325,285]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"178","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"with parameter $a \\in \\mathbb { C }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,286,881,411]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"178","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"One can show that $f _ { 2 }$ is mirror to $X _ { 2 }$ and $f _ { 3 }$ is mirror to $X _ { 3 }$ . In other words, a certain generating function for some Gromov–Witten invariants of $X _ { 2 }$ , called quantum period (see Sect. 3.1), is equal to a certain power series, called classical period (see Sect. 3.3), associated to $f _ { 2 }$ , and the same holds for $X _ { 3 }$ and $f _ { 3 }$ . Here we are using the formulation of the Mirror Symmetry correspondence between Fanos and Landau–Ginzburg models that is given in [8, 31, 32] and summarised in Sect. 3.4. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,412,881,483]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"178","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We will see that the Laurent polynomial $f _ { 2 }$ is closely related to the combinatorial input given by the Minkowski decomposition of the hexagon $F$ which is associated to the smoothing of $X$ to $X _ { 2 }$ . Analogously, $f _ { 3 }$ is closely related to the Minkowski decomposition of the hexagon $F$ which is associated to the smoothing of $X$ to $X _ { 3 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,530,412,549]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"178","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1.3 The General Picture ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,575,879,609]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"178","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"What we have described in the case of the projective cone over the del Pezzo surface of degree 6 is an instance of the following conjecture, which is still slightly vague. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,618,881,672]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"178","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conjecture 1.1 ([8]) Let $Q$ be a Fano polytope of dimension 3 and let $X _ { Q }$ be the corresponding toric Fano threefold. Assume that $X _ { Q }$ has Gorenstein singularities. From some “combinatorial input” on $Q$ one constructs ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,683,528,717]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"178","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(i) a smoothing $V$ of $X _ { Q }$ and (ii) a Laurent polynomial $f$ supported on $Q$ such that $f$ is mirror to $V$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,726,362,744]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"178","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,753,879,788]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"178","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The definition of “mirror” that we are using comes from [8, 31, 32] and is given in Definition 3.4. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,791,881,881]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"178","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If the toric variety $X _ { Q }$ is smooth (there are 18 cases), then the polytope $Q$ has only triangular facets which are standard simplices and $X _ { Q }$ is rigid. Thus $V = X _ { Q }$ and $f$ is uniquely determined by insisting that it has coefficient 1 on vertices of $Q$ and coefficient 0 on the origin. This case was already known by Givental [17, 18] who proved that $f$ is mirror to $X _ { Q }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,539,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"178","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An Example of Mirror Symmetry for Fano Threefolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"178","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"175 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,879,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"179","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the example considered in this note, the combinatorial input on $P$ is the choice of a maximal Minkowski decomposition of the facet $F$ of $P$ . There are two such choices which lead to two different smoothings of $X$ and to two different Laurent polynomials. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,162,881,394]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"179","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An interesting case, which is not too restrictive, is the following: the combinatorial input is the choice of a Minkowski decomposition of each facet of $Q$ into $A$ -triangles. Here an $A$ -triangle is either a unitary segment or a lattice triangle which is $\\mathbb { Z } ^ { 2 } \\rtimes \\mathbf { G L } _ { 2 } ( \\mathbb { Z } )$ -equivalent to the convex hull of the points $( 0 , 0 ) , ( 0 , 1 ) , ( \\ell , 0 )$ , for some integer $\\ell \\geq 1$ . For example, both maximal Minkowski decompositions of the hexagon $F$ are decompositions into $A$ -triangles. In these circumstances one can easily construct a Laurent polynomial $f$ which is supported on $Q$ and depends on the choice of the Minkowski decompositions of the facets of $Q$ (see [1], where such Laurent polynomial $f$ is called a Minkowski polynomial). In joint work with Corti and Hacking [11], we construct a smoothing $V$ of $X _ { Q }$ , under a slight additional assumption which is necessary by Petracci [29]. It is conjectured that $f$ is mirror to $V$ . However, even in this situation we completely lack a conceptual way to prove that $f$ is mirror to $V$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,396,881,448]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"179","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Unfortunately, there exist polytopes $Q$ which have facets without Minkowski decompositions into $A$ -triangles. So, at the moment, it is not clear what sort of combinatorial input we should consider on $Q$ in the general case. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,450,879,521]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"179","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Another approach to construct smoothings of the toric Fano variety $X _ { Q }$ is pursued by Coates et al. [10, 30]; they embed $X _ { Q }$ into a bigger toric variety $Z$ and try to deform it inside Z. This works very well in many explicit examples, but a general framework has yet to be discovered. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,522,879,575]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"179","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finally, it is worth mentioning that Conjecture 1.1 can be stated in all dimensions. Therefore, this might be a way to classify smooth Fano varieties that admit a toric degeneration. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,621,476,641]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"179","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1.4 Notation and Conventions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,665,881,737]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"179","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In a real vector space of finite dimension, a polyhedron is the intersection of finitely many closed half-spaces and a polytope is a compact polyhedron; equivalently, a polytope is the convex hull of a finite set. We denote by conv $\\{ \\cdot \\}$ the convex hull of a set. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[113,738,876,773]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"179","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"All varieties and schemes are defined over $\\mathbb { C }$ . We always use the following notation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,783,829,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"179","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\mathrm { d } \\mathrm { P } _ { 6 }$ the smooth del Pezzo surface of degree 6 \n$X$ the projective cone over the anticanonical embedding of $\\mathrm { d } \\mathrm { P } _ { 6 }$ \n$U$ the affine cone over the anticanonical embedding of $\\mathrm { d } \\mathrm { P } _ { 6 }$ \nX2 a general effective divisor of type $( 1 , 1 )$ in $\\mathbb { P } ^ { 2 } \\times \\mathbb { P } ^ { 2 }$ \n$X _ { 3 }$ $\\mathbb { P } ^ { \\bar { 1 } } \\times \\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 }$ \n$F$ the lattice polygon associated to $\\mathrm { d } \\mathrm { P } _ { 6 }$ (see (1) and the left part of Fig. 2) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,54,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"179","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[790,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"179","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. Petracci ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,799,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"180","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$P$ the lattice polytope associated to $X$ (see (2) and Fig. 1) $f _ { a }$ $z \\left( a + x + x y + y + x ^ { - 1 } + x ^ { - 1 } y ^ { - 1 } + y ^ { - 1 } \\right) + z ^ { - 1 }$ , for each $a \\in \\mathbb { C }$ $Q$ an arbitrary Fano polytope (see Definition 2.1) XQ the Fano toric variety associated to the Fano polytope $Q$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,205,389,226]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"180","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 The Geometry of $X$ ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,246,357,265]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"180","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.1 Toric Geometry ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,289,876,327]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"180","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We now recall the basics of Fano toric varieties. We refer the reader to [12, $\\ S 8 . 3 ]$ [15, p. 25], and [22]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,334,881,388]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"180","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 2.1 Let $N$ be a lattice of rank $n$ . A Fano polytope in $N$ is an $n$ -dimensional polytope $Q \\subseteq N _ { \\mathbb { R } }$ such that the origin $0 \\in N$ lies in the interior of $Q$ and every vertex of $Q$ is a primitive lattice point of $N$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,390,881,442]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"180","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The spanning fan of a Fano polytope $Q$ in $N$ is the complete fan whose cones are the cones over the proper faces of $Q$ . We denote by $X _ { Q }$ the toric variety associated to the spanning fan of a Fano polytope $Q$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,451,881,524]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"180","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For brevity, we say that $X _ { Q }$ is associated to $Q$ , and conversely. If $Q$ is a Fano polytope of dimension $n$ , then $X _ { Q }$ is an $n$ -dimensional complete toric variety which is Fano, i.e. its anticanonical divisor is \u001f2 3 2 3 2 3 $\\mathbb { Q }$ -Cartier and ample. Every Fano toric variety3 2 3 2 3! arises in this way from a Fano polytope. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,525,466,542]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"180","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For example, consider the hexagon ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[186,558,760,602]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0456/0a57f824ea6fe11db3fd6086ecaeae26c6dc96c10fd1b802234f30777549465f.jpg","list_items":null,"page_idx":"180","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nF = \\operatorname { c o n v } \\{ ( { \\binom { 1 } { 0 } } , ( { \\binom { 1 } { 1 } } ) , ( { \\binom { 0 } { 1 } } ) , ( { \\binom { - 1 } { 0 } } ) , ( { \\binom { - 1 } { - 1 } } ) , ( { \\binom { 0 } { - 1 } } ) \\} \\subseteq \\mathbb { R } ^ { 2 } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,618,881,689]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"180","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"which is depicted on the left of Fig. 2. It is clear that $F$ is a Fano polytope in $\\mathbb { Z } ^ { 2 }$ . The toric variety associated to its spanning fan is the smooth del Pezzo surface of degree 6, denoted by $\\mathrm { d } \\mathrm { P } _ { 6 }$ . The anticanonical map of $\\mathrm { d } \\mathrm { P } _ { 6 }$ is a closed embedding into $\\mathbb { P } ^ { 6 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,690,876,728]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"180","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now imagine to put the hexagon⎨ $F$ into the plane $\\mathbb { R } ^ { 2 } \\times \\{ 1 \\}$ in $\\mathbb { R } ^ { 3 }$ and create the⎬ pyramid over it with apex at the point ⎩⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ $( 0 , 0 , - 1 )$ : this is the polytope⎠ ⎝ ⎠ ⎝ ⎠ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[129,743,820,804]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0267/9bd01abfab559a2450127e4bf33db11a61fed6654fcfaf4491db5030b2a4917c.jpg","list_items":null,"page_idx":"180","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nP = \\mathrm { c o n v } \\{ ( { \\binom { 1 } { 1 } } , ( { \\binom { 1 } { 1 } } ) , ( { \\binom { 0 } { 1 } } ) , ( { \\binom { 1 } { 1 } } ) , ( { \\binom { - 1 } { 0 } } ) , ( { \\binom { - 1 } { 1 } } ) , ( { \\binom { 0 } { - 1 } } ) , ( { \\binom { 0 } { - 1 } } ) \\} \\subseteq \\mathbb { R } ^ { 3 }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,821,881,894]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"180","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and is depicted in Fig. 1. It is clear that $P$ is a Fano polytope in $\\mathbb { Z } ^ { 3 }$ . Let $X$ be the toric variety associated to the spanning fan of $P$ . Let $U$ be the affine toric open subscheme of $X$ associated to the hexagonal face of $P$ , i.e. $U$ is the affine toric variety associated to the cone $\\mathbb { R } _ { \\geq 0 } ( F \\times \\{ 1 \\} )$ . Hence $X$ (resp. $U$ ) is the projective (resp. affine) cone over the anticanonical embedding of $\\mathrm { d P } _ { 6 }$ . We have that $X$ is a Fano threefold with an isolated non-terminal canonical Gorenstein singularity at the vertex of the cone. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,539,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"180","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An Example of Mirror Symmetry for Fano Threefolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"180","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"177 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,879,142]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"181","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,189,298,210]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"181","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2 Equations ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,231,881,321]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"181","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The equations of the three closed embeddings $\\mathrm { d } \\mathsf { P } _ { 6 } \\subseteq \\mathbb { P } ^ { 6 }$ , $U \\subseteq \\mathbb { A } ^ { 7 }$ and $X \\subseteq \\mathbb { P } ^ { 7 }$ are the same and can be conveniently described in two ways. Here, $x _ { 1 } , \\ldots , x _ { 7 }$ denote the homogeneous coordinates of $\\mathbb { P } ^ { 6 }$ , the affine coordinates of $\\mathbb { A } ^ { 7 }$ and the last homogeneous coordinates of $\\mathbb { P } ^ { 7 }$ , as⎛ $x _ { 0 }$ is the remaining homogeneous coordinate⎞ of $\\mathbb { P } ^ { 7 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,324,296,340]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"181","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The first way is: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[389,358,605,418]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0908/6e3de3a9aabe7b0d3515cea1ea6baa828a2184803e0e5d7d3277768c45523f39.jpg","list_items":null,"page_idx":"181","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname { r a n k } { \\left( { \\begin{array} { l } { x _ { 7 } \\ x _ { 1 } \\ x _ { 2 } } \\\\ { x _ { 4 } \\ x _ { 7 } \\ x _ { 3 } } \\\\ { x _ { 5 } \\ x _ { 6 } \\ x _ { 7 } } \\end{array} } \\right) } \\leq 1 .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,433,881,506]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"181","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Note the repetition of $x _ { 7 }$ on the diagonal. If two of the $x _ { 7 }$ ’s had been two extra variables, these would have been the equations of the Segre embedding of $\\mathbb { P } ^ { 2 } \\times \\mathbb { P } ^ { 2 }$ in $\\mathbb { P } ^ { 8 }$ . This shows that $X$ is the intersection of the projective cone over the Segre embedding of $\\mathbb { P } ^ { 2 } \\times \\mathbb { P } ^ { 2 }$ with two hyperplanes of $\\mathbb { P } ^ { 9 }$ passing through the vertex. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,507,359,524]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"181","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now consider the cube ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[384,542,612,686]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0799/262d1ba485187f1012830774d38852135aadee08ca42a8c3a0b8019d718e2dab.jpg","list_items":null,"page_idx":"181","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[118,708,881,836]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"181","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where at the vertices there are the variables $x _ { 1 } , \\ldots , x _ { 7 }$ . Note the repetition of $x _ { 7 }$ . The second way to describe the equations is to consider the determinants of all rectangles which can be formed with edges of the cube or with diagonals of faces of the cube. If one of the $x _ { 7 }$ ’s had been an extra variable, these would have been the equations of the Segre embedding of $\\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 }$ into $\\mathbb { P } ^ { 7 }$ . This shows that $X$ is the intersection of the projective cone over the Segre embedding of $\\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 }$ with a hyperplane of $\\mathbb { P } ^ { 8 }$ passing through the vertex. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[790,54,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"181","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. Petracci ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"181","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"178 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,881,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"182","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The equations above also appear in [20, Example 3.3]. Moreover, these two ways of describing the equations of $X$ in $\\mathbb { P } ^ { 7 }$ are called Tom and Jerry, respectively, in [5]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,171,644,192]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"182","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.3 Minkowski Sums and Deformations of U ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,214,881,268]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"182","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We first define the notion of Minkowski sum of polyhedra (for instance see [34, $\\ S 1 . 1 ]$ . If $\\Pi _ { 1 } , \\ldots , \\Pi _ { r }$ are polyhedra in a real vector space, we define their Minkowski sum to be the polyhedron ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[218,286,781,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0104/4208566366ec8d72d405618e1b3bfc6769781789870a5a8dc15aaa0a571e3d61.jpg","list_items":null,"page_idx":"182","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\Pi _ { 1 } + \\cdot \\cdot \\cdot + \\Pi _ { r } : = \\{ p _ { 1 } + \\cdot \\cdot \\cdot + p _ { r } \\mid p _ { 1 } \\in \\Pi _ { 1 } , \\ldots , p _ { r } \\in \\Pi _ { r } \\} .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,322,881,414]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"182","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"When we have $\\Pi = \\Pi _ { 1 } + \\cdots + \\Pi _ { r }$ , we say that we have a Minkowski decomposition of the polyhedron $\\Pi$ . We consider Minkowski decompositions up to translation: for instance, we consider the Minkowski decomposition $( p + \\Pi _ { 1 } ) + ( - p + \\Pi _ { 2 } )$ to be equivalent to $\\Pi _ { 1 } + \\Pi _ { 2 }$ for every vector $p$ . Moreover, in what follows we require that the summands $\\Pi _ { j }$ are lattice polyhedra, i.e. their vertices belong to a fixed lattice.3 2 3! \u001f2 3 2 3! \u001f2 3 2 3! ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,414,879,448]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"182","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The hexagon $F$ has two maximal Minkowski decompositions (see Fig. 2): one into 3 unitary segments ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[150,463,799,509]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0159/ef1396ac6f942cb2aba9a2384f26a679c531e6db4d01c309b791a066a3fc05a2.jpg","list_items":null,"page_idx":"182","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nF = \\operatorname { c o n v } \\left\\{ { \\binom { 0 } { 0 } } , { \\binom { 1 } { 0 } } \\right\\} + \\operatorname { c o n v } \\left\\{ { \\binom { 0 } { 0 } } , { \\binom { 0 } { 1 } } \\right\\} + \\operatorname { c o n v } \\left\\{ { \\binom { 0 } { 0 } } , { \\binom { - 1 } { - 1 } } \\right\\}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,522,337,540]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"182","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and one into 2 triangles ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[179,555,769,599]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0815/ec2007de04de387266cc3ec966c01ca321b8c15bb6f8a64b71325e21e7369493.jpg","list_items":null,"page_idx":"182","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nF = \\operatorname { c o n v } \\left\\{ \\left( { 0 } _ { 0 } \\right) , \\left( { - 1 } _ { 0 } \\right) , \\left( { - 1 } _ { - 1 } \\right) \\right\\} + \\operatorname { c o n v } \\left\\{ \\left( { 0 } _ { 0 } \\right) , \\left( { 1 } _ { 0 } ^ { 1 } \\right) , \\left( { 1 } _ { 1 } ^ { 1 } \\right) \\right\\} .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,614,881,704]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"182","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Altmann [3] has noticed that Minkowski decompositions of polytopes induce deformations of affine toric varieties (see also [25] and [28]). More precisely, from a Minkowski decomposition of a polytope $\\Pi$ it is possible to construct an unobstructed deformation of the affine toric variety associated to the cone $\\mathbb { R } _ { \\geq 0 } ( \\Pi \\times \\{ 1 \\} )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,704,876,740]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"182","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the case at hand, the Minkowski decomposition (3) induces the deformation of $U$ over Spec $\\mathbb { C } [ u , v ]$ given by the equations⎝ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[350,756,646,816]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0455/fddfce9568643a6afc5185e8cdbac7c8bddfee39be16a4d5a1016515209be77c.jpg","list_items":null,"page_idx":"182","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname { r a n k } { \\left( \\begin{array} { l l l } { x _ { 7 } } & { x _ { 1 } } & { x _ { 2 } } \\\\ { x _ { 4 } } & { x _ { 7 } + u } & { x _ { 3 } } \\\\ { x _ { 5 } } & { x _ { 6 } } & { x _ { 7 } + v } \\end{array} \\right) } \\leq 1 .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,831,881,887]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"182","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Minkowski decomposition (4) induces the deformation of $U$ over Spec $\\mathbb { C } [ s ]$ given by the equations obtained by taking minors of rectangles on edges and diagonals of faces of the following cube. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,51,539,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"182","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An Example of Mirror Symmetry for Fano Threefolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"182","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"179 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[375,94,617,240]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0023/62aad225471d6ec5788405f62cd7ccc4e873d6bbae010dda87760b8a3e6da5be.jpg","list_items":null,"page_idx":"183","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[120,261,879,298]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"183","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moreover, Altmann [4] shows that the miniversal deformation of $U$ is (the completion of) the union of these two deformations and its base is $\\mathbb { C } [ [ s , u , v ] ] / ( s u , s v )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,342,478,364]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"183","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.4 The Two Smoothings of $X$ ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,387,505,405]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"183","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now we want to study deformations of $X$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,414,881,504]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"183","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 2.2 The base of the miniversal deformation of $X$ is $\\mathbb { C } [ [ s , u , v ] ] / ( s u , s v )$ and has two irreducible components. The 2-dimensional component $( s \\ = \\ 0$ ) is associated to the Minkowski decomposition (3) and deforms $X$ to a general divisor $X _ { 2 } \\in | O _ { \\mathbb { P } ^ { 2 } \\times \\mathbb { P } ^ { 2 } } ( 1 , 1 ) |$ . The $^ { l }$ -dimensional component $( u = v = 0$ ) is associated to the Minkowski decomposition (4) and deforms $X$ to $X _ { 3 } = \\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,513,881,585]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"183","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Consider the local-to-global spectral sequence for $\\mathrm { E x t } _ { X } ^ { \\bullet } ( \\Omega _ { X } , O _ { X } )$ : the second page is $E _ { 2 } ^ { p , q } = \\mathsf { H } ^ { q } ( X , \\mathcal { E } x t _ { X } ^ { p } ( \\Omega _ { X } , O _ { X } ) )$ . As $X$ has an isolated singularity, for all $p \\geq 1$ , the sheaf $\\mathcal { E } x t _ { X } ^ { p } ( \\Omega _ { X } , O _ { X } )$ is supported on the singular point of $X$ ; therefore, for all $p \\geq 1$ and $q \\geq 1$ , $E _ { 2 } ^ { p , q } = 0$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,587,881,678]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"183","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $j \\colon W \\ \\hookrightarrow \\ X$ be the smooth locus. The sheaves $\\mathcal { H } o m _ { X } ( \\Omega _ { X } , O _ { X } )$ and $j _ { * } \\Omega _ { W } ^ { 2 } \\otimes { \\cal O } _ { X } ( - K _ { X } )$ are the same, because they are both reflexive and coincide on $W$ . As $X$ is toric and $- K _ { X }$ is ample, by Bott–Steenbrink–Danilov vanishing [12, Theorem 9.3.1] (see also [6, 14, 26]) one has $E _ { 2 } ^ { 0 , q } = \\mathrm { H } ^ { q } ( X , \\mathcal { H } o m _ { X } ( \\Omega _ { X } , O _ { X } ) ) = 0$ for all $q \\geq 1$ . This argument comes from the proof of [33, Theorem 5.1]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,678,876,714]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"183","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Therefore $E _ { 2 }$ is zero outside the line $q = 0$ . This implies that, for all $p \\geq 0$ , the natural map ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[302,729,694,753]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0662/2abfa8124514a5d300dc43c54006eea783fe733c2a038d4e88065c9c6556fa94.jpg","list_items":null,"page_idx":"183","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname { E x t } _ { X } ^ { p } ( \\Omega _ { X } , O _ { X } ) \\to \\operatorname { H } ^ { 0 } ( X , { \\mathcal { E } } x t _ { X } ^ { p } ( \\Omega _ { X } , O _ { X } ) )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,768,881,822]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"183","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is an isomorphism. Since the unique singular point of $X$ is contained in $U$ and $U$ is affine, we have $\\mathrm { H } ^ { 0 } ( X , \\mathcal { E } x t _ { X } ^ { p } ( \\Omega _ { X } ^ { \\phantom { \\dagger } } , O _ { X } ) ) = \\mathrm { \\bar { E } x t } _ { U } ^ { p } ( \\Omega _ { U } ^ { \\phantom { \\dagger } } , O _ { U } )$ for all $p \\geq 1$ . This implies that, for all $p \\geq 1$ , the natural map ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[312,837,681,861]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0757/9b730caa7868713ab4d95bfc6530c6379e3879022883ac11fe8e22862ce4ffbd.jpg","list_items":null,"page_idx":"183","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\phi _ { p } \\colon \\mathrm { E x t } _ { X } ^ { p } ( \\Omega _ { X } , O _ { X } ) \\to \\mathrm { E x t } _ { U } ^ { p } ( \\Omega _ { U } , O _ { U } ) ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,876,296,894]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"183","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is an isomorphism. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,152,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"183","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"180 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[790,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"183","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. Petracci ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,270]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"184","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We now consider the functors of infinitesimal deformations of $X$ and $U \\colon D e f _ { X }$ and $D e f _ { U }$ , which are covariant functors from the category of local finite $\\mathbb { C }$ -algebras to the category of sets (see [24, §3]). There is an obvious map $\\phi \\colon D e f _ { X } D e f _ { U }$ , which restricts a deformation of $X$ to $U$ . Since $X$ is normal, $\\mathrm { E x t } _ { X } ^ { 1 } ( \\Omega _ { X } , O _ { X } )$ is the tangent space of $D e f _ { X }$ and $\\mathrm { E x t } _ { X } ^ { 2 } ( \\Omega _ { X } , { \\cal O } _ { X } )$ is an obstruction space for $D e f _ { X }$ , and a similar statement holds for $U$ . Since $\\phi _ { 1 }$ is bijective and $\\phi _ { 2 }$ is injective, we have that $\\phi$ induces an isomorphism on tangent spaces and an injection on obstruction spaces. Therefore, by [24, Remark 4.12], $\\phi$ is smooth and induces an isomorphism on tangent spaces. In particular, the two functors $D e f _ { X }$ and $D e f _ { U }$ have the same hull, i.e. the bases of the miniversal deformations of $X$ and $U$ are the same. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,271,879,360]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"184","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The equations of the two deformations of $U$ , given in Sect. 2.3, can be projectivised to construct deformations of $X$ : it is enough to replace $s , u$ and $v$ by $s x _ { 0 }$ , $u x _ { 0 }$ and $v x _ { 0 }$ . These are the two components of the miniversal deformation of $X$ . The fact that they are associated to the two Minkowski decompositions (3) and (4) of the hexagon $F$ follows from the discussion in Sect. 2.3. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,361,881,450]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"184","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From Sect. 2.2 we know that $X$ is the intersection of the projective cone over the Segre embedding of $\\mathbb { P } ^ { 2 } \\times \\mathbb { P } ^ { 2 }$ with two hyperplanes of $\\mathbb { P } ^ { 9 }$ passing through the vertex of the cone. On the component $( s = 0$ ), in the deformation we are moving these two hyperplanes away from the vertex. Therefore, the general fibre over this component is $X _ { 2 }$ , a general $( 1 , 1 )$ -divisor in $\\mathbb { P } ^ { 2 } \\times \\mathbb { P } ^ { 2 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,451,881,540]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"184","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recall that $X$ is the intersection of the projective cone over the Segre embedding of $\\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 }$ with a hyperplane of $\\mathbb { P } ^ { 8 }$ passing through the vertex. On the component $( u ~ = ~ v ~ = ~ 0 )$ ), in the deformation we are moving this hyperplane of $\\mathbb { P } ^ { 8 }$ away from the vertex. Therefore, the general fibre on this component is $X _ { 3 } = \\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 } \\times \\mathbb { P } ^ { 1 }$ . □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,587,373,608]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"184","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 Mirror Symmetry ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,626,735,647]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"184","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.1 Gromov–Witten Invariants and Quantum Periods ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,669,881,723]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"184","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The quantum period of a smooth Fano variety is a generating function for some genus zero Gromov–Witten invariants. The regularised quantum period is a slightly different version, which is convenient for our description of Mirror Symmetry. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,732,876,770]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"184","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3.1 ([8, 9, 31]) The quantum period and the regularised quantum period of a smooth Fano variety $V$ are the following power series ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[225,783,769,827]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0105/926a98c6f4278a20c488c4a094f73ffc6bf3076c79258a85dad835183b4dfdc5.jpg","list_items":null,"page_idx":"184","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nG _ { V } ( t ) = 1 + \\sum _ { \\beta \\in \\mathrm { H } _ { 2 } ( V , \\mathbb { Z } ) } \\langle [ \\mathsf { p t } ] \\psi ^ { - K _ { V } \\cdot \\beta - 2 } \\rangle _ { 0 , 1 , \\beta } ^ { V } t ^ { - K _ { V } \\cdot \\beta } \\in \\mathbb { Q } \\mathbb { I } t ]\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[173,833,824,876]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0772/7de5b065967ed9b99e153be5ae4e152ee58f317705b88ba0c259cad3c9df81f9.jpg","list_items":null,"page_idx":"184","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\widehat { G } _ { V } ( t ) = 1 + \\sum _ { \\beta \\in \\mathrm { H } _ { 2 } ( V , \\mathbb { Z } ) } ( - K _ { V } \\cdot \\beta ) ! \\langle [ \\mathrm { p t } ] \\psi ^ { - K _ { V } \\cdot \\beta - 2 } \\rangle _ { 0 , 1 , \\beta } ^ { V } t ^ { - K _ { V } \\cdot \\beta } \\in \\mathbb { Q } [ [ t ] ]\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,52,539,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"184","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An Example of Mirror Symmetry for Fano Threefolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"184","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,881,145]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"185","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where \u000f[pt]ψ−KV ·β−2\u0010 V denotes the 1-marked genus zero Gromov–Witten invariant of curve class $\\beta$ associated to the cohomology class of a point in $V$ and gravitational descendant of order $- K _ { V } \\cdot \\beta - 2$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,153,881,262]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"185","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Roughly speaking, $\\langle \\mathrm { [ p t ] } \\psi ^ { - K _ { V } \\cdot \\beta - 2 } \\rangle _ { 0 , 1 , \\beta } ^ { V }$ is the number of rational curves in $V$ of class $\\beta$ passing through a fixed general point of $V$ and satisfying a certain condition on their complex structure. Therefore, the quantum period $G _ { V }$ gives information about rational curves in $V$ . The series $G _ { V }$ is a symplectic invariant of $V$ , so it does not change if $V$ is deformed to another smooth Fano variety through a deformation with smooth fibres. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,264,881,316]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"185","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If the anticanonical line bundle $O _ { V } ( - K _ { V } )$ is divisible by a positive integer $m$ inside the Picard group of the smooth Fano variety $V$ , then only powers of $t ^ { m }$ appear in the (regularised) quantum period of $V$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,318,876,352]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"185","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is also possible to define quantum periods for Fano varieties with quotient singularities [27, $\\ S 3 . 3 ]$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,354,881,426]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"185","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is known how to compute the quantum period of smooth Fano varieties which are either toric or complete intersections in smooth Fano toric varieties [7, 17]. The quantum periods of all smooth Fano varieties of dimension $\\leq 3$ have been computed by Coates et al. [9]. In particular, we have the following formulae for $X _ { 2 }$ and $X _ { 3 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,433,876,469]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"185","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 3.2 ([9]) The quantum periods and the regularised quantum periods of $X _ { 2 }$ and $X _ { 3 }$ are the following. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[332,484,665,536]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0478/5a8b09583d2c2ae6de17b302875d7a9cf3bbf0bd0397044ba313b4dfb81a0e33.jpg","list_items":null,"page_idx":"185","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nG _ { X _ { 2 } } ( t ) = \\sum _ { l = 0 } ^ { \\infty } \\sum _ { m = 0 } ^ { \\infty } \\frac { ( l + m ) ! } { ( l ! ) ^ { 3 } ( m ! ) ^ { 3 } } t ^ { 2 l + 2 m }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[277,542,722,594]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0811/9faf04336bfe5a7d8ba7c258ef06919bd987a3ecbd2b7d4af312535c0a72a3ab.jpg","list_items":null,"page_idx":"185","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nG _ { X _ { 3 } } ( t ) = \\sum _ { l = 0 } ^ { \\infty } \\sum _ { m = 0 } ^ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { 1 } { ( l ! ) ^ { 2 } ( m ! ) ^ { 2 } ( n ! ) ^ { 2 } } t ^ { 2 l + 2 m + 2 n }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[186,603,810,663]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0951/2be59fb35ccf8833d81e668735bd05dcc03dfd73d220250d1c9b22030e3f9bc7.jpg","list_items":null,"page_idx":"185","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\widehat { G } _ { X _ { 2 } } ( t ) = 1 + 4 t ^ { 2 } + 6 0 t ^ { 4 } + 1 1 2 0 t ^ { 6 } + 2 4 2 2 0 t ^ { 8 } + 5 6 7 5 0 4 t ^ { 1 0 } + \\cdot \\cdot \\cdot } \\\\ & { \\widehat { G } _ { X _ { 3 } } ( t ) = 1 + 6 t ^ { 2 } + 9 0 t ^ { 4 } + 1 8 6 0 t ^ { 6 } + 4 4 7 3 0 t ^ { 8 } + 1 1 7 2 5 5 6 t ^ { 1 0 } + \\cdot \\cdot \\cdot } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,704,416,725]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"185","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.2 Laurent Polynomials ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,747,881,824]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"185","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $\\mathbb { C } [ x _ { 1 } ^ { \\pm } , \\ldots , x _ { n } ^ { \\pm } ]$ be the ring of Laurent polynomials in $n$ variables with coefficients in $\\mathbb { C }$ . To every monomial $x ^ { i } ~ = ~ x _ { 1 } ^ { i _ { 1 } } \\cdot \\cdot \\cdot x _ { n } ^ { i _ { n } }$ we associate the point $\\begin{array} { r l } { \\mathbf { \\boldsymbol { i } } } & { { } = } \\end{array}$ $( i _ { 1 } , \\ldots , i _ { n } ) \\in \\mathbb { Z } ^ { n }$ . The Newton polytope of a Laurent polynomial $f$ is the convex hull of the lattice points that correspond to the monomials that appear in $f$ , i.e. if ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[790,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"185","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. Petracci ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"185","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"182 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,321,111]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"186","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\textstyle f = \\sum _ { i \\in \\mathbb { Z } ^ { n } } a _ { i } x ^ { i }$ then ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[300,124,694,148]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0618/33726d82fe924ecd515c77e9d4e13e482b018945d21ff872f15be1f68abc2410.jpg","list_items":null,"page_idx":"186","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\operatorname { N e w t } ( f ) = \\operatorname { c o n v } \\left\\{ i \\in \\mathbb { Z } ^ { n } \\mid a _ { i } \\neq 0 \\right\\} \\subseteq \\mathbb { R } ^ { n } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,163,881,217]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"186","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $Q$ is a lattice polytope in $\\mathbb { Z } ^ { n }$ , we say that a Laurent polynomial $\\begin{array} { r l } { f } & { { } \\in } \\end{array}$ $\\mathbb { C } [ x _ { 1 } ^ { \\pm } , \\ldots , x _ { n } ^ { \\pm } ]$ is supported on $Q$ if every monomial appearing in $f$ corresponds to a lattice point of $Q$ , or equivalently if Newt $( f ) \\subseteq Q$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,219,881,325]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"186","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Given a Fano polytope $Q$ in $\\mathbb { Z } ^ { n }$ , Kasprzyk and Tveiten [21] have introduced and studied a particular class of Laurent polynomials supported on $Q$ ; they call them maximally mutable, because these behave well with respect to mutations of Fano polytopes [1]. The definition of maximally mutable Laurent polynomials in dimension 2 can be also found in [2]. We are not going to define maximally mutable Laurent polynomials here, we just mention some properties in a particular case. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,327,881,379]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"186","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In dimension 3, when the Fano toric threefold $X _ { Q }$ has Gorenstein singularities (equivalently $Q$ is a reflexive polytope of dimension 3), every maximally mutable Laurent polynomial on $Q$ is such that: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,388,881,462]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"186","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"• the coefficient of the monomial 1, corresponding to the origin of $\\mathbb { Z } ^ { 3 }$ , is 0; • the monomials corresponding to the vertices of $Q$ have coefficients equal to 1; • on the edges of $Q$ there are binomial coefficients. (For example, the 4 lattice points of an edge with lattice length 3 have coefficients 1, 3, 3, 1.) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,469,881,542]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"186","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the case of the polytope $P$ , a Laurent polynomial is supported on $P$ if and only if its monomials are among 1, 2 $x z$ , xyz, yz, $x ^ { - 1 } z , x ^ { - 1 } y ^ { - 1 } z , y ^ { - 1 } z , z ^ { - 1 }$ , which correspond to the lattice points of $P$ . The Laurent polynomials on $P$ which satisfy the three properties above form a 1-dimensional family ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[277,557,722,599]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0368/1144013c06073722afb57a378b845000f6f969f878125c98ee4cf9ef605efd53.jpg","list_items":null,"page_idx":"186","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nf _ { a } = z \\left( x + x y + y + { \\frac { 1 } { x } } + { \\frac { 1 } { x y } } + { \\frac { 1 } { y } } + a \\right) + { \\frac { 1 } { z } }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,612,881,684]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"186","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"with parameter $a \\in \\mathbb { C }$ . Here $a$ is the coefficient of the centre of the hexagonal facet of $P$ . Kasprzyk and Tveiten [21] show that there are exactly two maximally mutable Laurent polynomials on $P$ , namely $f _ { a }$ with $a = 2$ and $a = 3$ . One notices that, in these two cases, the restriction of $f _ { a }$ to the hexagonal facet of $P$ is reducible: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[275,698,724,753]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0608/d0526cd8fe4fb9405f399c15288f7be5217b96b63ee6ef2d1fd7d11a07560680.jpg","list_items":null,"page_idx":"186","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { c } { { f _ { 2 } = z ( 1 + x ) ( 1 + y ) ( 1 + x ^ { - 1 } y ^ { - 1 } ) + z ^ { - 1 } , } } \\\\ { { { } } } \\\\ { { f _ { 3 } = z ( 1 + x ^ { - 1 } y ^ { - 1 } + y ^ { - 1 } ) ( 1 + x y + y ) + z ^ { - 1 } . } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,770,881,861]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"186","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Newton polytopes of the three factors of $( 1 + x ) ( 1 + y ) ( 1 + x ^ { - 1 } y ^ { - 1 } )$ are the three unitary segments appearing in the Minkowski decomposition (3) of the hexagon $F$ . The Newton polytopes of the factors of $( 1 + x ^ { - 1 } y ^ { - \\bar { 1 } } + y ^ { - 1 } ) ( 1 + x y + y )$ are the two triangles appearing in the Minkowski decomposition (4) of the hexagon $F$ . The Laurent polynomials $f _ { 2 }$ and $f _ { 3 }$ are Minkowski polynomials in the sense of [1]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,51,542,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"186","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An Example of Mirror Symmetry for Fano Threefolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"186","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"183 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,375,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"187","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.3 Classical Periods ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[116,132,797,151]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"187","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We now define the classical period of a Laurent polynomial in $n$ variables. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,157,879,196]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"187","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3.3 ([1, 16]) The classical period of $f \\in \\mathbb { C } [ x _ { 1 } ^ { \\pm } , \\ldots , x _ { n } ^ { \\pm } ]$ is the power series ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[227,207,769,309]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0860/a3fe8c381cfb92a3260d2d60b91039e5a7631e9b3f1b5e4cfac56f26650f9277.jpg","list_items":null,"page_idx":"187","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { l } { \\displaystyle \\pi _ { f } ( t ) = \\left( \\frac { 1 } { 2 \\pi \\mathrm { i } } \\right) ^ { n } \\int _ { \\Gamma _ { \\varepsilon } } \\frac { 1 } { 1 - t f ( x _ { 1 } , \\dots , x _ { n } ) } \\frac { \\mathrm { d } x _ { 1 } \\wedge \\dots \\wedge \\mathrm { d } x _ { n } } { x _ { 1 } \\cdot \\cdot \\cdot x _ { n } } } \\\\ { = \\displaystyle \\sum _ { k = 0 } ^ { \\infty } \\mathrm { c o e f f } _ { 1 } ( f ^ { k } ) t ^ { k } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,321,881,394]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"187","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where in the first formula we are integrating a holomorphic $n$ -form of the torus $( \\mathbb { C } ^ { \\times } ) ^ { n } = \\operatorname { S p e c } \\mathbb { C } [ x _ { 1 } ^ { \\pm } , \\dots , x _ { n } ^ { \\pm } ]$ over the real torus $\\Gamma _ { \\varepsilon } = \\{ | x _ { 1 } | = \\cdot \\cdot \\cdot = | x _ { n } | = \\varepsilon \\} \\subseteq$ $( \\mathbb { C } ^ { \\times } ) ^ { n }$ , for some $0 < \\varepsilon \\ll 1$ , and coeff $( f ^ { k } ) \\in \\mathbb { C }$ is the coefficient of the monomial $1 = x _ { 1 } ^ { 0 } \\cdots x _ { n } ^ { 0 }$ - 1in the Laurent polynomial $f ^ { k }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,402,881,456]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"187","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The equality between the two formulae in the definition above comes from applying Cauchy’s integral formula $n$ times. The classical period $\\pi _ { f }$ is related to the Hodge theory of the fibres of $f \\colon ( \\mathbb { C } ^ { \\times } ) ^ { n } \\to \\mathbb { A } ^ { 1 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,457,774,475]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"187","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"One can see that the classical period of the Laurent polynomial $f _ { a }$ is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[170,486,826,573]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0752/6f56501d1ee209d7d7f3219318b6d0e3c0fedabf2bdd2eb183f7c0d0588374cd.jpg","list_items":null,"page_idx":"187","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { \\pi _ { f _ { a } } ( t ) = 1 + 2 a t ^ { 2 } + ( 6 a ^ { 2 } + 3 6 ) t ^ { 4 } + ( 2 0 a ^ { 3 } + 3 6 0 a + 2 4 0 ) t ^ { 6 } } \\\\ & { \\qquad + ( 7 0 a ^ { 4 } + 2 5 2 0 a ^ { 2 } + 3 3 6 0 a + 6 3 0 0 ) t ^ { 8 } } \\\\ & { \\qquad + ( 2 5 2 a ^ { 5 } + 1 5 1 2 0 a ^ { 3 } + 3 0 2 4 0 a ^ { 2 } + 1 1 3 4 0 0 a + 9 0 7 2 0 ) t ^ { 1 0 } + \\cdot \\cdot \\cdot } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,588,394,608]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"187","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for every $a \\in \\mathbb { C }$ . In particular, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[184,618,815,677]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0419/e49b42d699af52abaf8e2f1e63c273d348590e65b238c095ba5e64648855704c.jpg","list_items":null,"page_idx":"187","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { l } { { \\pi _ { f _ { 2 } } ( t ) = 1 + 4 t ^ { 2 } + 6 0 t ^ { 4 } + 1 1 2 0 t ^ { 6 } + 2 4 2 2 0 t ^ { 8 } + 5 6 7 5 0 4 t ^ { 1 0 } + \\cdots , } } \\\\ { { \\nonumber } } \\\\ { { \\pi _ { f _ { 3 } } ( t ) = 1 + 6 t ^ { 2 } + 9 0 t ^ { 4 } + 1 8 6 0 t ^ { 6 } + 4 4 7 3 0 t ^ { 8 } + 1 1 7 2 5 5 6 t ^ { 1 0 } + \\cdots . } } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,717,646,738]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"187","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.4 Fano/Landau–Ginzburg Correspondence ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,761,881,834]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"187","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Mirror Symmetry [8, 23] predicts that the mirror of a smooth Fano $n$ -fold $V$ is a pair $( Y , w )$ , called Landau–Ginzburg model, where $Y$ is an $n$ -fold and $w \\in \\Gamma ( Y , O _ { Y } )$ is a regular function. The Gromov–Witten theory of $V$ should be related to the Hodge theory of the fibres of $w \\colon Y \\ \\ \\mathbb { A } ^ { 1 }$ as follows: the regularised quantum period ${ \\widehat { G } } _ { V }$ (see Definition 3.1) of $V$ coincides with the period $\\pi _ { w }$ which is defined as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[790,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"187","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. Petracci ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,152,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"187","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"184 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,91,835,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"188","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[403,123,594,163]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0596/15ff247c7df311389ba442076adc66b9b4573b7168c6360296a8e5b141461945.jpg","list_items":null,"page_idx":"188","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\pi _ { w } ( t ) = \\int _ { \\Gamma } \\frac { \\Omega } { 1 - t w }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,178,879,213]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"188","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $\\Omega$ is an appropriate holomorphic $n$ -form on $Y$ and $\\Gamma \\in \\mathrm { H } _ { n } ( Y ; \\mathbb { Z } )$ is such that $\\begin{array} { r } { \\int _ { \\Gamma } \\Omega = 1 } \\end{array}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,214,881,340]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"188","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Under some circumstances (which conjecturally and experimentally should coincide with when there is a toric degeneration of $V$ ) there is an open subset of $Y$ that is isomorphic to the torus $( \\mathbb { C } ^ { \\times } ) ^ { n } = \\operatorname { S p e c } \\mathbb { C } [ x _ { 1 } ^ { \\pm } , \\dots , x _ { n } ^ { \\pm } ]$ . In this case the restriction of $w$ 1 to this open subset gives a Laurent polynomial $\\ddot { f } \\in \\mathbb { C } [ x _ { 1 } ^ { \\pm } , \\ldots , x _ { n } ^ { \\pm } ]$ . In this situation the period $\\pi _ { w }$ in (5), when $Y = ( \\mathbb { C } ^ { \\times } ) ^ { n }$ , $\\Gamma = \\{ | x _ { 1 } | = \\cdot \\cdot \\cdot =$ $\\left| x _ { n } \\right| = \\varepsilon \\}$ and $\\Omega = ( 2 \\pi \\mathbf { i } ) ^ { - n } ( x _ { 1 } \\cdot \\cdot \\cdot x _ { n } ) ^ { - 1 } \\mathrm { d } x _ { 1 } \\cdot \\cdot \\cdot \\cdot \\mathrm { d } x _ { n }$ , becomes the classical period of a Laurent polynomial (see Definition 3.3). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,340,874,376]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"188","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Thus, a down-to-earth formulation of Mirror Symmetry between smooth Fano varieties and Laurent polynomials is the following. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,384,881,439]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"188","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 3.4 ([8, 31, 32]) A Laurent polynomial $f \\in \\mathbb { C } [ x _ { 1 } ^ { \\pm } , \\ldots , x _ { n } ^ { \\pm } ]$ is mirror to a smooth Fano variety $V$ of dimension $n$ if the classical period of the former coincides with the regularised quantum period of the latter: 2 3 - $\\widehat { G } _ { V } ^ { ^ { \\bullet } } = \\pi _ { f }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[143,447,876,466]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"188","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The equality $\\widehat { G } _ { V } = \\pi _ { f }$ is equivalent to $G _ { V }$ being equal to the oscillatory integral ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[248,483,749,534]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0682/9ee20c42bb2767aa08f510215ce7027cadc87720dba56877467498b6a57a15bb.jpg","list_items":null,"page_idx":"188","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\left( { \\frac { 1 } { 2 \\pi \\mathrm { i } } } \\right) ^ { n } \\int _ { \\Gamma _ { \\varepsilon } } \\mathrm { e } ^ { t f } { \\frac { \\mathrm { d } x _ { 1 } \\wedge \\dots \\wedge \\mathrm { d } x _ { n } } { x _ { 1 } \\cdot \\cdot \\cdot x _ { n } } } = \\sum _ { k = 0 } ^ { \\infty } { \\frac { \\mathrm { c o e f f } _ { 1 } ( f ^ { k } ) } { k ! } } t ^ { k } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,549,881,605]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"188","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Moreover, the equality ${ \\widehat { G } } _ { V } ~ = ~ \\pi _ { f }$ can be upgraded to an equality between the Gauss–Manin connection on the middle cohomology of the fibres of $f$ and the Dubrovin connection of the quantum D-module of $V$ (see [19]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,614,876,651]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"188","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 3.5 ([1, 9]) The Laurent polynomial $f _ { 2 }$ (resp. $f _ { 3 }$ ) is mirror to the smooth Fano threefold $X _ { 2 }$ (resp. $X _ { 3 }$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,659,881,750]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"188","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Set $a = 2$ or $a = 3$ . We need to show that the two power series $\\pi _ { f _ { a } }$ and ${ \\widehat { G } } _ { X _ { a } }$ coincide. By comparing the formulae given in Proposition 3.2 and at the end of Sect. 3.3, one can check the equality of finitely many coefficients. In order to prove the equality of all coefficients, one has to show that $\\pi _ { f _ { a } }$ and ${ \\widehat { G } } _ { X _ { a } }$ satisfy the same linear differential equation; this is done in [1] and [9]. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,758,881,810]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"188","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Now we are ready to illustrate Conjecture 1.1 in the example of the projective cone over the smooth del Pezzo surface of degree 6, which is the running example of this note. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,812,879,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"188","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In Proposition 2.2 we saw that the Minkowski decomposition (3) of the hexagonal facet of $P$ into three unitary segments is associated to the smoothing of $X$ to $X _ { 2 }$ . In Sect. 3.2 we saw that the restriction of $f _ { 2 }$ to the hexagonal facet of $P$ is reducible and that the Newton polytopes of its three factors are the three unitary segments appearing in the Minkowski decomposition (3). By Proposition 3.5 we know that $f _ { 2 }$ is mirror to $X _ { 2 }$ . This is an instance of Conjecture 1.1: from the combinatorial input of the Minkowski decomposition of the hexagonal facet of $P$ into three unitary segments we have constructed the smoothing $X _ { 2 }$ of $X$ and the Laurent polynomial $f _ { 2 }$ which is mirror to $X _ { 2 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,51,542,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"188","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An Example of Mirror Symmetry for Fano Threefolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"188","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"185 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,879,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"189","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,180,879,250]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"189","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In a completely analogous manner, we can observe that the Minkowski decomposition (4) of the hexagonal facet of $P$ into two triangles induces the smoothing $X _ { 3 }$ of $X$ and the Laurent polynomial $f _ { 3 }$ . This provides another example for Conjecture 1.1 because $f _ { 3 }$ is mirror to $X _ { 3 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,252,881,394]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"189","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As mentioned in Sect. 1.3, given a reflexive polytope $Q$ of dimension 3, from the combinatorial datum given by the choice of a Minkowski decomposition of each facet of $Q$ into $A$ -triangles, one constructs an associated Laurent polynomial $f$ supported on $Q$ . From the same combinatorial datum on $Q$ (with a slight additional condition which we do not mention here), by Corti et al. [11] it is possible to construct a smoothing $V$ of the toric Fano threefold $X _ { Q }$ associated to $Q$ . It is conjectured that the smooth Fano threefold $V$ is mirror to the Laurent polynomial $f$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,396,879,486]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"189","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This circle of ideas should be considered as an approach to the problem of classifying smooth Fano varieties of dimension $\\geq 4$ . Indeed, computers can classify Fano polytopes; therefore, once one has developed a combinatorial technology for smoothing toric Fano varieties, one should be able to construct all smooth Fano varieties which admit a toric degeneration. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,487,881,665]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"189","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There is another difficulty: a smooth Fano variety may have many toric degenerations, hence may arise from several polytopes. For instance, $X _ { 3 } = \\bar { \\mathbb { P } } ^ { 1 } \\times \\bar { \\mathbb { P } } ^ { 1 } \\times \\bar { \\mathbb { P } } ^ { 1 }$ is itself toric and degenerates to the toric Fano $X$ . Conjecturally, these many toric degenerations of a smooth Fano correspond to many mirror Laurent polynomials; these Laurent polynomials are related via certain birational transformations of the torus $( \\mathbb { C } ^ { \\times } ) ^ { n }$ , which are called mutations [1, 13, 16] and preserve the classical periods. Therefore, it is conjectured that deformation families of smooth Fano varieties of dimension $n$ are in one-to-one correspondence with mutation-equivalence classes of some “special” Laurent polynomials in $n$ variables. We are not going to expand on this here because otherwise it would lead us far beyond the scope of this note. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,686,879,780]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"189","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acknowledgements I am indebted to Tom Coates, Alessio Corti, Paul Hacking, Alexander Kasprzyk, Thomas Prince, and the other members of the Fanosearch group for countless fruitful conversations about the topics of this note. I wish to thank the organisers of the conference “Birational geometry and moduli spaces”, held in Rome in June 2018, for giving me the opportunity to present a poster about this subject. Finally, I would like to thank Enrica Floris, Luigi Lunardon, and Diletta Martinelli for useful comments on a preliminary version of this note. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,776,724,791]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"189","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The author was funded by Kasprzyk’s EPSRC Fellowship EP/N022513/1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[790,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"189","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. Petracci ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,54,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"189","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,90,248,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"190","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,138,881,899]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"190","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Akhtar, M., Coates, T., Galkin, S., Kasprzyk, A.M.: Minkowski polynomials and mutations. SIGMA Symmetry Integrability Geom. Methods Appl. 8, Paper 094, 17 (2012) \n2. Akhtar, M., Coates, T., Corti, A., Heuberger, L., Kasprzyk, A., Oneto, A., Petracci, A., Prince, T., Tveiten, K.: Mirror symmetry and the classification of orbifold del Pezzo surfaces. Proc. Am. Math. Soc. 144(2), 513–527 (2016) \n3. Altmann, K.: Minkowski sums and homogeneous deformations of toric varieties. Tohoku Math. J. 47(2), 151–184 (1995) \n4. Altmann, K.: The versal deformation of an isolated toric Gorenstein singularity. Invent. Math. 128(3), 443–479 (1997) \n5. Brown, G., Kerber, M., Reid, M.: Fano 3-folds in codimension 4, Tom and Jerry. Part I. Compos. Math. 148(4), 1171–1194 (2012) \n6. Buch, A., Thomsen, J.F., Lauritzen, N., Mehta, V.: The Frobenius morphism on a toric variety. Tohoku Math. J. (2) 49(3), 355–366 (1997) \n7. Coates, T., Givental, A.: Quantum Riemann-Roch, Lefschetz and Serre. Ann. Math. (2) 165(1), 15–53 (2007) \n8. Coates, T., Corti, A., Galkin, S., Golyshev, V., Kasprzyk, A.: Mirror symmetry and Fano manifolds. In: European Congress of Mathematics, pp. 285–300. European Mathematical Society, Zürich (2013) \n9. Coates, T., Corti, A., Galkin, S., Kasprzyk, A.: Quantum periods for 3–dimensional Fano manifolds. Geom. Topol. 20(1), 103–256 (2016) \n10. Coates, T., Kasprzyk, A., Prince, T.: Laurent inversion. Preprint arXiv:1707.05842. https:// arxiv.org/1707.05842 \n11. Corti, A., Hacking, P., Petracci, A.: Smoothing toric Fano threefolds (in preparation) \n12. Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011) \n13. Cruz Morales, J.A., Galkin, S.: Upper bounds for mutations of potentials. SIGMA Symmetry Integrability Geom. Methods Appl. 9, Paper 005, 13 (2013) \n14. Fujino, O.: Multiplication maps and vanishing theorems for toric varieties. Math. Z. 257(3), 631–641 (2007) \n15. Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993). The William H. Roever Lectures in Geometry \n16. Galkin, S., Usnich, A.: Mutations of Potentials. IPMU, Kashiwa, pp. 10–0100 (2010). Preprint \n17. Givental, A.B.: Equivariant Gromov-Witten invariants. Int. Math. Res. Notices 13, 613–663 (1996) \n18. Givental, A.: A mirror theorem for toric complete intersections. In: Topological field theory, primitive forms and related topics (Kyoto, 1996). Progress in Mathematics, vol. 160, pp. 141– 175. Birkhäuser, Boston (1998) \n19. Golyshev, V.V.: Classification problems and mirror duality. In: Surveys in Geometry and Number Theory: Reports on Contemporary Russian Mathematics. London Mathematical Society Lecture Note Series, vol. 338, pp. 88–121. Cambridge University, Cambridge (2007) \n20. Jahnke, P., Radloff, I.: Terminal Fano threefolds and their smoothings. Math. Z. 269(3–4), 1129–1136 (2011) \n21. Kasprzyk, A., Tveiten, K.: Maximally mutable Laurent polynomials (in preparation) \n22. Kasprzyk, A.M., Nill, B.: Fano polytopes. In: Strings, Gauge Fields, and the Geometry Behind, pp. 349–364. World Science Publication, Hackensack (2013) \n23. Katzarkov, L., Przyjalkowski, V.: Landau-Ginzburg models—old and new. In: Proceedings of the Gökova Geometry-Topology Conference 2011, pp. 97–124. International Press, Somerville (2012) \n24. Manetti, M.: Differential graded Lie algebras and formal deformation theory. In: Algebraic geometry—Seattle 2005. Part 2. Proceedings of Symposia in Pure Mathematics, vol. 80, pp. 785–810. American Mathematical Society, Providence (2009) \n25. Mavlyutov, A.R.: Deformations of toric varieties via Minkowski sum decompositions of polyhedral complexes. Preprint arXiv:0902.0967. https://arxiv.org/abs/0902.0967 \n26. Musta¸ta, M.: Vanishing theorems on toric varieties. Tohoku Math. J. (2) ˘ 54(3), 451–470 (2002) \n27. Oneto, A., Petracci, A.: On the quantum periods of del Pezzo surfaces with ${ \\scriptstyle { \\frac { 1 } { 3 } } } ( 1 , 1 )$ singularities. Adv. Geom. 18(3), 303–336 (2018) \n28. Petracci, A.: Homogeneous deformations of toric pairs. Preprint arXiv:1801.05732. https:// arxiv.org/1801.05732 \n29. Petracci, A.: Some examples of non-smoothable Gorenstein Fano toric threefolds. Math. Z. https://doi.org/10.1007/s00209-019-02369-8 \n30. Prince, T.: Cracked polytopes and Fano toric complete intersections. Manuscripta Math. https:// doi.org/10.1007/s00229-019-01149-2 \n31. Przyjalkowski, V.: On Landau-Ginzburg models for Fano varieties. Commun. Number Theory Phys. 1(4), 713–728 (2007) \n32. Przyjalkowski, V.V.: Weak Landau-Ginzburg models of smooth Fano threefolds. Izv. Math. 77(4), 772–794 (2013) \n33. Totaro, B.: Jumping of the nef cone for Fano varieties. J. Algebraic Geom. 21(2), 375–396 (2012) \n34. Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,52,539,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"190","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"An Example of Mirror Symmetry for Fano Threefolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"190","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[116,90,883,376]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"191","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,54,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"191","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[790,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"191","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A. Petracci ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,88,738,115]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"192","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Chern Numbers of Uniruled Threefolds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,148,451,166]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"192","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Stefan Schreieder and Luca Tasin ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,288,881,376]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"192","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract In this paper we show that the Chern numbers of a smooth Mori fibre space in dimension three are bounded in terms of the underlying topological manifold. We also generalise a theorem of Cascini and the second named author on the boundedness of Chern numbers of certain threefolds to the case of negative Kodaira dimension. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,396,865,430]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"192","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords Minimal model program $\\cdot$ Mori fibre spaces $\\cdot$ Three-folds $\\cdot$ Topological properties of complex manifolds $\\cdot$ Characteristic classes and numbers ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,468,309,489]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"192","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,513,881,638]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"192","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"One of the most basic numerical invariants of a compact complex manifold are its Chern numbers. While these numbers depend only on the topological type of the complex structure of the tangent bundle, they are in general not invariants of the underlying topological manifold, but really depend on the complex structure. In fact, answering a question of Hirzebruch from 1954, all linear combinations of Chern and Hodge numbers which are topological invariants of smooth complex projective varieties have recently been determined in [9–11]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,639,881,746]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"192","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Generalising Hirzebruch’s question, Kotschick asked [8] (see also [12]) whether the topology of the underlying smooth manifold determines the Chern numbers of smooth complex projective varieties at least up to finite ambiguity. In [19], we have shown that in dimension at least four, this question has in general a negative answer. That is, there are smooth real manifolds that carry infinitely many complex algebraic structures such that the corresponding Chern numbers are unbounded, except for $c _ { n }$ , $c _ { 1 } c _ { n - 1 }$ and $c _ { 2 } ^ { 2 }$ which are known to be bounded (see [14] for the non-trivial one $c _ { 1 } c _ { n - 1 }$ ). This result left however open the case of threefolds, where it remains unknown whether $c _ { 1 } ^ { 3 }$ is determined up to finite ambiguity by the underlying smooth manifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,897,703,941]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"192","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_11 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[804,91,876,139]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"192","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,834,804,878]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"192","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"L. Tasin $( \\boxtimes )$ Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, Milano, Italy e-mail: luca.tasin@unimi.it ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,783,822,825]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"192","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"S. Schreieder Mathematisches Institut, Ludwig–Maximilians–Universität München, München, Germany e-mail: schreieder@math.lmu.de ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,899,876,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"192","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,91,879,163]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"193","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,165,881,363]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"193","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In [2], Cascini and the second named author started to investigate the boundedness question for Chern numbers via methods from the minimal model program, see also [16, 20] for further developments. In dimension three, the approach in [2] is motivated by the Miyaoka–Yau inequality, which implies that for a minimal smooth complex projective threefold of non-negative Kodaira dimension, $c _ { 1 } ^ { 3 }$ can be bounded in terms of the Betti numbers of $X$ , see e.g. [20, Proposition 9]. This observation makes it natural to approach the boundedness of $c _ { 1 } ^ { 3 } ( X )$ by trying to bound the effect on $c _ { 1 } ^ { 3 } ( X )$ of the steps in the minimal model program for $X$ . This leads to a positive answer for the boundedness question for many smooth projective threefolds of nonnegative Kodaira dimension whose minimal model program is a composition of blow-downs to points and smooth curves in smooth loci, see [2, Corollary 1.5]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,364,881,543]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"193","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this paper we focus on the case of threefolds of negative Kodaira dimension. The main difficulty that we face in this case is that the aforementioned Miyaoka– Yau inequality, which was essential for the case of non-negative Kodaira dimension, does not hold any longer. It is also known by examples of LeBrun [13], that the boundedness does not hold in the non-Kähler case. Nonetheless, for any smooth Kähler threefold $X$ we can run a minimal model program thanks to [6, 7]. If $X$ is uniruled then we arrive at a Mori fibre space $Y B$ , i.e. a Kähler threefold $Y$ with at most terminal singularities together with a morphism of relative Picard rank one with connected fibres to a complex Kähler variety $B$ of smaller dimension whose general fibre is Fano. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,545,555,561]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"193","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The first result of this paper is the following. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,572,879,626]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"193","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 1 Let $( X _ { i } ) _ { i \\geq 0 }$ be a sequence of Mori fibre spaces, where $X _ { i }$ are smooth Kähler threefolds. If each $X _ { i }$ is homeomorphic to $X _ { 0 }$ , then the sequence of Chern numbers $c _ { 1 } ^ { 3 } ( X _ { i } )$ is bounded. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,635,881,743]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"193","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The above result should be compared to the fact that all known examples of sequences of homeomorphic varieties with unbounded Chern numbers are Mori fibre spaces, and in fact projective bundles (see [19]). We therefore believe that together with the aforementioned results from [2], the above theorem puts forward strong evidence for the conjecture that the Chern numbers of smooth projective threefolds are determined up to finite ambiguity by the underlying smooth manifold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,743,881,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"193","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $X B$ is a Mori fibre space and $X$ is a smooth Kähler threefold, then there are three main cases to consider, depending on the dimension of $B$ . If $B$ is a point, then $X$ is a Fano variety and we conclude because Fano varieties of fixed dimension form a bounded family. If $B$ is a curve, then it is smooth projective and $X$ is also projective. Since the Pontryagin classes are up to torsion homeomorphism invariants by Novikov’s theorem [17], [20, Proposition 26] proves the above theorem in case all but finitely many of the $X _ { i }$ are Mori fibre spaces over points or curves. Using Novikov’s theorem [17] once again, Theorem 1 thus follows from the following more precise result about Mori fibre spaces over surfaces, where we denote by $H _ { t f } ^ { * } ( X , \\mathbb { Z } )$ the quotient of $H ^ { * } ( X , \\mathbb { Z } )$ by the subgroup of all torsion classes. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,54,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"193","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[671,54,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"193","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"S. Schreieder and L. Tasin ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[123,88,879,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"194","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,135,881,229]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"194","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 2 Let $( X _ { i } ) _ { i \\geq 0 }$ be a sequence of smooth Kähler threefolds admitting $a$ conic bundle structure $f _ { i } : X _ { i } \\longrightarrow S _ { i }$ of relative Picard number $^ { l }$ over a smooth Kähler surface $S _ { i }$ . If there is an isomorphism of graded rings $H _ { t f } ^ { * } ( X _ { i } , \\mathbb { Z } ) ~ \\simeq$ $H _ { t f } ^ { * } ( X _ { 0 } , \\mathbb { Z } )$ which respects the first Pontryagin classes, then the sequence of Chern numbers $c _ { 1 } ^ { 3 } ( X _ { i } )$ is bounded. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,237,881,364]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"194","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In view of Theorem 1, it is therefore natural to wonder if we can also bound the Chern numbers of certain threefolds of negative Kodaira dimension which are not necessarily Mori fibre spaces themselves. Our next result achieves this by generalising [2, Corollary 1.5] to the case of negative Kodaira dimension. To state it, recall that for any smooth complex projective threefold $X$ , there is a cubic form $F _ { X }$ on $H ^ { 2 } ( X , \\mathbb { Q } )$ , given by cup product. For technical reasons, we will assume that the discriminant $\\Delta _ { F _ { X } }$ of the cubic form is non-zero. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,372,881,462]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"194","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Theorem 3 Let $X$ be a smooth complex projective threefold which is uniruled and let $F _ { X }$ be its associated cubic. Assume that $\\Delta _ { F _ { X } } \\neq 0$ and that there exists $a$ birational morphism $f \\colon X \\to Y$ onto a Mori fibre space $Y$ , which is obtained as a composition of divisorial contractions to points and blow-downs to smooth curves in smooth loci. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,462,881,496]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"194","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Then there exists a constant $D$ depending only on the topology of the 6-manifold underlying $X$ such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[446,513,553,536]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0565/2fb2e96d403d75f6acaed640dfe0696554e2ad0a5fd50812a9946bae2cb199ae.jpg","list_items":null,"page_idx":"194","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n| K _ { X } ^ { 3 } | \\le D .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,552,881,642]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"194","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A major step in proving Theorem 3 is Proposition 8 (cf. [2, Theorem 1.3(2)]), where we show that in the assumptions of Theorem 3, most of the topological invariants of $Y$ are determined (up to finite ambiguity) a priori by the smooth manifold underlying $X$ . It would be interesting to understand to what extend this is true in general (see [4] for the case of Betti numbers): ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,651,879,723]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"194","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Question 4 Let $X$ be a smooth complex projective threefold with cubic form $F _ { X }$ and first Pontryagin class $p _ { 1 } ( X )$ . Let $\\mathcal { P }$ be the set of pairs $( F _ { Y } , p _ { 1 } ( Y ) )$ , taken up to isomorphism, such that there exists an MMP $X \\ \\ Y$ . Is the set $\\mathcal { P }$ determined by the pair of invariants $( F _ { X } , p _ { 1 } ( X ) )$ of $X$ up to finite ambiguity? ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,770,321,789]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"194","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1.1 Conventions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,813,881,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"194","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"All manifolds are closed and connected. A Kähler manifold is a complex manifold which admits a Kähler metric. For any (Kähler) manifold $X$ , we denote by $H _ { t f } ^ { * } ( X , \\mathbb { Z } )$ the quotient $H ^ { * } ( X , \\mathbb { Z } ) / H ^ { * } ( X , \\mathbb { Z } ) _ { t o r s }$ , where $H ^ { * } ( X , \\mathbb { Z } ) _ { t o r s }$ denotes the torsion subgroup of $H ^ { * } ( X , \\mathbb { Z } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,52,425,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"194","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Chern Numbers of Uniruled Threefolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,54,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"194","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,551,111]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"195","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 Mori Fibre Spaces Over Surfaces ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,132,687,151]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"195","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The starting point of our investigation is the following lemma. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,159,879,196]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"195","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 5 ([21, Sec. 7.1]) Let $f : X \\to S$ be a Mori fibre space such that $X$ is $a$ smooth projective threefold and $s$ is a surface. Then ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,205,881,334]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"195","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(i) $f : X \\to S$ is a standard (i.e. relative Picard number $I$ ) conic bundle and $s$ is smooth; \n(ii) the discriminant $D \\subset S$ of $f$ is either empty or a reduced curve with at worst ordinary double points; \n(iii) $e ( X ) = 2 ( e ( S ) - p _ { a } ( D ) + 1 )$ , $b _ { 1 } ( X ) = b _ { 1 } ( S )$ and $b _ { 3 } ( X ) = 2 ( b _ { 1 } ( X ) + p _ { a } ( D ) -$ 1); \n(iv) $D \\equiv - f _ { \\ast } K _ { X / S } ^ { 2 }$ and $- 4 K _ { S } \\equiv f _ { * } K _ { X } ^ { 2 } + D .$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[148,340,498,358]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"195","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We will also use the following lemma. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,367,876,403]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"195","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lemma 6 Let $f : X \\to S$ be a Mori fibre space such that $X$ is a smooth projective threefold and $s$ is a surface. Then, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[412,421,583,441]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0585/7e3142429574d533dcb200e3dc73560e8dc690a0e2c166a44baf5aa91d0bcf09.jpg","list_items":null,"page_idx":"195","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nf _ { * } p _ { 1 } ( X ) \\equiv - 3 D ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,456,876,493]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"195","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $p _ { 1 } ( X )$ is the first Pontryagin class of $X$ and $D \\subset S$ denotes the discriminant curve of $f$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,501,881,611]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"195","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Since $s$ is smooth projective by Lemma 5, the Néron–Severi group of $s$ is generated by very ample curves. Hence, it suffices to compute the intersection product with a general smooth projective curve $C \\subset S .$ The preimage $R : = f ^ { - 1 } ( C )$ is then the blow-up of a minimal ruled surface over $C$ in $C . D$ many points. The normal bundle of $R$ in $X$ is given by $\\mathcal { N } _ { R / X } = f ^ { * } \\mathcal { O } _ { S } ( C ) | _ { R }$ . Since $T _ { X } | _ { R } = T _ { R } \\oplus$ $\\textstyle \\mathcal { N } _ { R / X }$ , we get ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[138,626,856,650]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0238/e56417eb5cf6b0054884ade9b07af18ed6f782a27a1413d6b02f6828dcfd6cff.jpg","list_items":null,"page_idx":"195","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n( 1 + c _ { 1 } ( X ) \\vert _ { R } + c _ { 2 } ( X ) \\vert _ { R } ) ( 1 - c _ { 1 } ( \\mathcal { N } _ { R / X } ) + c _ { 1 } ^ { 2 } ( \\mathcal { N } _ { R / X } ) ) = 1 + c _ { 1 } ( R ) + c _ { 2 } ( R ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,666,507,684]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"195","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hence, using $\\mathcal { O } _ { X } ( R ) = f ^ { * } \\mathcal { O } _ { S } ( C )$ , we get ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[248,699,749,896]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0768/0d739b190cfc81ce7603c18ebe81666b61492906c72503e8ba6d18953db1fb73.jpg","list_items":null,"page_idx":"195","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { f _ { * } c _ { 2 } ( X ) . C = c _ { 2 } ( X ) . R } \\\\ & { \\qquad = c _ { 2 } ( X ) | _ { R } } \\\\ & { \\qquad = c _ { 2 } ( R ) - c _ { 1 } ^ { 2 } ( { \\mathcal { N } } _ { R / X } ) + c _ { 1 } ( { \\mathcal { N } } _ { R / X } ) c _ { 1 } ( X ) | _ { R } } \\\\ & { \\qquad = c _ { 2 } ( R ) - f ^ { * } { \\mathcal { O } } _ { S } ( C ) ^ { 3 } + f ^ { * } { \\mathcal { O } } _ { S } ( C ) ^ { 2 } c _ { 1 } ( X ) } \\\\ & { \\qquad = 2 - 4 g ( C ) + 2 + C . D + C ^ { 2 } \\cdot c _ { 1 } ( { \\mathbb { P } } ^ { 1 } ) } \\\\ & { \\qquad = - 2 K _ { S } . C - 2 C ^ { 2 } + C . D + 2 C ^ { 2 } } \\\\ & { \\qquad = - 2 K . C + C . D . } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[671,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"195","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"S. Schreieder and L. Tasin ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,152,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"195","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"192 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,742,111]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By Lemma 5, $f _ { * } c _ { 1 } ^ { 2 } ( X ) \\equiv - 4 K _ { S } - D$ . Using $p _ { 1 } = c _ { 1 } ^ { 2 } - 2 c _ { 2 }$ , we get ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[264,126,733,205]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0380/cf071459f8ed6df4f18980c8616644c75876aea5e170c5a280b4f7fcdfbc4977.jpg","list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array} { r l } & { f _ { * } p _ { 1 } ( X ) . C = f _ { * } c _ { 1 } ^ { 2 } ( X ) . C - 2 f _ { * } c _ { 2 } ( X ) . C } \\\\ & { \\phantom { f _ { * } p _ { 1 } ( X ) . C } = - 4 K _ { S } . C - D . C + 4 K _ { S } . C - 2 D . C } \\\\ & { \\phantom { f _ { * } p _ { 1 } ( X ) . C } = - 3 D . C , } \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,220,348,237]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"which proves the lemma. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,246,879,300]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof of Theorem 2 By [15, Theorem 1.1], any standard conic bundle $f : X \\to S$ , where $X$ is a smooth Kähler threefold, has an algebraic deformation. To bound $K _ { X _ { i } } ^ { 3 }$ it thus suffices to assume that $X _ { i }$ and $S _ { i }$ are projective for any $i$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,301,881,396]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By assumptions, there is an isomorphism $H _ { t f } ^ { 2 } ( X _ { i } , \\mathbb { Z } ) \\ \\simeq \\ H _ { t f } ^ { 2 } ( X _ { 0 } , \\mathbb { Z } ) $ which respects the trilinear forms given by cup products. We use this isomorphism to identify degree two cohomology classes of $X _ { i }$ with those of $X _ { 0 }$ (up to torsion). Using Poincaré duality, we further identify classes of $H _ { t f } ^ { 4 } ( X _ { i } , \\mathbb Z )$ with linear forms on $H _ { t f } ^ { 2 } ( X _ { i } , \\mathbb { Z } ) \\simeq H _ { t f } ^ { 2 } ( X _ { 0 } , \\mathbb { Z } )$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,399,881,453]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The codimension one linear subspace $f _ { i } { } ^ { * } \\mathbb { P } ( H ^ { 2 } ( S _ { i } , \\mathbb { Q } ) )$ of $\\mathbb { P } ( H ^ { 2 } ( X _ { 0 } , \\mathbb { Q } ) )$ is contained in the cubic hypersurface $\\{ \\alpha \\mid \\alpha ^ { 3 } = 0 \\}$ . Passing to a suitable subsequence we can therefore assume that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[369,466,630,490]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0193/1448cb259f7151ac30c9f493bf8c11b64f2738f5f6d48ae59d553c5b9485f2ea.jpg","list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nf _ { i } ^ { * } H ^ { 2 } ( S _ { i } , \\mathbb { Q } ) \\subset H ^ { 2 } ( X _ { 0 } , \\mathbb { Q } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,504,881,582]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"does not depend on $i$ . Let $\\ell _ { i } \\in H _ { t f } ^ { 4 } ( X _ { 0 } , \\mathbb { Z } )$ be the class of a fibre of $f _ { i }$ . The linear form determined by this class on $H ^ { 2 } ( X _ { 0 } , \\mathbb { Q } )$ has kernel $f _ { i } ^ { * } H ^ { 2 } ( S _ { i } , \\mathbb { Q } )$ , and so $\\ell _ { i } \\cdot \\mathbb { Q }$ is independent of $i$ . Since $\\ell _ { i }$ is an integral class with $K _ { X _ { i } } . \\ell _ { i } = - 2$ , we may after possibly passing to another subsequence assume that $\\ell _ { i } = \\ell$ does not depend on $i$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,582,876,618]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since the natural cup product pairing on $H ^ { 2 } ( S _ { i } , \\mathbb { Q } )$ can be recovered from the pairing ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,633,676,657]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0718/affcfae030c9eb2442a155661215b127046ddc5b5788e95b061e512cc13d7fca.jpg","list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nf _ { i } ^ { * } H ^ { 2 } ( S _ { i } , \\mathbb { Q } ) \\times f _ { i } ^ { * } H ^ { 2 } ( S _ { i } , \\mathbb { Q } ) { \\longrightarrow } \\ell \\mathbb { Q } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,671,874,707]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"we get that the pairing on $H ^ { 2 } ( S _ { i } , \\mathbb { Q } )$ is determined by the cubic form on $H ^ { 2 } ( X _ { 0 } , \\mathbb { Q } )$ and so it does not depend on $i$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,708,879,744]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since $f _ { i } ^ { * } H ^ { 2 } ( S _ { i } , \\bar { \\mathbb { Q } } ) \\subset H ^ { 2 } ( X _ { 0 } , \\mathbb { Q } )$ does not depend on $i$ , the same holds for the homomorphism ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[300,759,697,782]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0223/73e9fe583d810546c0f8787a2eee92ea7dc2001f42ede5d5a4b5bf4e6d7b0c5a.jpg","list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\psi _ { i } : H ^ { 2 } ( S _ { i } , \\mathbb { Q } ) { \\longrightarrow } \\mathbb { Q } , \\alpha \\longmapsto p _ { 1 } ( X _ { i } ) . f _ { i } ^ { * } \\alpha\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,798,881,834]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By the projection formula, we have $p _ { 1 } ( X _ { i } ) . f _ { i } ^ { * } \\alpha = ( f _ { i } ) _ { * } p _ { 1 } ( X _ { i } ) . \\alpha$ . Lemma 6 thus yields ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[412,852,580,872]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0235/94f8a6e09a9cfca2e6cfaf94ae604ff0957bf194c4c5435786521ade368e7ca2.jpg","list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\psi _ { i } ( \\alpha ) = - 3 D _ { i } . \\alpha ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,51,428,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Chern Numbers of Uniruled Threefolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,879,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"193 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[858,222,879,237]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"196","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"□ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,881,196]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $D _ { i }$ is the discriminant curve of $f _ { i }$ . This shows that the linear form determined by $[ D _ { i } ] \\in H ^ { 2 } ( S _ { i } , \\mathbb { Q } )$ on $H ^ { 2 } ( S _ { i } , \\mathbb { Q } )$ does not depend on i. Since the natural pairing $H ^ { 2 } ( S _ { i } , \\mathbb { Q } ) \\times H ^ { 2 } ( S _ { i } , \\mathbb { Q } ) \\mathbb { Q }$ is perfect by Poincaré duality, we get that the class $[ D _ { i } ] \\in H ^ { 2 } ( S _ { i } , \\mathbb { Q } )$ does not depend on $i$ . Using again the fact that we know the pairing on $H ^ { 2 } ( S _ { i } , \\mathbb { Q } )$ , we finally get that the self-intersection $D _ { i } ^ { 2 }$ does not depend on $i$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[132,196,835,216]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For any class $y \\in H ^ { 2 } ( X _ { 0 } , \\mathbb { Q } )$ , which does not lie in $f _ { i } ^ { * } H ^ { 2 } ( S _ { i } , \\mathbb { Q } )$ , we have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[116,229,879,252]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0524/4f5db11f1f0fc440a464f540be24889d7bd4d4a81a02a0c5a3cf6e8b3253b37b.jpg","list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n{ \\cal H } ^ { 2 } ( X _ { 0 } , \\mathbb { Q } ) = f _ { i } { } ^ { * } { \\cal H } ^ { 2 } ( S _ { i } , \\mathbb { Q } ) \\oplus y \\cdot \\mathbb { Q } \\mathrm { a n d } { \\cal H } ^ { 4 } ( X _ { 0 } , \\mathbb { Q } ) = f _ { i } { } ^ { * } { \\cal H } ^ { 2 } ( S _ { i } , \\mathbb { Q } ) \\cdot y \\oplus \\ell \\cdot \\mathbb { Q } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,267,881,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In particular, $y ^ { 2 } = u y + \\lambda \\ell$ for some $\\lambda \\in \\mathbb { Q }$ and $u \\in f _ { i } { ^ { * } H ^ { 2 } } ( S _ { i } , \\mathbb { Q } )$ . Replacing $y$ by a suitable multiple of $\\begin{array} { r } { y - \\frac { 1 } { 2 } u } \\end{array}$ , we may thus assume that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[293,322,699,345]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0201/1f06e5f6efe4418fc7a894d20ff77bbfec828984651172acaf71cf0a01e63282.jpg","list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\ny . \\ell = - 2 \\mathrm { ~ a n d ~ } y ^ { 2 } \\in { f _ { i } } ^ { * } H ^ { 4 } ( S _ { i } , \\mathbb { Q } ) = \\ell \\cdot \\mathbb { Q } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[145,361,369,379]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For any $X _ { i }$ , we then get ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[419,396,578,418]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0330/d1e077581be83014cda209bbf9de95039f2141edc0dbda9a997b4f0ae822d2e3.jpg","list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { X _ { i } } = y + f _ { i } ^ { * } z _ { i }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,433,410,451]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for some $z _ { i } \\in H ^ { 2 } ( S _ { i } , \\mathbb { Z } )$ . Since ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[416,465,578,490]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0331/05794fdd560ac1676de0bb611a334d55e9bb995285eb068384a2d7e09254ddd2.jpg","list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nK _ { X _ { i } } ^ { 3 } = y ^ { 3 } - 6 z _ { i } ^ { 2 } ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,507,503,527]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"it suffices to prove the boundedness of $z _ { i } ^ { 2 }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,528,876,563]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since $y \\cdot \\ell = - 2$ , the pushforward of $2 y f _ { i } ^ { * } z _ { i }$ via $f _ { i }$ yields $- 4 z _ { i }$ . Lemma 5 therefore implies that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[337,576,660,603]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0652/6e38a47a10af9d47ca8cdde3303b797578f366e7e25e5b34b0dd53caff8600cf.jpg","list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n( f _ { i } ) _ { * } K _ { X _ { i } } ^ { 2 } \\equiv - 4 z _ { i } \\equiv - 4 K _ { S _ { i } } - D _ { i } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,617,186,635]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hence, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[348,648,649,674]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0396/8935d8286be605f5b362d15dc2f23fa053a166b6402c86c0c0219aef92a96e9f.jpg","list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n1 6 z _ { i } ^ { 2 } = 1 6 K _ { S _ { i } } ^ { 2 } + 8 K _ { S _ { i } } D _ { i } + D _ { i } ^ { 2 } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,690,881,728]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since $D _ { i } ^ { 2 }$ does not depend on $i$ and $K _ { S _ { i } } ^ { 2 }$ is bounded in terms of the Betti numbers of $S _ { i }$ , the statement follows from the fact that $p _ { a } ( D _ { i } )$ is bounded by Lemma 5. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,773,398,794]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 Uniruled Threefolds ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,818,876,854]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Before we turn to the proof of Theorem 3, we state few preliminary facts about terminal $\\mathbb { Q }$ -factorial threefolds. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[671,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"S. Schreieder and L. Tasin ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"197","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"194 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,88,373,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"198","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.1 Invariant Triples ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,132,501,150]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"198","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $X$ be a terminal $\\mathbb { Q }$ -factorial threefold. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,151,876,187]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"198","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There exists a well-defined class $c _ { 2 } ( X ) \\ \\in \\ H ^ { 2 } ( X , \\mathbb { Z } ) ^ { \\vee } \\ = \\ \\mathrm { H o m } ( H ^ { 2 } ( Z , \\mathbb { Z } ) , \\mathbb { Z } )$ obtained in the following way (see page 411 in [22]). For any $\\alpha \\in H ^ { 2 } ( X , \\mathbb { Z } )$ set ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[389,201,605,225]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0262/04bb337b3f02b866b87ce809f69d77b86eda6ba62aa37100490a7fbe0e8235f7.jpg","list_items":null,"page_idx":"198","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nc _ { 2 } ( X ) . \\alpha = c _ { 2 } ( \\tilde { X } ) . f ^ { * } \\alpha ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[123,240,478,258]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"198","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $f : { \\tilde { X } } \\to X$ is a resolution of $X$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,259,881,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"198","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We then define the Pontryagin class $p _ { 1 } ( X ) \\in H ^ { 2 } ( X , \\mathbb { Q } ) ^ { \\vee }$ in terms of $c _ { 1 } ( X )$ and $c _ { 2 } ( X )$ in the same way as in the smooth case, where $c _ { 1 } ( X )$ is the class of $- K _ { X }$ in $H ^ { 2 } ( X , \\mathbb { Q } )$ : ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[366,327,630,351]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0441/3b5732e3624358f05c3057341db27ec957c68dcf4130fdd9c317a6a5f418d05e.jpg","list_items":null,"page_idx":"198","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\np _ { 1 } ( X ) : = c _ { 1 } ( X ) ^ { 2 } - 2 c _ { 2 } ( X ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,366,881,439]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"198","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We also associate to $X$ its cubic form $F _ { X } \\ \\in \\ S ^ { 3 } H ^ { 2 } ( X , \\mathbb { Z } ) ^ { \\vee }$ , which is induced by the cup product on $H ^ { 2 } ( X , \\mathbb { Z } )$ . In this way we can associate to $X$ the triple $( H _ { t . f . } ^ { 2 } ( X , \\mathbb { Z } ) , F _ { X } , p _ { 1 } ( X ) )$ . When $X$ is smooth, this triple encodes many geometrical properties of the 6-manifold underlying $X$ (see for instance [18] and [1]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,445,881,540]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"198","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Definition 7 We call $( H _ { t . f . } ^ { 2 } ( X , \\mathbb { Z } ) , F _ { X } , p _ { 1 } ( X ) )$ the invariant triple of $X$ . Two triples $( H , F , p )$ and $( H ^ { \\prime } , F ^ { \\prime } , p ^ { \\prime } )$ , where $H$ (resp. $H ^ { \\prime }$ ) is a free abelian group, $F \\in S ^ { 3 } H ^ { \\vee }$ (resp. $F \\in S ^ { 3 } H ^ { \\prime \\vee } )$ ) is a cubic form and $p$ is a linear form on $H \\otimes \\mathbb { Q }$ (resp. $p ^ { \\prime }$ is a linear form on $H ^ { \\prime } \\otimes \\mathbb { Q } )$ are isomorphic if there exists a linear isomorphism $T : H \\to H ^ { \\prime }$ which identifies $F$ with $F ^ { \\prime }$ and its $\\mathbb { Q }$ -extension identifies $p$ with $p ^ { \\prime }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,585,435,606]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"198","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.2 Terminal Singularities ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[125,630,842,648]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"198","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We now recall few known facts about terminal singularities in dimension three. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,650,881,774]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"198","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Let $( X , p )$ be the germ of a three-dimensional terminal singularity. The index of $p$ is the smallest positive integer $r$ such that $r K _ { X }$ is Cartier. It follows from the classification of terminal singularities, that there exists a deformation of $( X , p )$ into a space with $h \\geq 1$ terminal singularities $p _ { 1 } , \\ldots , p _ { h }$ which are isolated cyclic quotient singularities of index $r ( p _ { i } )$ (for details see [22, Remark 6.4]). The set $\\{ p 1 , \\ldots , p _ { h } \\}$ is called the basket $B ( X , p )$ of singularities of $X$ at $p$ . As in [5, Section 2.1], we define ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[396,788,601,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0574/5b864ccbd16bff3d0250136291b06708d86f187f8494ba22cd2cb732fbc324c6.jpg","list_items":null,"page_idx":"198","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\Xi ( X , p ) = \\sum _ { i = 1 } ^ { h } r ( p _ { i } ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,51,428,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"198","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Chern Numbers of Uniruled Threefolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"198","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"195 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,88,879,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"199","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Thus, if $X$ is a projective variety of dimension 3 with terminal singularities and Sing $X$ denotes the finite set of singular points of $X$ , we may define ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[375,139,621,183]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0186/54c91953419cff8c07bf0144af49d1d0563648feb2b7b53179aebfa70624b18a.jpg","list_items":null,"page_idx":"199","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\Xi ( X ) = \\sum _ { p \\in \\operatorname { S i n g } X } \\Xi ( X , p ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,223,405,244]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"199","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.3 Proof of Theorem 3 ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[118,268,881,324]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"199","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The following result is interesting by itself and leads naturally to the problem of understanding what kind of topological invariants are determined up to finite ambiguity during a running of an MMP, see Question 4. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,331,881,405]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"199","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 8 (Cf. [2, Theorem 1.3(2)]) Let $H$ be a finitely generated free abelian group of rank $n + 1$ , $F \\in S ^ { 3 } H ^ { \\vee }$ be a cubic form such that $\\Delta _ { F } \\neq 0$ and $p$ a linear form on $H$ . Consider the set $\\mathcal { P }$ of invariant triples $( H ^ { \\prime } , F ^ { \\prime } , p ^ { \\prime } )$ up to isomorphism, such that there exist ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,412,881,486]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"199","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(1) a terminal $\\mathbb { Q }$ -factorial threefold $X$ with associated triple $( H , F , p )$ ; (2) a terminal $\\mathbb { Q }$ -factorial threefold $Y$ with associated triple $( H ^ { \\prime } , F ^ { \\prime } , p ^ { \\prime } )$ ; (3) a birational morphism $f \\colon X \\to Y$ which is a divisorial contraction to a point or to a smooth curve contained in the smooth locus of $Y$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,495,330,512]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"199","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Then the set $\\mathcal { P }$ is finite. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,521,881,707]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"199","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Note that the proof of this case works also for $\\Delta _ { F } = 0$ . Consider the set $\\mathcal { A }$ of primitive elements $\\alpha \\in H$ such that $\\alpha$ is proportional to the exceptional divisor $E$ of some divisorial contraction to a point $f : X \\to Y$ as in the statement. The elements of $\\mathcal { A }$ are points of rank 1 for the Hessian of the cubic form $F$ and so they are finite by Cascini and Tasin [2, Proposition 3.3]. It follows from [2, Proposition 4.7] that for any sub-module $H ^ { \\prime } = f ^ { * } H _ { t . f . } ^ { 2 } ( Y , \\mathbb { Z } ) \\hookrightarrow H$ there is $\\alpha \\in { \\mathcal { A } }$ such that $\\alpha ^ { 2 } . H ^ { \\prime } = 0$ and such that the index of $H ^ { \\prime } \\oplus \\mathbb { Z } \\alpha$ in $H$ is at most $r ^ { n }$ , where $r = | \\alpha ^ { 3 } |$ . This implies that for all possible contractions to points $f \\colon X \\to Y$ as in the statement, the inclusion $f ^ { * } H _ { t . f . } ^ { 2 } ( Y , \\mathbb { Z } ) \\hookrightarrow H _ { t . f . } ^ { 2 } ( X , \\mathbb { Z } )$ : → is determined up to finite ambiguity. This determines also $F ^ { \\prime }$ up to finite ambiguity just restricting $F$ to $H ^ { \\prime }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,708,874,743]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"199","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To prove the finiteness of $p ^ { \\prime }$ consider a divisorial contraction to a point $f : X \\to$ $Y$ and write ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[382,759,612,780]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0167/678fb39e53889f14239d2f963b7c2c4524fbe2907dafbea0b24c652555de5bda.jpg","list_items":null,"page_idx":"199","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nc _ { 1 } ( X ) = f ^ { * } c _ { 1 } ( Y ) - c E ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,797,881,833]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"199","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $c$ is the discrepancy of the exceptional divisor $E$ . Since $c _ { 2 } ( X ) = f ^ { * } c _ { 2 } ( Y )$ we have that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[298,848,701,872]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0665/fcad983944705a1419887ea7cf0fb9e6dc4e8da91be742e1b2363f2a2d191a94.jpg","list_items":null,"page_idx":"199","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\np _ { 1 } ( X ) = f ^ { * } p _ { 1 } ( Y ) - 2 c f ^ { * } c _ { 1 } ( Y ) . E + c ^ { 2 } E ^ { 2 }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,52,152,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"199","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"196 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[671,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"199","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"S. Schreieder and L. Tasin ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,876,130]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and so $p _ { 1 } ( Y )$ is given by the restriction of $p _ { 1 } ( X )$ to $f ^ { * } H _ { t . f . } ^ { 2 } ( Y , \\mathbb { Q } )$ . This means that also $p ^ { \\prime }$ is determined up to finite ambiguity and we are done. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,130,881,166]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We now look at divisorial contractions to curves. Consider $\\mathcal { E }$ the set of pairs $( E , H ^ { \\prime } )$ where $E$ is a primitive element in $H$ and $H ^ { \\prime } \\subset H$ is a submodule such that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[419,181,578,202]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0202/6cb3760bda05462d6adc7bda016fdd694a1df98c9c8b479fc2badd8a4caa2459.jpg","list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nH = \\mathbb { Z } [ E ] \\oplus H ^ { \\prime }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,220,432,237]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and the cubic $F$ assumes the form ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[309,250,687,301]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0239/3391a63552f7f6a07bb7693845f026153400de7a2b21ec683702740f9debac93.jpg","list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nF = a x _ { 0 } ^ { 3 } + \\sum _ { i = 1 } ^ { n } b _ { i } x _ { 0 } ^ { 2 } x _ { i } + F ^ { \\prime } ( x _ { 1 } , \\dots , x _ { n } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,316,690,334]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"with respect to any basis $E , \\alpha _ { 1 } , \\ldots , \\alpha _ { n }$ with $\\alpha _ { 1 } , \\ldots , \\alpha _ { n } \\in H ^ { \\prime }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,336,881,423]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By Cascini and Tasin [2, Thm. 3.1] there are only finitely many possible nonequivalent reduced forms for $F$ . In particular, up to finite ambiguity, we can assume that the coefficients of $F$ in the expression (1) are fixed. Since the isotropy group of a cubic with non-zero discriminant is finite ([18, Thm. 4]), we deduce that $\\mathcal { E }$ is finite. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,424,881,478]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If $f \\colon X \\ \\to \\ Y$ is a divisorial contraction which contracts a divisor $E$ to a smooth curve $C$ in the smooth locus of $Y$ , then (see [18, Proposition 14] and [2, Proposition 4.8]) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[341,493,658,516]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0710/b04e8005c373a3ec17392b7df56337f74e7eb1ce8f3174153f24190cb5c64f51.jpg","list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nH ^ { 2 } ( X , \\mathbb { Z } ) = \\mathbb { Z } [ E ] \\oplus f ^ { * } H ^ { 2 } ( Y , \\mathbb { Z } )\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,534,157,549]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,564,676,588]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0470/73fd21445d408290f8772d76bc387788bb0cd8b7284340a7e9495330bddf6929.jpg","list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\np _ { 1 } ( X ) = f ^ { * } ( p _ { 1 } ( Y ) ) + E ^ { 2 } - 2 f ^ { * } ( C ) .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,603,879,659]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recalling that $E ^ { 2 } . f ^ { \\ast } ( \\alpha ) = - C . \\alpha$ for any $\\alpha \\in H ^ { 2 } ( Y , \\mathbb { Z } )$ we deduce that $p _ { 1 } ( Y )$ is determined by $p _ { 1 } ( X )$ , $E ^ { 2 }$ and by the inclusion $f ^ { * } H ( Y , \\mathbb { Z } ) \\hookrightarrow H ^ { 2 } ( X , \\mathbb { Z } )$ and we conclude using the finiteness of $\\mathcal { E }$ . □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,666,874,722]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 9 Let $( X _ { i } ) _ { i \\geq 0 }$ be a sequence of terminal $\\mathbb { Q }$ -factorial threefolds admitting a conic bundle structure $f _ { i } : X _ { i } \\longrightarrow S _ { i }$ of relative Picard number $I$ over $a$ surface $S _ { i }$ . Assume that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,731,774,785]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(1) the Euler characteristics $\\chi ( X _ { i } , { \\mathcal { O } } _ { X _ { i } } )$ are bounded and $b _ { 2 } ( X _ { i } ) = 2$ ; \n(2) the invariant triples of $X _ { 0 }$ and $X _ { i }$ are isomorphic for any $i$ ; \n(3) the sequence $\\Xi ( X _ { i } )$ is bounded. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,792,637,813]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Then the sequence of Chern numbers $c _ { 1 } ^ { 3 } ( X _ { i } )$ is bounded. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,821,881,876]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof Let $h$ be an ample generator of $\\operatorname { P i c } ( S _ { 0 } )$ and let $x \\in H ^ { 2 } ( X _ { 0 } , \\mathbb { Z } )$ be a primitive class proportional to $f _ { 0 } ^ { * } h$ . Then $x ^ { 3 } = 0$ , $x ^ { 2 } \\neq 0$ and letting $y = c _ { 1 } ( X _ { 0 } )$ we can write ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[364,891,630,914]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0798/16ddc0b06227043d5a9bb0f50beea19daea450798e143e3d9919f83b013c219f.jpg","list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nH ^ { 2 } ( X _ { 0 } , \\mathbb { Q } ) = x \\cdot \\mathbb { Q } \\oplus y \\cdot \\mathbb { Q } .\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,51,428,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Chern Numbers of Uniruled Threefolds ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[847,52,876,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"200","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"197 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[118,90,881,201]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From now on we will use the isomorphism $H ^ { 2 } ( X _ { i } , \\mathbb { Q } ) \\simeq H ^ { 2 } ( X _ { 0 } , \\mathbb { Q } )$ to think about $x$ and $y$ as basis elements of $H ^ { 2 } ( X _ { i } , \\mathbb { Q } )$ . Note that $x ^ { 2 } y \\neq 0$ since already $x ^ { 3 } = 0$ and $x ^ { 2 } \\neq 0$ . Moreover, the space of elements in $H ^ { 2 } ( X _ { 0 } , \\mathbb { C } )$ with zero cube is a union of three lines (through 0) and so we may assume without loss of generality that for each $i$ , the pullback of the generator of $H ^ { 2 } ( S _ { i } , \\mathbb { Z } )$ to $X _ { i }$ is a multiple of $x$ . In particular, $x ^ { 2 }$ is a multiple of the class of the general fibre of $X _ { i } \\longrightarrow S _ { i }$ for all $i$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,201,230,217]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[384,235,610,255]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0658/99b60351af9776feed6a69a37cb46eed7d6d9a4a154da150b8f466e463ef3914.jpg","list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nc _ { 1 } ( X _ { i } ) = a _ { i } \\cdot x + b _ { i } \\cdot y ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[118,271,881,325]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"for some $a _ { i } , b _ { i } \\in \\mathbb { Q }$ . Since $\\Xi ( X _ { i } )$ is bounded, there is a positive integer $r$ such that $r K _ { X _ { i } }$ is Cartier for any $i$ . In particular, $r a _ { i } , r b _ { i } \\in \\mathbb { Z }$ . Since $K _ { X _ { i } } . C = - 2$ where $C$ is a general fibre, we deduce that the sequence of $b _ { i }$ is bounded. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,327,876,361]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We are going to bound the sequence of $a _ { i }$ . By the singular version of Riemann– Roch [22, Corollary 10.3] we get ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[275,376,722,400]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0742/fbb255b268257f25a58c2bf4be066e1196ae54108ae47de8fedeb0472a30b72d.jpg","list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n4 8 \\chi ( X _ { i } , \\mathcal { O } _ { X _ { i } } ) = c _ { 1 } ( X _ { i } ) . p _ { 1 } ( X _ { i } ) - c _ { 1 } ( X _ { i } ) ^ { 3 } + T _ { i }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[120,417,179,432]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[362,445,633,493]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0668/05ce1a22c31711934356c14526b433d73b5b5e09b98f63a8d92e16a876d2d0e0.jpg","list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nT _ { i } = \\sum _ { p _ { \\alpha } } \\left( r ( p _ { \\alpha } ) - \\frac { 1 } { r ( p _ { \\alpha } ) } \\right) ,\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[123,507,879,545]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and the sum runs over all the points of all the baskets of $X _ { i }$ . Note that $T _ { i }$ is a bounded sequence since $\\Xi ( X _ { i } )$ is bounded. This implies that ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[123,558,874,582]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"0273/8de54bd10fc79689bd8447dc3898083f6ee2ace40cc4e72023340181c17c08da.jpg","list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n4 8 \\chi ( X _ { i } , \\mathcal { O } _ { X _ { i } } ) = - 3 a _ { i } ^ { 2 } b _ { i } x ^ { 2 } y + a _ { i } x ( 2 p _ { 1 } ( X _ { i } ) - 3 b _ { i } ^ { 2 } y ^ { 2 } ) + b _ { i } ^ { 3 } y ^ { 3 } + b _ { i } y p _ { 1 } ( X _ { i } ) + T _ { i }\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[116,597,833,617]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and so the $a _ { i }$ are also bounded, since $b _ { i } x ^ { 2 } y \\neq 0$ and $\\chi ( X _ { i } , { \\mathcal { O } } _ { X _ { i } } )$ are bounded. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,624,881,696]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proof of Theorem 3 Let $f : X \\to Y$ be the birational contraction as in Theorem 3. By the proof of [2, Corollary 1.5], we know that $| K _ { X } ^ { 3 } - K _ { Y } ^ { 3 } |$ is bounded by a constant depending only on the Betti numbers of $X$ and on the cubic form $F _ { X }$ . To conclude we need to bound $K _ { Y } ^ { 3 }$ in terms of the topology of $X$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,698,881,786]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since $Y$ is a Mori fibre space and $\\Delta _ { F _ { Y } } \\neq 0$ , we deduce that either $Y$ is a Fano variety or $Y$ has a conic bundle structure over a surface with second Betti number 1 (otherwise there would be an element in $H ^ { 2 } ( X , \\mathbb { C } )$ with square zero, which would imply that $\\{ F = 0 \\}$ has a singular point and so $\\Delta _ { F _ { Y } } = 0 $ ). Since terminal Fano threefolds are bounded, we are left with the conic bundle case. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[118,788,881,878]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Proposition 8 assures us that the invariant triple of $Y$ is determined up to finite ambiguity by the invariant triple of $X$ . Moreover, the Euler characteristic $\\chi ( Y , { \\mathcal { O } } _ { Y } ) = \\chi ( X , { \\mathcal { O } } _ { X } )$ is bounded in terms of the Betti numbers of $X$ and by Cascini and Zhang [3, Prop. 3.3] we also have a bound for $\\Xi ( Y )$ depending only on $b _ { 2 } ( X )$ . The result follows then from Proposition 9. □ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[671,52,876,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"S. Schreieder and L. Tasin ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,52,150,64]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"198 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[861,602,879,614]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"201","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"□ ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,91,879,166]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"202","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acknowledgements We thank P. Cascini for many useful conversations on the topic of this manuscript. The first author is supported by DFG grant “Topologische Eigenschaften von Algebraischen Varietäten” (project nr. 416054549). The second author was supported by the DFG grant “Birational Methods in Topology and Hyperkähler Geometry” and is a member of the GNSAGA group of INdAM. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[120,213,248,232]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"202","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[120,246,881,858]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"202","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Bisi, C., Cascini, P., Tasin, L.: A remark on the Ueno-Campana’s threefold. Michigan Math. J. 65(3), 567–572 (2016) \n2. Cascini, P., Tasin, L.:On the Chern numbers of a smooth threefold. Trans. Am. Math. Soc. 370(11), 7923–7958 (2018) \n3. Cascini, P., Zhang, D.-Q. Effective finite generation for adjoint rings. Ann. Inst. Fourier (Grenoble) 64(1), 127–144 (2014) \n4. Chen, H.-K.: Betti Numbers in Three Dimensional Minimal Model Program. Preprint arXiv:1605.04372 \n5. Chen, J.A., Hacon, C.D.: Factoring 3-fold flips and divisorial contractions to curves. J. Reine Angew. Math. 657, 173–197 (2011) \n6. Höring, A., Peternell, T.: Mori fibre spaces for Kähler threefolds. J. Math. Sci. Univ. Tokyo 22(1), 219–246 (2015) \n7. Höring, A., Peternell, T.: Minimal models for Kähler threefolds. Invent. Math. 203(1), 217–264 (2016) \n8. Kotschick, D.: Chern numbers and diffeomorphism types of projective varieties. J. Topol. 1, 518–526 (2008) \n9. Kotschick, D.: Characteristic numbers of algebraic varieties. Proc. Natl. Acad. Sci. U. S. A. 106, 10114–10115 (2009) \n10. Kotschick, D.: Topologically invariant Chern numbers of projective varieties. Adv. Math. 229, 1300–1312 (2012) \n11. Kotschick, D., Schreieder, S.: The Hodge ring of Kähler manifolds. Compositio Math. 149, 637–657 (2013) \n12. Lafont, J.-F., Sorcar, G., Zheng, F.: Some Kähler structures on products of 2-spheres. L’Enseign. Math. 64, 127–142 (2018) \n13. LeBrun, C.: Topology versus Chern numbers of complex 3-folds. Pac. J. Math. 191, 123–131 (1999) \n14. Libgober, A.S., Wood, J.W.: Uniqueness of the complex structure on Kähler manifolds of certain homotopy types. J. Differ. Geom. 32, 139–154 (1990) \n15. Lin, H.-Y.: Algebraic Approximations of Compact Kähler Threefolds. Preprint arXiv:1710.01083 \n16. Martinelli, D., Schreieder, S., Tasin, L.: On the number and boundedness of minimal models of general type, to appear in Annales scientifiques de l’ENS \n17. Novikov, S.P.: On manifolds with free abelian fundamental group and their application. Izv. Akad. Nauk SSSR Ser. Mat. 30, 207–246 (1966) \n18. Okonek, C., Van de Ven, A.: Cubic forms and complex 3-folds. Enseign. Math. (2) 41(3–4), 297–333 (1995) \n19. Schreieder, S., Tasin, L.: Algebraic structures with unbounded Chern numbers. J. Topology 9, 849–860 (2016) \n20. Schreieder, S., Tasin, L.: Kähler structures on spin 6-manifolds. Math. Ann. 373(1–2), 397–419 (2019) \n21. Shafarevich, I.R.: (ed.) Algebraic Geometry. V. Fano Varieties. Encyclopaedia of Mathematical Sciences, vol. 47, pp. $\\mathrm { i } \\mathbf { v } { + } 2 4 7$ . Springer, Berlin (1999) \n22. Reid, M.: Young person’s guide to canonical singularities. In: Algebraic Geometry, Bowdoin, 1985, pp. 345–414. Proceedings of Symposia in Pure Mathematics, vol. 46, Part 1. 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All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,106,751,166]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,183,353,243]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,183,751,243]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,259,353,319]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,669,881,750]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,752,876,791]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,791,876,858]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,872,813,899]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,596,701,657]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ISSN 2281-518X ISSN 2281-5198 (electronic) Springer INdAM Series ISBN 978-3-030-37113-5 ISBN 978-3-030-37114-2 (eBook) https://doi.org/10.1007/978-3-030-37114-2 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,239,112]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Preface \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,342,876,376]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This volume collects contributions from speakers at the INdAM Workshop “Birational Geometry and Moduli Spaces”, which was held in Rome on 11–15 June 2018. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,379,881,430]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The workshop was devoted to the interplay between birational geometry and moduli spaces, two central topics in Algebraic Geometry that have always attracted the interest of national and international researchers. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,433,881,666]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A longstanding problem in geometry is the classification of geometric objects. The starting point is a geometric object, such as a curve, a variety or a sheaf, defined by some equations or given by some conditions. Then, one can deform it to get a new geometric object related to the old one. This expresses the concept of a family of an object and the main purpose is the classification of these families, in such a way that the classifying space is a reasonable geometric space. This space is the so-called moduli space and its geometric points parametrise the objects that we are considering. One is interested in classifying geometric objects up to isomorphism, so that the moduli space has a variety structure, or geometric objects with their automorphisms, thus meaning that the moduli space has a stack structure. In order to understand the local and global structure of moduli spaces, it is important to handle well techniques from deformation theory. Indeed, deformation theory can be regarded as a tool to understand the local geometry of moduli spaces. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,666,879,755]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There is also interest in classifying geometric objects up to birational morphism. Many results are known for the birational study of algebraic surfaces, while for higher dimension there are many open problems that can be tackled via the minimal model program. Great progress has been made in recent years, supporting the development of new techniques and innovative ideas. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,758,879,810]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The workshop focused on these research aspects and their interactions and offered the possibility to disseminate the knowledge of advanced results and key techniques used to solve open problems in the area. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,812,879,882]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This volume covers many topics in the wide research area of birational geometry and moduli spaces, and includes both surveys and original research papers. In particular, it contains works on irreducible holomorphic symplectic manifolds, Severi varieties, degeneration of Calabi–Yau varieties, uniruled threefolds, toric Fano threefolds, mirror symmetry, canonical bundle formula, the Lefschetz principle, birational transformation, and deformation of diagrams of algebras. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[865,923,876,932]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,876,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,126,881,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We are indebted to Francesco Bastianelli and Antonio Rapagnetta for their help in organising the workshop and to Emilia Mezzetti for her support as a member of the Scientific Committee. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,180,874,214]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On behalf of the Scientific and Organising Committees, we would like to thank the Foundation Compositio Mathematica for their support. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,216,874,250]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We are all very grateful to the Istituto Nazionale di Alta Matematica “Francesco Severi” (INdAM) for the generous support and help that made all this possible. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,252,881,304]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally, we would like to offer special thanks to all the participants and speakers of the INdAM Workshop, and especially to the authors who accepted the invitation to publish here. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,324,241,414]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Milano, Italy Trieste, Italy Pavia, Italy Bari, Italy Pisa, Italy \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[694,324,881,414]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Elisabetta Colombo Barbara Fantechi Paola Frediani Donatella Iacono Rita Pardini \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[817,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Preface \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,54,136,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,261,112]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Contents \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,342,881,394]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Negative Rational Curves and Their Deformations on Hyperkähler Manifolds. . \\nEkaterina Amerik \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,405,881,457]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moduli Spaces of Cubic Threefolds and of Irreducible Holomorphic Symplectic Manifolds . 13 Chiara Camere \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,468,881,522]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Note on Severi Varieties of Nodal Curves on Enriques Surfaces . 29 \\nCiro Ciliberto, Thomas Dedieu, Concettina Galati, \\nand Andreas Leopold Knutsen \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,531,876,566]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Travel Guide to the Canonical Bundle Formula 37 Enrica Floris and Vladimir Lazic´ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,576,881,630]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model . 57 Anne-Sophie Kaloghiros \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,639,879,675]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras . 77 Emma Lepri and Marco Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,684,881,738]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties . 109 César Lozano Huerta and Alex Massarenti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,747,879,800]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? . 133 Luigi Lunardon \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,810,876,846]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Examples of Irreducible Symplectic Varieties 151 Arvid Perego \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[854,920,879,933]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"vii \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,876,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An Example of Mirror Symmetry for Fano Threefolds. 173 Andrea Petracci \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,135,876,169]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Chern Numbers of Uniruled Threefolds . 189 Stefan Schreieder and Luca Tasin \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,148,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"viii \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[806,54,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Contents \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,88,403,112]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"About the Editors \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,342,881,394]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Elisabetta Colombo is Associate Professor of Geometry at the University of Milan. Her research field is complex algebraic geometry, and she studies mainly curves and abelian varieties and their moduli. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,417,881,469]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Barbara Fantechi is Full Professor in Geometry at SISSA-ISAS in Trieste. Her research interests include deformation theory, derived algebraic geometry and stacks. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,490,881,545]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Paola Frediani is Associate Professor of Geometry at the University of Pavia. Her research area is algebraic geometry, in particular moduli spaces of curves and abelian varieties and Hodge theory. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,567,879,621]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Donatella Iacono is a Researcher in Geometry at the University of Bari. Her research focuses on deformation theory and differential graded Lie algebras in algebraic geometry. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,642,881,696]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Rita Pardini is Full Professor of Geometry at the University of Pisa. Her research area is algebraic geometry, especially surfaces and their moduli, irregular varieties and coverings. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[861,921,876,932]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,758,141]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Negative Rational Curves and Their Deformations on Hyperkähler Manifolds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,177,300,193]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ekaterina Amerik \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,291,881,343]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract We survey some results about rational curves on hyperkähler manifolds, explaining how to prove a certain deformation-invariance statement for loci covered by rational curves with negative Beauville–Bogomolov square. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,361,740,379]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords Hyperkähler manifold $\\\\cdot$ Rational curve $\\\\cdot$ MBM classes \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,417,879,451]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This text is an expanded version of a talk which I gave at the INDAM workshop in Rome in June 2018. It largely follows the paper [2] and our other joint works. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,496,437,518]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 Hyperkähler Manifolds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,543,881,668]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A compact Kähler manifold $X$ is irreducible holomorphic symplectic (IHSM), or hyperkähler (see next section for a brief explanation of the term), if it is simplyconnected and $H ^ { 2 , 0 } ( X ) = \\\\mathbb { C } \\\\sigma$ for some nowhere degenerate (“symplectic”) form $\\\\sigma$ . Such a manifold is even-dimensional (we set $2 n \\\\ = \\\\ d i m ( X ) )$ and has trivial canonical bundle. Such manifolds can be seen as higher-dimensional analogues of K3 surfaces. The most-studied IHSM are actually closely related to K3 surfaces, namely those are the deformations of their punctual Hilbert schemes; we call them \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,897,690,941]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_1 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[804,91,876,139]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,719,881,779]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The results presented at the INDAM workshop and in this survey have been obtained jointly with M. Verbitsky. Partially supported by Brazilian-French network in Mathematics, partially supported by the Russian Academic Excellence Project ‘5-100’. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[865,899,876,909]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,797,863,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Amerik $\\\\boxtimes ,$ Université Paris-Sud, Laboratoire de Mathématiques d’Orsay, Orsay, France National Research University Higher School of Economics, Laboratory of Algebraic Geometry and Its Applications, Moscow, Russia e-mail: ekaterina.amerik@math.u-psud.fr \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,879,142]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IHSM of K3 type. By Bogomolov’s decomposition theorem, together with the abelian and the Calabi–Yau manifolds the hyperkähler manifolds are building blocks for the varieties with zero first Chern class. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,144,881,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The main feature that the hyperkähler manifolds share with surfaces is the presence of a natural integral quadratic form $q$ on their second cohomology: the Beauville–Bogomolov form. It can be defined by integrating the differential forms, but we omit the explicit definition and only mention the following Fujiki’s formula, which stresses the topological nature of $q$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[416,247,578,270]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0799/45e10753cbdff0ab0291e311cd409131ffc0081d11e0c3d87c84ac2d3eb9978f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n( q ( \\\\alpha ) ) ^ { n } = C \\\\alpha ^ { 2 n } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,286,881,412]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $C$ is a positive constant which only depends on the deformation type of $X$ . It is well-known that this form has the same signature as the intersection form on a surface would have, that is, $( 3 , b _ { 2 } - 3 )$ on $H ^ { 2 } ( X , \\\\mathbb { R } )$ and $( 1 , b _ { 2 } - 3 )$ on $H ^ { 1 , 1 } ( X , \\\\mathbb { R } )$ . We refer to [5] or [6] for details (another classical reference [13] is suggested by the referee). The cone of classes of strictly positive square in $H ^ { 1 , 1 } ( X , \\\\mathbb { R } )$ has two connected components. We call the positive cone $\\\\mathsf { P o s } ( X )$ the one containing the Kähler classes. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,459,353,480]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 Rational Curves \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,503,881,630]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On a K3 surface S, for every integral $( 1 , 1 )$ -class $z$ of square $^ { - 2 }$ , either $z$ or $- z$ is effective by Riemann–Roch formula. If $z$ is represented by an irreducible curve, we call it a $^ { - 2 }$ -curve. By adjunction, such a curve is smooth and rational. These curves are important for the understanding of the geometry of $s$ : for instance, by the work of Pyatetski-Shapiro and Shafarevich [14] in the algebraic setting, or Looijenga and Peters [9] in the Kähler one, the ample cone of $s$ is bounded, within the positive cone, by the orthogonals to $^ { - 2 }$ -curves. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,675,683,696]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.1 Lower Bound for the Dimension of a Family \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,719,881,827]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A sufficiently general (in particular non-projective) hyperkähler manifold has no curves at all, indeed the deformation theory implies that it has no integral classes of type (1, 1). When it does, one would like to understand the deformations of such a rational curve $C \\\\subset X$ inside and outside of $X$ . A lower bound for the dimension of the deformation space of maps $f : \\\\mathbb { P } ^ { 1 } \\\\to X$ is provided by the Euler characteristics of the pullback of the tangent space, computed by the Riemann–Roch formula \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[300,845,697,864]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0334/61b0683a9d5fce8fea3fe5f9b50e204e30761956234db923de2d69d805028f8a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\chi ( f ^ { * } T _ { X } ) = - K _ { X } C + d i m ( X ) = d i m ( X )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[794,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Amerik \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,54,132,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,90,876,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"as soon as $K _ { X }$ is trivial, so that $C$ deforms in a family of dimension $2 n - 3$ (where 3 accounts for the automorphisms of $\\\\mathbb { P } ^ { 1 }$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,126,881,324]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Z. Ran noticed in [15] that on a hyperkähler $X$ , there is a better bound: a rational curve deforms in a family of dimension at least $2 n - 2$ . We have learned the following beautiful proof of this fact from Eyal Markman. Recall first that a hyperkähler manifold comes together with a natural deformation of complex structure, called the twistor deformation. It appears as follows: by Yau’s theorem, on an irreducible holomorphic symplectic manifold there is a hyperkähler metric, that is a Riemannian metric $g$ with three complex structures $I , J , K$ , which are Kähler with respect to $g$ (i.e. the corresponding forms are Kähler) and multiply as quaternions. For any $( a , b , c )$ with $a ^ { 2 } + b ^ { 2 } + c ^ { 2 } = 1 , a I + b J + c K$ is also a Kähler structure, so that $X$ is a member of a family $\\\\mathcal { X } _ { t }$ , $t \\\\in S ^ { 2 } , \\\\mathcal { X }$ is in fact a complex manifold and its projection $\\\\pi$ to $S ^ { 2 } = \\\\mathbb { C P } ^ { 1 }$ is holomorphic. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,324,881,466]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now Markman’s idea is as follows: let $C \\\\subset X$ be a rational curve, that is the image of some (say generically injective) map $f : \\\\mathbb { P } ^ { 1 } \\\\to X$ . By Riemann–Roch again, it must move within $\\\\mathcal { X }$ in a family of dimension at least $- K _ { \\\\mathcal { X } } C + d i m ( \\\\mathcal { X } ) -$ $3 = 2 n - 2$ . But it is well-known that the neighbouring fibers $X _ { t }$ of $X$ in $\\\\mathcal { X }$ do not contain any curves at all, whereas a deformation of a curve in a fiber of $\\\\pi$ obviously must be contained in a fiber of $\\\\pi$ (the intersection number of a curve and a fiber is zero when the curve is contained in a fiber and strictly positive when not). So in fact all deformations of $C$ in $\\\\mathcal { X }$ remain in $X$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,513,307,534]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2 Minimality \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,557,881,719]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If there is only a finite number of rational curves through a general point of the locus they cover, then this locus is of dimension $2 n - 1$ , that is, a divisor (remark that $X$ itself is not uniruled). However in general this does not have to be true. Already on manifolds of K3 type examples show that the uniruled loci can have any dimension between $n$ and $2 n - 1$ . As for the dimension of the family of deformations of a rational curve $C$ , it can be arbitrarily large if one does not impose some minimality assumption, for example, that $C$ is of minimal degree among the rational curves through the general point of the corresponding locus. Behind such a minimality assumption is of course the famous bend-and-break lemma. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,729,876,764]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 2.2.1 (Mori) In any one-parameter family of rational curves through two distinct points $x$ and $y$ , there is a reducible member. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,773,712,791]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"So the minimality of the degree is equivalent to the following. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,800,881,836]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition A rational curve $C$ is called minimal if there is only a finite number of deformations of $C$ through two general points of the locus they cover. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,846,879,899]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By bend-and-break lemma, any irreducible uniruled subvariety $Z$ of $X$ is generically covered by a family of minimal rational curves, which are deformations of some minimal curve $C \\\\subset Z$ . We say that $Z$ is a locus of $C$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,719,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Negative Rational Curves and Their Deformations on Hyperkähler Manifolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[865,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,879,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $C$ is a minimal rational curve, let $Z$ be a locus of $C$ and consider the rational quotient $r : Z \\\\to Q$ . The restriction of the form $\\\\sigma$ to $Z$ at its general point has kernel along the fiber of $r$ , since there are no holomorphic $k$ -forms on a rationally connected manifold for $k > 0$ . One derives the following \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,169,879,223]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 2.2.2 The codimension $k$ of $Z$ is equal to the relative dimension of $r$ , and any minimal rational curve deforms in a family of dimension exactly $2 n - 2$ in $X$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,234,881,304]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Indeed since $\\\\sigma$ is non-degenerate on $X$ , its kernel in restriction to $Z$ is at most of rank $k$ , on the other hand we must have at least a $k - 1$ -parameter family of rational curves through a general point of $Z$ , so the fibers of rational quotient are at least $k$ -dimensional. This also shows that $Z$ is coisotropic. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,315,874,349]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Applying Markman’s trick to the local universal family instead of the twistor deformation, one gets the following \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,358,876,396]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 2.2.3 A minimal rational curve in $X$ deforms to a neighbouring IHSM $X _ { t }$ if and only if its cohomology class remains of type $( 1 , 1 )$ on $X _ { t }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,405,881,548]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is interesting for several reasons to understand the loci of minimal rational curves and their behaviour under global deformations. One encounters two problems here. First of all, under a global deformation a minimal curve can cease to be minimal. Secondly, the family of deformations of $C$ may have several components, $C$ being minimal in some of them but not in the others. So the geometry of uniruled loci is still not well-understood even on much studied IHSM. For instance one conjectures but does not know that a projective deformation of the Hilbert square of a K3 contains a rational surface. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,549,881,620]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The point of the present talk is to indicate that all such problems have a satisfactory solution in the case when the Beauville–Bogomolov square of $C$ is negative. Here the Beauville–Bogomolov square of a curve is a rational number, obtained by considering $H _ { 2 } ( X , \\\\mathbb { Z } )$ as an overlattice of $H ^ { 2 } ( X , \\\\mathbb { Z } )$ by duality. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,621,876,656]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The loci covered by minimal rational curves of negative square are especially interesting as these are loci of birational contractions, at least in the projective case. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,702,396,723]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 Deformation Spaces \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,741,448,761]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.1 Teichmüller and Torelli \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,785,879,893]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Deformations of IHSM are unobstructed by the work of Todorov [17]. Thus the local deformation space $D e f ( X )$ of $X$ is identified with a neighbourhood of zero in $H ^ { 1 } ( X , T _ { X } )$ , which is in turn identified with $H ^ { 1 } ( X , \\\\Omega _ { X } ^ { 1 } )$ by means of the symplectic form. The locus of deformations where a class $\\\\lambda$ remains of type $( 1 , 1 )$ is the hyperplane orthogonal to $\\\\lambda$ with respect to the Beauville–Bogomolov form. This local picture globalizes as follows. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[794,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Amerik \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,54,132,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"13\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $M$ be the underlying differentiable manifold and ${ \\\\mathsf { C o m p } } ( M )$ the space of complex structures of Kähler type (by Kodaira–Spencer stability theorem, this is an open subset in the space of all complex structures). We consider the Teichmüller space \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[289,175,703,199]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0163/d4cba24dccf1a57edd455e4e326d6e8bb23cd4d58550536d28c57537ff3125b2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathsf { T e i c h } = \\\\mathsf { T e i c h } ( M ) = \\\\mathsf { C o m p } ( M ) / \\\\mathsf { D i f f } ^ { 0 } ( M ) ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,214,881,342]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\mathsf { D i f f } ^ { 0 } ( M )$ is the group of isotopies: the action of the whole diffeomorphism group $\\\\mathsf { D i f f } ( M )$ on ${ \\\\mathsf { C o m p } } ( M )$ is “too bad” for taking any reasonable quotient. Instead we consider only the action of the mapping class group $\\\\Gamma =$ ${ \\\\dot { \\\\mathsf { D i f f } } } ( M ) / { \\\\mathsf { D i f f } } ^ { 0 } ( M )$ on Teich $( M )$ . Huybrechts in [7] proved that Teich $( M )$ has only finitely many connected components when $M$ carries a hyperkähler structure. By abuse of notation, we shall use the notation Teich $( M )$ for the component containing our chosen complex structure. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,342,881,396]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The space Teich $( M )$ is only generically Hausdorff. Verbitsky in [18] proved the Torelli theorem for hyperkähler manifolds, namely that the Hausdorff quotient Teich $_ { b } ( M )$ is isomorphic to the period domain. The latter is defined as follows: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[255,409,742,433]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0336/5d47af8b1a207f2c16557848d22273016fb4a11195f921055a3ab13c3d3f7996.jpg\",\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathsf { P e r } ( M ) = \\\\{ l \\\\in \\\\mathbb { P } H ^ { 2 } ( M , \\\\mathbb { C } ) | q ( l , l ) = 0 , q ( l , \\\\bar { l } ) > 0 \\\\}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,448,879,504]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and the period map per from $\\\\mathsf { T e i c h } ( M )$ to ${ \\\\mathsf { P e r } } ( M )$ , realizing the up-tononhausdorffness isomorphism, is defined by sending the parameter point for a complex manifold $X$ to the complex line generated by the class of $\\\\sigma$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,504,879,576]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is often convenient (e.g. in Verbitsky’s proof) to see ${ \\\\mathsf { P e r } } ( M )$ as the grassmannian of positive (with respect to q) 2-planes in the real cohomology $G r _ { + + } ( 2 , H ^ { 2 } ( M , \\\\mathbb { R } ) )$ (the transition from the open subset of the complex quadric as above is simply by taking the real and the imaginary part). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,576,881,666]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Define the subspace $\\\\mathsf { T e i c h } _ { z } = \\\\mathsf { T e i c h } _ { z } ( M ) \\\\subset \\\\mathsf { T e i c h } ( M )$ as the locus where an integral class $z$ remains of type $( 1 , 1 )$ . It is easy to see that this is the inverse image by per of the hyperplane $z ^ { \\\\bot }$ (the $q$ -orthogonal to $z$ ). It is clear from the description in the following section that ${ \\\\mathsf { T e i c h } } _ { z }$ is not Hausdorff even generically and splits naturally in two “halves”. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,711,594,732]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.2 Kähler Chambers and MBM Classes \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,756,876,792]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The following theorem summarizes the main results of [1] (similar results to some of those have been independently obtained by Mongardi in [11]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,800,879,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3.2.1 The Kähler cone $K ( X )$ is a connected component of the complement to a union of hyperplanes $z ^ { \\\\bot }$ , $z \\\\in R \\\\cap H ^ { 1 , 1 } ( X )$ , in $\\\\mathsf { P o s } ( X )$ , where $R$ is the set of negative integral classes in $H ^ { 2 } ( X , \\\\mathbb { Z } )$ , called MBM classes, which can be characterized by the following equivalent conditions: z is MBM iff ${ \\\\mathsf { T e i c h } } _ { z }$ containes no twistor curve iff a rational multiple of z is represented by a rational curve on a “generic $^ { \\\\circ 1 }$ deformation of $X$ within ${ \\\\mathsf { T e i c h } } _ { z }$ (that is, such deformation that only the multiples of z are integral classes of type (1, 1)). The orthogonal hyperplane to an MBM class of type (1, 1) supports the wall of the Kähler cone of $X$ , or of $a$ birational model $X ^ { \\\\prime }$ , or the monodromy image of such a wall. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,719,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Negative Rational Curves and Their Deformations on Hyperkähler Manifolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[865,54,879,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"14\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,91,879,163]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,172,879,226]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The orthogonals to MBM classes of type (1, 1) thus divide the positive cone into chambers, called Kähler-Weil chambers by Markman. Each chamber is the Kähler cone of a birational model $X ^ { \\\\prime }$ or its monodromy translate. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,235,879,289]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3.2.2 (Markman) The points in $p e r ^ { - 1 } ( p e r ( X ) )$ , that is, the inseparable points of Teich one of which corresponds to $X$ , are in one-to-one correspondence with the Kähler-Weil chambers of $\\\\mathsf { P o s } ( X )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,298,881,370]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In particular, ${ \\\\mathsf { T e i c h } } _ { z }$ is non-separated even at its general point: at such a point $z$ is the only MBM class of type $( 1 , 1 )$ and $z ^ { \\\\bot }$ divides the positive cone in two chambers. Choosing an orientation, we naturally separate ${ \\\\mathsf { T e i c h } } _ { z }$ in two halves $\\\\mathtt { T e i c h } _ { z } ^ { \\\\pm }$ (the complex structures which have the Kähler cone “to the left” and “to the right” of $z ^ { \\\\bot }$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,372,881,570]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The next step is to take the part of, say, $\\\\mathtt { T e i c h } _ { z } ^ { + }$ corresponding to chambers adjacent to $z ^ { \\\\bot }$ (that is, such that $z ^ { \\\\bot }$ contains a wall, rather than a lower-dimensional face, of the chamber in question). We denote this part by $\\\\mathsf { T e i c h } _ { z } ^ { m i n }$ . Its separable quotient coincides with $\\\\mathtt { T e i c h } _ { z } ^ { + }$ itself, but fibers over some points of Per are slightly smaller than those in $\\\\mathtt { T e i c h } _ { z } ^ { + }$ : indeed we only keep the chambers adjacent to $z ^ { \\\\bot }$ , so that on the varieties parameterized by $\\\\mathsf { T e i c h } _ { z } ^ { m i n }$ , a multiple of the class $z$ is represented by a minimal rational curve (as $z ^ { \\\\bot }$ gives a part of the boundary of the Kähler cone with non-empty interior), whereas on the varieties in the complement $\\\\mathsf { T e i c h } _ { z } ^ { + } - \\\\mathsf { T e i c h } _ { z } ^ { m i n }$ this shall be true only for some monodromy transform of $z$ and only on some birational model, whereas $z$ itself may be e.g. the class of a reducible rational curve. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,572,876,608]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We are going to be concerned with deformations of rational curves in the class $z$ over Teichmin . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,653,305,672]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 Main Result \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,696,881,824]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $X$ is in $\\\\mathsf { T e i c h } _ { z } ^ { m i n }$ we consider the locus $Z$ covered by curves whose class is proportional to $z$ . When $X$ is projective, it follows from Kawamata’s base point free theorem that $Z$ is the contraction locus of a birational morphism. Indeed take a sufficiently general integral class in that part of $z ^ { \\\\bot }$ which is a wall of the Kähler cone of $X$ . By Lefschetz theorem it is the class of a line bundle $L$ which is nef since it is on the boundary of the Kähler cone, big since by definition its square is strictly positive, and therefore eventually base-point-free by Kawamata’a theorem. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,869,876,900]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1On an arbitrary deformation, an MBM class would be, up to a multiple, represented by a possibly reducible rational curve, but the converse is in general false. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[794,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Amerik \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,54,132,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"15\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,879,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The morphism given by the sections of the line bundle $L ^ { \\\\otimes m }$ for $m$ large enough contracts exactly the curves in the classes proportional to z. In particular $Z$ has only finitely many irreducible components, and also $Z$ is uniruled since all contraction loci are uniruled (see e.g. [8]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,163,881,252]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The finiteness (and, post factum, the uniruledness) carries over to the non-projective case, since the loci $Z _ { t } ~ \\\\subset ~ X _ { t }$ of curves in a class proportional to z on $X _ { t }$ are deformations of $Z \\\\subset X$ . We note that a recent work of Bakker and Lehn [4] seems to imply a much less obvious fact that $Z$ is contractible even if $X$ is non-projective. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,253,585,270]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our main results can be summarized as follows. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,280,881,424]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 4.1 Let $\\\\mathcal { X }$ be the universal family of IHSM with second Betti number greater than 5 over $\\\\mathsf { T e i c h } _ { z } ^ { m i n }$ and $\\\\mathcal { Z }$ the subvariety covered by curves of class proportional to z. All fibers $Z _ { t }$ of $\\\\mathcal { Z }$ are homeomorphic and stratified diffeomorphic, with the possible exception of t corresponding to the complex structures with the maximal Picard number. Moreover the homeomorphisms (and stratified diffeomorphisms) can be chosen in such a way that they respect rational curves, so that they induce maps between Zariski-open subsets of fibers of rational quotients on $Z _ { t }$ and $Z _ { t } ^ { \\\\prime } , t , t ^ { \\\\prime } \\\\in \\\\mathsf { T e i c h } _ { z } ^ { m i n }$ . These maps are bimeromorphisms. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,433,881,560]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 4.2 The complex manifolds $X _ { t }$ with maximal Picard number are isolated points of $\\\\mathsf { T e i c h } _ { z } ^ { m i n }$ . Indeed the image of such a complex structure by the period map is in the intersection of $h ^ { \\\\perp }$ for all $h \\\\in P i c ( X _ { t } )$ . The results of Bakker and Lehn seem to imply a strengthening of ours, eliminating the nonmaximality assumption and eventually giving a real analytic isomorphism of the loci $Z _ { t }$ inducing a biholomorphism of fibers of the rational quotient. This is a work in progress which we hope to describe in a forthcoming paper. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,605,284,626]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 Ergodicity \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,648,879,720]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our main tool is Verbitsky’s ergodicity theorem for the action of the mapping class group on the Teichmüller space ([19, 20]). An action on a space with a measure is ergodic if any invariant measurable subset or its complement has measure zero. Most of the orbits of such an action on a manifold are dense. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,729,876,767]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 5.1 (Verbitsky) Let $X$ be an IHSM with $b _ { 2 } ( X ) ~ \\\\geq ~ 5$ . There are three types of orbits of the action of the mapping class group $\\\\Gamma$ on Teich: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,774,876,810]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(1) closed orbits of complex structures with rational period plane, i.e. such that Picard number is maximal; \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,812,881,846]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(2) dense orbits of complex structures such that the period plane contains no rational vector; \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,719,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Negative Rational Curves and Their Deformations on Hyperkähler Manifolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[865,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"16\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,879,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"17\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(3) “intermediate orbits”: orbits of complex structures whose period planes contain a single rational vector v. The orbit closure then consists of all period planes containing v and is totally real, that is, neither contained in a complex subvariety nor contains a positive-dimensional one, even locally. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,171,881,349]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"17\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sketch of proof It is done in two steps. The mapping class group acts on the period domain and one first proves these statements for period planes (the orbit of a period plane is dense whenever the plane contains no rational vector, etc). This is obtained by Ratner theory which describes actions of groups generated by unipotents on homogeneous spaces $\\\\Gamma \\\\backslash G$ , where $\\\\Gamma$ is a lattice in a simple Lie group $G$ . Indeed another result by Verbitsky, a corollary of a general theorem by Sullivan, affirms that the image of the mapping class group $\\\\Gamma$ in $G = S O ^ { + } ( 3 , b _ { 2 } - 3 )$ (the automorphism group of the second cohomology of $X$ ) is a lattice. Let $G = S O ^ { + } ( 3 , b _ { 2 } - 3 )$ and $H = S O ^ { + } ( 1 , b _ { 2 } - 3 )$ , so that $G / H$ is fibered in $S O ( 2 )$ over Per and $H$ is generated by unipotents. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,351,881,512]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"17\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ratner theory (see [16], but also [12] for a very accessible exposition) states that $\\\\overline { { x H } }$ is again an orbit under a closed intermediate subgroup $s$ , also generated by unipotents and in which $\\\\Gamma \\\\cap S$ is a lattice (this particular result is known as Ratner’s orbit closure theorem and applies to actions of subgroups generated by unipotents on a quotient of a Lie group by a lattice). From the study of Lie group structure on $G$ we derive that the subgroup must be either $H$ itself (the orbit is closed), or the whole of $G$ (the orbit is dense), or the stabilizer of an extra vector $\\\\cong S O ^ { + } ( 2 , b _ { 2 } - 3 )$ (the third case). One concludes passing in an obvious way (via the double quotient) from an $H$ -action on $\\\\Gamma \\\\backslash G$ to a $\\\\Gamma$ -action on $G / H$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,515,881,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"17\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The other step is to prove that the period map commutes with taking orbit closures, in the following way. A point of Teich can be seen as a pair of a complex structure and a Kähler chamber. Introduce the space Teich $K$ which consists of pairs $( I , \\\\omega )$ where $I \\\\in$ Teich and $\\\\omega$ is a Kähler form of square 1. The period map is injective on $\\\\mathsf { T e i c h } _ { K }$ . In other words, $\\\\mathsf { T e i c h } _ { K }$ is naturally embedded in $\\\\mathsf { P e r } _ { K }$ , the homogeneous manifold of all pairs consisting of a period point $I \\\\in$ Per and an element $\\\\omega$ of square one in its positive cone (which indeed depends only on the period point, not on the complex structure itself). The latter is a homogeneous space, so we can try to apply Ratner theory to prove the following result, which clearly implies what we need: for any $I$ , the closure of the $\\\\Gamma$ -orbit of $( I , K ( I ) ) \\\\subset$ Teich $\\\\kappa \\\\subset \\\\mathsf { P e r } _ { K }$ contains the orbit of (PerI, Pos(I )) (here by an orbit of the subset we mean the union of its translates and $K ( I )$ denotes the Kähler cone of the complex structure $I$ ). Now one can construct orbits of one-parameter subgroups which are entirely contained in $( I , K ( I ) )$ and such that the closure of their projection to $\\\\mathsf { P e r } _ { K } / \\\\Gamma$ contains the projection of the positive cone. Indeed, one deduces from the non-maximality of the Picard number that $K ( I )$ has a “round part”. This is used to find many horocycles in $K ( I )$ tangent to the round part of the boundary. The horocycle is an orbit of a one-parameter unipotent subgroup. Applying Ratner theory to a sufficiently general horocycle of this type, one sees that the closure of its image in $\\\\mathsf { P e r } _ { K } / \\\\Gamma$ contains an entire $S O ( H ^ { 1 , 1 } ( I ) )$ -orbit ([20], Proposition 3.5), which is the positive cone $\\\\mathsf { P o s } ( I )$ . □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[794,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"17\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Amerik \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,54,132,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"17\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,91,881,219]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For our purposes, we are concerned with the space $\\\\mathsf { T e i c h } _ { z } ^ { m i n }$ which maps onto $\\\\mathsf { P e r } _ { z }$ , that is, the hyperplane $z ^ { \\\\bot }$ in Per. On these spaces there is an action of $\\\\Gamma _ { z }$ , the subgroup of the mapping class group whose action on the second cohomology fixes z. The first step of the argument, concerning the orbits of period planes, carries over verbatim, we only have to suppose $b _ { 2 } > 5$ rather than $b _ { 2 } \\\\geq 5$ as we are now working in a hyperplane. The nature of the considerations in the second step is such that they naturally carry over to $\\\\mathsf { T e i c h } _ { z } ^ { m i n }$ rather than ${ \\\\mathsf { T e i c h } } _ { z }$ or $\\\\mathtt { T e i c h } _ { z } ^ { + }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,226,879,301]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 5.2 Assume $b _ { 2 } ( M ) > 5$ and let $z \\\\in H ^ { 2 } ( M , \\\\mathbb { Z } )$ be an MBM class and $\\\\Gamma _ { z }$ the subgroup of the mapping class group consisting of all elements whose action on the second cohomology fixes z. Then $\\\\Gamma _ { z }$ acts on $\\\\mathsf { T e i c h } _ { z } ^ { \\\\mathrm { m i n } }$ ergodically, and there are the same three types of orbits of this action as above. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,310,876,346]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof It proceeds along the same lines. We introduce the spaces $\\\\mathsf { P e r } _ { K , z }$ consisting of pairs \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[298,360,699,384]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0694/7f84924342304367e0b2d8463dd48b7cca1f6ea309791e13464cf04938c51e51.jpg\",\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\{ ( \\\\mathsf { P e r } ( I ) , \\\\omega \\\\in \\\\mathsf { P o s } ( I ) \\\\cap z ^ { \\\\perp } ) , q ( \\\\omega , \\\\omega ) = 1 \\\\}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,399,881,653]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and $\\\\mathsf { T e i c h } _ { K , z }$ consisting of pairs $( I \\\\in \\\\mathsf { T e i c h } _ { z } ^ { \\\\operatorname* { m i n } } , \\\\omega ) \\\\in \\\\mathsf { P e r } _ { K , z }$ where $\\\\omega$ belongs to the wall of the Kähler cone given by $z ^ { \\\\bot }$ . We denote such a wall by $K ( I ) _ { z }$ , though of course its elements are not Kähler forms in the complex structure $I$ , but rather semipositive limits of those. Since the complex structures in $\\\\mathsf { T e i c h } _ { z } ^ { \\\\mathrm { m i n } }$ which have the same period point are in one-to-one correspondence with the walls of the Kähler chambers in which the other MBM classes partition $z ^ { \\\\bot }$ , $\\\\mathsf { T e i c h } _ { K , z }$ again embeds naturally in $\\\\mathsf { P e r } _ { K , z }$ . We fix a complex structure $I$ with non-maximal Picard number. We need to prove that the closure of the $\\\\Gamma _ { z }$ -orbit of the subset $( I , K ( I ) _ { z } )$ contains the orbit of $( \\\\mathsf { P e r } ( I ) , \\\\mathsf { P o s } ( I ) \\\\cap z ^ { \\\\perp } )$ . This is done exactly in the same way as in Verbitsky’s theorem. We take a general three-dimensional subspace $W$ in $z ^ { \\\\bot }$ , the intersection of $W$ with our wall $K ( I ) _ { z }$ contains horocycles, and we deduce from Ratner orbit closure theorem and Proposition 3.5 of [20] that the closure of the projection of such a horocycle to $\\\\mathsf { P e r } _ { K , z } / \\\\Gamma _ { z }$ is large, containing an $S O ( H ^ { 1 , 1 } ( I ) \\\\cap$ $z ^ { \\\\bot }$ )-orbit, which is the projection of ${ \\\\mathsf { P o s } } ( I ) \\\\cap z ^ { \\\\perp }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,698,451,719]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 Proof of the Main Result \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,741,881,851]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We are concerned with the family of smooth manifolds $\\\\mathcal { X }$ over $\\\\mathsf { T e i c h } _ { z } ^ { \\\\mathrm { m i n } }$ and its analytic subset $\\\\mathcal { Z }$ , and we wish to show that $\\\\mathcal { Z }$ has no degenerations except possibly over points of maximal Picard number. It is well-known that an analytic subset of a complex manifold admits a “nice” stratification (Whitney stratification or Thom-Mather stratification; see [10]). Recall also the following first isotopy lemma by Thom (we refer to [10] for precise definitions and proofs). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,719,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Negative Rational Curves and Their Deformations on Hyperkähler Manifolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[865,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"18\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,879,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma ([10], Proposition 11.1) Let $f : Y B$ be a smooth mapping of smooth manifolds and W a closed subset of $Y$ admitting Whitney stratification, such that $f : W \\\\to B$ is proper. If the restriction of f to each stratum of $W$ is a submersion then $W$ is locally trivial over $B$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,171,881,334]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In our setting we deduce that the family $\\\\mathcal { Z }$ is locally trivial on the complement $B \\\\subset \\\\mathsf { T e i c h } _ { z } ^ { m i n }$ to a union (possibly countable, but finite in a neighbourhood of any point in the base) $P \\\\subset \\\\mathsf { T e i c h } _ { z } ^ { m i n }$ of proper analytic subsets. If $x \\\\in \\\\mathsf { T e i c h } _ { z } ^ { m i n }$ is a point with dense $\\\\Gamma$ -orbit, then its orbit hits $B$ and therefore $Z _ { x }$ is naturally stratified diffeomorphic to $Z _ { b }$ , $b \\\\in B$ . If the orbit of $x \\\\in \\\\mathsf { T e i c h } _ { z } ^ { m i n }$ does not hit $B$ , itself and its closure must be contained in $P$ . But we know about the orbit closures that even locally they are not contained in a proper complex subvariety, except possibly when $x$ corresponds to a manifold $X$ with maximal Picard number. Thus the degenerations in the family $\\\\mathcal { Z }$ can only happen over such points. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,334,879,460]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The same is true for other families over $\\\\mathsf { T e i c h } _ { z } ^ { m i n }$ coming from the geometry of IHSM over it. We obtain stratified diffeomorphisms preserving rational curves by applying the $\\\\Gamma$ -orbit argument to Barlet spaces parameterizing rational curves with cohomology class proportinal to $z$ , and then to the incidence varieties. Note that these spaces are not naturally embedded into smooth manifolds, so it is not quite the first isotopy lemma which applies, but there are versions of it suitable for our situation (e.g. [21], corollaire 5.1). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,462,881,515]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally, at a general point of a fiber of the rational quotient, the holomorphic tangent space is generated by the tangents to rational curves. A diffeomorphism preserving the holomorphic tangent space is a biholomorphism. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,524,879,614]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note Added in Proof During the INDAM workshop, A. Rapagnetta attracted our attention to a recent work by Bakker and Lehn [4], which in the meantime allowed us to considerably strengthen the above result. The main point is that it indeed follows from Bakker and Lehn’s work that minimal rational curves can be contracted even in the non-projective case. An interested reader may consult our recent preprint [3]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,660,248,680]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[127,702,881,882]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Amerik, E., Verbitsky, M.: Rational curves on hyperkähler manifolds. Int. Math. Res. Not. 2015(23), 13009–13045 (2015) \\n2. Amerik, E., Verbitsky, M.: MBM loci in families of hyperkahler manifolds and centers of birational contractions (2018). arXiv:1804.00463 \\n3. Amerik, E., Verbitsky, M.: Contraction centers in families of hyperkähler manifolds (2019). arXiv:1903.04884 \\n4. Bakker, B., Lehn, C.: A global Torelli theorem for singular symplectic varieties (2016). arXiv:1612.07894 \\n5. Beauville, A.: Variétés dont la première classe de Chern est nulle. J. Differ. Geom. 18, 755–782 (1983) \\n6. Huybrechts, D.: Compact hyperkähler manifolds: basic results. Invent. Math. 135, 63–113 (1999) \\n7. Huybrechts, D.: Finiteness results for compact hyperkähler manifolds. J. Reine Angew. Math. 558, 15–22 (2003) \\n8. Kawamata, Y.: On the length of an extremal rational curve. Invent. Math. 105(3), 609–611 (1991) \\n9. Looijenga, E., Peters, C.: Torelli theorems for Kähler K3 surfaces. Compos. Math. 42(2), 145– 186 (1980/1981) \\n10. Mather, J.: Notes on Topological Stability. Harvard University, Cambridge (1970) \\n11. Mongardi, G.: A note on the Kähler and Mori cones of hyperkähler manifolds. Asian J. Math. 19(4), 583–591 (2015) \\n12. Morris, D.W.: Ratner’s Theorems on Unipotent Flows. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (2005) \\n13. O’Grady, K.: Higher-dimensional analogues of K3 surfaces. In: Current Developments in Algebraic Geometry. Mathematical Sciences Research Institute Publications, vol. 59, pp. 257– 293. Cambridge University Press, Cambridge (2012) \\n14. Pjateckii-Shapiro, I.I., Shafarevich, I.R.: Torelli’s theorem for algebraic surfaces of type K3. Izv. Akad. Nauk SSSR Ser. Mat. 35, 530–572 (1971) \\n15. Ran, Z.: Hodge theory and deformations of maps. Compos. Math. 97(3), 309–328 (1995) \\n16. Ratner, M.: Raghunathan’s topological conjecture and distributions of unipotent flows. Duke Math. J. 63(1), 235–280 (1991) \\n17. Todorov, A.: The Weil-Petersson geometry of the moduli space of $S U ( n \\\\geq 3 )$ ) (Calabi–Yau) manifolds. I. Commun. Math. Phys. 126(2), 325–346 (1989) \\n18. Verbitsky, M.: Mapping class group and a global Torelli theorem for hyperkähler manifolds. Duke Math. J. 162(15), 2929–2986 (2013) \\n19. Verbitsky, M.: Ergodic complex structures on hyperkähler manifolds. Acta Math. 215(1), 161– 182 (2015) \\n20. Verbitsky, M.: Ergodic complex structures on hyperkahler manifolds: an erratum (2017). arXiv:1708.05802 \\n21. Verdier, J.-L.: Stratifications de Whitney et théorème de Bertini-Sard. Invent. Math. 36, 295– 312 (1976) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,54,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[794,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"19\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Amerik \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,881,527]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"20\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,719,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"20\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Negative Rational Curves and Their Deformations on Hyperkähler Manifolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"20\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,667,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moduli Spaces of Cubic Threefolds and of Irreducible Holomorphic Symplectic Manifolds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,202,273,220]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Chiara Camere \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,286,881,412]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract In this survey, based on joint work of the author and S. Boissière and A. Sarti, we will describe an isomorphism between the moduli space of smooth cubic threefolds, as described by Allcock, Carlson and Toledo, and the moduli space of fourfolds of $K 3 ^ { [ 2 ] }$ -type with a special non-symplectic automorphism of order three; then, I will show some consequences of this isomorphism concerning degenerations of non-symplectic automorphisms. Finally we will explore possible generalizations of the problem to higher dimensions and other moduli spaces of cubic threefolds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,432,861,521]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords Irreducible holomorphic symplectic manifold $\\\\cdot$ Hyperkähler manifold $\\\\cdot$ Non-symplectic automorphism $\\\\cdot$ Ball quotient $\\\\cdot$ Cubic threefold $\\\\cdot$ Moduli space of cubic threefold $\\\\cdot$ Occult period map $\\\\cdot$ Pfaffian cubic threefold $\\\\cdot$ Degeneration of irreducible holomorphic symplectic manifold $\\\\cdot$ Degeneration of automorphism $\\\\cdot$ Fano variety of a cubic fourfold \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,558,309,579]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,602,879,656]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The aim of this survey is to illustrate the results of a joint work by the author and S. Boissière and A. Sarti [17]; as a consequence, almost all the material here is neither original nor new, with the exception of a part of Sect. 4. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,657,879,764]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Irreducible holomorphic symplectic (denoted $I H S$ ) manifolds are smooth compact Kähler manifolds $X$ which are simply connected and which carry a holomorphic symplectic two-form $\\\\omega _ { X }$ such that $H ^ { 2 , 0 } ( X ) = \\\\mathbb { C } \\\\omega _ { X }$ . These manifolds, which also carry a hyperkähler structure, are one of the building blocks of Beauville– Bogomolov decomposition theorem [8], and in particular they have $c _ { 1 } ( X ) _ { \\\\mathbb { R } } = 0$ . One of the main properties of these manifolds is the fact that the group $H ^ { 2 } ( X , \\\\mathbb { Z } )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,897,690,941]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_2 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,812,845,879]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Camere $( \\\\boxtimes )$ Max Planck Institute of Mathematics, Bonn, Germany Present Address: Dipartimento di Matematica, Università degli Studi di Milano, Milano, Italy e-mail: chiara.camere@mpim-bonn.mpg.de; camere@dima.unige.it; chiara.camere@unimi.it \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[804,91,876,139]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,899,876,909]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"21\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,881,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"can be endowed with the structure of an integral non-degenerate indefinite lattice of signature $( 3 , b _ { 2 } ( X ) - 3 )$ by means of the so-called Beauville–Bogomolov quadratic form $q _ { X }$ . The global Torelli theorem of Huybrechts [25], Markman [33], and Verbitsky [40] asserts that this lattice controls in many senses the geometry of the manifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,180,881,324]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"One deformation family of IHS manifolds is that of fourfolds of $K 3 ^ { [ 2 ] }$ -type: these are deformations of the Hilbert scheme of two points of a smooth $K 3$ surface. In particular, all these fourfolds $X$ have $b _ { 2 } ( X ) = 2 3$ , the maximal one for an IHS in dimension four, and the Beauville–Bogomolov lattice is isometric to $L : = L _ { K 3 } \\\\oplus$ $\\\\langle - 2 \\\\rangle$ , where $L _ { K 3 }$ is the $K 3$ lattice ${ \\\\cal U } ^ { \\\\oplus 3 } \\\\oplus { \\\\cal E } _ { 8 } ( - 1 ) ^ { \\\\oplus 2 }$ . Among the projective elements of this deformation family, Beauville and Donagi show in [11] that the Fano variety $F ( Y )$ of lines contained in a smooth cubic fourfold $Y \\\\subset \\\\mathbb { P } ^ { 5 }$ is a fourfold of $K 3 ^ { [ 2 ] }$ - type which always carries a polarization of degree six. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,324,881,503]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An automorphism $\\\\sigma \\\\in \\\\operatorname { A u t } ( X )$ of an IHS manifold $X$ induces an isometry $\\\\sigma ^ { * } \\\\in$ $O ( H ^ { 2 } ( X , \\\\mathbb { Z } ) )$ , more precisely one induced by a monodromy operator on it (see [33, Theorem 1.3]), and an isomorphism of Hodge structures on $H ^ { 2 } ( X , \\\\mathbb { C } )$ . As a consequence, the line $H ^ { 2 , 0 } ( X )$ is invariant under $\\\\sigma ^ { * }$ : if $\\\\sigma ^ { * } \\\\omega _ { X } = \\\\omega _ { X }$ , we say that $\\\\sigma$ is symplectic; else, if $\\\\sigma ^ { * } \\\\omega _ { X } = \\\\xi \\\\omega _ { X }$ with $\\\\xi$ a non-trivial root of unity, we say that $\\\\sigma$ is non-symplectic. From work of Beauville [7, Proposition 6] we know that the existence of such a non-symplectic automorphism forces the manifold $X$ to be projective (there exists an invariant ample class); moreover, the order $n$ of $\\\\sigma$ , if it is finite, is bounded by the inequality $\\\\varphi ( n ) \\\\leq b _ { 2 } ( X ) - \\\\rho ( X )$ , where $\\\\varphi$ denotes Euler’s totient function and $\\\\rho ( X )$ is the Picard number of $X$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,506,881,737]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Starting with the paper [14], we have been interested in classifying nonsymplectic automorphisms of prime order acting on fourfolds of $K 3 ^ { [ 2 ] }$ -type. Because of the properties just mentioned above, when $X$ is a projective fourfold of $K 3 ^ { [ 2 ] }$ -type and $\\\\sigma ~ \\\\in ~ \\\\operatorname { A u t } ( X )$ is non-symplectic of prime order $p$ , we have $2 \\\\leq p \\\\leq 2 3$ . The key idea of [14] is to classify the isometries of $L$ induced by such automorphisms, exactly as it is done in the case of $K 3$ surfaces by Nikulin [36] and by Artebani–Sarti–Taki [5]. As in the case of $K 3$ surfaces, in the case of fourfolds $X$ of $K 3 ^ { [ 2 ] }$ -type the map $\\\\operatorname { A u t } ( X ) \\\\to O ( H ^ { 2 } ( X , \\\\mathbb { Z } ) )$ is injective, so that $\\\\sigma ^ { * }$ is non-trivial if $\\\\sigma$ is. Building up on fundamental work contained in [13], we show that these actions are classified by a pair of primitive sublattices: the invariant lattice $T : = H ^ { 2 } ( X , \\\\mathbb { Z } ) ^ { \\\\sigma ^ { * } }$ and its orthogonal complement $s$ inside $L$ . This allows the partial classification contained in [14]: we could obtain all the cases with $p \\\\neq 5 , 2 3$ ; $p = 2 3$ was then solved in [15] and $p = 5$ in [39]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[143,738,756,756]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Nowadays, the state of the art of the classification is the following: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,765,881,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"• Most cases can be constructed starting from a non-symplectic automorphism $\\\\tau$ of the same order $p$ on a smooth $K 3$ surface $\\\\Sigma$ , either by considering the natural automorphism $\\\\sigma = \\\\tau ^ { [ 2 ] }$ on $\\\\Sigma ^ { [ 2 ] }$ (see for example [12] for further details), or by considering the (twisted) induced automorphism on a moduli space of stable (twisted) sheaves on $\\\\Sigma$ (see [35, 38] and [19] for further details). For $p = 2$ , there is the first (historically) remarkable exception to the above cases when $T = \\\\langle 2 \\\\rangle$ : this is Beauville’s famous non-natural non-symplectic involution defined on the Hilbert scheme of two points of a smooth quartic surface not containing a line [7]. This example is in fact a special instance of the nonsymplectic cover involution on O’Grady’s double EPW sextics (see [22, 37] and references therein). There are a few more cases which cannot come from $K 3$ surfaces and are degenerations of this one. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[792,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,54,141,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"22\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[143,90,879,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,180,881,267]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"• For $p = 3$ , two exceptions are given by $T = \\\\langle 6 \\\\rangle$ and $T = \\\\langle 6 \\\\rangle \\\\oplus E _ { 6 } ^ { * } ( 3 ) $ : in both cases the only examples that we can explicitly construct are obtained on Fano varieties of lines of cubic fourfolds endowed with additional symmetries. The first case is the main object of [17], whereas the second one has still to be further understood. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,270,881,378]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"• For $p = 2 3$ , the only possibility is $T = \\\\langle 4 6 \\\\rangle$ : in [15] we show that there exists only one fourfold of $K 3 ^ { [ 2 ] }$ -type carrying such a non-symplectic automorphism of order 23 inside the 20-dimensional family of $\\\\langle 4 6 \\\\rangle$ -polarized fourfolds of $K 3 ^ { [ 2 ] }$ - type once we fix the action on $L$ . Unfortunately, the proof is not constructive but uses in a crucial way the global Torelli theorem for IHS manifolds, and the explicit construction of such a fourfold remains an open question. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,387,881,530]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As already mentioned, the main object of [17], and of this survey as well, is the case $p = 3$ , $T = \\\\langle 6 \\\\rangle$ : we will see that in this case we obtain a relation between the moduli space of marked fourfolds of $K 3 ^ { [ 2 ] }$ -type with such a non-symplectic automorphism of order three and the classical GIT moduli space of smooth cubic threefolds. The existence of such a relation is not so surprising once one sees the first example of such a non-symplectic automorphism, perhaps more surprising is the fact that there is an isomorphism between these two moduli spaces; indeed, both are uniformized by a ten-dimensional complex ball quotient. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,531,881,620]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In Sect. 2 we will explain how to obtain the isomorphism and some of the immediate consequences, after illustrating how the two moduli spaces appearing are constructed, respectively in [16] and in [2]. More precisely, we will see that both moduli spaces are isomorphic to the complement of a hyperplane arrangement $\\\\mathcal { H }$ inside a ten-dimensional complex ball quotient. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,621,881,764]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Section 3 then deals with understanding what happens on hyperplanes inside $\\\\mathcal { H } \\\\mathrm { : \\\\Omega }$ this is completely understood in the case of cubic threefolds, since we obtain semistable points in the GIT quotient, corresponding either to nodal cubics or to the chordal cubic, as shown in [2]. What we investigate in Sect. 3 is what happens to the non-symplectic automorphism of order three and how it degenerates. The final outcome of this analysis will be that the two subarrangements are in fact birational to moduli spaces of $K 3 ^ { [ 2 ] }$ -fourfolds carrying a non-symplectic automorphism of order three of different type in the classification. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,765,879,854]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The picture that we obtained is thus the analogue in higher dimension of the isomorphism of the moduli space of cubic surfaces with the moduli space of $K 3$ surfaces with a special non-symplectic automorphism of order three, illustrated by Dolgachev, van Geemen and Kondo in [¯ 21] building on the construction of Allcock, Carlson and Toledo in [1] (see also [9] for a nice survey on this subject). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,855,876,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally, in Sect. 4, we explore related problems. As we observed above, nonnatural automorphisms on Hilbert schemes of $K 3$ surfaces are extremely difficult to construct; in [17], we also proved that there exist Pfaffian cyclic cubic fourfolds $Y \\\\subset \\\\mathbb { P } ^ { 5 }$ such that $\\\\psi : F ( Y ) \\\\simeq \\\\Sigma ^ { [ 2 ] }$ for a certain $K 3$ surface $\\\\Sigma$ of degree 14 not containing a line, by means of Beauville–Donagi’s isomorphism [11]. On $\\\\Sigma ^ { [ 2 ] }$ we then obtain a non-natural automorphism by conjugation with $\\\\psi$ of the automorphism of order three, which we construct in Sect. 2, on the Fano variety of any cyclic cubic fourfold. Another interesting question is what happens to higher dimensional manifolds of $K 3 ^ { [ n ] }$ -type related to cyclic cubic fourfolds, for example Lehn–Lehn– Sorger–van Straten’s eightfolds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,571,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moduli Spaces of Cubic Threefolds and of IHS Manifolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"23\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"24\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,279,567,300]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"24\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 An Isomorphism of Moduli Spaces \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,324,881,394]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"24\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section we will review the main constructions of [2] and [16] of the moduli spaces, and then prove the isomorphism among the two moduli spaces. Before describing the two moduli spaces involved, we first of all recall the main example of a non-symplectic automorphism of order three which we will consider here. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,439,642,462]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"24\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.1 Fano Varieties of Cyclic Cubic Fourfolds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,483,879,666]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"24\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recall that, for any smooth cubic fourfold $Y \\\\subset \\\\mathbb { P } ^ { 5 }$ , Beauville and Donagi show in [11] that the Fano variety $F ( Y )$ of lines contained in $Y$ is of $K 3 ^ { [ 2 ] }$ -type; moreover, they prove that the Plücker polarization $H _ { F }$ has degree six, so that $\\\\operatorname* { P i c } ( F ( Y ) ) \\\\supset \\\\langle 6 \\\\rangle$ , and that the Abel–Jacobi map $A : H ^ { 4 } ( Y , \\\\mathbb { Z } ) \\\\ { \\\\stackrel { \\\\cdot } { \\\\to } } \\\\ H ^ { 2 } ( F ( Y ) , \\\\mathbb { Z } )$ maps $H _ { Y } ^ { 2 }$ , where $H _ { Y }$ is the ample class on $Y$ , to $H _ { F }$ and induces an anti-isometry on primitive cohomology (the orthogonal of the polarization), i.e. an isometry $A _ { 0 } : H _ { 0 } ^ { 4 } ( Y , \\\\mathbb { Z } ) ( - 1 ) \\\\to H _ { 0 } ^ { 2 } ( F ( Y ) , \\\\mathbb { Z } )$ . As a consequence, $H _ { 0 } ^ { 2 } ( F ( Y ) , \\\\mathbb { Z } ) \\\\cong S : =$ $U ^ { \\\\oplus 2 } \\\\oplus E _ { 8 } ( - 1 ) ^ { \\\\oplus 2 } \\\\oplus A _ { 2 } ( - 1 )$ as lattices, where $U$ denotes the standard hyperbolic plane and $E _ { 8 } ( - 1 )$ , $A _ { 2 } ( - 1 )$ are the negative definite lattices associated to the Dynkin diagrams $E _ { 8 }$ and $A _ { 2 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,666,881,900]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"24\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $Y \\\\subset \\\\mathbb { P } ^ { 5 }$ be the triple cover of $\\\\mathbb { P } ^ { 4 }$ branched along a smooth cubic threefold $C \\\\subset \\\\mathbb { P } ^ { 4 }$ ; such a fourfold is what is usually called a cyclic cubic fourfold. If $G ( X _ { 0 } :$ $\\\\dots : X _ { 4 } ) = 0$ is an equation defining $C$ , then $Y$ is the zero locus defined by $X _ { 5 } ^ { 3 } +$ $G = 0$ , and it is smooth if and only if $C$ is smooth. We choose $\\\\sigma \\\\in \\\\operatorname { A u t } ( Y )$ to be the cover automorphism obtained as restriction to $Y$ of the map $X _ { 5 } \\\\mapsto \\\\xi X _ { 5 }$ , for $\\\\xi$ a fixed (for the rest of the paper) primitive third root of unity. Let $\\\\sigma$ be the induced automorphism on $F ( Y )$ : in [14, Example 4.6] we show that it is a non-symplectic automorphism of order three, that the fixed locus is the Fano surface $F ( C )$ of lines contained in the cubic threefold $C$ and that the invariant lattice of $\\\\sigma$ is $T : = \\\\langle 6 \\\\rangle$ , embedded in $L$ in such a way that its orthogonal complement is isometric to $s$ (this is the so-called embedding of non-split type of an ample class of square six, see [23] for further details). It is clear that such an automorphism cannot be natural nor induced on a moduli space of stable sheaves on a K3 surface by a non-symplectic automorphism of the underlying K3 surface, since these can only exist on moduli spaces $X$ of stable sheaves on K3 surfaces with $\\\\rho ( X ) \\\\ge 2$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[792,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"24\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,54,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"24\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,879,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,171,703,192]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2 An “Occult Period Map” for Cubic Threefolds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,214,881,268]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We now start to recall Allcock–Carlson–Toledo’s construction of a period map for the moduli space of smooth cubic threefolds, which has been later on defined “occult” by Kudla and Rapoport in [27]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,270,881,468]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $C _ { 3 } ^ { s } : = | O _ { \\\\mathbb { P } ^ { 4 } } ( 3 ) | \\\\ / \\\\ P G L _ { 5 } ( \\\\mathbb { C } )$ be the GIT moduli space of stable cubic threefolds; it contains as an open subset the moduli space $C _ { 3 } ^ { s m }$ of smooth cubic threefolds. Classical work of Clemens and Griffiths gives a map ${ C } _ { 3 } ^ { s m } \\\\ \\\\ { \\\\mathcal { A } } _ { 5 }$ by associating to $C$ its Intermediate Jacobian $I J ( C ) : = H ^ { 2 , 1 } ( C , \\\\mathbb { C } ) ^ { * } / H ^ { 3 } ( C , \\\\mathbb { Z } )$ , which is a ppav of dimension 5; the problem is that these two moduli spaces are respectively of dimension 10 and 15, so there is no hope to get an isomorphism. This is the reason for which Allcock, Carlson and Toledo came up with a different idea to construct a period map for smooth cubic threefolds by using the period map of smooth cubic fourfolds, and in particular by showing that it gives an isomorphism with a 10-dimensional complex ball quotient. We will now review the main ideas of their construction. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,468,881,600]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $C \\\\in C _ { 3 } ^ { s m }$ and let $Y \\\\subset \\\\mathbb { P } ^ { 5 }$ be the associated cyclic cubic fourfold. Recall that a marking of any cubic fourfold $Y$ is an isometry $\\\\eta _ { 0 } : H _ { 0 } ^ { 4 } ( Y , \\\\mathbb { Z } ) \\\\to S ( - 1 )$ and that the period of the marked pair $( Y , \\\\eta _ { 0 } )$ is $[ \\\\eta _ { 0 } ( H ^ { 3 , 1 } ( Y ) ) ] \\\\in \\\\mathbb { P } ( S ( - 1 ) \\\\otimes \\\\mathbb { C } )$ . The key idea is now to use the restriction of the period map to cyclic cubic fourfolds to define a period map for cubic threefolds: when $Y$ is cyclic, its period is an eigenvector for the action of $\\\\sigma ^ { * }$ , so that $[ \\\\eta _ { 0 } ( H ^ { 3 , 1 } ( Y ) ) ] \\\\in \\\\mathbb { P } ( S ( - 1 ) _ { \\\\xi } )$ , where $S ( - 1 ) _ { \\\\xi }$ is the eigenspace of $\\\\eta _ { 0 } \\\\circ \\\\sigma ^ { * } \\\\circ \\\\eta _ { 0 } ^ { - 1 }$ relative to $\\\\xi$ inside $S ( - 1 ) \\\\otimes \\\\mathbb { C }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,602,879,693]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Fix as initial data a smooth cubic threefold $\\\\bar { C } \\\\in C _ { 3 } ^ { s m }$ , and let $\\\\bar { Y } , \\\\bar { \\\\sigma }$ and $\\\\bar { \\\\eta _ { 0 } }$ be respectively the cyclic cubic fourfold associated to $\\\\bar { C }$ , the cover automorphism as defined above and a chosen marking of $\\\\bar { Y }$ . We define $\\\\rho _ { 0 } : = \\\\bar { \\\\eta _ { 0 } } \\\\circ \\\\bar { \\\\sigma } \\\\circ \\\\bar { \\\\eta _ { 0 } } ^ { - 1 } \\\\in$ $O ( S ( - 1 ) )$ to be the abstract isometry induced by $\\\\bar { \\\\sigma }$ , and we use it to move the action around the locally trivial family of cubic fourfolds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,695,881,767]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A framing of $C \\\\in C _ { 3 } ^ { s m }$ is the equivalence class of a marking $\\\\eta _ { 0 }$ of $Y$ such that $\\\\eta _ { 0 } \\\\circ \\\\rho _ { 0 } = \\\\rho _ { 0 } \\\\circ \\\\eta _ { 0 }$ , up to the action of $\\\\mu _ { 6 } : = \\\\{ \\\\pm \\\\mathrm { i d } _ { S } , \\\\pm \\\\rho _ { 0 } , \\\\pm \\\\rho _ { 0 } ^ { 2 } \\\\}$ : indeed, $S ( - 1 )$ has a structure of $\\\\mathbb { Z } [ \\\\xi ]$ -module induced by $\\\\rho _ { 0 }$ , and $\\\\mu _ { 6 }$ is exactly the group of units in $\\\\mathbb { Z } [ \\\\xi ]$ . We define the moduli space of framed pairs up to isomorphism: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[277,783,715,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0527/6fa3f67a6e9a6877d8918ce4ea919605bf78359bb15d5000dc1ad7ad29a4ae4c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathcal { F } _ { 3 } ^ { s m } : = \\\\{ ( C , \\\\eta _ { 0 } ) | C \\\\in C _ { 3 } ^ { s m } , \\\\ \\\\eta _ { 0 } \\\\ \\\\mathrm { i s ~ a ~ f r a m e } \\\\} / \\\\simeq .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,821,881,894]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The group $\\\\Gamma _ { 0 } : = \\\\{ \\\\gamma \\\\in O ( S ( - 1 ) ) | \\\\gamma \\\\circ \\\\rho _ { 0 } = \\\\rho _ { 0 } \\\\circ \\\\gamma \\\\}$ acts on a framed pair by $( C , \\\\eta _ { 0 } ) \\\\mapsto ( C , \\\\gamma \\\\circ \\\\eta _ { 0 } )$ , but because of our definition of framings as equivalence classes, the action is not free. We thus introduce $\\\\mathbb { P } \\\\Gamma : = \\\\Gamma _ { 0 } / \\\\mu _ { 6 }$ , and we recover the moduli space of smooth cubic threefolds as the quotient $C _ { 3 } ^ { s m } \\\\simeq \\\\mathcal { F } _ { 3 } ^ { s m } / \\\\mathbb { P } \\\\Gamma$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,571,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moduli Spaces of Cubic Threefolds and of IHS Manifolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"25\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We can now study the map induced by the period map of cubic fourfolds on $\\\\mathcal { F } _ { 3 } ^ { s m } \\\\colon$ as we already observed, the period $[ \\\\eta _ { 0 } ( H ^ { 3 , 1 } ( Y ) ) ] \\\\in \\\\mathbb { P } ( S ( - 1 ) _ { \\\\xi } )$ and it is welldefined even for the framing, since composing with any element in $\\\\mu _ { 6 }$ corresponds to multiplying the period for a non-zero scalar. A standard argument shows that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[250,175,744,199]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0484/f80fbd1d3a1732d31e9ee59c08523959d4235c0cf3c50af9b351e0ef3276a1ef.jpg\",\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n[ \\\\eta _ { 0 } ( H ^ { 3 , 1 } ( Y ) ) ] \\\\in \\\\mathbb { B } _ { 1 0 } \\\\simeq \\\\{ x \\\\in \\\\mathbb { P } ( S ( - 1 ) _ { \\\\xi } ) | ( x , \\\\bar { x } ) < 0 \\\\} ;\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,214,728,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"equivalently, the period domain is a ten-dimensional complex ball. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,240,881,297]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 1 The period map sending $( C , \\\\eta _ { 0 } ) \\\\in \\\\mathcal { F } _ { 3 } ^ { s m } t o \\\\left[ \\\\eta _ { 0 } ( H ^ { 3 , 1 } ( Y ) ) \\\\right] \\\\in \\\\mathbb { B } _ { 1 0 }$ defines an isomorphism onto the image and is equivariant with respect to the action of P\\b, so that we have the following commutative diagram: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[375,309,626,399]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0713/79eefc3d726829e8adea7325e84cacda846c203ecbf1e503e4cd877826db8945.jpg\",\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { l l } { \\\\mathcal { F } _ { 3 } ^ { s m } \\\\xrightarrow { \\\\quad \\\\simeq \\\\quad \\\\simeq \\\\quad \\\\gg \\\\mathbb { B } _ { 1 0 } \\\\setminus \\\\mathcal { H } \\\\quad } } \\\\\\\\ { \\\\bigg \\\\downarrow \\\\quad } \\\\\\\\ { C _ { 3 } ^ { s m } \\\\xrightarrow { \\\\quad \\\\geqslant \\\\quad ( \\\\mathbb { B } _ { 1 0 } \\\\setminus \\\\mathcal { H } ) / \\\\mathbb { P } \\\\Gamma } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,418,876,454]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\mathcal { H }$ is the union of hyperplanes in $\\\\mathbb { B } _ { 1 0 }$ orthogonal to classes $\\\\delta \\\\in S ( - 1 )$ with $\\\\delta ^ { 2 } = 2$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,501,744,522]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.3 Moduli Spaces of $( \\\\pmb { \\\\rho } , \\\\pmb { T } )$ -Polarized IHS Fourfolds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,545,881,653]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section, we recall the constructions of [16] and [17] of moduli spaces of IHS manifolds of $K 3 ^ { [ n ] }$ -type carrying a non-symplectic automorphism of prime order associated to a given pair of sublattices $( T , S )$ , but instead of working in full generality we illustrate everything in the case of interest here. We fix a primitive embedding $j : S \\\\hookrightarrow L$ and we call $\\\\theta \\\\in L$ a generator of $j ( S ) ^ { \\\\perp }$ , which is of degree six as we have already discussed. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,653,881,801]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A marking of a fourfold of $K 3 ^ { [ 2 ] }$ -type $X$ is an isometry $\\\\eta ~ : ~ H ^ { 2 } ( X , \\\\mathbb { Z } ) \\\\ \\\\to$ $L$ . Let $\\\\boldsymbol { \\\\mathcal { M } } _ { L }$ be the moduli space of marked pairs $( X , \\\\eta )$ of $K 3 ^ { [ 2 ] }$ -type modulo isomorphism. The choice of the initial data made in Sect. 2.2 produces also initial data for this construction: indeed, we consider $F ( { \\\\bar { Y } } )$ the Fano variety of the cyclic cubic fourfold $\\\\bar { Y }$ and we define a marking by extending $\\\\bar { \\\\eta _ { 0 } } \\\\circ A ^ { - 1 }$ to a map $\\\\bar { \\\\eta } ~ : { \\\\cal H } ^ { 2 } ( F ( \\\\bar { Y } ) , \\\\mathbb { Z } ) { \\\\cal L }$ which sends $H _ { F ( \\\\bar { Y } ) }$ to $\\\\theta$ . In this way we obtain a point of $\\\\boldsymbol { \\\\mathcal { M } } _ { L }$ ; let $\\\\mathcal { M } _ { L } ^ { + }$ be the connected component containing $( F ( { \\\\bar { Y } } ) , { \\\\bar { \\\\eta } } )$ . We also get a non-symplectic automorphism $\\\\bar { \\\\sigma } \\\\in \\\\mathop { \\\\mathrm { A u t } } ( F ( \\\\bar { Y } ) )$ of order three and a corresponding isometry $\\\\rho : = \\\\bar { \\\\eta } \\\\circ \\\\bar { \\\\sigma } ^ { \\\\ast } \\\\circ \\\\bar { \\\\eta } ^ { - 1 } \\\\in O ( L )$ of order three which satisfies the following additional properties: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[792,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Camere \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"26\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"18 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,879,129]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,139,881,211]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. $\\\\rho _ { \\\\lvert S } = \\\\rho _ { 0 }$ ; \\n2. $\\\\dot { \\\\rho \\\\mathbf { \\\\rho } } \\\\in \\\\mathbf { M o n } ^ { 2 } ( L ) : = \\\\bar { \\\\eta } \\\\circ \\\\mathbf { M o n } ^ { 2 } ( F ( \\\\bar { Y } ) \\\\circ \\\\bar { \\\\eta } ^ { - 1 }$ is a monodromy operator on $L$ , so by [23, §3] it has real spinor norm $\\\\mathbf { s n } _ { L } ^ { \\\\mathbb { R } } ( \\\\rho ) = 1$ (see loc. cit. for the definition of $\\\\mathrm { s n } _ { L } ^ { \\\\mathbb { R } } .$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,219,881,274]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 1 A $\\\\left( \\\\rho , \\\\langle 6 \\\\rangle \\\\right)$ -polarized pair $( X , \\\\eta )$ is the data of a marked pair $( X , \\\\eta ) \\\\in$ $\\\\boldsymbol { M } _ { L } ^ { + }$ and $\\\\sigma ~ \\\\in ~ \\\\operatorname { A u t } ( X )$ respectively, for which there exists $\\\\iota : \\\\langle 6 \\\\rangle \\\\hookrightarrow \\\\operatorname { P i c } ( X )$ a primitive embedding of lattices such that $\\\\eta \\\\circ \\\\iota = j$ and such that $\\\\eta \\\\circ \\\\sigma \\\\circ \\\\eta ^ { - 1 } = \\\\rho$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,282,881,337]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This definition turns out to be the counterpart of the notion of a framing for a cubic threefold. We define $M _ { \\\\rho , \\\\langle 6 \\\\rangle }$ to be the subspace of isomorphism classes of $\\\\left( \\\\rho , \\\\langle 6 \\\\rangle \\\\right)$ -polarized pairs inside $\\\\boldsymbol { M } _ { L } ^ { + }$ . The stabilizer of $M _ { \\\\rho , \\\\langle 6 \\\\rangle }$ is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[287,352,708,376]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0593/19359532945b4fe42c96d38010c07c85a0da285e3e5a0babab372eee85ac63b8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf { M o n } ^ { 2 } ( L ) _ { \\\\rho } : = \\\\{ g \\\\in \\\\mathbf { M o n } ^ { 2 } ( L ) | g \\\\circ \\\\rho = \\\\rho \\\\circ g \\\\} ;\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,391,660,411]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"we define the quotient space $N _ { \\\\rho , \\\\langle 6 \\\\rangle } : = M _ { \\\\rho , \\\\langle 6 \\\\rangle } / \\\\operatorname { M o n } ^ { 2 } ( L ) _ { \\\\rho }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,412,881,465]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $\\\\Gamma _ { \\\\rho } : = r _ { S } ( { \\\\mathbf { M o n } } ^ { 2 } ( L ) _ { \\\\rho } )$ be the image of $\\\\mathsf { M o n } ^ { 2 } ( L ) _ { \\\\rho }$ via the restriction map $r _ { S } :$ $O ( L ) \\\\to O ( S )$ , which acts on the periods in $\\\\mathbb { P } ( S \\\\otimes \\\\mathbb { C } )$ . One of the key ingredients of our main result is the following \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,474,712,493]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 1 There is an isomorphism of groups $\\\\Gamma _ { \\\\rho } \\\\simeq \\\\Gamma _ { 0 } / \\\\{ \\\\pm \\\\mathrm { i d } _ { S } \\\\}$ . \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,500,881,627]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The period map of marked pairs of $K 3 ^ { [ 2 ] }$ -type restricts to a holomorphic map $\\\\begin{array} { r l r } { M _ { \\\\rho , \\\\langle 6 \\\\rangle } } & { { } \\\\to } & { { \\\\mathbb { B } } _ { 1 0 } } \\\\end{array}$ , since again $[ \\\\eta ( H ^ { 2 , 0 } ( X ) ) ] ~ \\\\in ~ \\\\mathbb { P } ( S _ { \\\\xi } )$ (this is nothing but the equivariance of the Abel–Jacobi map of a cyclic cubic fourfold with respect to the action of $\\\\sigma$ ). In order to describe the image we need crucially the notion of MBM class $\\\\delta \\\\ \\\\in \\\\ L$ introduced by Amerik and Verbitsky in [3], which we do not recall in detail here since, as we will see in Sect. 2.4, it is possible to characterize these classes lattice-theoretically. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,635,881,708]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2 The period map restricts to an isomorphism of the subspace of isomorphism classes of $\\\\left( \\\\rho , \\\\langle 6 \\\\rangle \\\\right)$ -polarized pairs $M _ { \\\\rho , \\\\langle 6 \\\\rangle }$ onto its image inside $\\\\mathbb { B } _ { 1 0 }$ , and it is equivariant with respect to the action of $\\\\mathsf { M o n } ^ { 2 } ( L ) _ { \\\\rho }$ , so that it descends to the quotients. We have the following commutative diagram: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[362,722,637,809]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"1004/3a4449a02372326ccef05c0351e404712027be078254b8c236a836260abf37d4.jpg\",\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } { M _ { \\\\rho , \\\\langle 6 \\\\rangle } \\\\xrightarrow { \\\\quad \\\\simeq \\\\quad \\\\mathbb { B } _ { 1 0 } \\\\setminus \\\\mathcal { H } ^ { \\\\prime } } } & { { } } \\\\\\\\ { \\\\bigg \\\\downarrow } & { { } } \\\\\\\\ { N _ { \\\\rho , \\\\langle 6 \\\\rangle } \\\\xrightarrow { \\\\quad \\\\geqslant \\\\quad \\\\mathbb { P } _ { 6 } \\\\quad \\\\qquad } \\\\mathbb { V } ^ { \\\\prime } ) / \\\\Gamma _ { \\\\rho } } & { { } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[113,828,842,848]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\mathcal { H }$ is the union of hyperplanes in $\\\\mathbb { B } _ { 1 0 }$ orthogonal to MBM classes $\\\\delta \\\\in S$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,51,571,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moduli Spaces of Cubic Threefolds and of IHS Manifolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,52,879,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"27\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"19 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,382,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.4 The Isomorphism \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,132,879,186]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We now conclude the proof of the main result: we are left with comparing the two hyperplane arrangements, since it immediately follows from Lemma 1 that the two arithmetic subgroups $\\\\mathbb { P } \\\\Gamma$ and $\\\\Gamma _ { \\\\rho }$ have the same orbits on $\\\\mathbb { B } _ { 1 0 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,187,881,259]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is known from work of Bayer–Hassett–Tschinkel [6] and Mongardi [34] that MBM classes for fourfolds of $K 3 ^ { [ 2 ] }$ -type are exactly classes $\\\\delta \\\\in L$ either with $\\\\delta ^ { 2 } = - 2$ or with $\\\\delta ^ { 2 } = - 1 0$ and $\\\\mathrm { d i v } _ { L } ( \\\\delta ) = 2$ , where the divisibility of $x$ in a lattice $L$ is defined as the generator of the ideal $( x , L )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,267,719,286]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 2 All MBM classes $\\\\delta \\\\in S$ satisfy $\\\\delta ^ { 2 } = - 2$ , and $\\\\mathcal { H } = \\\\mathcal { H }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,295,881,331]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This follows easily from the fact that $\\\\mathrm { d i v } _ { S } ( \\\\delta )$ is a common divisor of $\\\\operatorname* { d e t } ( S ) = 3$ and of $\\\\operatorname* { d e t } ( L ) = 2$ , so it cannot be two. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,340,881,414]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3 There is an isomorphism between the moduli spaces of smooth cubic threefolds $C _ { 3 } ^ { s m }$ and the arithmetic quotient $N _ { \\\\rho , \\\\langle 6 \\\\rangle }$ of the subspace of isomorphism classes of $( \\\\rho , \\\\langle 6 \\\\rangle )$ -polarized pairs inside $\\\\mathcal { M } _ { L } ^ { + }$ , such that the following diagram commutes: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[314,427,681,513]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0724/2515834a508a5c10dcd1285223b35c055a819a788f447427c217e256bf785113.jpg\",\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[123,534,879,570]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Whilst a priori we did not know whether any element of $N _ { \\\\rho , \\\\langle 6 \\\\rangle }$ was exactly the Fano variety of a cyclic cubic fourfold, this follows immediately from Theorem 3. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,579,876,615]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 1 Any $( X , \\\\eta ) \\\\in { \\\\cal M } _ { \\\\rho , \\\\langle 6 \\\\rangle }$ is of the form $( F ( Y ) , \\\\eta )$ for some $Y$ cyclic cubic fourfold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,624,876,660]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A second interesting consequence is that we obtain a more classically-flavoured period map also in the case of fourfolds of $K 3 ^ { [ 2 ] }$ -type. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,669,881,705]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 2 There is an open embedding of $N _ { \\\\rho , \\\\langle 6 \\\\rangle }$ inside ${ \\\\mathcal { A } } _ { 5 }$ given by sending $( X , \\\\eta )$ into ${ \\\\mathrm { A l b } } ( X ^ { \\\\sigma } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,713,790,732]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Indeed, this follows immediately from the fact that ${ \\\\mathrm { A l b } } ( F ( C ) ) \\\\cong I J ( C )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,777,328,798]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 Degenerations \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,821,881,894]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Degenerations of smooth cubic threefolds are well-studied; in particular it is proven in [2] that the hyperplane arrangement $\\\\mathcal { H }$ is the union of two sub-arrangements ${ \\\\mathcal { H } } _ { n }$ and $\\\\mathcal { H } _ { c }$ respectively corresponding to stable cubics with at least one nodal singularity and to the semistable chordal cubic. From a lattice theoretical viewpoint, the two cases correspond respectively to nodal and chordal roots $\\\\delta \\\\in S ( - 1 )$ of square 2. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[792,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Camere \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"28\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"20 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,90,879,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"29\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,126,881,288]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"29\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the case of cubic threefolds, it is nowadays very well-understood what happens when the period hits these hyperplanes; here, we want to answer the same question in the case of IHS manifolds. In particular the aim of this section is to explain what happens to the non-symplectic automorphism of order three when the period of $( X , \\\\eta )$ is in $\\\\mathcal { H } .$ This has been already explained by Dolgachev and Kondo in ¯ [20] for $K 3$ surfaces. We know that in the fiber of the period map of fourfolds of $K 3 ^ { [ 2 ] }$ -type over a point in the hyperplane arrangement $\\\\mathcal { H }$ we can still find smooth fourfolds; in order to understand what happens to the $( \\\\rho , T )$ -polarization we apply [33, Theorem 1.6] and obtain the following statement: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,295,881,388]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"29\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 1 Let $\\\\omega \\\\in \\\\delta ^ { \\\\perp } \\\\subset \\\\mathcal { H }$ and let $( X , \\\\eta )$ be any element in $\\\\mathcal { P } _ { 6 } ^ { - 1 } ( \\\\omega )$ ; then there exists $\\\\beta \\\\ \\\\in \\\\ \\\\operatorname { B i r } ( X )$ and $w \\\\in W _ { \\\\mathrm { e x c } } ( X )$ such that $\\\\eta ^ { - 1 } \\\\circ \\\\rho \\\\circ \\\\eta \\\\ : = \\\\ : w \\\\circ \\\\beta ^ { * }$ , where $W _ { \\\\mathrm { e x c } } ( X ) \\\\subset O ( H ^ { 2 } ( X , \\\\mathbb { Z } ) )$ is the subgroup generated by reflections in prime exceptional divisors, (i.e. reduced and irreducible effective divisors of negative degree). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,396,881,487]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"29\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As a consequence of Proposition 1, one might be tempted to expect that the isometry $\\\\rho$ no longer induces an automorphism but only a birational map of order three with a non-empty indeterminacy locus. In fact this is not the case, and the automorphism changes in the sense that it has a different action on the lattice. In order to explain this phenomenon we need to recall some definitions from [20]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,495,876,531]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"29\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 2 Given $\\\\omega \\\\in \\\\mathcal { H } .$ , the degeneracy sublattice of $\\\\omega$ is the primitive sublattice of $s$ spanned by MBM classes which are orthogonal to $\\\\omega$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,540,881,650]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"29\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The key remark is that for any MBM class in $s$ , we have $\\\\delta ^ { \\\\perp } \\\\cap \\\\mathbb { P } ( S _ { \\\\xi } ) = \\\\rho ( \\\\delta ) ^ { \\\\perp } \\\\cap$ $\\\\mathbb { P } ( S _ { \\\\xi } )$ . From now on we suppose $\\\\omega \\\\in H \\\\subset { \\\\mathcal { H } }$ for only one hyperplane $H = \\\\delta ^ { \\\\perp }$ ; in this case the degeneracy lattice is $R _ { \\\\delta } : = \\\\langle \\\\delta , \\\\rho ( \\\\delta ) \\\\rangle \\\\cong A _ { 2 } ( - 1 )$ ; this admits, up to isometry, two primitive embeddings into $s$ , according to the two cases $R _ { \\\\delta } ^ { \\\\perp } =$ $U \\\\oplus U ( a ) \\\\oplus E _ { 8 } ( - 1 ) ^ { \\\\oplus 2 }$ for $a = 1$ , 3; we speak respectively about chordal and nodal roots $\\\\delta$ in the two cases $a = 1 , 3$ , since the following holds: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,657,874,695]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"29\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 3 Let $\\\\delta \\\\in S$ have square $^ { - 2 }$ and let $\\\\omega \\\\in \\\\delta ^ { \\\\perp } \\\\subset \\\\mathcal { H }$ be very general; then, $R _ { \\\\delta } ^ { \\\\perp }$ is unimodular if and only if $\\\\omega \\\\in \\\\mathcal { H } _ { c }$ , if and only if $a = 1$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,704,879,740]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"29\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We now want to look at how the isometry $\\\\rho$ degenerates along the hyperplanes. We call $T _ { \\\\delta }$ the saturation of $\\\\langle 6 \\\\rangle \\\\oplus R _ { \\\\delta }$ in $L$ , so that $\\\\overline { { ( T _ { \\\\delta } ) _ { L } ^ { \\\\perp } } } \\\\subset S _ { \\\\delta } : = \\\\overline { { ( R _ { \\\\delta } ) _ { S } ^ { \\\\perp } } }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,749,879,785]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"29\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 4 The lattice $T _ { \\\\delta }$ is respectively isometric to $T _ { n } : = U ( 3 ) \\\\oplus \\\\langle - 2 \\\\rangle \\\\cong \\\\langle 6 \\\\rangle \\\\oplus$ $A _ { 2 } ( - 1 )$ for $\\\\omega \\\\in \\\\mathcal { H } _ { n }$ (case $a = 3$ ), and to $T _ { c } : = U \\\\oplus \\\\langle - 2 \\\\rangle$ for $\\\\omega \\\\in \\\\mathcal { H } _ { c }$ (case $a = 1$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,794,876,828]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"29\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We define $\\\\rho _ { \\\\delta }$ as the isometry of $L$ obtained by gluing $\\\\mathrm { i d } _ { T _ { \\\\delta } }$ and $\\\\rho _ { | S _ { \\\\delta } }$ : it is the degeneration of the isometry $\\\\rho$ to $\\\\delta ^ { \\\\perp }$ , and by construction we have $L ^ { \\\\rho _ { \\\\delta } } = T _ { \\\\delta }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,837,879,881]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"29\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 2 The isometry $\\\\rho _ { \\\\delta } \\\\in \\\\mathbf { M o n } ^ { 2 } ( L )$ and we have $\\\\rho _ { \\\\delta } = r _ { \\\\rho ( \\\\delta ) } \\\\circ r _ { \\\\delta } \\\\circ \\\\rho$ , where $r _ { v } \\\\in O ( L )$ denotes the reflection in $v$ , given by $\\\\begin{array} { r } { x \\\\mapsto x - 2 \\\\frac { ( x , v ) } { ( v , v ) } v } \\\\end{array}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,571,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"29\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moduli Spaces of Cubic Threefolds and of IHS Manifolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"29\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,876,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In order to distinguish the two cases, we denote by $\\\\rho _ { n }$ and $\\\\rho _ { c }$ the isometries $\\\\rho _ { \\\\delta }$ respectively when $\\\\delta$ is a nodal and a chordal root. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,171,375,190]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.1 The Nodal Locus \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,214,881,360]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A semistable nodal cubic threefold, whose period maps to the very general $\\\\omega \\\\in \\\\delta ^ { \\\\perp }$ with $\\\\delta$ a nodal root, is of the form $C : X _ { 0 } Q + M _ { 3 } = 0$ with an $A _ { 1 }$ -singularity in $p = [ 1 : 0 : \\\\ldots : 0 ]$ ; hence, the associated cubic fourfold has equation $Y : X _ { 0 } Q +$ $M _ { 3 } + X _ { 3 } ^ { 5 } = 0$ with an $A _ { 2 }$ -singularity in $p = [ 1 : 0 : \\\\ldots : 0 ]$ . Allcock, Carlson and Toledo extend $\\\\mathcal { P } _ { 3 }$ to semistable points by using the period of $\\\\Sigma : \\\\ d Q = M _ { 3 } + X _ { 5 } ^ { 3 } = 0$ : when the singularity of $C$ is $A _ { 1 }$ , this is a smooth $K 3$ surface, endowed with a nonsymplectic automorphism $\\\\tau$ of order three, with invariant lattice $U ( 3 ) \\\\subset H ^ { 2 } ( \\\\Sigma , \\\\mathbb { Z } )$ (see [4]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,361,881,450]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Indeed, it is a classical fact that $F ( Y )$ is singular in this situation and that the singular locus is contained in $F ( Y , p )$ , defined as the surface of lines contained in $Y$ which pass through one of the isolated singularities $p$ (compare with [30, 37] and [41]). The surface $\\\\Sigma$ is the minimal resolution of $F ( Y , p )$ in this case, and Lehn shows in [30] that $\\\\Sigma ^ { [ 2 ] } F ( Y )$ is a resolution of singularities. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,451,881,506]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moreover, the natural automorphism $\\\\tau ^ { \\\\left[ 2 \\\\right] }$ on $\\\\Sigma ^ { [ \\\\bar { 2 } ] }$ is induced by the original automorphism of $\\\\mathbb { P } ^ { 5 }$ which we had chosen at the beginning, and its invariant sublattice is exactly $T _ { n }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,513,881,551]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 3 There exists a marking $\\\\eta : H ^ { 2 } ( \\\\Sigma ^ { [ 2 ] } , \\\\mathbb { Z } ) \\\\to L$ such that $( \\\\Sigma ^ { [ 2 ] } , \\\\eta )$ is $( \\\\rho _ { n } , T _ { n } )$ -polarized. In particular, $\\\\eta ^ { - 1 } \\\\circ \\\\rho _ { n } \\\\circ \\\\eta = ( \\\\tau ^ { [ 2 ] } ) ^ { * }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,558,881,680]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We now choose a very general point $( \\\\Sigma ^ { [ 2 ] } , \\\\eta )$ as above and such that $\\\\operatorname { P i c } ( \\\\Sigma ^ { [ 2 ] } ) \\\\cong$ $T _ { n }$ , and we define $K ( T _ { n } ) : = \\\\eta ( \\\\mathcal { K } _ { \\\\Sigma ^ { [ 2 ] } } )$ . Following the construction of [16] and the techniques first developed in [26], we say that $( X , \\\\eta ) \\\\in \\\\mathcal { M } _ { \\\\rho _ { n } , T _ { n } } ^ { \\\\xi }$ is $K ( T _ { n } )$ -general if $\\\\eta ( \\\\mathcal { K } _ { X } ) \\\\cap ( T _ { n } ) _ { \\\\mathbb { R } } = K ( T _ { n } )$ ; the moduli space of $K ( T _ { n } )$ -general $( \\\\rho _ { T _ { n } } , T _ { n } )$ -polarized K ∩ = fourfolds of K3[2]-type is MξK(Tn),ρn a nd its quotient for the action of $\\\\mathbf { M o n } ^ { 2 } ( L , \\\\rho _ { n } )$ is NξK(Tn),ρn . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[143,680,763,696]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Making the above reasoning global, in [17] we obtain the following $\\\\Delta _ { 3 } ^ { s } = C _ { 3 } ^ { s } \\\\setminus C _ { 3 } ^ { s m }$ is birational to ${ \\\\cal N } _ { K ( T _ { n } ) , \\\\rho _ { n } } ^ { \\\\xi }$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,705,772,728]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,770,400,791]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.2 The Chordal Locus \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,815,881,887]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the case of the stable chordal locus $\\\\Xi _ { 3 } ^ { s }$ the picture is more complicated: associated to one semistable chordal cubic we find a hyperplane arrangement. The fact is, as Allcock, Carlson and Toledo show, that the variation of Hodge structure along a 1- parameter family hitting $\\\\mathcal { H } _ { c }$ depends on the path chosen, so that from one semistable point in the moduli space one gets in fact an entire hyperplane arrangement. Again, they show that the variety realizing the degeneration of the Hodge structure is a $K 3$ surface $\\\\Sigma$ carrying a non-symplectic automorphism $\\\\tau$ of order three having fixed lattice isometric to $U$ (see [4]). The final result, though through a less geometrical proof, is the following \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[792,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Camere \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"30\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"22 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,881,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,189,731,210]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 5 The stable chordal locus $\\\\Xi _ { 3 } ^ { s }$ is birational to $\\\\surd _ { K ( T _ { c } ) , \\\\rho _ { c } } ^ { \\\\xi }$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,217,881,292]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It remains an open question to understand the geometry of the (singular) Fano variety of lines of the cyclic fourfold corresponding to the chordal cubic threefold, and which relation, if any, it has with the Hilbert schemes of $K 3$ surfaces in NξK(Tc),ρc . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,292,879,363]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorems 4 and 5 in some sense shed also new light on why smooth cubic threefolds are degenerating to $K 3$ surfaces: in dimension four the picture becomes much more uniform and the degeneration is explained as degeneration of nonsymplectic automorphisms of order three on fourfolds of $K 3 ^ { [ 2 ] }$ -type. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,408,669,429]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 Further Developments and Open Questions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,447,542,468]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4.1 Pfaffian Cyclic Cubic Fourfolds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,490,881,618]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Other natural loci to study inside the arithmetic quotient $N _ { \\\\rho , \\\\left. 6 \\\\right. } = M _ { \\\\rho , \\\\left. 6 \\\\right. } / \\\\mathrm { M o n } ^ { 2 } ( L ) _ { \\\\rho }$ are Heegner divisors, and in particular those coming from the intersection among the family of cyclic cubic fourfolds and the so-called Hassett’s divisors $\\\\mathcal { H } _ { d } \\\\in C _ { 4 } ^ { s m }$ (see [24] for the exact definition). In [17] we have studied the closely-related question about the existence of a cyclic Pfaffian cubic fourfold not containing a plane (indeed, the closure of the Pfaffian locus is $\\\\mathcal { H } _ { 1 4 }$ , and cubics containing a plane are exactly the elements of $\\\\mathcal { H } _ { 8 }$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,618,881,707]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recall the following construction from classical Pfaffian geometry, as described in [11]: let $V$ be a six-dimensional vector space and let ${ \\\\mathcal P } f \\\\subset \\\\mathbb P ( \\\\Lambda ^ { 2 } V ^ { * } )$ be the hypersurface of degenerate skew-symmetric 2-forms. Any Pfaffian cubic fourfold $Y$ is then of the form $\\\\mathcal P f \\\\cap \\\\mathbb { P } ( W )$ for $W \\\\subset \\\\Lambda ^ { 2 } V ^ { * }$ six-dimensional. The dual $K 3$ surface of $Y$ is $\\\\Sigma : = G ( 2 , V ) \\\\cap \\\\mathbb { P } ( W ^ { \\\\perp } )$ , of degree 14. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,717,881,770]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 4 There exists a smooth cyclic Pfaffian cubic fourfold $Y \\\\subset \\\\mathbb { P } ^ { 5 }$ defined over $\\\\mathbb { Q }$ which does not contain a plane, and such that the dual $K 3$ surface $\\\\Sigma$ does not contain a line. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,780,879,852]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The interest of Proposition 4 is that Beauville and Donagi proved in [11] that under these special assumptions, the Fano variety of a Pfaffian cubic fourfold is isomorphic to the Hilbert scheme of two points on $\\\\Sigma$ . As a consequence, the automorphism is defined on $\\\\Sigma ^ { [ 2 ] }$ and we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,861,876,896]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 3 There exists a non-natural non-symplectic automorphism of order three on $\\\\Sigma ^ { [ 2 ] }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,571,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moduli Spaces of Cubic Threefolds and of IHS Manifolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"31\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,180]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"See [17] for a geometrical description. We remark that more in general, one obtains a non-natural automorphism on a Hilbert square whenever there exists a $K 3$ surface $\\\\Sigma$ such that $\\\\Sigma ^ { [ 2 ] } \\\\cong F ( Y )$ for a cyclic cubic fourfold Y , but the question of studying the intersection, which happens in positive codimension, among Hassett’s divisors and the locus of cyclic cubic fourfolds remains open. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,225,589,246]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4.2 Higher Dimensional IHS Manifolds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,267,881,358]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Fourfolds of $K 3 ^ { [ 2 ] }$ -type are not the only IHS manifolds that can be associated to cubic fourfolds: a recent example are the Lehn–Lehn–Sorger–van Straten $( L L S \\\\nu S )$ eightfolds of $K 3 ^ { [ 4 ] }$ -type, defined as contraction $Z$ of the Hilbert scheme of generalized twisted cubic curves contained in a smooth cubic fourfold $Y$ which does not contain a plane [31]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,360,881,504]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In [18, §6] we show that there is a non-symplectic automorphism $\\\\zeta$ of order three on the LLSvS eightfold $Z$ associated to a smooth cyclic cubic fourfold not containing a plane; we can thus define the isometry $\\\\rho _ { Z } \\\\ { \\\\in \\\\ \\\\mathrm { { M o n } } ^ { 2 } } ( L _ { K 3 } \\\\oplus \\\\langle - 6 \\\\rangle )$ associated to his action (starting from a fixed initial example as above) the invariant lattice of $\\\\zeta$ is $\\\\langle 2 \\\\rangle$ and its orthogonal complement is again $s$ inside $L _ { K 3 } \\\\oplus \\\\langle - 6 \\\\rangle$ . Exactly the same theory illustrated in 2.3 following [16] leads to the construction of the moduli space ${ \\\\cal M } _ { \\\\rho _ { Z } , \\\\langle 2 \\\\rangle }$ of $( \\\\rho _ { Z } , \\\\langle 2 \\\\rangle )$ -polarized eightfolds of $K 3 ^ { [ 4 ] }$ -type; in particular, [16, Theorem 1.2] implies the following \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,512,883,549]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 4 Let $\\\\mathcal { H } ^ { \\\\prime }$ be the hyperplane arrangement in $\\\\mathbb { P } ( S \\\\otimes \\\\mathbb { C } )$ consisting of hyperplanes orthogonal to MBM classes in S. The period map \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[373,564,624,587]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0371/8e6dff16679b107355845dfe60157d0d48cd0e8edf9a24e2bed58f082fe634f0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathcal { P } _ { 2 } : M _ { \\\\rho _ { Z } , \\\\langle 2 \\\\rangle } \\\\to \\\\mathbb { B } _ { 1 0 } \\\\setminus \\\\mathcal { H } ^ { \\\\prime \\\\prime }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,602,876,621]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is an isomorphism, equivariant with respect to the action of $\\\\mathbf { M o n } ^ { 2 } ( L _ { K 3 } \\\\oplus \\\\langle - 6 \\\\rangle ) _ { \\\\rho _ { Z } }$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,629,881,756]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It follows from [34] that in dimension eight the MBM classes that can be contained in $s$ are classes $\\\\delta \\\\in S$ either such that $\\\\delta ^ { 2 } = - 2$ or such that $\\\\delta ^ { 2 } = - 6$ or $\\\\delta ^ { 2 } = - 2 4$ and $\\\\mathrm { d i v } = 3$ . As a consequence, $\\\\mathcal { H } ^ { \\\\prime }$ is strictly bigger than $\\\\mathcal { H } _ { \\\\mathrm { ~ ~ } }$ , though this was to be expected because the construction of the LLSvS eighfold is only possible in the open subset of cubic fourfolds not containing a plane. Another difference which arises with the increase in dimension is that the group of isometries induced by monodromy operators is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,770,876,794]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0061/4e263f94e28fb64d45c164706e03ebd86e2a02736cfc49c6f0d37b63eb9f6bf6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf { M o n } ^ { 2 } ( L _ { K 3 } \\\\oplus \\\\langle - 6 \\\\rangle ) = \\\\{ g \\\\in O ( L _ { K 3 } \\\\oplus \\\\langle - 6 \\\\rangle ) | \\\\operatorname { s n } ^ { \\\\mathbb { B } } ( g ) = 1 , \\\\bar { g } = \\\\mathrm { i d } \\\\in O ( q _ { A _ { L _ { K 3 } \\\\oplus \\\\langle - 6 \\\\rangle } } \\\\} ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,809,537,827]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"so a priori it could be of index two inside $\\\\mathbb { P } \\\\Gamma$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[143,828,624,845]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In any case, ideally one obtains a finite rational map \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[302,860,694,884]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0506/e9d7b8fae03d0c7c2d02951f9077fb4365d06d06e4614df11c36466eb2a5f8dd.jpg\",\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left. M _ { \\\\rho _ { Z } , \\\\langle 2 \\\\rangle } / \\\\left. M o n ^ { 2 } ( L _ { K 3 } \\\\oplus \\\\langle - 6 \\\\rangle ) _ { \\\\rho _ { Z } } \\\\right. \\\\right. - \\\\left. C _ { 3 } ^ { s m } . \\\\right.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[792,54,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Camere \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,54,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"32\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,214]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moreover, as it was explained in another talk of this workshop by Lahoz, forthcoming work of Bayer, Lahoz, Macrì, Neuer, Perry and Stellari includes the construction of many other families of higher dimensional polarized manifolds of $K 3 ^ { [ n ] }$ -type associated to cubic fourfolds (see [32, Theorem 5.11]), so it is expected that there is a “tower” of rational maps from moduli spaces of manifolds of $K 3 ^ { [ n ] }$ -type with a non-symplectic automorphism of order three having invariant lattice of rank one to the moduli space of smooth cubic threefolds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,261,334,280]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4.3 Another Case \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,306,881,468]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the recent work [29], the authors use the same techniques of Allcock, Carlson and Toledo in order to study the moduli space $\\\\mathcal { W }$ of pairs $( C , H )$ consisting of a smooth cubic threefold $C \\\\in C _ { 3 } ^ { s m }$ and of a hyperplane $H$ . In particular, if $G =$ 0 is an equation of $C$ and $L \\\\ = \\\\ 0$ is an equation of $H$ , they consider the cubic fourfold $Y$ defined by the equation $G + X _ { 5 } ^ { 2 } L \\\\ = \\\\ 0$ , and they consider the open subset $\\\\mathcal { W } ^ { 0 } \\\\subset \\\\mathcal { W }$ given by pairs for which $Y$ is smooth, so that they can study the restriction of the period map $\\\\mathcal { P } _ { 4 }$ to it. Again, this construction allows to relate the GIT compactification $\\\\bar { \\\\mathcal W }$ of $\\\\mathcal { W } ^ { 0 }$ and the Baily–Borel compactification of the period domain. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,471,881,650]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Again, if one considers the Fano variety of lines $F ( Y )$ , this has a non-symplectic involution $\\\\iota$ induced by $X _ { 5 } ~ \\\\mapsto ~ - X _ { 5 }$ , as observed in [10]. The computations contained in [29] on the transcendental part of $H ^ { 4 } ( Y , \\\\mathbb { Z } )$ , which is isomorphic to $U ^ { \\\\oplus 2 } \\\\oplus D _ { 4 } ^ { \\\\oplus 3 }$ , shed new light on this involution: since $F ( Y )$ is general, by dimension count, in the family carrying this kind of involution, we have that the orthogonal complement of the invariant lattice $M$ of $\\\\iota$ is $N : = U ^ { \\\\oplus 2 } \\\\oplus D _ { 4 } ( - 1 ) ^ { \\\\oplus 3 }$ and that $M$ is a 2-elementary overlattice of $\\\\langle 6 \\\\rangle \\\\oplus E _ { 6 } ( - 2 )$ . A standard computation shows that $M \\\\cong \\\\langle 2 \\\\rangle \\\\oplus \\\\langle - 2 \\\\rangle ^ { \\\\oplus 6 }$ : we remark that in [14, Figure 2] this case is erroneously omitted, though it cannot be realized by a natural automorphism, because $N$ has no primitive embedding in $L _ { K 3 }$ (see [28, Theorem 3.4]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,651,879,776]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Fix a primitive embedding $j _ { M } : M \\\\subset L$ such that $j _ { M } ( M ) ^ { \\\\perp } \\\\cong N$ ; the fourfolds of $K 3 ^ { [ 2 ] }$ -type with such a non-symplectic involution fixing $M$ are exactly all the lattice polarized fourfolds $( X , \\\\eta ) \\\\in \\\\mathcal { M } _ { L }$ such that there exists $i : M \\\\subset H ^ { 2 } ( X , \\\\mathbb { Z } )$ primitive with $\\\\eta \\\\circ i = j _ { M }$ and such that $i ( M ) \\\\cap { \\\\mathcal { K } } _ { X } \\\\neq \\\\emptyset$ , as explained in [26]. We denote by $\\\\textstyle { \\\\mathcal { M } } _ { M } \\\\subset { \\\\mathcal { M } } _ { L }$ the subspace of these lattice-polarized pairs and by $M _ { M , K }$ the connected component containing a pair $( F ( Y ) , \\\\eta )$ with $Y$ smooth as above such that $\\\\operatorname { P i c } ( F ( Y ) ) \\\\cong M$ , where $K : = \\\\eta ( \\\\mathcal { K } _ { F ( Y ) } \\\\cap \\\\operatorname { P i c } ( F ( Y ) ) )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,777,879,828]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Forthcoming work of the author will try to answer the question whether $\\\\mathcal { W }$ and $M _ { M , K }$ are isomorphic, and, if the answer is positive, to check whether the isomorphism extends to the compactifications. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,851,879,909]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acknowledgements The author wants to express her gratitude to S. Boissière and A. Sarti for reading a draft of this survey and to the Max Planck Institute of Mathematics for providing a great working environment. The comments of the anonymous referee, which the author gratefully appreciated, have greatly helped to improve and clarify the exposition. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,571,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moduli Spaces of Cubic Threefolds and of IHS Manifolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"33\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,250,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,138,881,899]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Allcock, D., Carlson, J.A., Toledo, D.: The complex hyperbolic geometry of the moduli space of cubic surfaces. J. Algebraic Geom. 11(4), 659–724 (2002) \\n2. Allcock, D., Carlson, J.A., Toledo, D.: The moduli space of cubic threefolds as a ball quotient. Mem. Am. Math. Soc. 209(985), xi $+ 7 0$ (2011) \\n3. Amerik, E., Verbitsky, M.: Rational curves on hyperkähler manifolds. Int. Math. Res. Not. 2015(23), 13009–13045 (2015) \\n4. Artebani, M., Sarti, A.: Non-symplectic automorphisms of order 3 on $K 3$ surfaces. Math. Ann. 342(4), 903–921 (2008) \\n5. Artebani, M., Sarti, A., Taki, S.: $K 3$ surfaces with non-symplectic automorphisms of prime order. Math. Z. 268(1–2), 507–533 (2011). With an appendix by Shigeyuki Kondo¯ \\n6. Bayer, A., Hassett, B., Tschinkel, Y.: Mori cones of holomorphic symplectic varieties of K3 type. Ann. Sci. Éc. Norm. Supér. (4) 48(4), 941–950 (2015) \\n7. Beauville, A.: Some remarks on Kähler manifolds with $c _ { 1 } = 0$ . In: Classification of Algebraic and Analytic Manifolds. Progress in Mathematics, vol. 39, pp. 1–26. Katata (1982) \\n8. Beauville, A.: Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18(4), 755–782 (1983) \\n9. Beauville, A.: Moduli of cubic surfaces and Hodge theory (after Allcock, Carlson, Toledo). In: Géométries à courbure négative ou nulle, groupes discrets et rigidités. Séminar Congress, vol. 18, pp. 445–466 (2009) \\n10. Beauville, A.: Antisymplectic involutions of holomorphic symplectic manifolds. J. Topol. 4(2), 300–304 (2011) \\n11. Beauville, A., Donagi, R.: La variété des droites d’une hypersurface cubique de dimension 4. C. R. Acad. Sci. Paris Sér. I Math. 301(14), 703–706 (1985) \\n12. Boissière, S.: Automorphismes naturels de l’espace de Douady de points sur une surface. Can. J. Math. 64(1), 3–23 (2012) \\n13. Boissière, S., Nieper-Wißkirchen, M., Sarti, A.: Smith theory and irreducible holomorphic symplectic manifolds. J. Topol. 6(2), 361–390 (2013) \\n14. Boissière, S., Camere, C., Sarti, A.: Classification of automorphisms on a deformation family of hyper-Kähler four-folds by $p$ -elementary lattices. Kyoto J. Math. 56(3), 465–499 (2016) \\n15. Boissière, S., Camere, C., Mongardi, G., Sarti, A.: Isometries of ideal lattices and hyperkähler manifolds. Int. Math. Res. Not. 2016(4), 963–977 (2016) \\n16. Boissière, S., Camere, C., Sarti, A.: Complex ball quotients from manifolds of $K 3 ^ { [ n ] }$ -type. J. Pure Appl. Algebra, 223(3), 1123–1138 (2019). https://doi.org/10.1016/j.jpaa.2018.05.017 \\n17. Boissière, S., Camere, C., Sarti, A.: Cubic threefolds and hyperkähler manifolds uniformized by the 10-dimensional complex ball. Ann. 373(3–4), 1429–1455 (2019). https://doi.org/10. 1007/s00208-018-1766-9 \\n18. Camere, C., Cattaneo, A.: Non-symplectic automorphisms of odd prime order on manifolds of $K 3 ^ { [ n ] }$ -type. Manuscripta Math. (2019). https://doi.org/10.1007/s00229-019-01163-4 \\n19. Camere, C., Kapustka, G., Kapustka, M., Mongardi, G.: Verra four-folds, twisted sheaves, and the last involution. Int. Math. Res. Not. 2019(21), 6661–6710 (2018). https://doi.org/10.1093/ imrn/rnx327 \\n20. Dolgachev, I.V., Kondo, S.: Moduli of ¯ $K 3$ surfaces and complex ball quotients. In: Arithmetic and Geometry Around Hypergeometric Functions. Progress in Mathematics, vol. 260, pp. 43– 100. Birkhäuser, Basel (2007) \\n21. Dolgachev, I., van Geemen, B., Kondo, S.: A complex ball uniformization of the moduli space ¯ of cubic surfaces via periods of $K 3$ surfaces. J. Reine Angew. Math. 588, 99–148 (2005) \\n22. Ferretti, A.: Special subvarieties of EPW sextics. Math. Z. 272(3–4), 1137–1164 (2012) \\n23. Gritsenko, V., Hulek, K., Sankaran, G.K.: Moduli of K3 surfaces and irreducible symplectic manifolds. In: Handbook of Moduli. Vol. I. Advanced Lectures in Mathematics, vol. 24, pp. 459–526. International Press, Somerville (2013) \\n24. Hassett, B.: Special cubic fourfolds. Compos. Math. 120(1), 1–23 (2000) \\n25. Huybrechts, D.: Compact hyper-Kähler manifolds: basic results. Invent. Math. 135(1), 63–113 (1999) \\n26. Joumaah, M.: Non-symplectic involutions of irreducible symplectic manifolds of $K 3 ^ { [ n ] }$ -type. Math. Z. 283(3–4), 761–790 (2016) \\n27. Kudla, S., Rapoport, M.: On occult period maps. Pac. J. Math. 260(2), 565–581 (2012) \\n28. Laza, R.: Maximally algebraic potentially irrational cubic fourfolds (2018). arXiv:1805.04063 \\n29. Laza, R., Pearlstein, G., Zhang, Z.: On the moduli space of pairs consisting of a cubic threefold and a hyperplane. Adv. Math. 340, 684–722 (2018) \\n30. Lehn, C.: Twisted cubics on singular cubic fourfolds—on Starr’s fibration. Math. Z. 290(1), 379–388 (2018) \\n31. Lehn, C., Lehn, M., Sorger, C., van Straten, D.: Twisted cubics on cubic fourfolds. J. Reine Angew. Math. 731, 87–128 (2017) \\n32. Macrì, E., Stellari, P.: Lectures on non-commutative K3 surfaces, Bridgeland stability, and moduli spaces. In: Birational Geometry of Hypersurfaces. Lecture Notes of the Unione Matematica Italiana. Springer, Berlin (2019). arXiv:1807.06169 \\n33. Markman, E.: A survey of Torelli and monodromy results for holomorphic-symplectic varieties. In: Complex and Differential Geometry. Springer Proceedings in Mathematics, vol. 8, pp. 257–322 (2011) \\n34. Mongardi, G.: A note on the Kähler and Mori cones of hyperkähler manifolds. Asian J. Math. 19(4), 583–591 (2015) \\n35. Mongardi, G., Wandel, M.: Induced automorphisms on irreducible symplectic manifolds. J. Lond. Math. Soc. (2) 92(1), 123–143 (2015) \\n36. Nikulin, V.V.: Integral symmetric bilinear forms and some of their applications. Math. USSR Izv. 14, 103–167 (1980) \\n37. O’Grady, K.G.: Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics. Duke Math. J. 134(1), 99–137 (2006) \\n38. Ohashi, H., Wandel, M.: Non-natural non-symplectic involutions on symplectic manifolds of $K 3 ^ { [ 2 ] }$ -type (2013). arXiv:1305.6353 \\n39. Tari, K.: Automorphismes des variétés de Kummer généralisées, Ph.D. thesis, Université de Poitiers, 2015 \\n40. Verbitsky, M.: Mapping class group and a global torelli theorem for hyperkähler manifolds. Duke Math. J. 162(15), 2929–2986 (2013). https://doi.org/10.1215/00127094-2382680 \\n41. Yamagishi, R.: Singularities of Fano varieties of lines on singular cubic fourfolds (2017). arXiv:1712.03546 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[792,54,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Camere \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,54,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"34\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[116,87,881,605]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,571,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moduli Spaces of Cubic Threefolds and of IHS Manifolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"35\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,669,141]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Note on Severi Varieties of Nodal Curves on Enriques Surfaces \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,175,742,211]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ciro Ciliberto, Thomas Dedieu, Concettina Galati, and Andreas Leopold Knutsen \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,286,881,430]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract Let $| L |$ be a linear system on a smooth complex Enriques surface $s$ whose general member is a smooth and irreducible curve of genus $p$ , with $L ^ { 2 } > 0$ , and let $V _ { | L | , \\\\delta } ( S )$ be the Severi variety of irreducible $\\\\delta$ -nodal curves in $| L |$ . We denote by $\\\\pi : X S$ the universal covering of $s$ . In this note we compute the dimensions of the irreducible components $V$ of $V _ { | L | , \\\\delta } ( S )$ . In particular we prove that, if $C$ is the curve corresponding to a general element $[ C ]$ of $V$ , then the codimension of $V$ in $| L |$ is $\\\\delta$ if $\\\\pi ^ { - 1 } ( C )$ is irreducible in $X$ and it is $\\\\delta - 1$ if $\\\\pi ^ { - 1 } ( C )$ consists of two irreducible components. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,450,792,466]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords Severi varieties $\\\\cdot$ Moduli $\\\\cdot$ Nodal curves $\\\\cdot$ Enriques surfaces \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,487,728,506]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2010 Mathematics Subject Classification 14H20, 14H10, 14J28 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,897,692,941]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_3 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[804,91,876,139]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,834,649,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. L. Knutsen Department of Mathematics, University of Bergen, Bergen, Norway e-mail: andreas.knutsen@math.uib.no \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,648,699,690]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Ciliberto Dipartimento di Matematica, Università di Roma Tor Vergata, Roma, Italy e-mail: cilibert@axp.mat.uniroma2.it \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,768,867,825]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Galati $\\\\bigstar$ ) Dipartimento di Matematica e Informatica, Università della Calabria, Arcavacata di Rende (CS), Italy e-mail: galati@mat.unical.it \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,701,856,758]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"T. Dedieu Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS IMT, Toulouse, France e-mail: thomas.dedieu@math.univ-toulouse.fr \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,899,876,909]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"36\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,312,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,133,881,186]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $s$ be a smooth complex projective surface and $L$ a line bundle on $s$ such that the complete linear system $| L |$ contains smooth, irreducible curves (such a line bundle, or linear system, is often called a Bertini system). Let \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[334,202,660,238]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0418/5cb3f1506ca78496fb8afbf95dd857eeae6f02832c660a8112e588861c9393f3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\np : = p _ { a } ( L ) = \\\\frac { 1 } { 2 } L \\\\cdot ( L + K _ { S } ) + 1 ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,253,514,270]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"be the arithmetic genus of any curve in $| L |$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,271,876,307]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For any integer $0 ~ \\\\le ~ \\\\delta ~ \\\\le ~ p$ , consider the locally closed, functorially defined subscheme of $| L |$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[375,324,624,345]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0357/ab71e43db2d15c7057a95ca1386c99f61257397280bee695140da81ed312c141.jpg\",\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nV _ { | L | , \\\\delta } ( S ) \\\\mathrm { o r s i m p l y } V _ { | L | , \\\\delta }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,360,881,414]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"parameterizing irreducible curves in $| L |$ having only $\\\\delta$ nodes as singularities; this is called the Severi variety of $\\\\delta$ -nodal curves in $| L |$ . We will let $g : = p - \\\\delta$ , the geometric genus of the curves in $V _ { | L | , \\\\delta }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,415,879,468]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is well-known that, if $V _ { | L | , \\\\delta }$ is non-empty, then all of its irreducible components $V$ have dimension $\\\\mathrm { d i m } ( V ) \\\\geq \\\\mathrm { d i m } | L | - \\\\delta .$ . More precisely, the Zariski tangent space to $V _ { | L | , \\\\delta }$ at the point corresponding to $C$ is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[330,484,667,509]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0010/8bb758794d0ee67e3dcbb90dd9a46c92e50520d44f96cecbd8dd4fbc81455b20.jpg\",\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nT _ { [ C ] } V _ { | L | , \\\\delta } \\\\simeq H ^ { 0 } ( L \\\\otimes \\\\mathcal { T } _ { N } ) / < C > ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,524,881,632]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\mathcal { T } _ { N } = \\\\mathcal { T } _ { N | S }$ is the ideal sheaf of subscheme $N$ of $s$ consisting of the $\\\\delta$ nodes of $C$ (see, e.g., [4, $\\\\ S 1 ]$ ). Thus, $V _ { | L | , \\\\delta }$ is smooth of dimension dim $| L | - \\\\delta$ at $[ C ]$ if and only if the set of nodes $N$ imposes independent conditions on $| L |$ . In this case, $V _ { | L | , \\\\delta }$ is said to be regular at $[ C ]$ . An irreducible component $V$ of $V _ { | L | , \\\\delta }$ will be said to be regular if the condition of regularity is satisfied at any of its points, equivalently, if it is smooth of dimension $\\\\dim | L | - \\\\delta$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,633,881,794]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The existence and regularity problems of $V _ { | L | , \\\\delta } ( S )$ have been studied in many cases and are the most basic problems one may ask on Severi varieties. We only mention some of known results. In the case $S \\\\simeq \\\\mathbb { P } ^ { 2 }$ , Severi proved the existence and regularity of $V _ { | L | , \\\\delta } ( S )$ in [13]. The description of the tangent space is due to Severi and later to Zariski [15]. The existence and regularity of $V _ { | L | , \\\\delta } ( S )$ when $s$ is of general type has been studied in [4] and [3]. Further regularity results are provided in [10]. More recently Severi varieties on K3 surfaces have received a lot of attention for many reasons. In this case Severi varieties are known to be regular (cf. [14]) and are nonempty on general K3 surfaces by Mumford and Chen (cf. [2, 12]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,795,881,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As far as we know, Severi varieties on Enriques surfaces have not been studied yet, apart from [8, Thm. 4.12] which limits the singularities of a general member of the Severi variety $V _ { | L | } ^ { g }$ of irreducible genus $g$ curves in $| L |$ , and gives a sufficient condition for the density of the latter in the Severi variety $V _ { | L | , p - g }$ of $( p - g )$ -nodal curves. In particular, the existence problem is mainly open and we intend to treat it in a forthcoming article. The result of this paper is Proposition 1, which answers the regularity question for Severi varieties of nodal curves on Enriques surfaces. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[740,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Ciliberto et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"37\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"30 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,763,111]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 Regularity of Severi Varieties on Enriques Surfaces \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,133,879,186]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $s$ be a smooth Enriques surface, i.e. a smooth complex surface with nontrivial canonical bundle $\\\\omega _ { S } \\\\not \\\\cong O _ { S }$ , such that $\\\\omega _ { S } ^ { \\\\otimes 2 } \\\\simeq { \\\\cal O } _ { S }$ and $H ^ { 1 } ( O _ { S } ) = 0$ . We denote linear (resp. numerical) equivalence by $\\\\sim ( \\\\mathrm { r e s p . } \\\\equiv )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,187,881,331]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $L$ be a line bundle on $s$ such that $L ^ { 2 } \\\\ > \\\\ 0$ . It is well-known that $| L |$ contains smooth, irreducible curves if and only if it contains irreducible curves (see [5, Thm. 4.1 and Prop. 8.2]); in other words, on Enriques surfaces the Bertini linear systems are the linear systems that contain irreducible curves. Moreover, by [6, Prop. 2.4], this is equivalent to $L$ being nef and not of the form $L \\\\sim P + R$ , with $| P |$ an elliptic pencil and $R$ a smooth rational curve such that $P \\\\cdot R = 2$ (in which case $p = 2 ,$ ). If $| L |$ is a Bertini linear system, the adjunction formula, the Riemann– Roch theorem, and Mumford vanishing yield that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[318,345,678,369]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0092/d3d819fb5bdb8cac17274936d8635a2afe56709562c7efaa8707d4787ad7f0a6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nL ^ { 2 } = 2 ( p - 1 ) \\\\mathrm { a n d } \\\\mathrm { d i m } | L | = p - 1\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,385,275,402]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(see, e.g., [5, 7]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[143,403,728,421]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $K _ { S }$ be the canonical divisor. It defines an étale double cover \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[435,438,562,456]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"1019/5d1003755db4fd418dbb82d15fb62f642d48daea415321f12be5728f18310cb9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\pi : X \\\\longrightarrow S\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,474,881,546]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $X$ is a smooth, projective $K 3$ surface (that is, $\\\\omega _ { X } \\\\simeq O _ { X }$ and $H ^ { 1 } ( O _ { X } ) = 0 ;$ ), endowed with a fixed-point-free involution $\\\\iota$ , which is the universal covering of $s$ . Conversely, the quotient of any- $K 3$ surface by a fixed-point-free involution is an Enriques surface. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,548,881,620]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $C \\\\subset S$ be a reduced and irreducible curve of genus $g \\\\ge 2$ . We will henceforth denote by $\\\\nu _ { C } : { \\\\widetilde { C } } \\\\to C$ the normalization of $C$ and define $\\\\eta _ { \\\\cal C } : = { \\\\cal O } _ { \\\\cal C } ( K _ { \\\\cal S } ) =$ $O _ { C } ( - K _ { S } )$ , a nontrivial 2-torsion element in $\\\\operatorname { P i c } ^ { 0 } C$ , and $\\\\eta _ { \\\\widetilde { C } } : = \\\\nu _ { C } ^ { * } \\\\eta _ { C }$ . The fact that $\\\\eta _ { C }$ is nontrivial follows from the cohomology of the restriction sequence \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[232,636,758,656]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0746/e008f4d9bc83d24c3fec86e6de875445b9a2d2e770bb4b5a1388f6b8a1a8092b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n0 \\\\longrightarrow O _ { S } ( K _ { S } - C ) \\\\longrightarrow O _ { S } ( K _ { S } ) \\\\longrightarrow \\\\eta _ { C } \\\\longrightarrow 0 ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,677,881,714]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"which yields $h ^ { 0 } ( \\\\eta _ { C } ) = h ^ { 1 } ( K _ { S } - C ) = h ^ { 1 } ( C ) = 0$ , the latter vanishing as $C$ is big and nef. One has the fiber product \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[382,729,615,825]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0528/b3df77c5d4ed74a669184ebace98cdba893f2d192aed18400449e36787d1a32f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l r } & { } & { ( \\\\pi ^ { - 1 } C ) \\\\times _ { C } \\\\widetilde { C } \\\\longrightarrow \\\\widetilde { C } } \\\\\\\\ & { } & { \\\\downarrow } \\\\\\\\ & { } & { ( \\\\pi ^ { - 1 } C ) \\\\xrightarrow [ ] { } { \\\\times _ { C } \\\\widetilde { C } } \\\\longrightarrow { \\\\cal C } , } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,52,412,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Severi Varieties on Enriques Surfaces \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"38\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[612,88,876,539]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 1 $\\\\\\\\eta _ { \\\\\\\\tilde { C } } = \\\\\\\\nu _ { C } ^ { * } ( \\\\\\\\eta _ { C } ) \\\\\\\\neq 0$ \\\",\\\"Fig. 2 $\\\\\\\\eta _ { \\\\\\\\tilde { C } } = \\\\\\\\nu _ { C } ^ { * } ( \\\\\\\\eta _ { C } ) = 0$ \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0534/faac3f0347d90c1ac08b8026dbe25ba5e2be076d355ba4c2ab0c965008a9b639.jpg\",\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[118,567,881,623]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where π| 1 and the upper horizontal map are the double coverings induced respectively by- - $\\\\eta _ { C }$ and $\\\\eta _ { \\\\widetilde { C } }$ . By standard results on coverings of complex manifolds (cf. [1, Sect. I.17]), two cases may happen:- - \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,630,881,704]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"• $\\\\eta _ { \\\\widetilde { C } } \\\\nsimeq O _ { \\\\widetilde { C } }$ and $\\\\pi ^ { - 1 } C$ is irreducible, as in Fig. 1; $\\\\eta _ { \\\\widetilde { C } } \\\\simeq O _ { \\\\widetilde { C } }$ and $\\\\pi ^ { - 1 } C$ consists of two irreducible components conjugated by the involution $\\\\iota$ . These two components are not isomorphic to $C$ , as $\\\\eta _ { C }$ is nontrivial, as in Fig. 2 (each component of $\\\\tilde { C }$ is a partial normalization of $C$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,713,881,785]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As mentioned in the Introduction, it is well-known that any irreducible component of a Severi variety on a $K 3$ surface is regular when nonempty (see, e.g., [4, Ex. 1.3]; see also [8, $\\\\ S 4 . 2 ]$ ). The corresponding result on Enriques surfaces is the following. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,785,879,821]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"First note that, in the above notation, the dimension of the Severi variety of genus\\u0003 \\u0004 $g = p _ { g } ( C )$ curves in $| L | = | C |$ at the point $[ C ]$ satisfies the inequalities \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[179,837,767,888]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0002/c245b900dc7bed6c46ab74675404cfb34f8c79576bf895ba0f1e6a34ecf380e7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\ng - 1 \\\\leq \\\\dim _ { { [ C ] } } \\\\bigl ( V _ { | L | } ^ { g } \\\\bigr ) \\\\leq h ^ { 0 } ( \\\\omega _ { \\\\tilde { C } } \\\\otimes \\\\eta _ { \\\\tilde { C } } ) = \\\\left\\\\{ g - 1 \\\\begin{array} { l l } { \\\\mathrm { i f } \\\\eta _ { \\\\tilde { C } } \\\\not \\\\simeq O _ { \\\\tilde { C } } } \\\\\\\\ { g } & { \\\\mathrm { i f } \\\\eta _ { \\\\tilde { C } } \\\\simeq O _ { \\\\tilde { C } } } \\\\end{array} \\\\right.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,54,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[740,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"39\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Ciliberto et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,142]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(see [8, ineq. (2.6) and Lem. 2.3]). Our result implies that the second inequality in (3) is in fact an equality when $C$ is nodal, and gives a concrete geometric description of the situation in both cases. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,150,881,225]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 1 Let $L$ be a Bertini linear system, with $L ^ { 2 } > 0$ , on a smooth Enriques surface $S$ . Then the Severi variety $V _ { | L | , \\\\delta } ( S )$ is smooth and every irreducible component $V \\\\subseteq V _ { | L | , \\\\delta } ( S )$ has either dimension $g \\\\mathrm { ~ - ~ } 1$ or $g$ ; in the former case the component is regular. Furthermore, with the notation introduced above, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,232,876,270]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. for any curve $C$ in a $( g - 1 )$ -dimensional irreducible component $V$ , $\\\\pi ^ { - 1 } C$ is irreducible (whence an element in $V _ { | \\\\pi ^ { * } L | , 2 \\\\delta } ( X ) )$ ; \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,270,881,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. for any $g$ -dimensional component $V$ , there is a line bundle $L ^ { \\\\prime }$ on $X$ with $( L ^ { \\\\prime } ) ^ { 2 } =$ $2 ( p - d ) - 2$ and $L ^ { \\\\prime } \\\\cdot \\\\iota ^ { * } L ^ { \\\\prime } = 2 d$ for some integer $d$ satisfying \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[432,319,587,358]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0288/2241f5f8760b7d9848a9fe07a9a32b6838ae862918284b550be5e29695b24063.jpg\",\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\frac { p - 1 } { 2 } \\\\leq d \\\\leq \\\\delta ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,370,879,462]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"such that $\\\\pi ^ { * } L \\\\simeq L ^ { \\\\prime } \\\\otimes \\\\iota ^ { * } L ^ { \\\\prime }$ , and the curves parametrized by $V \\\\subseteq V _ { | L | , \\\\delta } ( S )$ are the birational images by $\\\\pi$ of the curves in $V _ { | L ^ { \\\\prime } | , \\\\delta - d } ( X )$ intersecting their conjugates by $\\\\iota$ transversely (in 2d points). In other words, for any $[ C ] \\\\in V$ , we have $\\\\pi ^ { - 1 } C = Y + \\\\iota ( Y )$ , with $[ Y ] \\\\in V _ { | L ^ { \\\\prime } | , \\\\delta - d } ( X )$ and $[ \\\\iota ( Y ) ] \\\\in V _ { | \\\\iota ^ { * } L ^ { \\\\prime } | , \\\\delta - d } ( X )$ intersecting transversely. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,462,876,500]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Furthermore, if $L ^ { \\\\prime } \\\\simeq \\\\iota ^ { * } L ^ { \\\\prime }$ , which is the case if $s$ is general in moduli, then $\\\\begin{array} { r } { d = \\\\frac { p - 1 } { 2 } } \\\\end{array}$ and $L \\\\sim 2 M$ , for some $M \\\\in \\\\operatorname { P i c } S$ such that $M ^ { 2 } = d$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,507,881,615]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We will henceforth refer to components of dimension $g - 1$ as regular and the ones of dimension $g$ as nonregular. Note however that from a parametric perspective the Severi variety has the expected dimension and is smooth in both cases, as the fact that (3) is an equality indicates; we do not dwell on this here, and refer to [8] for a discussion of the differences between the parametric and Cartesian points of view (the latter is the one we adopted in this text). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,617,881,669]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note that Proposition 1 does not assert that the Severi variety $V _ { | L | , \\\\delta }$ is necessarilynon-empty: in such a situation, $V _ { | L | , \\\\delta }$ does not have any irreducible component and the statement is empty. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,677,881,734]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Pick any curve $C$ in an irreducible component $V$ of $V _ { | L | , \\\\delta } ( S )$ . Let $f : \\\\widetilde { S } S$ be the blow-up of $s$ at $N$ , the scheme of the δ nodes of - $C$ , denote by e the (total) exceptional divisor and by - $\\\\widetilde { C }$ the strict transform of $C$ . Thus $f _ { | \\\\widetilde { C } } = \\\\nu _ { C }$ and we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[318,749,678,773]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0339/0caaee88fbc3f402e816aa356fb2677a6f93be4fdf90e2615f11532335421a62.jpg\",\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { \\\\widetilde { S } } \\\\sim f ^ { * } K _ { S } + \\\\mathfrak { e } \\\\mathrm { ~ a n d ~ } \\\\widetilde { C } \\\\sim f ^ { * } C - 2 \\\\mathfrak { e } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,791,394,807]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From the restriction sequence \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[232,822,763,849]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0577/e2ee572338439d2dcdbe19fdf8f41126cd202252c9bfaeb4632a0d1cfc8b2b7a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n0 \\\\longrightarrow { \\\\cal O } _ { \\\\widetilde { S } } ( \\\\mathfrak { c } ) \\\\longrightarrow { \\\\cal O } _ { \\\\widetilde { S } } ( \\\\widetilde { C } + \\\\mathfrak { c } ) \\\\longrightarrow \\\\omega _ { \\\\widetilde { C } } ( \\\\eta _ { \\\\widetilde { C } } ) \\\\longrightarrow 0 \\\\longrightarrow 0 \\\\longrightarrow 0 \\\\longrightarrow 0 .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,52,412,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Severi Varieties on Enriques Surfaces \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"40\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,191,106]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"we find \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[150,121,849,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0063/85f1c7aba00ced4bd9583d643d0931cf6b442681bb43488d29eb54e13dccfd9c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\dim T _ { [ C ] } V _ { | L | , \\\\delta } ( S ) = \\\\dim | L \\\\otimes \\\\mathcal { T } _ { N } | = h ^ { 0 } ( L \\\\otimes \\\\mathcal { T } _ { N } ) - 1 = h ^ { 0 } ( f ^ { * } L - \\\\mathfrak { e } ) - 1 } \\\\\\\\ & { \\\\qquad = h ^ { 0 } ( O _ { \\\\widetilde { S } } ( \\\\widetilde { C } + \\\\mathfrak { e } ) ) - 1 = h ^ { 0 } ( \\\\omega _ { \\\\widetilde { C } } ( \\\\eta _ { \\\\widetilde { C } } ) ) } \\\\\\\\ & { \\\\qquad = \\\\left\\\\{ \\\\begin{array} { l l } { g - 1 , } & { \\\\mathrm { i f ~ } \\\\eta _ { \\\\widetilde { C } } \\\\not \\\\cong O _ { \\\\widetilde { C } } , } \\\\\\\\ { g , } & { \\\\mathrm { i f ~ } \\\\eta _ { \\\\widetilde { C } } \\\\simeq O _ { \\\\widetilde { C } } . } \\\\end{array} \\\\right. } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[123,244,879,280]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the upper case, by (1), we have that $V _ { | L | , \\\\delta }$ is smooth at $[ C ]$ of dimension $g - 1 = p - \\\\delta - 1 = \\\\dim | L \\\\otimes \\\\mathcal { T } _ { N } |$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,282,881,426]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Assume next that we are in the lower case. Then, by the discussion prior to the proposition, we have $\\\\pi ^ { - 1 } C = Y + \\\\iota ( Y )$ for an irreducible curve $Y$ on $X$ , such that $\\\\pi$ maps both $Y$ and $\\\\iota ( Y )$ birationally, but not isomorphically, to $C$ . In particular, $Y$ and $\\\\iota ( Y )$ have geometric genus $p _ { g } ( Y ) = p _ { g } ( \\\\iota ( Y ) ) = p _ { g } ( C ) = p - \\\\delta = g$ . Set $L ^ { \\\\prime } : =$ $O _ { X } ( Y )$ and $2 d : = Y \\\\cdot \\\\iota ( Y )$ . Note that $d$ is an integer because, if $y = \\\\iota ( x ) \\\\in Y \\\\cap \\\\iota ( Y )$ , then $\\\\iota ( y ) = x \\\\in Y \\\\cap \\\\iota ( Y )$ . Since $Y \\\\simeq \\\\iota ( Y )$ and $\\\\pi$ is étale, both $Y$ and $\\\\iota ( Y )$ are nodal with $\\\\delta - d$ nodes and they intersect transversely at $2 d$ points, which are pairwise conjugate by $\\\\iota$ , and therefore map to $d$ nodes of $C$ . Hence $d \\\\leq \\\\delta$ . We have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[223,442,774,462]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0892/5edff3eaff0bbee0d27868c9384c5acc798c5203f477f482200070673a1fff53.jpg\",\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\np _ { a } ( Y ) = p _ { a } ( \\\\iota ( Y ) ) = g + \\\\delta - d = p - \\\\delta + \\\\delta - d = p - d ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,480,193,496]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"whence \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[391,512,608,534]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0943/aa64b6913dfda5f8ba5c87ace187ea3c7acaf2f6b9542199447597c63dadf740.jpg\",\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n( L ^ { \\\\prime } ) ^ { 2 } = 2 ( p - 1 - d ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,552,471,569]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By the Hodge index theorem, we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[198,584,797,620]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0017/ca4ccb7c9352563f8a82820575e165e78b8fb93f19d59f699d6f9053c44ec009.jpg\",\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n4 ( p - 1 - d ) ^ { 2 } = \\\\left( ( L ^ { \\\\prime } ) ^ { 2 } \\\\right) ^ { 2 } = ( L ^ { \\\\prime } ) ^ { 2 } ( \\\\iota ^ { * } L ^ { \\\\prime } ) ^ { 2 } \\\\leq \\\\left( L ^ { \\\\prime } \\\\cdot \\\\iota ^ { * } L ^ { \\\\prime } \\\\right) ^ { 2 } = 4 d ^ { 2 } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,635,309,651]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"whence $p - 1 \\\\leq 2 d$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,651,881,761]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By the regularity of Severi varieties on $K 3$ surfaces, any irreducible component of $V _ { | L ^ { \\\\prime } | , \\\\delta - d } ( X )$ has dimension $\\\\dim | L ^ { \\\\prime } | - ( \\\\delta - d ) \\\\ : = \\\\ : p _ { g } ( Y ) \\\\ : = \\\\ : g$ . Hence, $V$ is $g$ -dimensional; more precisely, the curves parameterized by $V$ are the (birational) images by $\\\\pi$ of the curves in an irreducible component of $V _ { | L ^ { \\\\prime } | , \\\\delta - d } ( X )$ intersecting their conjugates by $\\\\iota$ transversely (in $2 d$ points). By (4), it also follows that dim $V =$ dim $T _ { [ C ] } V _ { | L | , \\\\delta } ( S )$ , so that $[ C ]$ is a smooth point of $V _ { | L | , \\\\delta } ( S )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,761,881,851]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To prove the final assertion of the proposition, observe that, by the regularity of Severi varieties on $K 3$ surfaces, we may deform $Y$ and $\\\\iota ( Y )$ on $X$ to irreducible curves $Y ^ { \\\\prime }$ and $\\\\iota ( Y ^ { \\\\prime } )$ with any number of nodes $\\\\leq \\\\delta - d$ and intersecting transversally in $2 d$ points; in particular, we may deform $Y$ and $\\\\iota ( Y )$ to smooth curves $Y ^ { \\\\prime }$ and $\\\\iota ( Y ^ { \\\\prime } )$ . Thus, $C ^ { \\\\prime } : = \\\\pi ( Y ^ { \\\\prime } )$ is a member of $V _ { | L | , d }$ , whence of geometric genus $p - d$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"34 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[740,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"41\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Ciliberto et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since d $\\\\operatorname* { l i m } | Y ^ { \\\\prime } | = p _ { a } ( Y ^ { \\\\prime } ) = p _ { g } ( C ^ { \\\\prime } ) = p _ { a } ( C ^ { \\\\prime } ) - d = p - d$ , the component of $V _ { | L | , d }$ containing $[ C ^ { \\\\prime } ]$ has dimension $\\\\mathrm { d i m } | L | - d + 1 = p - d$ . We thus have di $\\\\mathrm { m } | { \\\\cal L } \\\\otimes { \\\\mathcal { T } } _ { N ^ { \\\\prime } } | = \\\\dim | { \\\\cal L } | - d + 1$ , where $N ^ { \\\\prime }$ is the set of $d$ nodes of $C ^ { \\\\prime }$ , hence $N ^ { \\\\prime }$ does not impose independent conditions on $| L |$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,163,881,270]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Assume now that $L ^ { \\\\prime } \\\\simeq \\\\iota ^ { * } L ^ { \\\\prime }$ , which—as is well-known (see, e.g., [9, $\\\\ S 1 1 ]$ )—is the case occurring for generic $s$ , as then Pic $X$ is precisely the invariant part under $\\\\iota$ of $H _ { 2 } ( X , \\\\mathbb { Z } )$ . Then $2 d = L ^ { \\\\prime } \\\\cdot \\\\iota ^ { * } L ^ { \\\\prime } = ( L ^ { \\\\prime } ) ^ { 2 } = 2 ( p - 1 - d )$ , so that $p - 1 = 2 d$ . Since $L ^ { 2 } = 2 ( p - 1 ) = 4 d$ and $N ^ { \\\\prime }$ does not impose independent conditions on $| L |$ , by [11, Prop. 3.7] there is an effective divisor $D \\\\subset S$ containing $N ^ { \\\\prime }$ satisfying $L - 2 D \\\\geq 0$ and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[341,285,656,322]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0983/f35e67c5f67d551b8cf4ab47a5ddbbbfef7837e5cb2b7ff86a6bc3f43b391b41.jpg\",\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nL \\\\cdot D - d \\\\leq D ^ { 2 } \\\\leq \\\\frac { ( i ) } { 2 } L \\\\cdot D \\\\stackrel { ( i i ) } { \\\\leq } d ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,334,879,408]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"with equality in (i) or (ii) only if $L \\\\equiv 2 D$ ; moreover, since $L - 2 D \\\\ge 0$ , the numerical equivalence $L \\\\equiv 2 D$ implies the linear equivalence $L \\\\sim 2 D$ . Now since $N ^ { \\\\prime } \\\\subset D$ , we must have $L \\\\cdot D = C ^ { \\\\prime } \\\\cdot D \\\\geq 2 d$ , hence the inequalities in (6) are all equalities, and thus $D ^ { 2 } = d$ and $L \\\\sim 2 D$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,415,881,469]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The following corollary is a straightforward consequence of Prop. 1 and the fact that the nodes on curves in a regular component in a Severi variety (on any surface and in particular on a K3 surface) can be independently smoothened. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,480,879,534]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 1 If a Severi variety $V _ { | L | , \\\\delta }$ on an Enriques surface has a regular (resp., nonregular) component, then for any $0 \\\\leq \\\\delta ^ { \\\\prime } \\\\leq \\\\delta$ (resp., $d \\\\leq \\\\delta ^ { \\\\prime } \\\\leq \\\\delta$ , with $d$ as in Prop. 1), also $V _ { | L | , \\\\delta ^ { \\\\prime } }$ contains a regular (resp., nonregular) component. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,581,248,599]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[127,621,881,893]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Barth, W., Hulek, K., Peters, C., van de Ven, A.: Compact Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3rd edn., Folge, vol. 4. Springer, Berlin (2004) \\n2. Chen, X.: Rational curves on $K 3$ surfaces. J. Algebraic Geom. 8, 245–278 (1999) \\n3. Chiantini, L., Ciliberto, C.: On the Severi varieties of surfaces in $\\\\mathbb { P } ^ { 3 }$ . J. Algebraic Geom. 8, 67–83 (1999) \\n4. Chiantini, L., Sernesi, E.: Nodal curves on surfaces of general type. Math. Ann. 307, 41–56 (1997) \\n5. Cossec, F.R.: Projective models of Enriques surfaces. Math. Ann. 265, 283–334 (1983) \\n6. Cossec, F.R.: On the Picard group of Enriques surfaces. Math. Ann. 271, 577–600 (1985) \\n7. Cossec, F.R., Dolgachev, I.V.: Enriques Surfaces. I. Progress in Mathematics, vol. 76. Birkhäuser, Boston (1989) \\n8. Dedieu, T., Sernesi, E.: Equigeneric and equisingular families of curves on surfaces. Publ. Mat. 61, 175–212 (2017) \\n9. Dolgachev, I.V., Kondo, S.: Moduli of $K 3$ surfaces and complex ball quotients. In: Arithmetic and Geometry Around Hypergeometric Functions. Progress in Mathematics, vol. 260, pp. 43– 100. Birkhäuser, Basel (2007) \\n10. Flamini, F.: Some results of regularity for Severi varieties of projective surfaces. Commun. Algebra 29, 2297–2311 (2001) \\n11. Knutsen, A.L.: On kth-order embeddings of $K 3$ surfaces and Enriques surfaces. Manuscr. Math. 104, 211–237 (2001) \\n12. Mori, S., Mukai, S.: The uniruledness of the moduli space of curves of genus 11. In: Proceedings of Algebraic Geometry. Tokyo/Kyoto. Lecture Notes in Mathematics, vol. 1016, pp. 334–353. Springer, Berlin (1983) \\n13. Severi, F., Vorlesungen über algebraische Geometrie. Teubner, Leipzig (1921) \\n14. Tannenbaum, A.: Families of curves with nodes on K3 surfaces. Math. Ann. 260, 239–253 (1982) \\n15. Zariski, O.: Dimension theoretic characterization of maximal irreducible sistems of plane nodal curves. Am. J. Math. 104, 209–226 (1982) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,412,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Severi Varieties on Enriques Surfaces \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"42\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,91,881,240]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"43\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,54,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"43\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[740,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"43\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Ciliberto et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,744,139]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Travel Guide to the Canonical Bundle Formula \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,175,444,193]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Enrica Floris and Vladimir Lazic´ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,291,879,325]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract We survey known results on the canonical bundle formula and its applications in algebraic geometry. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,345,876,361]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords Canonical bundle formula $\\\\cdot$ Moduli divisor $\\\\cdot$ Minimal Model Program \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,399,309,420]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,444,881,496]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Minimal Model Program (MMP) predicts that every projective pair with mild singularities is birationally built out of three classes of pairs: those whose log canonical classes are ample, numerically trivial or anti-ample. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,498,879,588]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"More precisely, let $( X , \\\\Delta )$ be a log canonical pair. Then there should exist a birational contraction $\\\\varphi \\\\colon ( X , \\\\Delta ) \\\\quad \\\\longrightarrow \\\\quad ( X _ { \\\\mathrm { m i n } } , \\\\Delta _ { \\\\mathrm { m i n } } )$ together with a fibration $f \\\\colon ( X _ { \\\\mathrm { m i n } } , \\\\Delta _ { \\\\mathrm { m i n } } ) \\\\ \\\\ X _ { \\\\mathrm { c a n } }$ so that $K _ { X _ { \\\\operatorname* { m i n } } } + \\\\Delta _ { \\\\operatorname* { m i n } } \\\\sim _ { \\\\mathbb { Q } } f ^ { * } A$ , for a suitable ample $\\\\mathbb { Q }$ -divisor $A$ on $X _ { \\\\mathrm { { c a n } } }$ . Note that the Iitaka dimension of $K _ { X } + \\\\Delta$ restricted to a general fibre of the composed map $f \\\\circ \\\\varphi$ is zero. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,588,876,623]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is a natural and important question to determine whether singularities of the MMP are preserved in this process. The singularities of $( X _ { \\\\mathrm { m i n } } , \\\\Delta _ { \\\\mathrm { m i n } } )$ are the same as those of $( X , \\\\Delta )$ . On the other hand, it remains an open problem whether there exists a boundary divisor $\\\\Delta _ { \\\\mathrm { { c a n } } }$ on $X _ { \\\\mathrm { { c a n } } }$ such that $( X _ { \\\\mathrm { { c a n } } } , \\\\Delta _ { \\\\mathrm { { c a n } } } )$ is log canonical and $K _ { X _ { \\\\mathrm { m i n } } } + \\\\Delta _ { \\\\mathrm { m i n } } \\\\sim _ { \\\\mathbb { Q } } f ^ { * } ( X _ { \\\\mathrm { c a n } } + \\\\Delta _ { \\\\mathrm { c a n } } ) ;$ ; in other words, whether singularities of $( X , \\\\Delta )$ descend to the canonical model $X _ { \\\\mathrm { { c a n } } }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,897,690,941]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_4 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,768,815,825]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Floris $( \\\\boxtimes )$ 0 Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348, Futuroscope Chasseneuil Cedex, France e-mail: enrica.floris@univ-poitiers.fr \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[804,91,876,139]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,836,724,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"V. Lazic´ Fachrichtung Mathematik, Universität des Saarlandes, Saarbrücken, Germany e-mail: lazic@math.uni-sb.de \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,675,881,750]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lazic was supported by the DFG-Emmy-Noether-Nachwuchsgruppe “Gute Strukturen in der ´ höherdimensionalen birationalen Geometrie”. During the initial stage of this project Floris was funded by the Max Planck Institute for Mathematics in Bonn. We would like to thank A. Golota for pointing out the reference [17] and for useful conversations and the referee for his/her careful reading. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,899,876,909]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"44\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,879,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,162,881,270]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"When the singularities of $( X , \\\\Delta )$ are klt, it is known by a work of Ambro and Kawamata that such a divisor exists; this result has already had numerous consequences in birational geometry. However, the proof is not constructive: more precisely, one loses control of the coefficients of $\\\\Delta$ at the last step. It is desirable that the singularities of $( X _ { \\\\mathrm { { c a n } } } , \\\\Delta _ { \\\\mathrm { { c a n } } } )$ reflect in a canonical way the singularities of $( X , \\\\Delta )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[143,270,610,286]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In general, with notation as above, it is known that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[353,306,640,325]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"1018/93228e2e75606fecb04b90befe10517ae58ef01df6bd9a23f71ddd8799193147.jpg\",\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nA \\\\sim _ { \\\\mathbb { Q } } K _ { X _ { \\\\mathrm { c a n } } } + B _ { X _ { \\\\mathrm { c a n } } } + M _ { X _ { \\\\mathrm { c a n } } } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,340,881,412]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $B _ { X _ { \\\\mathrm { c a n } } }$ —the discriminant—is closely related to the singularities of $f$ , and the divisor $M _ { X _ { \\\\mathrm { c a n } } }$ —the moduli divisor—conjecturally carries information on the birational variation of the fibres of $f$ . A formula of this form is called the canonical bundle formula. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,414,881,468]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This paper is an attempt to give an account of all the known results on the canonical bundle formula, and to serve as a guide to those wishing to study this important subject. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,513,410,534]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 Lc-Trivial Fibrations \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,557,879,593]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We work over $\\\\mathbb { C }$ . We denote ${ \\\\mathsf { b y } } \\\\equiv , \\\\sim$ and ${ \\\\sim } _ { \\\\mathbb { Q } }$ the numerical, linear and $\\\\mathbb { Q }$ -linear equivalence of divisors respectively. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,594,881,683]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For a Weil $\\\\mathbb { Q }$ -divisor $D = \\\\sum d _ { i } D _ { i }$ , for a real number $r$ we denote $D ^ { \\\\leq r } : =$ $\\\\textstyle \\\\sum _ { d _ { i } \\\\leq r } d _ { i } D _ { i }$ . If $f \\\\colon X \\\\ \\\\to \\\\ Y$ is a proper surjective morphism between normal varieties and $D$ is a Weil $\\\\mathbb { R }$ -divisor on $X$ , then $D _ { v }$ and $D _ { h }$ denote the vertical and the horizontal part of $D$ with respect to $f$ . In this setup, we say that $D$ is $f$ -exceptional if codimY Supp $f ( D ) \\\\geq 2$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,684,879,737]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section we introduce the main topic of this survey—lc-trivial fibrations. In this section we define them and give some examples which will accompany us through the paper. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,738,879,773]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We first need to introduce singularities of pairs. This is by now a standard topic in higher dimensional birational geometry, and good references are [27] and [26]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,774,881,827]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A pair $( X , \\\\Delta )$ consists of a normal variety $X$ and a Weil\\u0003 $\\\\mathbb { Q }$ -divisor $\\\\Delta$ such that\\u0004 $K _ { X } + \\\\Delta$ is $\\\\mathbb { Q }$ -Cartier. A pair $( X , \\\\Delta )$ is log smooth if $X$ is smooth and the support of $\\\\Delta$ is a simple normal crossings divisor. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,828,879,882]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A log resolution of a pair $( X , \\\\Delta )$ is a birational morphism $f \\\\colon Y \\\\to X$ such that the exceptional locus $\\\\operatorname { E x c } ( f )$ is a divisor and the pair $\\\\left( Y , \\\\operatorname { S u p p } ( f _ { * } ^ { - 1 } \\\\Delta + \\\\operatorname { E x c } ( f ) \\\\right)$ is log smooth. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[706,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Floris and V. Lazic´ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,54,141,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"45\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,881,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $( X , \\\\Delta )$ be a pair and if $\\\\pi \\\\colon Y X$ is a birational morphism with $Y$ normal, we can write \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[284,138,710,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0003/b91d006a5f4a05012ff26f47da28110b52d79ca533f33856578bf96bd12e4eac.jpg\",\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { Y } \\\\sim _ { \\\\mathbb { Q } } \\\\pi ^ { * } ( K _ { X } + \\\\Delta ) + \\\\sum a ( E _ { i } , X , \\\\Delta ) \\\\cdot E _ { i } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,180,881,270]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $E _ { i } ~ \\\\subseteq ~ Y$ are distinct prime divisors and the numbers $a ( E _ { i } , X , \\\\Delta ) \\\\in \\\\mathbb { Q }$ are called discrepancies. The order of vanishing at the generic point of each $E _ { i }$ defines a geometric valuation on $\\\\mathbb { C } ( X )$ . The pair $( X , \\\\Delta )$ is $k l t$ , respectively log canonical, if $a ( E , X , \\\\Delta ) > - 1$ , respectively $a ( E , X , \\\\Delta ) \\\\geq - 1$ , for every geometric valuation $E$ over $X$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,271,881,325]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Much of what we say in this paper can be generalised to pairs $( X , \\\\Delta )$ , where $\\\\Delta$ is allowed to have real coefficients. We stick to rational divisors mostly for reasons of clarity and simplicity. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,370,523,391]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.1 Definition and First Examples \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,414,876,450]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The objects for which we can write a canonical bundle formula are called lc-trivial fibrations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,459,876,496]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 2.1 Let $( X , \\\\Delta )$ be a pair. A morphism $f \\\\colon ( X , \\\\Delta ) \\\\ \\\\ Y$ to a normal projective variety $Y$ is a klt-trivial, respectively $l c$ -trivial, fibration if: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,504,881,578]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(a) $f$ is a surjective morphism with connected fibres, \\n(b) $( X , \\\\Delta )$ has klt, respectively log canonical, singularities over the generic point of $Y$ , \\n(c) there exists a $\\\\mathbb { Q }$ -Cartier $\\\\mathbb { Q }$ -divisor $D$ on $Y$ such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[428,594,605,615]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0247/7674071a4942d50a218e901c1182f42569f0f3fde4cfa5b00797c67cf5dd4db7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { X } + \\\\Delta \\\\sim _ { \\\\mathbb { Q } } f ^ { * } D ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,630,881,698]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(d) there exists a log resolution $\\\\pi ^ { \\\\prime } \\\\colon X ^ { \\\\prime } \\\\to X$ of $( X , \\\\Delta )$ such that, if $\\\\mathcal { E }$ is the set of all geometric valuations over $X$ which are defined by a prime divisor $E$ on $X ^ { \\\\prime }$ such that $a ( E , X , \\\\Delta ) > - 1$ , and if we denote $\\\\Xi ^ { \\\\prime } = \\\\sum _ { E \\\\in { \\\\mathcal { E } } } a ( E , X , \\\\Delta ) \\\\cdot E$ , then \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[389,713,646,734]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0643/0f5aa705ab313d8e8912b7e405c4d0eeb0a73eeef8ed7081e2175704007fcb78.jpg\",\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf { r k } ( f \\\\circ \\\\pi ^ { \\\\prime } ) _ { * } \\\\mathcal { O } _ { X ^ { \\\\prime } } ( [ \\\\Xi ^ { \\\\prime } ] ) = 1 .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,750,881,786]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Terminology 2.2 In [2], klt-trivial fibrations as in Definition 2.1 are called lc-trivial fibrations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,797,879,833]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 2.3 We make a few comments on the condition (d) in Definition 2.1. For simplicity, assume that the pair $( X , \\\\Delta )$ is klt. Note that then \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[364,848,635,870]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0082/1e0b1e4e8f91b949478ef068aab22b91588ce0f464416353c012ad821b714e7b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\Xi ^ { \\\\prime } \\\\sim _ { \\\\mathbb { Q } } K _ { X ^ { \\\\prime } } - \\\\pi ^ { \\\\prime * } ( K _ { X } + \\\\Delta ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,51,505,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Travel Guide to the Canonical Bundle Formula \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,52,879,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"46\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"39 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,876,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The divisor $\\\\lceil \\\\Xi ^ { \\\\prime } \\\\rceil$ is effective on the generic fibre of the morphism $f \\\\circ \\\\pi ^ { \\\\prime }$ by (b), hence $\\\\mathbf { r k } ( f \\\\circ \\\\pi ^ { \\\\prime } ) _ { * } \\\\mathcal { O } _ { X ^ { \\\\prime } } ( \\\\lceil \\\\Xi ^ { \\\\prime } \\\\rceil ) \\\\geq 1 .$ . Therefore, the point of (d) is the opposite inequality. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,127,881,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The most important case to keep in mind is that when the divisor $\\\\Delta$ is effective on the generic fibre: indeed, in that case the divisor $\\\\lceil \\\\Xi ^ { \\\\prime } \\\\rceil$ is an effective exceptional divisor on the generic fibre of $f \\\\circ \\\\pi ^ { \\\\prime }$ , and the condition (d) is immediate. However, in order to be able to study lc-trivial fibrations by applying basic operations of birational geometry in Sect. 2.2, it is crucial to allow divisors $\\\\Delta$ with negative coefficients. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,234,881,325]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"One more thing to notice is that if (d) holds for a log resolution $\\\\pi ^ { \\\\prime }$ , then it holds on any log resolution $\\\\pi ^ { \\\\prime \\\\prime } \\\\colon X ^ { \\\\prime \\\\prime } \\\\to X$ which factors through $\\\\pi ^ { \\\\prime }$ . Indeed, define the divisor $\\\\Xi ^ { \\\\prime \\\\prime }$ on $X ^ { \\\\prime \\\\prime }$ analogously as in Definition 2.1, and let $\\\\theta \\\\colon X ^ { \\\\prime \\\\prime } X ^ { \\\\prime }$ be the induced morphism. Since $X ^ { \\\\prime }$ and $X ^ { \\\\prime \\\\prime }$ are smooth, there exists an integral effective divisor $E$ such that $K _ { X ^ { \\\\prime \\\\prime } } \\\\sim \\\\theta ^ { * } K _ { X ^ { \\\\prime } } + E$ . Thus, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[170,339,826,391]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0067/f9de043feebd0fa0a7c0778ac2207454d73e3c125db7c138f34bdf38279d6902.jpg\",\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { f _ { * } \\\\pi _ { * } ^ { \\\\prime \\\\prime } \\\\mathcal { O } _ { X ^ { \\\\prime \\\\prime } } ( \\\\lceil \\\\Xi ^ { \\\\prime \\\\prime } \\\\rceil ) = f _ { * } \\\\pi _ { * } ^ { \\\\prime \\\\prime } \\\\mathcal { O } _ { X ^ { \\\\prime \\\\prime } } ( \\\\lceil \\\\theta ^ { * } \\\\Xi ^ { \\\\prime } + E \\\\rceil ) = f _ { * } \\\\pi _ { * } ^ { \\\\prime \\\\prime } \\\\mathcal { O } _ { X ^ { \\\\prime \\\\prime } } ( \\\\lceil \\\\theta ^ { * } \\\\Xi ^ { \\\\prime } \\\\rceil + E ) } \\\\\\\\ & { \\\\qquad \\\\subseteq f _ { * } \\\\pi _ { * } ^ { \\\\prime \\\\prime } \\\\mathcal { O } _ { X ^ { \\\\prime \\\\prime } } ( \\\\theta ^ { * } \\\\lceil \\\\Xi ^ { \\\\prime } \\\\rceil + E ) = f _ { * } \\\\pi _ { * } ^ { \\\\prime } \\\\mathcal { O } _ { X ^ { \\\\prime } } ( \\\\lceil \\\\Xi ^ { \\\\prime } \\\\rceil ) , } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,405,840,423]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where we used that $\\\\lceil \\\\theta ^ { * } \\\\Xi ^ { \\\\prime } \\\\rceil \\\\leq \\\\theta ^ { * } \\\\lceil \\\\Xi ^ { \\\\prime } \\\\rceil$ . Therefore, $\\\\mathrm { r k } ( f \\\\circ \\\\pi ^ { \\\\prime \\\\prime } ) _ { * } \\\\mathcal { O } _ { X ^ { \\\\prime \\\\prime } } ( [ \\\\Xi ^ { \\\\prime \\\\prime } ] ) = 1$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,424,876,459]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In general, one can show that (d) is independent of the choice of the resolution by using [14, Lemma 2.7]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,468,876,504]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now we can formulate the canonical bundle formula associated to an lc-trivial fibration. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,513,881,567]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 2.4 Let $f \\\\colon ( X , \\\\Delta ) \\\\to Y$ be an lc-trivial fibration such that $K _ { X } + \\\\Delta \\\\sim _ { \\\\mathbb { Q } }$ $f ^ { * } D$ for some $\\\\mathbb { Q }$ -Cartier $\\\\mathbb { Q }$ -divisor $D$ on $Y$ . If $P \\\\subseteq Y$ is a prime divisor, the log canonical threshold of $f ^ { * } P$ with respect to $( X , \\\\Delta )$ is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[129,584,861,605]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\gamma _ { P } = \\\\operatorname* { s u p } \\\\{ t \\\\in \\\\mathbb { R } \\\\mid ( X , \\\\Delta + t f ^ { * } P )$ is log canonical over the generic point of $P$ }. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,621,881,675]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The condition that $( X , \\\\Delta + t f ^ { * } P )$ is log canonical over the generic point of \\b $P$ means that for every geometric valuation $E$ over $X$ which surjects onto $P$ , we have $a ( E , X , \\\\Delta + t f ^ { * } P ) \\\\geq - 1 .$ . The discriminant of $f$ is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[394,692,603,714]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0353/273f1a6aed8b5d9d93e60297e9cf431d523c61012550f61429123a2a7a7e748b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { B _ { Y } = \\\\sum _ { P } ( 1 - \\\\gamma _ { P } ) P . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,731,879,767]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Fix $\\\\varphi \\\\in \\\\mathbb { C } ( X )$ and the smallest positive integer $r$ such that $\\\\begin{array} { r } { K _ { X } + \\\\Delta + \\\\frac { 1 } { r } \\\\operatorname { d i v } \\\\varphi = f ^ { * } D } \\\\end{array}$ . Then there exists a unique Weil $\\\\mathbb { Q }$ -divisor $M _ { Y }$ , the moduli part of $f$ , such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[296,780,701,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0908/a973c4f0cbdc1c11231df3236029e9b41719bcb74042a76755d49ee7d603cac0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { K _ { X } + \\\\Delta + \\\\frac 1 r \\\\operatorname { d i v } \\\\varphi = f ^ { * } ( K _ { Y } + B _ { Y } + M _ { Y } ) . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,821,685,839]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This formula is the canonical bundle formula associated to $f$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,846,881,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 2.5 The definition of the discriminant as above first appeared in [23, Theorem 2]. The discriminant is a Weil $\\\\mathbb { Q }$ -divisor on $Y$ , and it is effective if $\\\\Delta$ is effective. We notice that the discriminant is defined as an actual divisor, while the moduli part is defined only up to $\\\\mathbb { Q }$ -linear equivalence: it depends on the choice of $D$ . Further, if $G$ is a $\\\\mathbb { Q }$ -divisor on $Y$ , then $f \\\\colon ( X , \\\\Delta + f ^ { * } G ) \\\\to Y$ is an lc-trivial fibration with discriminant $B _ { Y } + G$ and moduli divisor $M _ { Y }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"40 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[706,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"47\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Floris and V. Lazic´ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,879,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,162,879,216]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note that if we are interested in proving properties of the moduli divisor of an lc-trivial fibrations as above, we may always assume that the pair $( X , \\\\Delta )$ is log canonical by [12, Remark 3.6], although we may not assume that $\\\\Delta$ is effective. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,223,879,261]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 2.6 Assume that $X$ is smooth, that $\\\\Delta = 0$ , that $Y$ is a curve, that $f ^ { - 1 } ( P )$ is smooth and that $f ^ { * } P = m f ^ { - 1 } ( P )$ is a multiple fibre. Then $\\\\begin{array} { r } { \\\\gamma _ { P } = \\\\frac { 1 } { m } } \\\\end{array}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,268,881,378]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 2.7 This example is historically the first example of a canonical bundle formula. Let $f \\\\colon X \\\\to C$ be an elliptic fibration, that is, $X$ is a smooth surface, $C$ is a smooth curve and a general fibre of $f$ is a smooth elliptic curve. We assume furthermore that $f$ is relatively minimal: there are no $( - 1 )$ -curves contained in the fibres of $f$ . Kodaira in [24, Theorem 6.2] classified the singular fibres of $f$ . Kodaira’s canonical bundle formula reads as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[364,394,630,414]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"1023/a66b8185be1265725ffd0b9be6f35b123f34b1db532124960a3973172faffff4.jpg\",\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { X } \\\\sim f ^ { * } ( K _ { C } + B _ { C } + M _ { C } ) ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,432,881,486]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $B _ { C }$ is defined in terms of the classification of the singular fibres and by [24, 32] we have $1 2 M _ { C } = j ^ { * } \\\\mathcal { O } _ { \\\\mathbb { P } ^ { 1 } } ( 1 )$ , with $j \\\\colon C \\\\ \\\\to \\\\ \\\\mathbb { P } ^ { 1 }$ being the $j$ -invariant. For a detailed account on Kodaira’s canonical bundle formula see [4, Chapter V, $\\\\ S 7 \\\\mathrm { - } \\\\ S 1 3 ]$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,492,881,569]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 2.8 Let $\\\\ b X = \\\\mathbb P ^ { 1 } \\\\ b X \\\\mathbb P ^ { 1 }$ and let $D$ be a reduced divisor of bidegree $( d , k )$ with $d \\\\geq 2 .$ Let $\\\\begin{array} { r } { \\\\Delta = \\\\frac { 2 } { d } D } \\\\end{array}$ and let $f \\\\colon ( X , \\\\Delta ) \\\\to \\\\mathbb { P } ^ { 1 }$ be the projection onto the second factor. Then $f$ is an lc-trivial fibration. Indeed, $K _ { X } + \\\\Delta$ has bidegree $( - 2 , - 2 ) +$ $\\\\textstyle { \\\\frac { 2 } { d } } ( d , k ) = ( 0 , - 2 + { \\\\frac { 2 k } { d } } )$ and therefore is the pullback of a divisor from $\\\\mathbb { P } ^ { 1 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,612,435,633]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2 Base Change Property \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,656,881,710]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this subsection we investigate how canonical bundle formulas behave under base change. This will help improve the properties of the moduli part of a canonical bundle formula, at least on a sufficiently high birational model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,711,876,746]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $f \\\\colon ( X , \\\\Delta ) \\\\ \\\\to \\\\ Y$ is a klt-trivial fibration (respectively lc-trivial), and if we consider a base change diagram \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[398,765,601,855]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0331/59564a581e7264aa550e72c51003e72a949a1703a2882a2b2053349f92fb20b4.jpg\",\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l r } & { ( X ^ { \\\\prime } , \\\\Delta ^ { \\\\prime } ) \\\\xrightarrow { \\\\tau } ( X , \\\\Delta ) } & \\\\\\\\ & { f ^ { \\\\prime } \\\\downarrow } & \\\\\\\\ & { Y ^ { \\\\prime } \\\\xrightarrow { \\\\rho } ( \\\\begin{array} { c c } { \\\\rho } & { \\\\sqrt { f } } \\\\\\\\ { \\\\rho } & { \\\\rho } \\\\end{array} ) } & \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,505,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"48\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Travel Guide to the Canonical Bundle Formula \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,876,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"49\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\rho$ is a proper generically finite morphism, $X ^ { \\\\prime }$ is the normalisation of the fibre product and $\\\\Delta ^ { \\\\prime }$ is defined so that we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[373,142,621,163]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0635/70142ecbf933f57f99715597ebde0dbbea79ac019740c9142e909802c8e5c0b4.jpg\",\n \"list_items\": null,\n \"page_idx\": \"49\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { X ^ { \\\\prime } } + \\\\Delta ^ { \\\\prime } = \\\\tau ^ { * } ( K _ { X } + \\\\Delta ) ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,180,879,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"49\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"then $f ^ { \\\\prime } \\\\colon ( X ^ { \\\\prime } , \\\\Delta ^ { \\\\prime } ) \\\\to Y ^ { \\\\prime }$ is also a klt-trivial (respectively lc-trivial) fibration. In the rest of the paper, we implicitly refer to this klt-trivial fibration when writing $M _ { Y ^ { \\\\prime } }$ and $B _ { Y ^ { \\\\prime } }$ for the moduli part and discriminant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,235,876,270]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"49\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $\\\\rho$ is birational, then $\\\\rho _ { * } M _ { Y ^ { \\\\prime } } = M _ { Y }$ and $\\\\rho _ { * } B _ { Y ^ { \\\\prime } } = B _ { Y }$ ; in other words, these collections of divisors form $^ b$ -divisors, see for instance [2, Section 1.2]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[143,271,810,288]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"49\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The following is the base change property of canonical bundle formulas. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,297,881,351]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"49\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2.9 Let $f \\\\colon ( X , \\\\Delta ) \\\\to Y$ be an lc-trivial fibration. Then there exists $a$ proper birational morphism $Y ^ { \\\\prime } Y$ such that for every proper birational morphism $\\\\pi \\\\colon Y ^ { \\\\prime \\\\prime } \\\\to Y ^ { \\\\prime }$ we have: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,360,785,396]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"49\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(i) $K _ { Y ^ { \\\\prime } } + B _ { Y ^ { \\\\prime } }$ is a $\\\\mathbb { Q }$ -Cartier divisor and $K _ { Y ^ { \\\\prime \\\\prime } } + B _ { Y ^ { \\\\prime \\\\prime } } = \\\\pi ^ { * } ( K _ { Y ^ { \\\\prime } } + B _ { Y ^ { \\\\prime } } ) ,$ (ii) $M _ { Y ^ { \\\\prime } }$ is a nef $\\\\mathbb { Q }$ -Cartier divisor and $M _ { Y ^ { \\\\prime \\\\prime } } = \\\\pi ^ { * } M _ { Y ^ { \\\\prime } }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,405,881,495]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"49\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The first version of Theorem 2.9 is [23, Theorem 2], which essentially shows the nefness of the moduli part; this is also the point of the proof where the condition (d) in Definition 2.1 is used. Theorem 2.9 has been proved in this form for klt-trivial fibrations by Ambro [2, Theorem 0.2]. For lc-trivial fibrations, it was proved by Kollár [25, Theorem 8.3.7] and [18, Theorem 3.6], with an alternative proof in [11]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,496,879,549]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"49\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the context of the previous theorem, we say that $M _ { Y ^ { \\\\prime } }$ descends to $Y ^ { \\\\prime }$ , and we call any such $Y ^ { \\\\prime }$ , where additionally $B _ { Y ^ { \\\\prime } }$ has simple normal crossings support, an Ambro model for $f$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,551,881,818]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"49\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We give a brief sketch of the proof of the nefness of the moduli divisor in the previous theorem; very good references are [2, Lemma 5.2] and especially [25, Theorem 8.5.1 and $\\\\ S 8 . 1 0 ]$ , where many more details are given. By the proof of [25, Theorem 8.5.1], it suffices to show the claim after making a suitable generically finite base change and taking the cyclic cover of $X$ associated to $\\\\sqrt [ r ] { \\\\varphi }$ . The base change as in (1) that we are aiming for is a composition of a log resolution with a Kawamata cover such that, on an open subset of $U ^ { \\\\prime } \\\\subseteq Y ^ { \\\\prime }$ whose complement has codimension at least 2 in $Y ^ { \\\\prime }$ , the local systems $R ^ { i } f _ { * } ^ { \\\\prime } \\\\mathbb { C } _ { X ^ { \\\\prime } } | _ { U ^ { \\\\prime } }$ have unipotent monodromies; this is the content of [25, 8.10.7–8.10.10]. We may also assume that $M _ { Y ^ { \\\\prime } }$ is a Cartier divisor. If $f$ is a klt-trivial fibration, then by [25, Theorem 8.5.1] the line bundle $\\\\mathcal { O } _ { Y ^ { \\\\prime } } ( M _ { Y ^ { \\\\prime } } )$ is a quotient of the locally free sheaf $f _ { * } ^ { \\\\prime } \\\\omega _ { X ^ { \\\\prime } / Y ^ { \\\\prime } }$ . Then one applies [22, Theorem 5], which asserts that $f _ { * } ^ { \\\\prime } \\\\omega _ { X ^ { \\\\prime } / Y ^ { \\\\prime } }$ is the canonical extension of the bottom piece of the Hodge filtration of $R ^ { \\\\dim X - \\\\dim Y } f _ { * } ^ { \\\\prime } \\\\mathbb { C } _ { X ^ { \\\\prime } } | _ { U ^ { \\\\prime } }$ , and hence its quotients are nef by the same result. A similar statement holds also for lc-trivial fibrations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,821,879,873]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"49\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recently, a stronger statement was shown in [17, Theorem 1.1]. The result shows that [22, Theorem 5] can be improved to show that not only any quotient of $f _ { * } ^ { \\\\prime } \\\\omega _ { X ^ { \\\\prime } / Y ^ { \\\\prime } }$ is nef, but moreover, it carries a singular metric whose Lelong numbers are all zero. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[706,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"49\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Floris and V. Lazic´ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"49\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"42 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,90,879,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This is much stronger than being nef, as it implies, in particular, that the multiplier ideal associated to this metric is trivial. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,126,533,142]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We summarise this in the following result. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,151,881,205]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2.10 Let $f \\\\colon ( X , \\\\Delta ) \\\\to Y$ be a klt-trivial fibration. Then there exists $a$ proper birational morphism $Y ^ { \\\\prime } Y$ such that for every proper birational morphism $\\\\pi \\\\colon Y ^ { \\\\prime \\\\prime } \\\\to Y ^ { \\\\prime }$ we have: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,214,881,270]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(i) $K _ { Y ^ { \\\\prime } } + B _ { Y ^ { \\\\prime } }$ is a $\\\\mathbb { Q }$ -Cartier divisor and $K _ { Y ^ { \\\\prime \\\\prime } } + B _ { Y ^ { \\\\prime \\\\prime } } = \\\\pi ^ { * } ( K _ { Y ^ { \\\\prime } } + B _ { Y ^ { \\\\prime } } ) .$ , (ii) $M _ { Y ^ { \\\\prime } }$ is a $\\\\mathbb { Q }$ -Cartier divisor carrying a singular metric whose all Lelong numbers are zero, and $M _ { Y ^ { \\\\prime \\\\prime } } = \\\\pi ^ { * } M _ { Y ^ { \\\\prime } }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,277,881,351]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The proof of part (ii) of the theorem is the same as the sketch of the proof of Theorem 2.9(ii) above, since a $\\\\mathbb { Q }$ -divisor carries a singular metric whose all Lelong numbers are zero if and only if its pullback by a proper surjective map carries a singular metric whose all Lelong numbers are zero by [7, Corollary 4]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,396,451,415]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.3 Inversion of Adjunction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,439,881,584]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In order to appreciate the following result and to see why base change property is important, let us go back to the construction of the canonical bundle formula. Recall that the discriminant divisor was constructed in terms of local log canonical thresholds, that is, log canonical thresholds over the generic point of a prime divisor; in particular, with notation from Definition 2.4, for some prime divisor $P$ on $Y$ , the pair $( X , \\\\Delta + \\\\gamma _ { P } f ^ { * } P )$ does not have to be globally log canonical. However, the following inversion of adjunction [2, Theorem 3.1] states that this is precisely what happens on an Ambro model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,593,879,665]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2.11 Let $f \\\\colon ( X , \\\\Delta ) \\\\ \\\\to \\\\ Y$ be an lc-trivial fibration, and assume that $Y$ is an Ambro model for $f$ . Then $( Y , B _ { Y } )$ has klt, respectively log canonical, singularities in a neighbourhood of a point $y ~ \\\\in ~ Y$ if and only if $( X , \\\\Delta )$ has $k l t$ , respectively log canonical, singularities in a neighbourhood of $f ^ { - 1 } ( y )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,674,876,710]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note that Theorem 2.11 is stated for klt-trivial fibrations in [2], but the proof extends verbatim to the lc-trivial case by using Theorem 2.9. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,711,876,746]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We finish this subsection with the following nice result [3, Proposition 3.1] which we will apply in the proof of Theorem 2.16 below. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,755,879,810]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2.12 Let $f \\\\colon ( X , \\\\Delta ) \\\\to Y$ be a klt-trivial fibration, and assume that $Y$ is an Ambro model for $f$ . Then for every base change by a proper generically finite morphism $w \\\\colon W \\\\to Y$ we have $K _ { W } + B _ { W } \\\\sim _ { \\\\mathbb { Q } } w ^ { * } ( K _ { Y } + B _ { Y } )$ and $M _ { W } \\\\sim _ { \\\\mathbb { Q } } w ^ { * } M _ { Y }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,505,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"50\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Travel Guide to the Canonical Bundle Formula \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,567,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.4 Coefficients of the Moduli Divisor \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,132,876,169]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Often in applications one needs to bound the denominators of $M _ { Y }$ . For instance, such bounds were used in [10, Theorem 1.2]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,178,876,213]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2.13 For each nonnegative integer b there exists an integer N depending on $b$ such that the following holds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,214,881,288]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $f \\\\colon ( X , \\\\Delta ) \\\\ \\\\to \\\\ Y$ be a klt-trivial fibration with a general fibre $F$ , and let $r$ be the smallest positive integer such that $r ( K _ { F } + \\\\Delta | _ { F } ) \\\\sim 0 .$ . Let $E F$ be the associated $r$ -th cyclic cover and let $\\\\overline { { E } }$ be a resolution of singularities of $E$ . If $\\\\dim { H ^ { \\\\dim \\\\overline { { E } } } } ( \\\\overline { { E } } , \\\\mathbb { C } ) = \\\\bar { b }$ , then the divisor $N M _ { Y }$ is integral. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,297,876,333]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The result was proved in [19, Theorem 3.1] when $\\\\Delta = 0$ , but the same proof works for klt-trivial fibrations [10, Theorem 5.1]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,334,876,369]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A more refined result holds when a general fibre is a rational curve, [9, Theorem 1.6(2)]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,378,874,414]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2.14 Fix a positive integer r. For a prime number $q$ set $s ( q ) = \\\\operatorname* { m a x } \\\\{ s \\\\mid$ $q ^ { s } \\\\leq 2 r \\\\}$ and define \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[416,427,580,472]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0342/b57e001ad36d49cdbb1bfad8fc3ab7e1be31d1f824e27fb151953355237bad97.jpg\",\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nN = \\\\prod _ { q p r i m e } q ^ { s ( q ) } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,484,881,540]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(a) Let $f \\\\colon ( X , \\\\Delta ) \\\\ \\\\to \\\\ Y$ be an lc-trivial fibration whose general fibre $F$ is a rational curve, and assume that $r$ is the smallest positive integer such that $r ( K _ { F } + \\\\Delta | _ { F } ) \\\\sim 0 .$ . Then the divisor $N M _ { Y }$ is integral. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,540,881,594]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(b) Assume $r$ is odd. Then there exists an lc-trivial fibration $f \\\\colon ( X , \\\\Delta ) \\\\to Y$ such that if v is the smallest integer for which the divisor $v M _ { Y }$ is integral, then $v =$ $N / r$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,639,510,660]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.5 Goodness of Moduli Divisors \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,683,881,773]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now we come to the central topic of this survey, already announced in the introduction: the descent of singularities. Since we already know the nefness of the moduli divisor by Theorem 2.9, if it were additionally big, then this would allow to conclude in many cases. Bigness is too much to ask; however, the following result of Ambro [3, Theorem 3.3 and Proposition 4.4] turns out to be almost as good. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[706,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Floris and V. Lazic´ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"51\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"44 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,144]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2.15 Let $f \\\\colon ( X , \\\\Delta ) \\\\ \\\\to \\\\ Y$ be a klt-trivial fibration between normal projective varieties such that $\\\\Delta$ is effective over the generic point of Y . Then there exists a diagram \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[334,160,669,247]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0740/541e9d4cad5c4759ad43625a780c5daa43b37865df1e0a5e3f14798ed275b1aa.jpg\",\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { c c c } { { ( X , \\\\Delta ) } } & { { } } & { { ( X ^ { + } , \\\\Delta ^ { + } ) } } \\\\\\\\ { { f \\\\Bigg \\\\downarrow } } & { { } } & { { \\\\Bigg \\\\downarrow f ^ { + } } } \\\\\\\\ { { Y \\\\longleftarrow \\\\quad } } & { { W \\\\longleftrightarrow { Y ^ { + } } } } & { { } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,268,211,285]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"such that: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,294,881,385]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(i) $f ^ { + } \\\\colon ( X ^ { + } , \\\\Delta ^ { + } ) \\\\to Y ^ { + }$ is a klt-trivial fibration, \\n(ii) $\\\\tau$ is generically finite and surjective, and $\\\\rho$ is surjective, \\n(iii) if $M _ { Y }$ and $M _ { Y ^ { + } }$ are the moduli divisors of $f$ and $f ^ { + }$ respectively, then $M _ { Y ^ { + } }$ is big and, after possibly a birational base change, we have $\\\\tau ^ { * } M _ { Y } = \\\\rho ^ { * } M _ { Y ^ { + } }$ , \\n$( i \\\\nu )$ there exists a non-empty open set $U \\\\subseteq W$ and an isomorphism \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[291,399,751,487]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0515/f685aa053a04842dc32fb72bf44ac243c403958fd93a56062cc32085289ca2d8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } { ( X , \\\\Delta ) \\\\times _ { Y } U \\\\xrightarrow { \\\\cong } } & { { } \\\\xrightarrow { \\\\cong \\\\qquad ( X ^ { + } , \\\\Delta ^ { + } ) \\\\times _ { Y ^ { + } } U } } \\\\\\\\ { \\\\qquad } & { { } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[129,503,441,521]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(v) if there exists an isomorphism \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[359,537,683,557]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0836/32e71a41d6f5b71c4d6ec6b5c0d5e663967328eebc90a2709dcc3b156ba2ed40.jpg\",\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\Phi \\\\colon ( X , \\\\Delta ) \\\\times _ { Y } U ( F , \\\\Delta _ { F } ) \\\\times U\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[157,573,879,591]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"over a non-empty open subset $U \\\\subseteq Y$ , then $\\\\Phi$ extends to an isomorphism over \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[293,605,749,630]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0949/737c4824ed20eb8bb59693e4115e16392a654c68d3fe56c0c4bd145774246909.jpg\",\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nY ^ { 0 } = Y \\\\setminus \\\\big ( \\\\mathrm { S u p p } B _ { Y } \\\\cup \\\\mathrm { S i n g } ( Y ) \\\\cup f ( \\\\mathrm { S u p p } \\\\Delta _ { v } ^ { \\\\le 0 } ) \\\\big ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[148,645,858,663]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note that in (v) one does not need that $\\\\Delta$ is effective on the generic fibre of $f$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,665,874,701]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For us, the most important part of this result is (iii). Its immediate consequence is the descent of klt singularities [3, Theorem 0.2]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,708,881,782]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2.16 Let $( X , \\\\Delta )$ be a projective klt pair with $\\\\Delta$ effective, and let $f \\\\colon X \\\\to$ $Y$ be a surjective morphism to a normal projective variety such that $K _ { X } + \\\\Delta \\\\sim _ { \\\\mathbb { Q } }$ $f ^ { * } D$ for some $\\\\mathbb { Q }$ -Cartier $\\\\mathbb { Q }$ -divisor $D$ on $Y$ . Then there exists an effective $\\\\mathbb { Q }$ -divisor $\\\\Delta \\\\gamma$ on $Y$ such that the pair $( Y , \\\\Delta _ { Y } )$ is klt and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[364,798,633,819]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0834/8179d8ed163b308051bffd1ac6b952d578ae43084ab60771f87b76446fe71220.jpg\",\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { X } + \\\\Delta \\\\sim _ { \\\\mathbb { Q } } f ^ { * } ( K _ { Y } + \\\\Delta _ { Y } ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,51,505,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Travel Guide to the Canonical Bundle Formula \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,52,879,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"52\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"45 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,879,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof We use the notation from Theorem 2.15, which clearly applies in our situation. We have the base change diagram \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[387,144,608,231]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0936/d84ae0b5ad4dbbf60dd8688ce6d46f6c7771033bb03d9206ae88957936de44a2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { c } { { ( W , \\\\Delta _ { W } ) \\\\xrightarrow { w } > ( X , \\\\Delta ) } } \\\\\\\\ { { f _ { W } \\\\Bigg \\\\downarrow } } \\\\\\\\ { { W \\\\xrightarrow [ \\\\tau ] { } \\\\qquad \\\\forall . } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,252,881,325]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We may additionally assume that $\\\\tau$ factors through an Ambro model $\\\\pi \\\\colon Y ^ { \\\\prime } \\\\to Y$ of $f$ , and denote by $\\\\sigma : W \\\\to Y ^ { \\\\prime }$ the induced morphism. By replacing $W$ by a suitable log resolution, by an easy argument with the Stein factorisation of $\\\\sigma$ together with Theorem 2.12 we may assume that $W$ is an Ambro model of $f _ { W }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,325,879,415]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now, $\\\\tau ^ { * } D \\\\ \\\\sim _ { \\\\mathbb { Q } } \\\\ K _ { W } \\\\ + \\\\ B _ { W } \\\\ + \\\\ M _ { W }$ , and by the inversion of adjunction, Theorem 2.11, the pair $( W , B _ { W } )$ is klt. Since the divisor $M _ { Y ^ { + } }$ is nef and big, by using a version of Kodaira’s trick [27, Proposition 2.61] together with Bertini’s theorem, we may find an effective $\\\\mathbb { Q }$ -divisor $E _ { W }$ on $W$ such that $M _ { W } \\\\ \\\\sim _ { \\\\mathbb { Q } } \\\\ E _ { W }$ and such that the pair $( W , B _ { W } + E _ { W } )$ is klt. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,415,876,454]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By Theorem 2.12 we have $K _ { W } + B _ { W } \\\\sim _ { \\\\mathbb { Q } } \\\\sigma ^ { * } ( K _ { Y ^ { \\\\prime } } + B _ { Y ^ { \\\\prime } } )$ and $M _ { W } \\\\sim _ { \\\\mathbb { Q } } \\\\sigma ^ { * } M _ { Y ^ { \\\\prime } }$ Hence, if we denote $\\\\begin{array} { r } { E _ { Y ^ { \\\\prime } } = \\\\frac { 1 } { \\\\deg \\\\sigma } \\\\sigma _ { * } E } \\\\end{array}$ , we have $M _ { Y ^ { \\\\prime } } \\\\sim _ { \\\\mathbb { Q } } E _ { Y ^ { \\\\prime } }$ and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[298,471,699,492]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0672/81052eaf079ca0c706dbc63f61ec79ef8913114c8cb6ec79a87b1aec22ee9e34.jpg\",\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { W } + B _ { W } + E \\\\sim _ { \\\\mathbb { Q } } \\\\sigma ^ { * } ( K _ { Y ^ { \\\\prime } } + B _ { Y ^ { \\\\prime } } + E _ { Y ^ { \\\\prime } } ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,507,756,527]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Then the pair $( Y ^ { \\\\prime } , B _ { Y ^ { \\\\prime } } + E _ { Y ^ { \\\\prime } } )$ is klt by [27, Proposition 5.20]. Setting \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[328,543,667,563]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0126/b8cdff66a61c9733f11302b4bdeba41be663b58b08d5ed4805266d4e03744cfb.jpg\",\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\Delta _ { Y } = \\\\pi _ { * } ( B _ { Y ^ { \\\\prime } } + E _ { Y ^ { \\\\prime } } ) = B _ { Y } + \\\\pi _ { * } E ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[123,579,876,617]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"we have $K _ { Y } + \\\\Delta _ { Y } \\\\sim _ { \\\\mathbb { Q } } D$ , and $( Y , \\\\Delta _ { Y } )$ is a klt pair. Since $\\\\Delta$ is effective, the divisor $B _ { Y }$ is effective, thus $\\\\Delta \\\\gamma$ is effective. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,624,881,696]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally, we mention that Theorem 2.15(i)–(iii) was generalised to lc-trivial fibrations where $\\\\Delta \\\\geq 0$ and $( X , \\\\Delta )$ is log canonical in [18, Lemma 1.1]. The proof involves running a careful MMP in order to reduce to a situation where one has a klt-trivial fibration. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,705,881,813]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 2.17 Theorem 2.15(iv) does not hold for lc-trivial fibrations. For instance, let $\\\\boldsymbol { X } \\\\ : = \\\\ : \\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 }$ , let $f$ be the second projection, let $\\\\delta$ be the diagonal and set $\\\\begin{array} { r } { \\\\Delta = \\\\delta + \\\\frac 1 2 \\\\{ 0 \\\\} \\\\times \\\\mathbb { P } ^ { 1 } + \\\\frac 1 2 \\\\{ \\\\infty \\\\} \\\\times \\\\mathbb { P } ^ { 1 } } \\\\end{array}$ . Then $f \\\\colon ( X , \\\\Delta ) \\\\to \\\\mathbb { P } ^ { 1 }$ is an lc-trivial fibration with discriminant supported on $\\\\{ 0 \\\\} \\\\cup \\\\{ \\\\infty \\\\}$ . By considering log resolutions, one calculates\\u0003 \\u0004 that the discriminant is equal to $\\\\textstyle { \\\\frac { 1 } { 2 } } 0 + { \\\\frac { 1 } { 2 } } \\\\infty$ , hence the moduli divisor is torsion. Indeed, we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[296,828,699,852]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0920/1b9cca47006faf1fb1e6d7980d7db7536527d6eb7edbe8d816130a49fc7e2a1a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { K _ { X } + \\\\Delta \\\\sim _ { \\\\mathbb { Q } } f ^ { * } \\\\big ( K _ { \\\\mathbb { P } ^ { 1 } } + \\\\frac { 1 } { 2 } 0 + \\\\frac { 1 } { 2 } \\\\infty + M _ { \\\\mathbb { P } ^ { 1 } } \\\\big ) , } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,867,544,887]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"but the divisor $K _ { X } + \\\\Delta$ has bi-degree $( 0 , - 1 )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[706,51,879,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Floris and V. Lazic´ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"53\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"46 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,879,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"However, $f$ is not birational to a product. Indeed, it induces a family of rational curves with 4 marked points parametrised by $\\\\mathbb { P } ^ { 1 } \\\\colon$ for $t \\\\in \\\\mathbb { P } ^ { 1 }$ , set $p _ { 1 } ( t ) = 0$ , $p _ { 2 } ( t ) =$ 1, $p _ { 3 } ( t ) = p _ { 4 } ( t ) = t$ . This family of marked curves is not trivial, therefore the fibration cannot be locally a product. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,207,514,226]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 B-Semiampleness Conjectures \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,250,881,358]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We saw above in Theorem 2.16 that klt singularities descend along a klt-trivial fibration. However, even if one had the full analogue of Theorem 2.15 in the log canonical setting one could not follow the same strategy to show that log canonical singularities descend. Moreover, one sees that the use of Kodaira’s trick in the proof of Theorem 2.16 obliterated the connection of the coefficients of the divisor $\\\\Delta \\\\gamma$ to that of the divisor $\\\\Delta$ . In order to remedy the situation, something more is needed. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,360,881,412]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The main conjecture on the canonical bundle formula predicts something much stronger: that the moduli part is semiample on an Ambro model of the fibration. We discuss it in this section. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,414,876,450]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There are two versions of the conjecture; the stronger one was proposed in [31, Conjecture 7.13.3]. We start with the stronger, effective version. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,459,881,548]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Effective B-Semiampleness Conjecture Fix positive integers d and r. Then there exists an integer m depending only on $d$ and $r$ , such that for any $l c$ -trivial fibration $f \\\\colon ( X , B ) \\\\to Y$ with the generic fibre $F$ , if dim $F = d$ and $r$ is the smallest positive integer such that $r ( K _ { F } + B | _ { F } ) \\\\sim 0$ , there exists an Ambro model $Y ^ { \\\\prime }$ of $f$ such that $m M _ { Y ^ { \\\\prime } }$ is base point free. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,558,879,611]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"More generally, any conjecture as above, in which $m$ depends on some invariants of the generic fibre of the fibration, goes under the name of effective $^ b$ - semiampleness. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,612,879,683]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the original statement [31, Conjecture 7.13.3], the constant m depended on $\\\\mathrm { d i m } X$ and $r$ . The main result of [10] is that it suffices to show the Effective B-Semiampleness Conjecture in the case where $Y$ is a curve. This led to the formulation of the Effective B-Semiampleness Conjecture above. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,684,874,719]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The conjecture is widely open. We list below the cases where it is known. They all make use of some notion of moduli space for the fibres. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,729,876,764]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3.1 The Effective $B$ -Semiampleness Conjecture holds in the following cases: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,773,881,881]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(1) if general fibres are elliptic curves by [24, 32]; we have $m = 1 2$ ; \\n(2) if general fibres are rational curves [31, Theorem 8.1]; \\n(3) if general fibres are $K 3$ surfaces or abelian varieties of dimension d by [13, Theorem 1.2]; then we have $m = 1 9 k$ , respectively $m = k ( d + 1 )$ , where $k$ is a weight associated to the Baily-Borel-Satake compactification of the period domain. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,503,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"54\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Travel Guide to the Canonical Bundle Formula \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[145,90,820,106]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the weaker version of the conjecture we lose control on the constant $m$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,115,876,153]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"B-Semiampleness Conjecture Let $f \\\\colon ( X , B ) \\\\to Y$ be an lc-trivial fibration. Then there exists an Ambro model $Y ^ { \\\\prime }$ of $f$ such that $M _ { Y ^ { \\\\prime } }$ is semiample. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,162,881,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"More is known about this conjecture than about its effective version above, although the progress has been limited to the cases where either the bases $Y$ or the fibres of $f$ are low dimensional. We summarise the situation for low dimensional fibres in the following result. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,241,876,279]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3.2 Apart from the cases listed in Theorem 3.1, the B-Semiampleness Conjecture holds in the following cases: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,286,881,360]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(1) if the fibres are surfaces of Kodaira dimension 0 by [13, Lemma 4.1 and Corollary 6.4] and [29, Part I, (5.15.9)(ii)]; \\n(2) if $f$ is a klt-trivial fibration and the generic fibre is a uniruled surface not isomorphic to $\\\\mathbb { P } ^ { 2 }$ by [8, Theorem 1.7]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,367,876,402]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The B-Semiampleness Conjecture holds for another important family of fibrations: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,412,879,468]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3.3 Let $f \\\\colon ( X , \\\\Delta ) \\\\to Y$ be an lc-trivial fibration, and assume that the \\nmoduli part $M _ { Y }$ descends to Y . If $M _ { Y } \\\\equiv 0$ , then $M _ { Y } \\\\sim _ { \\\\mathbb { Q } } 0$ . In particular, if dim $Y = 1$ , then $M _ { Y }$ is semiample. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,475,881,530]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3.3 is [3, Theorem 3.5] for klt-trivial fibrations and [10, Theorem 1.3] for lc-trivial fibrations. Theorem 1.2 in [10] states that Effective B-Semiampleness Conjecture is true for klt-trivial fibrations with numerically trivial moduli part. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,531,879,585]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Another partial result is [5, Theorem 3.2]. They prove that a small perturbation of the moduli part in a specific direction is semiample, under some effectivity hypotheses for $K _ { Y } + B _ { Y }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,630,441,650]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.1 Restrictions to Divisors \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,674,881,728]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As we saw in the previous subsection, the progress on the B-Semiampleness Conjecture has been concentrated on the cases of either the low dimension of the base $Y$ or the low dimension of the generic fibre of the fibration $f$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,729,881,800]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In [12, Theorem A] we obtained the following result towards the conjecture valid in every dimension. Note that the phrase the $B$ -semiampleness Conjecture in dimension $n$ means that we consider the conjecture in the case when the dimension of the base $Y$ is $n$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,809,817,827]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3.4 Assume the $B$ -Semiampleness Conjecture in dimension $n - 1$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,828,881,881]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $( X , \\\\Delta )$ be a log canonical pair and let $f \\\\colon ( X , \\\\Delta ) \\\\ \\\\to \\\\ Y$ be an lc-trivial fibration to an $n$ -dimensional variety $Y$ , where the divisor $\\\\Delta$ is effective over the generic point of Y . Assume that $Y$ is an Ambro model for $f$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[706,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Floris and V. Lazic´ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,54,141,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"55\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,879,144]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Then for every birational model $\\\\pi \\\\colon Y ^ { \\\\prime } \\\\to Y$ and for every prime divisor $T$ on $Y ^ { \\\\prime }$ with the normalisation $T ^ { \\\\nu }$ and the induced morphism $\\\\nu \\\\colon T ^ { \\\\nu } \\\\to Y ^ { \\\\prime }$ , the divisor $\\\\nu ^ { * } \\\\pi ^ { * } M _ { Y }$ is semiample on $T ^ { \\\\nu }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,153,879,205]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As a corollary, combining with Theorem 3.3, we obtain that the restriction of the moduli part to every prime divisor on every sufficiently high birational model of $Y$ is semiample if $Y$ is a surface. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,208,881,441]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We comment on the proof of Theorem 3.4, as it will be useful in the following subsection. We first apply a base change to $Y$ and modify $( X , \\\\Delta )$ by blowing up suitably, but we try to remember $( X , \\\\Delta )$ along the proof. We then run a suitable relative MMP over $Y$ , which contracts many “bad” components of $\\\\Delta$ (in particular, those with negative coefficients); as a result, we obtain a new lc-trivial fibration $g \\\\colon ( W , \\\\Delta _ { W } ) \\\\to Y$ with $\\\\Delta _ { W } \\\\geq 0$ and with the same moduli divisor $M _ { Y }$ . Choosing a minimal log canonical centre $s$ of $( W , \\\\Delta _ { W } )$ which surjects onto $T$ , we obtain an induced $k l t$ -trivial fibration $g | _ { S } \\\\colon ( S , \\\\Delta _ { S } ) \\\\to T ^ { \\\\prime }$ , where $T ^ { \\\\prime }$ is obtained from the Stein factorisation of the morphism $S \\\\to T$ . Then we first show that $M _ { Y } | _ { T ^ { \\\\prime } }$ is almost $M _ { T ^ { \\\\prime } }$ . Even though at this step we may not deduce equality between these two divisors, we can control their difference in a very precise manner. After a suitable further blowup of $Y$ , we can force this difference to disappear and we conclude by induction on the dimension. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,487,375,507]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.2 Reduction Result \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,531,879,584]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As we mentioned above, in the setup of lc-trivial fibrations $f \\\\colon ( X , \\\\Delta ) \\\\to Y$ one does not assume that $\\\\Delta$ is effective. Furthermore, often it is much more difficult to work with lc-trivial fibrations than with klt-trivial fibrations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,587,876,621]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In [12] the B-Semiampleness Conjecture is reduced to the following conjecture with much weaker hypotheses. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,630,881,684]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conjecture 3.5 Let $( X , \\\\Delta )$ be a log canonical pair and let $f \\\\colon ( X , \\\\Delta ) \\\\to Y$ be $a$ klt-trivial fibration to an $n$ -dimensional variety Y . If $Y$ is an Ambro model of $f$ and if the moduli divisor $M _ { Y }$ is big, then $M _ { Y }$ is semiample. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,693,419,711]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We show in [12, Theorem E]: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,720,876,756]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3.6 Assume Conjecture 3.5 in dimensions at most n. Then the $B$ - Semiampleness Conjecture holds in dimension $n$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,765,738,783]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The proof is similar to the proof of Theorem 3.4 sketched above. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,505,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"56\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Travel Guide to the Canonical Bundle Formula \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,348,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"57\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.3 Generalisation \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,132,879,222]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"57\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the papers [16, 20] the authors consider slc-trivial fibrations. Those are completely analogous to lc-trivial fibrations, the difference being that the ambient space $X$ is not irreducible, but the pair $( X , \\\\Delta )$ is slc on the generic fibre of the fibration; such a setup appears occasionally in inductive proofs. The precise statement is [16, Definition 4.1]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,223,881,294]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"57\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Then one can define, as in the case of lc-trivial fibrations, the moduli divisor and the discriminant. Then [16, Theorem 1.2] proves the analogue of Theorem 2.9 for slc-trivial fibrations, and [20, Theorem 1.3] shows the analogue of Theorem 3.3 in this context. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,340,403,361]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"57\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 Parabolic Fibrations \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,385,879,420]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"57\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally, in this section we discuss a more general situation than that of lc-trivial fibrations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,421,881,546]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"57\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $g \\\\colon ( X , \\\\Delta ) \\\\to Z$ be a surjective morphism, where $( X , \\\\Delta )$ is a klt projective pair and \\u0003 $Z$ is a projective variety. Assume that \\u0004 $g _ { * } \\\\mathcal { O } _ { X } \\\\big ( m ( K _ { X } + \\\\Delta ) \\\\big ) \\\\neq 0$ for some positive integer $m$ , and consider the relative Iitaka fibration $f \\\\colon X \\\\to Y$ associated to $K _ { X } + \\\\Delta$ . Possibly by blowing up further, one may assume that $( X , \\\\Delta )$ is log smooth and that $f$ is a morphism. Then if $F$ is a general fibre of $f$ , we have $\\\\kappa \\\\big ( F , ( K _ { X } + \\\\Delta ) | _ { F } \\\\big ) = 0$ , however $K _ { X } + \\\\Delta$ is not necessarily a pullback from $Y$ . One still wonders if there is a canonical bundle formula for the map $f$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,548,876,582]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"57\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The resulting formula is the canonical bundle formula of Fujino and Mori [19]. We first need a definition, which is justified from the setup above.\\u0003 \\u0004 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,591,879,648]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"57\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 4.1 A parabolic fibration is a fibration $f \\\\colon ( X , \\\\Delta ) \\\\to Y$ , where $( X , \\\\Delta )$ is a projective klt pair, $Y$ is a smooth projective variety and if $F$ is the generic fibre of $f$ , then $\\\\kappa \\\\big ( F , ( K _ { X } + \\\\Delta ) | _ { F } \\\\big ) = 0$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,656,876,690]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"57\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The following is [19, Section 4], the canonical bundle formula of Fujino and Mori. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,699,879,735]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"57\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 4.2 Let $f \\\\colon ( X , \\\\Delta ) \\\\to Y$ be a parabolic fibration, where $\\\\Delta$ is effective. Then there is a commutative diagram \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[428,752,569,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0253/4c866d0d811e56238dad330e6f3e6a516b17bf61e123e8c40e9129cd7f46c02e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"57\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { c } { { X ^ { \\\\prime } \\\\hfill ~ \\\\frac { \\\\tau ^ { \\\\prime } } { \\\\tau } X } } \\\\\\\\ { { f ^ { \\\\prime } \\\\hfill \\\\sqrt { \\\\hfill } } } \\\\\\\\ { { Y ^ { \\\\prime } \\\\hfill ~ \\\\frac { \\\\displaystyle \\\\vphantom { \\\\Bigg | } \\\\tau Y , } { \\\\tau Y , } } } \\\\end{array} \\\\} f \\\\hfill ~ \\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,864,876,900]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"57\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\tau$ and $\\\\tau ^ { \\\\prime }$ are birational, $X ^ { \\\\prime }$ and $Y ^ { \\\\prime }$ are smooth, and $f ^ { \\\\prime }$ has connected fibres, such that the following holds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"57\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"50 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[706,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"57\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Floris and V. Lazic´ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,876,127]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"58\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There exist effective $\\\\mathbb { Q }$ -divisors $B ^ { + }$ and $B ^ { - }$ on $X ^ { \\\\prime }$ without common components, a $\\\\mathbb { Q }$ -divisor $\\\\Delta ^ { \\\\prime } \\\\geq 0$ on $X ^ { \\\\prime }$ and $\\\\mathbb { Q }$ -divisors $B _ { Y ^ { \\\\prime } }$ and $M _ { Y ^ { \\\\prime } }$ on $Y$ such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[259,142,735,165]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0843/9c19159b852b3966883dfd7e24124ed04baf7fa748c5922d4259beb364b119d6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"58\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { X ^ { \\\\prime } } + \\\\Delta ^ { \\\\prime } + B ^ { - } \\\\sim _ { \\\\mathbb { Q } } f ^ { \\\\prime * } ( K _ { Y ^ { \\\\prime } } + B _ { Y ^ { \\\\prime } } + M _ { Y ^ { \\\\prime } } ) + B ^ { + } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,181,391,199]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"58\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"with the following properties: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[127,207,881,243]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"58\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(i) the pair $( { X } ^ { \\\\prime } , { \\\\Delta } ^ { \\\\prime } )$ is klt and log smooth, and there exists an effective exceptional divisor $E$ on $X ^ { \\\\prime }$ such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[364,259,678,282]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0553/a9fc79ed289bb35cdac51eebe2c39469b4ccbcbaba25bec1b98106f91b68c09d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"58\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { X ^ { \\\\prime } } + \\\\Delta ^ { \\\\prime } \\\\sim _ { \\\\mathbb { Q } } \\\\tau ^ { \\\\prime * } ( K _ { X } + \\\\Delta ) + E ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,297,883,406]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"58\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(ii) $f _ { * } ^ { \\\\prime } \\\\mathcal { O } _ { X ^ { \\\\prime } } ( \\\\lfloor n B ^ { + } \\\\rfloor ) \\\\simeq \\\\mathcal { O } _ { Y ^ { \\\\prime } }$ for all $n \\\\in \\\\mathbb { N }$ , \\n(iii) $B ^ { - }$ is $f ^ { \\\\prime }$ -exceptional and $\\\\tau ^ { \\\\prime }$ -exceptional, \\n$( i \\\\nu )$ the induced map $f ^ { \\\\prime } \\\\colon ( X ^ { \\\\prime } , \\\\Delta ^ { \\\\prime } + B ^ { - } - B ^ { + } ) \\\\to Y ^ { \\\\prime }$ is a klt-trivial fibration, and $B _ { Y ^ { \\\\prime } }$ and \\u0003 $M _ { Y ^ { \\\\prime } }$ are the corresponding discriminant and moduli divisors,\\u0004 \\u0003 \\u0004 \\u0003 \\n(v) the pair $( Y ^ { \\\\prime } , B _ { Y ^ { \\\\prime } } )$ is klt, $B _ { Y ^ { \\\\prime } } \\\\geq 0$ and $M _ { Y ^ { \\\\prime } }$ is nef, \\n$( \\\\nu i )$ for every $n \\\\in \\\\mathbb { N }$ sufficiently divisible we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[164,420,876,444]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0092/711db7d2151f1e4cad393e4a8d76e2268f01bc4664da3f2694a172cffa488ba3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"58\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nH ^ { 0 } \\\\big ( X , n ( K _ { X } + \\\\Delta ) \\\\big ) \\\\simeq H ^ { 0 } \\\\big ( X ^ { \\\\prime } , n ( K _ { X ^ { \\\\prime } } + \\\\Delta ^ { \\\\prime } ) \\\\big ) \\\\simeq H ^ { 0 } \\\\big ( Y ^ { \\\\prime } , n ( K _ { Y ^ { \\\\prime } } + B _ { Y ^ { \\\\prime } } + M _ { Y ^ { \\\\prime } } ) \\\\big ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,459,881,513]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"58\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There are several non-trivial parts of this formula which do not follow from considerations in the previous sections: the existence of divisors $B ^ { + }$ and $B ^ { - }$ with the properties (ii) and (iii) above, as well as the fact that $B _ { Y ^ { \\\\prime } }$ is effective. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,515,881,603]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"58\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Part (vi) follows immediately from (i), (ii) and (iii). We sketch how (iv) follows from (i), (ii) and (iii), following [1, Lemma 4.2]. Let $F ^ { \\\\prime }$ be a general fibre of\\u0003 $f ^ { \\\\prime }$ , and we define the divisor\\u0004 $\\\\Xi ^ { \\\\prime }$ with respect to $f ^ { \\\\prime } \\\\colon ( X ^ { \\\\prime } , \\\\Delta ^ { \\\\prime } + B ^ { - } - B ^ { + } ) \\\\to Y ^ { \\\\prime }$ as in Definition 2.1(d). We may assume that $\\\\Xi ^ { \\\\prime } = - \\\\Delta ^ { \\\\prime } - B ^ { - } + B ^ { + }$ , and $B ^ { - } | _ { F ^ { \\\\prime } } = 0$ by (iii). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,603,876,641]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"58\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We have $( K _ { X ^ { \\\\prime } } + \\\\Delta ^ { \\\\prime } + B ^ { - } - B ^ { + } ) | _ { F ^ { \\\\prime } } \\\\sim _ { \\\\mathbb { Q } } 0$ by construction and $\\\\kappa \\\\big ( F ^ { \\\\prime } , ( K _ { X ^ { \\\\prime } } +$ $\\\\Delta ^ { \\\\prime } ) | _ { F ^ { \\\\prime } } ) = 0$ by (i), hence \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[293,657,701,680]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0165/941c11613f306845af51c5651e8fb215ffe8d7feef117c90c83126016e63639c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"58\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\kappa ( F ^ { \\\\prime } , B ^ { + } | _ { F ^ { \\\\prime } } ) = \\\\kappa \\\\big ( F ^ { \\\\prime } , ( B ^ { + } - B ^ { - } ) | _ { F ^ { \\\\prime } } \\\\big ) = 0 .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,695,879,732]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"58\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since there exists a positive integer $b$ such that $\\\\lceil B ^ { + } \\\\rceil | _ { F ^ { \\\\prime } } \\\\leq b B ^ { + } | _ { F ^ { \\\\prime } }$ , this implies $\\\\kappa \\\\left( F ^ { \\\\prime } , \\\\lceil B ^ { + } \\\\rceil | _ { F ^ { \\\\prime } } \\\\right) = 0$ . Therefore,\\u0003 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[218,746,776,831]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0140/518eb59d144e683f55fa675c84c05e422bd574b3355f4506e633b02e55cdee65.jpg\",\n \"list_items\": null,\n \"page_idx\": \"58\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\kappa \\\\big ( F ^ { \\\\prime } , \\\\lceil \\\\Xi ^ { \\\\prime } \\\\rceil | _ { F ^ { \\\\prime } } \\\\big ) = \\\\kappa \\\\big ( F ^ { \\\\prime } , \\\\lceil - \\\\Delta ^ { \\\\prime } - B ^ { - } + B ^ { + } \\\\rceil | _ { F ^ { \\\\prime } } \\\\big ) } \\\\\\\\ & { \\\\qquad \\\\leq \\\\kappa \\\\big ( F ^ { \\\\prime } , \\\\lceil - \\\\Delta ^ { \\\\prime } \\\\rceil | _ { F ^ { \\\\prime } } + \\\\lceil - B ^ { - } \\\\rceil | _ { F ^ { \\\\prime } } + \\\\lceil B ^ { + } \\\\rceil | _ { F ^ { \\\\prime } } \\\\big ) } \\\\\\\\ & { \\\\qquad \\\\leq \\\\kappa \\\\big ( F ^ { \\\\prime } , \\\\lceil B ^ { + } \\\\rceil | _ { F ^ { \\\\prime } } \\\\big ) = 0 . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,845,642,863]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"58\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This shows part (d) of Definition 2.1, and the rest is easy. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,51,505,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"58\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Travel Guide to the Canonical Bundle Formula \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"58\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,879,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"59\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In order to apply the previous result, we recall that for a log canonical pair $( X , \\\\Delta )$ , the ring \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[289,139,710,181]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0231/d195f4cce33a933c5ba0c15726082bd2dd8fd0c88c959dd3ec7334704bba9e08.jpg\",\n \"list_items\": null,\n \"page_idx\": \"59\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nR ( X , K _ { X } + \\\\Delta ) = \\\\bigoplus _ { n \\\\in \\\\mathbb { N } } H ^ { 0 } \\\\big ( X , \\\\lfloor n ( K _ { X } + \\\\Delta ) \\\\rfloor \\\\big )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,196,881,256]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"59\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is the canonical ring of $( X , \\\\Delta )$ . We also recall that for a graded ring $R = \\\\bigoplus _ { \\\\mathrm { ~ \\\\tiny ~ R ~ } } R _ { n }$ , the $d$ -th Veronese subring of $R$ is defined as $R ^ { ( d ) } : = \\\\bigoplus _ { n \\\\in \\\\mathbb { N } } R _ { d n } .$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,256,881,328]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"59\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An immediate consequence of Theorem 4.2 is the following result [19, Theorem 5.2], which has widespread use in the Minimal Model Program. It often allows to pass from a pair $( X , \\\\Delta )$ with $\\\\kappa ( X , K _ { X } + \\\\Delta ) \\\\geq 0$ to a pair $( { X } ^ { \\\\prime } , { \\\\Delta } ^ { \\\\prime } )$ on which $K _ { X ^ { \\\\prime } } + \\\\Delta ^ { \\\\prime }$ is big. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,336,881,391]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"59\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 4.3 Let $( X , \\\\Delta )$ be a projective klt pair with $\\\\kappa ( K _ { X } + \\\\Delta ) = \\\\ell \\\\geq 0 .$ . Then there exist an $\\\\ell$ -dimensional klt pair $( { X } ^ { \\\\prime } , { \\\\Delta } ^ { \\\\prime } )$ with $\\\\kappa ( X ^ { \\\\prime } , K _ { X ^ { \\\\prime } } + \\\\Delta ^ { \\\\prime } ) = \\\\ell$ and positive integers $d$ and $d ^ { \\\\prime }$ such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[302,406,697,429]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0971/ba45c7011bbe9528c5fa28e9ef14ede116aa2307de3080269fe0c38d59a51e23.jpg\",\n \"list_items\": null,\n \"page_idx\": \"59\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nR ( X , K _ { X } + \\\\Delta ) ^ { ( d ) } \\\\simeq R ( X ^ { \\\\prime } , K _ { X ^ { \\\\prime } } + \\\\Delta ^ { \\\\prime } ) ^ { ( d ^ { \\\\prime } ) } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,445,876,481]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"59\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The proof follows immediately from Theorem 4.2(vi), by combining it with the\\u0004 proof of Theorem 2.16; see also the proof of Theorem 4.4 below. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,483,881,590]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"59\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Assume now that $( X , \\\\Delta )$ has simple normal crossings and let $\\\\Delta ^ { + }$ and $\\\\Delta ^ { - }$ be effective divisors without common components such that $\\\\Delta = \\\\Delta ^ { + } - \\\\Delta ^ { - }$ . Then if $\\\\kappa \\\\left( F , ( K _ { X } + \\\\Delta ^ { + } ) | _ { F } \\\\right) = 0$ for a general fibre $F$ of $f$ , and if there exists a good model of $( F , \\\\Delta ^ { + } | _ { F } )$ , then it was shown in [15, Theorem 3.13] that the moduli b-divisor is b-nef and good in the sense of [3, Definition 3.2]; this is an application of MMP techniques from [18] and [3, Theorem 3.3]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,591,881,680]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"59\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We finish the paper with the following result, which can sometimes be used in order to avoid running a Minimal Model Program; for instance, compare the proofs of [21, Lemma 4.4] and [28, Theorem 5.3]. Note that $\\\\nu ( X , L )$ denotes the numerical dimension of a divisor $L$ on a projective variety $X$ , see for instance [28, $\\\\ S 2 . 2 ]$ for basic properties and related references. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,689,881,744]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"59\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 4.4 Let $( X , \\\\Delta )$ be a projective klt pair and let $f \\\\colon ( X , \\\\Delta ) \\\\ \\\\to \\\\ Y$ be $a$ parabolic fibration such that $\\\\nu \\\\big ( F , ( K _ { X } + \\\\Delta ) | _ { F } \\\\big ) = 0$ for $a$ general fibre $F$ of $f$ . Then there exists a commutative diagram \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[428,758,571,849]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0297/34fa4a108075e61e5bd2eacc5c6d0e26854bf285eef3075ff45df24761205874.jpg\",\n \"list_items\": null,\n \"page_idx\": \"59\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { c } { { X ^ { \\\\prime } \\\\xrightarrow { \\\\pi ^ { \\\\prime } } \\\\times X } } \\\\\\\\ { { f ^ { \\\\prime } \\\\Biggl \\\\downarrow } } \\\\\\\\ { { Y ^ { \\\\prime } \\\\xrightarrow { } Y , } } \\\\end{array} \\\\downarrow { } f\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[706,51,879,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"59\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Floris and V. Lazic´ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"59\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"52 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,881,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $X ^ { \\\\prime }$ and $Y ^ { \\\\prime }$ are smooth, $f ^ { \\\\prime }$ has connected fibres, $\\\\pi$ and $\\\\pi ^ { \\\\prime }$ are birational, and such that, if we write \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[337,142,658,163]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0955/d8b8f377c8bbf9f04a7e88057bf289988fffeaff7f789d391de4a1bcdd3e8056.jpg\",\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { X ^ { \\\\prime } } + \\\\Delta ^ { \\\\prime } \\\\sim _ { \\\\mathbb { Q } } \\\\pi ^ { \\\\prime * } ( K _ { X } + \\\\Delta ) + E ^ { \\\\prime } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,180,608,198]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\Delta ^ { \\\\prime }$ and $E ^ { \\\\prime }$ have no common components, then: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[129,208,246,225]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(i) we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[314,241,724,264]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0136/38422ac441c437e60567f406b2b700a711b908590e4afd766882d839a4c5d84a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { X ^ { \\\\prime } } + \\\\Delta ^ { \\\\prime } + B ^ { - } \\\\sim _ { \\\\mathbb { Q } } f ^ { \\\\prime * } ( K _ { Y ^ { \\\\prime } } + \\\\Delta _ { Y ^ { \\\\prime } } ) + B ^ { + } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[159,279,881,315]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where the pair $( Y ^ { \\\\prime } , \\\\Delta _ { Y ^ { \\\\prime } } )$ is klt, and the divisors $B ^ { + }$ and $B ^ { - }$ are effective and have no common components, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,315,692,352]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(ii) $B ^ { - }$ is $\\\\pi ^ { \\\\prime }$ -exceptional and $f ^ { \\\\prime }$ -exceptional, (iii) we have $f _ { * } ^ { \\\\prime } { \\\\mathcal { O } } _ { X ^ { \\\\prime } } ( \\\\lfloor \\\\ell B ^ { + } \\\\rfloor ) \\\\simeq { \\\\mathcal { O } } _ { Y ^ { \\\\prime } }$ for all positive integers $\\\\ell$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[111,360,808,379]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moreover, if $K _ { X } + \\\\Delta$ is pseudoeffective, then \\u0004 \\u0003 $K _ { Y ^ { \\\\prime } } + \\\\Delta _ { Y ^ { \\\\prime } }$ is pseudoeffective.\\u0004 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,387,701,405]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof By [6, Corollaire 3.4] and [30, Corollary V.4.9] we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[280,420,717,444]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0084/1febfd3b0e665f2973bd8b9762e8f794e5ed3a4e78508be12a0d0d9933cdd552.jpg\",\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { \\\\kappa \\\\Big ( F , ( K _ { X } + \\\\Delta ) | _ { F } \\\\Big ) = \\\\nu \\\\Big ( F , ( K _ { X } + \\\\Delta ) | _ { F } \\\\Big ) = 0 . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,459,881,495]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By Theorem 4.2 there exists a diagram as in the theorem such that (ii) and (iii) hold, as well as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[259,510,735,534]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0970/79d349015cc553890e431e6ae3cfe7f175a587c49860f5ea063032fdb813b606.jpg\",\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { X ^ { \\\\prime } } + \\\\Delta ^ { \\\\prime } + B ^ { - } \\\\sim _ { \\\\mathbb { Q } } f ^ { \\\\prime * } ( K _ { Y ^ { \\\\prime } } + B _ { Y ^ { \\\\prime } } + M _ { Y ^ { \\\\prime } } ) + B ^ { + } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,548,881,621]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $( Y ^ { \\\\prime } , B _ { Y ^ { \\\\prime } } )$ is klt and $M _ { Y ^ { \\\\prime } }$ is the moduli part of the associated klt-trivial fibration\\u0003 \\u0004 $f ^ { \\\\prime } \\\\colon ( X ^ { \\\\prime } , \\\\Delta ^ { \\\\prime } + B ^ { - } - B ^ { + } ) \\\\to Y ^ { \\\\prime }$ . Then analogously as in the proof of Theorem 2.16 one shows that there exists an effective $\\\\mathbb { Q }$ -divisor $\\\\Delta _ { Y ^ { \\\\prime } } \\\\sim _ { \\\\mathbb { Q } } B _ { Y ^ { \\\\prime } } + M _ { Y ^ { \\\\prime } }$ such that the pair $( Y ^ { \\\\prime } , \\\\Delta _ { Y ^ { \\\\prime } } )$ is klt, which gives (i). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,621,881,747]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally, if $F ^ { \\\\prime }$ is a general fibre of $f ^ { \\\\prime }$ , we have $\\\\nu \\\\big ( F ^ { \\\\prime } , ( K _ { X ^ { \\\\prime } } + \\\\Delta ^ { \\\\prime } ) | _ { F ^ { \\\\prime } } \\\\big ) = 0$ by (2) and by [28, Lemma 2.3]. Therefore, there exists a good model of $( F ^ { \\\\prime } , \\\\Delta ^ { \\\\prime } | _ { F ^ { \\\\prime } } )$ by [6, Corollaire 3.4] and [30, Corollary V.4.9], hence $( K _ { X ^ { \\\\prime } } + \\\\Delta ^ { \\\\prime } ) | _ { F ^ { \\\\prime } }$ is geometrically abundant in the sense of [30, Definition V.2.23]. Then by [30, Lemma V.2.27] for an ample divisor\\u0003 $A$ on $Y ^ { \\\\prime }$ and for any positive rational number\\u0003 $\\\\varepsilon$ , the divisor\\u0004 $K _ { X ^ { \\\\prime } } +$ $\\\\Delta ^ { \\\\prime } + \\\\varepsilon f ^ { \\\\prime * } A$ is geometrically abundant. In particular, $\\\\kappa ( X ^ { \\\\prime } , K _ { X ^ { \\\\prime } } + \\\\Delta ^ { \\\\prime } + \\\\varepsilon f ^ { \\\\prime * } A ) \\\\ge 0$ , and hence by (i) and by (iii) we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[198,761,799,815]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0360/0a4f5ed4d27a6b4dff6bee6f897f0eb56b0ee59484bd385aa542511b663fa0d6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\kappa \\\\big ( Y ^ { \\\\prime } , K _ { Y ^ { \\\\prime } } + \\\\Delta _ { Y ^ { \\\\prime } } + \\\\varepsilon A ) = \\\\kappa \\\\big ( X ^ { \\\\prime } , f ^ { \\\\prime * } ( K _ { Y ^ { \\\\prime } } + \\\\Delta _ { Y ^ { \\\\prime } } + \\\\varepsilon A ) + B ^ { + } \\\\big ) } \\\\\\\\ & { ~ } \\\\\\\\ & { ~ = \\\\kappa ( X ^ { \\\\prime } , K _ { X ^ { \\\\prime } } + \\\\Delta ^ { \\\\prime } + B ^ { - } + \\\\varepsilon f ^ { \\\\prime * } A ) \\\\ge 0 . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,830,881,866]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since this holds for any positive rational number $\\\\varepsilon$ , we conclude that $K _ { Y ^ { \\\\prime } } + \\\\Delta _ { Y ^ { \\\\prime } }$ is pseudoeffective, as desired. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,51,505,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Travel Guide to the Canonical Bundle Formula \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,52,879,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"60\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"53 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,250,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"61\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,136,883,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"61\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Ambro, F.: Nef dimension of minimal models. Math. Ann. 330(2), 309–322 (2004) \\n2. Ambro, F.: Shokurov’s boundary property. J. Differ. Geom. 67(2), 229–255 (2004) \\n3. Ambro, F.: The moduli $^ b$ -divisor of an lc-trivial fibration. Compos. Math. 141(2), 385–403 (2005) \\n4. Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. In: Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 4. Springer, Berlin (1984) \\n5. Birkar, C., Chen, Y.: Images of manifolds with semi-ample anti-canonical divisor. J. Algebraic Geom. 25(2), 273–287 (2016) \\n6. Druel, S.: Quelques remarques sur la décomposition de Zariski divisorielle sur les variétés dont la première classe de Chern est nulle. Math. Z. 267(1–2), 413–423 (2011) \\n7. Favre, C.: Note on pull-back and Lelong number of currents. Bull. Soc. Math. France 127(3), 445–458 (1999) \\n8. Filipazzi, S.: On a generalized canonical bundle formula and generalized adjunction. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2018, to appear). arXiv:1807.04847 \\n9. Floris, E.: Bounds on the denominators in the canonical bundle formula. Ann. Inst. Fourier (Grenoble) 63(5), 1951–1969 (2013) \\n10. Floris, E.: Inductive approach to effective b-semiampleness. Int. Math. Res. Not. IMRN 2014(6), 1465–1492 (2014) \\n11. Floris, E.: Nefness: generalization to the lc case. C. R. Math. Acad. Sci. Paris 352(4), 333–337 (2014) \\n12. Floris, E., Lazic, V.: On the B-semiampleness conjecture. Épijournal Géom. Algébrique ´ 3, Article No. 12 (2019) \\n13. Fujino, O.: A canonical bundle formula for certain algebraic fiber spaces and its applications. Nagoya Math. J. 172, 129–171 (2003) \\n14. Fujino, O.: Theory of non-lc ideal sheaves: basic properties. Kyoto J. Math. 50(2), 225–245 (2010) \\n15. Fujino, O.: Some remarks on the minimal model program for log canonical pairs. J. Math. Sci. Univ. Tokyo 22(1), 149–192 (2015) \\n16. Fujino, O.: Fundamental properties of basic slc-trivial fibrations (2018). arXiv:1804.11134 \\n17. Fujino, O., Fujisawa, T.: On semipositivity theorems (2017). arXiv:1701.02039 \\n18. Fujino, O., Gongyo, Y.: On the moduli b-divisors of lc-trivial fibrations. Ann. Inst. Fourier 64(4), 1721–1735 (2014) \\n19. Fujino, O., Mori, S.: A canonical bundle formula. J. Differ. Geom. 56(1), 167–188 (2000) \\n20. Fujino, O., Fujisawa, T., Liu, H.: Fundamental properties of basic slc-trivial fibrations, II (2018). arXiv:1808.10604 \\n21. Gongyo, Y., Lehmann, B.: Reduction maps and minimal model theory. Compos. Math. 149(2), 295–308 (2013) \\n22. Kawamata, Y.: Characterization of abelian varieties. Compos. Math. 43(2), 253–276 (1981) \\n23. Kawamata, Y.: Subadjunction of log canonical divisors. II. Am. J. Math. 120(5), 893–899 (1998) \\n24. Kodaira, K.: On compact complex analytic surfaces. II. Ann. Math. (3) 77, 563–626 (1963) \\n25. Kollár, J.: Kodaira’s canonical bundle formula and adjunction. In: Corti, A. (ed.) Flips for 3- Folds and 4-Folds. Oxford Lecture Series in Mathematics and Its Applications, vol. 35, Oxford University Press, Oxford, pp. 134–162 (2007) \\n26. Kollár, J.: Singularities of the minimal model program. Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge (2013). With a collaboration of Sándor Kovács \\n27. Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge (1998) \\n28. Lazic, V., Peternell, Th.: On Generalised Abundance, I. Publ. Res. Inst. Math. Sci. (2018, to ´ appear). arXiv:1808.00438 \\n29. Mori, S.: Classification of higher-dimensional varieties. In: Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proceedings of Symposia in Pure Mathematics, vol. 46, pp. 269–331. American Mathematical Society, Providence (1987) \\n30. Nakayama, N.: Zariski-Decomposition and Abundance. MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo (2004) \\n31. Prokhorov, Yu.G., Shokurov, V.V.: Towards the second main theorem on complements. J. Algebraic Geom. 18(1), 151–199 (2009) \\n32. Ueno, K.: Classification of algebraic varieties. I. Compos. Math. 27, 277–342 (1973) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,54,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"61\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[706,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"61\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Floris and V. Lazic´ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,881,213]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"62\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,505,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"62\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Travel Guide to the Canonical Bundle Formula \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"62\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,747,166]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"63\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,202,362,220]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"63\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Anne-Sophie Kaloghiros \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,288,879,358]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"63\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract It is known that a maximal intersection log canonical Calabi–Yau surface pair is crepant birational to a toric pair. This does not hold in higher dimension: this article presents some examples of maximal intersection Calabi–Yau pairs that admit no toric model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,378,738,394]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"63\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords Calabi–Yau $\\\\cdot$ Fano varieties $\\\\cdot$ Minimal model program \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,432,496,453]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"63\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 Introduction and Motivation \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,477,876,512]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"63\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Calabi–Yau (CY) pair $( X , D _ { X } )$ consists of a normal projective variety $X$ and a reduced sum of integral Weil divisors $D _ { X }$ such that $K _ { X } + D _ { X } \\\\sim _ { \\\\mathbb { Z } } 0$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,513,881,602]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"63\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The class of CY pairs arises naturally in a number of problems and comprises examples with very different birational geometry. Indeed, on the one hand, a Gorenstein Calabi–Yau variety $X$ can be identified with the CY pair $( X , 0 )$ . On the other hand, if $X$ is a Fano variety, and if $D _ { X }$ is an effective reduced anticanonical divisor, then $( X , D _ { X } )$ is also a CY pair. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,611,234,629]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"63\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 1 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,639,881,732]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"63\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(a) A pair $( X , D _ { X } )$ is (t, dlt) (resp. $\\\\displaystyle ( \\\\mathbf { t } , \\\\mathbf { l } \\\\mathbf { c } ) )$ if $X$ is $\\\\mathbb { Q }$ -factorial, terminal and $( X , D _ { X } )$ divisorially log terminal (resp. log canonical). \\n(b) A birational map $( X , D _ { X } ) \\\\stackrel { \\\\varphi } { \\\\quad - - } ( Y , D _ { Y } )$ is volume preserving or crepant\\b birational if $a _ { E } ( K _ { X } + D _ { X } ) = a _ { E } ( K _ { Y } + D _ { Y } )$ for every geometric valuation $E$ with centre on $X$ and on $Y$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,743,881,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"63\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The dual complex of a dlt pair $( Z , D z = \\\\sum D _ { i } )$ is the regular cell complex obtained by attaching an $( | I | - 1 )$ -dimensional cell for every irreducible component of a non-empty intersection $\\\\cap _ { i \\\\in I } D _ { i }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,834,769,879]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"63\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A.-S. Kaloghiros $( \\\\boxtimes )$ Department of Mathematics, Brunel University London, Uxbridge, UK e-mail: anne-sophie.kaloghiros@brunel.ac.uk; annesophie.kaloghiros@brunel.ac.uk \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,897,690,941]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"63\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_5 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[804,91,876,139]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"63\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,899,876,909]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"63\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,876,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"64\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The dual complex encodes the combinatorics of the lc centres of a dlt pair and [5]- shows that its PL homeomorphism class is a volume preserving birational invariant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,126,879,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"64\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By [3, Theorem 1.9], a (t, lc) CY pair $( X , D _ { X } )$ has a volume preserving (t, dlt) modification $( \\\\widetilde { X } , D _ { \\\\tilde { X } } ) ( X , D _ { X } )$ , and the birational map between two such modifications is volume preserving. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,180,876,216]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"64\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abusing notation, I call dual complex the following volume preserving birational invariant of a (t, lc) CY pair $( X , D _ { X } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,223,876,261]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"64\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 2 ${ \\\\mathcal { D } } ( X , D _ { X } )$ is the PL homeomorphism class of the dual complex of a volume preserving (t, dlt) modification of $( X , D _ { X } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,270,881,342]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"64\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As the underlying varieties of CY pairs range from CY to Fano varieties, they can have very different birational properties. However, $X$ being Fano is not a volume preserving birational invariant of the pair $( X , D _ { X } )$ . Following [14], I consider the following volume preserving birational invariant notion: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,351,881,385]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"64\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3 A (t, lc) CY pair $( X , D _ { X } )$ has maximal intersection if dim $\\\\phantom { } _ { 1 } \\\\mathscr { D } ( X , D _ { X } )$ $= \\\\dim X - 1$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,394,881,468]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"64\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In other words, $( X , D _ { X } )$ has maximal intersection if there is a volume preserving (t, dlt) modification of $( X , D _ { X } )$ with a 0-dimensional log canonical centre. Maximal intersection CY pairs have some Fano-type properties; Kollár and $\\\\mathrm { X u }$ show the following: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,475,756,495]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"64\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 1 Let $( X , D _ { X } )$ be a dlt maximal intersection CY pair, then: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,503,881,561]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"64\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. [14, Proposition 19] $X$ is rationally connected, \\n2. [14, Theorem 21] there is a volume preserving map $( X , D _ { X } ) \\\\mathrel { - } \\\\stackrel { \\\\varphi } { } ( Z , D _ { Z } )$ such that $D _ { Z }$ fully supports a big and semiample divisor. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,570,881,642]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"64\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark $I$ The expression “Fano-type” should be understood with a pinch of salt. Having maximal intersection is a degenerate condition: a general (t, lc) CY pair $( X , D _ { X } )$ with $X$ Fano and $D _ { X }$ a reduced anticanonical section need not have maximal intersection. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,651,881,686]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"64\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 4 A toric pair $( X , D _ { X } )$ is a (t, lc) CY pair formed by a toric variety and the reduced sum of toric invariant divisors. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,689,760,705]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"64\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A toric model is a volume preserving birational map to a toric pair. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,716,211,732]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"64\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 2 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,743,881,815]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"64\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(a) If $X$ is a normal toric variety and $D _ { X }$ the reduced sum of toric invariant divisors, $( X , D _ { X } )$ is log canonical [4, Corollary 11.4.25], so that $( X , D _ { X } )$ is a toric pair precisely when $X$ is terminal and $\\\\mathbb { Q }$ -factorial. \\n(b) If a CY pair $( X , D _ { X } )$ has a toric model, $X$ is rational. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[744,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"64\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A.-S. Kaloghiros \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,141,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"64\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[141,108,765,127]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"65\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 1 A CY pair with a toric model has maximal intersection. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,163,876,199]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"65\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 3 In dimension 2, the converse holds: maximal intersection CY surface pairs are precisely those with a toric model [7, Proposition 1.3]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,208,881,298]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"65\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The characterisation of CY pairs with a toric model is an open and difficult problem. A characterisation of toric pairs was conjectured by Shokurov and is proved in [1], but it is not clear how to refine it to get information on the existence of a toric model. A motivation to better understand the birational geometry of CY pairs and their relation to toric pairs comes from mirror symmetry. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,300,881,408]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"65\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The mirror conjecture extends from a duality between Calabi–Yau varieties to a correspondence between Fano varieties and Landau-Ginzburg models, i.e. noncompact Kähler manifolds endowed with a superpotential. Most known constructions of mirror partners rely on toric features such as the existence of a toric model or of a toric degeneration. In an exciting development, Gross, Hacking and Keel conjecture the following construction for mirrors of maximal intersection CY pairs. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,415,879,487]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"65\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conjecture 1 ([7, Section 0.4] [8, Conjecture 1.9]) Let $( Y , D _ { Y } )$ be a simple normal crossings maximal intersection CY pair and denote by \\u000e $U$ the complement \\u000f $Y \\\\setminus D _ { Y }$ . Assume that $D _ { Y }$ supports an ample divisor, let $R$ be the ring $k [ { \\\\mathrm { P i c } } ( Y ) ^ { \\\\times } ] ,$ $\\\\Omega$ the canonical volume form on $U$ and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,498,851,531]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0867/5b003161e76c67b2060fc053d7b234f011a484f598aff3ffe5c409abb7152469.jpg\",\n \"list_items\": null,\n \"page_idx\": \"65\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nU ^ { \\\\mathrm { t r o p } } ( \\\\mathbb { Z } ) = \\\\{ \\\\substack { \\\\mathrm { d i v i s o r i a l ~ v a l u a t i o n s } \\\\upsilon : k ( U ) \\\\setminus \\\\{ 0 \\\\} \\\\mathbb { Z } \\\\mathrm { ~ w i t h ~ } \\\\upsilon ( \\\\Omega ) < 0 } \\\\} \\\\cup \\\\{ 0 \\\\} .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,546,879,600]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"65\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Then, the free $R$ -module $V$ with basis $U ^ { \\\\mathrm { t r o p } } ( \\\\mathbb { Z } )$ has a natural finitely generated $R$ - algebra structure whose structure constants are non-negative integers determined by counts of rational curves on $U$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[143,602,690,618]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"65\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Denote by $K$ the torus $\\\\operatorname { K e r } \\\\{ \\\\operatorname { P i c Y } \\\\to \\\\operatorname { P i c } ( U ) \\\\}$ . The fibration \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[339,636,658,657]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0675/33c5f4dfcf39a537e9a6b19ceaa30009ff90052d9633bb8865129f69c0cd2df2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"65\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\np \\\\colon \\\\operatorname { S p e c } ( V ) \\\\to \\\\operatorname { S p e c } ( R ) = T _ { \\\\operatorname { P i c } ( Y ) }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,672,879,708]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"65\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is a $T _ { K }$ -equivariant flat family of affine maximal intersection log CY varieties. The quotient \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[384,726,612,747]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0146/477cacd943c482502b25c241ef98c37229fb3dcb940d25065ea0bb10b1273b5a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"65\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname { S p e c } ( V ) / T _ { K } \\\\to T _ { \\\\operatorname { P i c } ( U ) }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,762,571,780]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"65\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"only depends on $U$ and is the mirror family of $U$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,789,879,825]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"65\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Versions of Conjecture 1 are proved for cluster varieties in [9], but relatively few examples are known. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,827,881,899]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"65\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This note presents examples of maximal intersection CY pairs that do not admit a toric model and for which one can hope to construct the mirror partner proposed in Conjecture 1. Table 1 summarises the properties of the Examples of (t, lc) CY pairs constructed in Sects. 3 and 4. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,52,854,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"65\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"65\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,111,879,331]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0832/ed83a89cfa1b694271cb8e4c7388662f31b055540605ac00b85f89cfaa45c5ed.jpg\",\n \"list_items\": null,\n \"page_idx\": \"66\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=1 colspan=1>Boundary</td><td rowspan=1 colspan=1>Reference</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>Sing(Dx)</td><td rowspan=1 colspan=1>dim D(X,Dx)</td></tr><tr><td rowspan=5 colspan=1>Normal</td><td rowspan=1 colspan=1>Example 2</td><td rowspan=1 colspan=1>Smooth X4 C P4</td><td rowspan=1 colspan=1>Cusp T4,4.4</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=2 colspan=1>Example 3</td><td rowspan=1 colspan=1>Nodal X4 C P4</td><td rowspan=1 colspan=1>Cusp T3,3.4 and</td><td rowspan=2 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>#Sing(X)=3</td><td rowspan=1 colspan=1>3odps</td></tr><tr><td rowspan=1 colspan=1>Example 4</td><td rowspan=1 colspan=1>Smooth X4 C P4</td><td rowspan=1 colspan=1>Cusp T3.4.4</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>Example 8</td><td rowspan=1 colspan=1>Smooth X4 C P4</td><td rowspan=1 colspan=1>Two cusps T2.3.6</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=6 colspan=1>Non-normal</td><td rowspan=2 colspan=1>Example 5</td><td rowspan=2 colspan=1>Smooth X3 C P4</td><td rowspan=1 colspan=1>Nodal plane cubic curve</td><td rowspan=2 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>Double pinch point at node</td></tr><tr><td rowspan=2 colspan=1>Example 6</td><td rowspan=1 colspan=1>Nodal X4 C P4</td><td rowspan=1 colspan=1>LUL'</td><td rowspan=2 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>#Sing(X)=6</td><td rowspan=1 colspan=1>Double pinch point at L ∩ L'</td></tr><tr><td rowspan=2 colspan=1>Example 7</td><td rowspan=1 colspan=1>Nodal X4 C P4</td><td rowspan=1 colspan=1>LUL'</td><td rowspan=2 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>#Sing(X）=6</td><td rowspan=1 colspan=1>LnL'=0</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 1 Examples of (t, lc) CY pairs that do not admit a toric model \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[118,363,635,384]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"66\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 Auxiliary Results on Threefold CY Pairs \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,406,881,534]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"66\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The examples in Sect. 3 are threefold maximal intersection CY pairs whose underlying varieties are birationally rigid. In particular, such pairs admit no toric model; this shows that the results in [7] on maximal intersection surface CY pairs do not extend to higher dimensions. In this section, I first recall some results on birational rigidity of Fano threefolds. Then, I introduce the (t, dlt) modifications suited to the construction outlined in Conjecture 1 and discuss the singularities of the boundary $D _ { X }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,579,394,600]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"66\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.1 Birational Rigidity \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,623,879,659]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"66\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $X$ be a terminal $\\\\mathbb { Q }$ -factorial Fano threefold. When $X$ has Picard rank 1, $X$ is a Mori fibre space, i.e. an end product of the classical MMP. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,669,879,710]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"66\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 5 A birational map $Y / S \\\\stackrel { \\\\varphi } { \\\\mathrm { \\\\tiny ~ - - } } \\\\mathrm { ~ \\\\nabla ~ } Y ^ { \\\\prime } / S ^ { \\\\prime }$ between Mori fibre spaces $Y / S$ and $Y ^ { \\\\prime } / S ^ { \\\\prime }$ is square if it fits into a commutative square \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[398,726,599,807]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0829/b0aaca31191cdf991d02203f69613254d38a4b95a6b4f832b72c0d68878a092a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"66\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { l } { Y \\\\mathrm { ~ --- ~ } - \\\\mathrm { ~ ! ~ } \\\\underline { { \\\\varphi } } \\\\mathrm { ~ --- ~ } - \\\\mathrm { ~ ! ~ } Y ^ { \\\\prime } } \\\\\\\\ { \\\\Big \\\\downarrow } \\\\\\\\ { \\\\begin{array} { r } { \\\\underline { { \\\\begin{array} { r } { \\\\begin{array} { r } { \\\\begin{array} { r } { \\\\begin{array} { r } { \\\\end{array} } } \\\\end{array} } \\\\end{array} } } } \\\\\\\\ { S \\\\mathrm { ~ --- ~ } - \\\\mathrm { ~ ! ~ } \\\\underline { { g } } \\\\mathrm { ~ --- ~ } - \\\\mathrm { ~ } \\\\underline { { \\\\succ } } \\\\mathrm { ~ ^ { \\\\prime } ~ } } \\\\end{array} } } \\\\end{array} \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,831,879,873]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"66\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $g$ is birational and the restriction $Y _ { \\\\eta } \\\\stackrel { \\\\varphi _ { \\\\eta } } { \\\\ - - \\\\ + \\\\ } Y _ { \\\\eta } ^ { \\\\prime }$ is biregular, where $\\\\eta$ is the function field of the base $k ( S )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[744,54,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"66\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A.-S. Kaloghiros \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"66\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"60 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,94,879,154]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"67\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Mori fibre space $Y / S$ is (birationally) rigid if for every birational map $Y / S \\\\stackrel { \\\\varphi } { - - } $ $Y ^ { \\\\prime } / S ^ { \\\\prime }$ to another Mori fibre space, there is a birational self map $Y / S \\\\stackrel { \\\\alpha } { \\\\longrightarrow } X / S$ such that $\\\\varphi \\\\circ \\\\alpha$ is square. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,162,876,198]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"67\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In particular, if $X$ is a rigid Mori fibre space, then $X$ is non-rational and no (t, lc) CY pair $( X , D _ { X } )$ admits a toric model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,199,879,253]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"67\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Non-singular quartic hypersurfaces $X _ { 4 } ~ \\\\subset ~ \\\\mathbb { P } ^ { 4 }$ are probably the most famous examples of birationally rigid threefolds [11]. Some mildly singular quartic hypersurfaces are also known to be birationally rigid, in particular, we have: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,261,879,316]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"67\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 1 ([2, 18]) Let $X _ { 4 } \\\\subset \\\\mathbb { P } ^ { 4 }$ be a quartic hypersurface with no worse than ordinary double points. If $| \\\\mathrm { S i n g } ( X ) | \\\\leq 8$ , then $X$ is $\\\\mathbb { Q }$ -factorial (in particular, $X$ is a Mori fibre space) and is birationally rigid. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,361,517,382]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"67\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2 Singularities of the Boundary \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,405,881,498]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"67\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"I now state some results on the singularities of the boundary of a threefold (t, lc) CY pair. Let $( X , D _ { X } )$ be a threefold (t, lc) CY pair and $( \\\\widetilde { X } , \\\\overbrace { D _ { \\\\widetilde { X } } } )$ a (t, dlt) modification. A stratum of $( \\\\widetilde { X } , D _ { \\\\widetilde { X } } )$ is an irreducible component of a non-empty intersection of\\u0003 \\u0004 components of $D _ { \\\\tilde { X } }$ . Given a stratum $W$ , there is a divisor $\\\\mathrm { D i f f } _ { W } D _ { \\\\tilde { X } }$ on $W$ such that $( W , \\\\mathrm { D i f f } _ { W } D _ { \\\\tilde { X } } )$ is a lc CY pair and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[328,513,671,539]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0203/2f1e8cb4d65ef07b4e148bc4c0b7d80cc8892f4692ea581278ee737e533f34fe.jpg\",\n \"list_items\": null,\n \"page_idx\": \"67\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { W } + \\\\cal { D } \\\\mathrm { i f f } _ { W } { \\\\cal D } _ { \\\\tilde { X } } \\\\sim _ { \\\\mathbb { Q } } \\\\left( K _ { \\\\tilde { X } } + { \\\\cal D } _ { \\\\tilde { X } } \\\\right) _ { | W } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,552,876,590]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"67\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"When $K _ { \\\\tilde { X } } + D _ { \\\\tilde { X } }$ is Cartier and $D _ { \\\\tilde { X } }$ reduced, $\\\\mathrm { D i f f } _ { W } D _ { \\\\tilde { X } }$ is the sum of the restrictions of the components of $D _ { \\\\tilde { X } }$ that do not contain $W$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,590,879,660]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"67\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In particular, for any irreducible component $s$ of $D _ { \\\\tilde { X } }$ , the link of S in ${ \\\\mathcal { D } } ( X , D _ { X } )$ is the dual complex $\\\\mathcal { D } ( S , \\\\mathrm { D i f f } _ { S } D _ { \\\\tilde { X } } )$ . Therefore, if $( X , D _ { X } )$ has maximal intersection, so does $( S , \\\\mathrm { D i f f } _ { S } D _ { \\\\tilde { X } } )$ . By the results of [7], $( S , { \\\\mathrm { D i f f } } _ { S } D _ { X } )$ then has a toric model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,662,881,750]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"67\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As $X$ has terminal singularities, $X$ is normal and Cohen–Macaulay. Any Cartier component $s$ of the boundary $D _ { X }$ is Cohen–Macaulay and satisfies Serre’s condition $S _ { 2 }$ . By [15, Proposition 16.9], $( S , { \\\\mathrm { D i f f } } _ { S } D _ { X } )$ is semi log canonical (slc). As a special case, if $X$ is Gorenstein and $D _ { X }$ is irreducible, this shows that $D _ { X }$ has slc singularities. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,752,881,804]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"67\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"I am particularly interested in producing examples of (t, lc) CY pairs for which- - the mirror partners proposed in Conjecture 1 can be constructed. The discussion in-[10] motivates the following definition: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,812,881,852]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"67\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 6 A (t, dlt) modification - $( \\\\widetilde { X } , D _ { \\\\tilde { X } } ) ( X , D _ { X } )$ is good if $( \\\\widetilde { X } , D _ { \\\\widetilde { X } } )$ is a log smooth variety (in the sense of log geometry), i.e. if $( \\\\widetilde { X } , D _ { \\\\widetilde { X } } )$ is a toroidal pair. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,860,879,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"67\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 4 The properties of (t, dlt) pairs and those of toroidal pairs are somehow different. On the one hand, $( \\\\widetilde { X } , D _ { \\\\widetilde { X } } )$ being dlt implies that the pair has simple normal crossings at the generic point of each stratum, which is not in general satisfied by- - toroidal pairs. On the other hand, as $\\\\widetilde { X }$ has terminal singularities and is analytically isomorphic to a toric variety, by [4, Theorem 11.4.8], $\\\\widetilde { X }$ only has finite quotient singularities, i.e. $\\\\widetilde { X }$ is an orbifold (in particular, $\\\\widetilde { X }$ has no Gorenstein singularities). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,51,872,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"67\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model 61 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[854,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"67\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,879,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"68\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,169,881,279]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"68\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Normal Singularities Let $p \\\\in \\\\mathrm { S i n g } ( D _ { i } )$ be an isolated singularity lying on a single component of the boundary. Since $K _ { X } + D _ { X } \\\\sim _ { \\\\mathbb { Z } } 0$ , the indices of $K _ { X }$ and of $D _ { X }$ at $p$ are the same. After taking a canonical cover, assume that $X$ is Gorenstein at $p$ , and without loss of generality (for the purpose of this discussion) assume that $D _ { X } = D _ { i }$ is irreducible and that $\\\\mathrm { S i n g } ( D _ { X } ) = \\\\{ p \\\\}$ . Then, since $D _ { X }$ is Cartier, by [15, Proposition 16.4], we have $K _ { D _ { X } } \\\\sim _ { \\\\mathbb { Q } } ( K _ { X } + D _ { X } ) _ { | D _ { X } }$ , so that $D _ { X }$ is Gorenstein. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,280,881,334]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"68\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $f \\\\colon ( { \\\\widetilde { X } } , D _ { \\\\tilde { X } } ) \\\\ \\\\ ( X , D _ { X } )$ is a good modification, $\\\\widetilde { X }$ is smooth (because it is Gorenstein) and up to composing with the minimal resolution of $D _ { \\\\tilde { X } }$ , we may assume that $f _ { | D _ { \\\\tilde { X } } } \\\\colon D _ { \\\\tilde { X } } \\\\to D _ { X }$ is a resolution. Write: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[268,349,728,372]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0237/778a246c4b729342a440a488777a0f0ad1d0d81e9961b44de560e7fd016079e2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"68\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { D _ { \\\\tilde { X } } } = ( K _ { \\\\tilde { X } } + D _ { \\\\tilde { X } } ) _ { } { \\\\mid { \\\\cal D } _ { \\\\tilde { X } } } = ( f _ { \\\\mid { \\\\cal D } _ { \\\\tilde { X } } } ) ^ { \\\\ast } ( K _ { \\\\cal D _ { X } } ) - E _ { \\\\mid { \\\\cal D } _ { \\\\tilde { X } } }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[123,388,879,426]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"68\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $E$ is defined by $K _ { \\\\tilde { X } } + f _ { * } ^ { - 1 } D _ { X } + E = f ^ { * } ( K _ { X } + D _ { X } )$ . We see that $p$ is canonical if $E \\\\cap D _ { \\\\tilde { X } } = \\\\beta$ , and elliptic otherwise. Indeed, let \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[410,442,589,469]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0356/31a5eac5a88e503d5f5cae3196a929be71e057dd00ed9859efd2427dd0d7e7d6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"68\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nD _ { \\\\tilde { X } } \\\\stackrel { q } { \\\\to } \\\\overline { { { D } } } _ { X } \\\\stackrel { \\\\mu } { \\\\to } D _ { X }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,484,881,629]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"68\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"be the factorisation of $f _ { | D _ { \\\\tilde { X } } }$ through the minimal resolution of $( p \\\\in D _ { X } )$ . Then, $q$ is either an isomorphism or an isomorphism at the generic point of each component of $E _ { | D _ { \\\\tilde { X } } }$ by [3, Lemma 2.5] because $f$ is volume preserving. We have: $K _ { \\\\overline { { D } } _ { X } } \\\\ =$ $\\\\mu ^ { * } K _ { D _ { X } } - Z$ , where the effective cycle $Z = q _ { * } ( E _ { D _ { \\\\tilde { X } } } )$ is either empty (and $p$ is canonical) or a reduced sum of $\\\\mu$ -exceptional curves (and $p$ is elliptic). In the second case, $Z \\\\sim - K _ { D _ { \\\\tilde { X } } }$ is the fundamental cycle of $( p \\\\in D _ { X } )$ . If $Z$ is irreducible, it is reduced and has genus 1; if not, every irreducible component of $Z$ is a smooth rational curve of self-intersection $^ { - 2 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,629,881,701]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"68\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"When $p$ is elliptic, $Z$ is reduced and $p$ is a Kodaira singularity [12, Theorem 2.9], i.e. a resolution is obtained by blowing up points of the singular fibre in a degeneration of elliptic curves; further, in Arnold’s terminology, the singularity $p$ is uni or bimodal. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,702,881,809]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"68\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Further, $p \\\\in D _ { i }$ is a hypersurface singularity (resp. a codimension 2 complete intersection, resp. not a complete intersection) when $- 3 \\\\leq Z ^ { 2 } \\\\leq - 1$ (resp. $\\\\bar { Z } ^ { 2 } =$ $^ { - 4 }$ , resp. $Z ^ { 2 } \\\\leq - 5 )$ [16]. When $- 4 \\\\leq Z ^ { 2 } \\\\leq - 1$ , normal forms are known for $p \\\\in D _ { i }$ : Table 2 lists normal forms of slc hypersurface singularities, while normal forms of codimension 2 complete intersections elliptic singularities are given in [20]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[744,54,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"68\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A.-S. Kaloghiros \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,54,141,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"68\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,112,881,453]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0388/bf0fb97d5d67f8ac311e4c2e4a267065c8feb139c7057062de6167784b9e3abc.jpg\",\n \"list_items\": null,\n \"page_idx\": \"69\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=1 colspan=1>Type</td><td rowspan=1 colspan=1>Name</td><td rowspan=1 colspan=1>Symbol</td><td rowspan=1 colspan=2>Equationf ∈C[x,y,z]</td><td rowspan=1 colspan=1>multof</td></tr><tr><td rowspan=1 colspan=1>Terminal</td><td rowspan=1 colspan=1>Smooth</td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=2>x</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=5 colspan=1>Canonical</td><td rowspan=5 colspan=1>du Val</td><td rowspan=1 colspan=1>An</td><td rowspan=1 colspan=1>x²+y²+zn+1</td><td rowspan=1 colspan=1>n≥1</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>D</td><td rowspan=1 colspan=1>x²+z（y²+n-2）</td><td rowspan=1 colspan=1>n≥4</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>E6</td><td rowspan=1 colspan=1>x²+y³+4</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>x²+y³+yz³</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>E8</td><td rowspan=1 colspan=1>x²+y³+5</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=4 colspan=1>1c</td><td rowspan=3 colspan=1>Simple elliptic</td><td rowspan=1 colspan=1>X1.0</td><td rowspan=1 colspan=1>x²+y4+z4+λxyz</td><td rowspan=1 colspan=1>4≠64</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>J2.0</td><td rowspan=1 colspan=1>x²+y³++λxyz</td><td rowspan=1 colspan=1>≠432</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>T3.3.3</td><td rowspan=1 colspan=1>x³+y³+³+λxyz</td><td rowspan=1 colspan=1>x≠-27</td><td rowspan=1 colspan=1>3</td></tr><tr><td rowspan=1 colspan=1>Cusp</td><td rowspan=1 colspan=1>Tp.q,r</td><td rowspan=1 colspan=1>xP+yq+z+xyz</td><td rowspan=1 colspan=1>++<1</td><td rowspan=1 colspan=1>2or3</td></tr><tr><td rowspan=7 colspan=1>slc</td><td rowspan=1 colspan=1>Normal crossing</td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=1>x²+y²</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>Pinch point</td><td rowspan=1 colspan=1>D</td><td rowspan=1 colspan=1>x²+y²</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=5 colspan=1>Degenerate cusp</td><td rowspan=1 colspan=1>T2.0.8</td><td rowspan=1 colspan=1>x²+y²+z²</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>T2.q，8</td><td rowspan=1 colspan=1>x²+y²（z²+y9-2)</td><td rowspan=1 colspan=1>q≥3</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>T.80.8</td><td rowspan=1 colspan=1>xyz</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>3</td></tr><tr><td rowspan=1 colspan=1>Tp.0.8</td><td rowspan=1 colspan=1>xyz+xp</td><td rowspan=1 colspan=1>p≥3</td><td rowspan=1 colspan=1>③</td></tr><tr><td rowspan=1 colspan=1>Tp.q.</td><td rowspan=1 colspan=1>xyz+xP+yq</td><td rowspan=1 colspan=1>q≥p≥3</td><td rowspan=1 colspan=1>3</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 2 Dimension 2 slc hypersurface singularities [17] \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[118,483,801,524]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"69\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 Examples of Rigid Maximal Intersection Threefold CY Pairs \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,549,881,621]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"69\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"All the examples below are (t, lc) CY pairs $( X , D _ { X } )$ which admit no toric model. Except for Example 5, all underlying varieties $X$ are birationally rigid quartic hypersurfaces by Proposition 1; the underlying variety in Example 5 is a smooth cubic threefold, and therefore non-rational. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,666,562,689]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"69\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.1 Examples with Normal Boundary \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[143,731,854,767]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"69\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 2 Consider the CY pair $( X , D _ { X } )$ where $X$ is the nonsingular quartic hypersurface \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[255,782,742,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0452/d6facd7c312fad3dfa673b87e49ba44604203b16cb5f6ecffd8b0f5e73418d10.jpg\",\n \"list_items\": null,\n \"page_idx\": \"69\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nX = \\\\{ x _ { 1 } ^ { 4 } + x _ { 2 } ^ { 4 } + x _ { 3 } ^ { 4 } + x _ { 0 } x _ { 1 } x _ { 2 } x _ { 3 } + x _ { 4 } ( x _ { 0 } ^ { 3 } + x _ { 4 } ^ { 3 } ) = 0 \\\\}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,821,580,840]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"69\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and $D _ { X }$ is its hyperplane section $X \\\\cap \\\\{ x _ { 4 } = 0 \\\\}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,794,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"69\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[769,855,879,872]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"69\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(continued) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"69\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[148,109,364,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 2 (continued) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,126,854,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The quartic surface $D _ { X }$ has a unique singular point $p = ( 1 { : } 0 { : } 0 { : } 0 { : } 0 )$ , and using the notation of Table 1, $p$ is locally analytically equivalent to a $T _ { 4 , 4 , 4 }$ cusp \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[350,192,644,217]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0985/07950970fe2f04c55b0cbcff23008e88ce58cfd24d7a54e1aaa454fca84cc1b6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n0 \\\\in \\\\{ x ^ { 4 } + y ^ { 4 } + z ^ { 4 } + x y z = 0 \\\\} .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[148,232,801,250]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$D _ { X }$ is easily seen to be rational: the projection from the triple point $p$ is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[414,264,580,291]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0787/1fc1e846e1539c6e71b852b5a53a47449a9b73a0c16db2b9688817bd159ac451.jpg\",\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nD _ { X } \\\\ { \\\\mathrm { ~ -- } } \\\\to \\\\mathbb { P } _ { x _ { 1 } , x _ { 2 } , x _ { 3 } } ^ { 2 } ;\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[148,306,854,343]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"this map is the blowup of the 12 points $\\\\{ x _ { 1 } ^ { 4 } + x _ { 2 } ^ { 4 } + x _ { 3 } ^ { 4 } = x _ { 1 } x _ { 2 } x _ { 3 } = 0 \\\\}$ , of which 4 lie on each coordinate line $L _ { i } = \\\\{ x _ { i } = 0 \\\\}$ , for $i = 1 , 2 , 3$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,345,851,379]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"I treat this example in detail and construct explicitly a good (t, dlt) modification of the pair $( X , D _ { X } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,381,851,433]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $f \\\\colon X _ { p } \\\\ \\\\to \\\\ X$ be the blowup of $p$ , then $X _ { p }$ is non-singular, the exceptional divisor $E$ satisfies $( E , { \\\\cal O } _ { E } ( E ) ) \\\\ = \\\\ ( { \\\\mathbb P } ^ { \\\\hat { 2 } } , { \\\\cal O } _ { \\\\mathbb P ^ { 2 } } ( - 1 ) )$ , and if $D$ denotes the proper transform of $D _ { X }$ , we have: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[348,450,649,472]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0386/17249a1e7cbe376503dafc6402674c1b8353b935ed3d3cc90e4bc29daa799873.jpg\",\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { X _ { p } } + D + E = f ^ { * } ( K _ { X } + D ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,489,851,545]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Explicitly, the blowup $\\\\mathcal { F } \\\\to \\\\mathbb { P } ^ { 4 }$ of $\\\\mathbb { P } ^ { 4 }$ at $p$ is the rank 2 toric variety⎞ $\\\\mathrm { T V } ( I , A )$ , where $ I ~ = ~ ( u , x _ { 0 } ) \\\\cap ( x _ { 1 } , . . . , x _ { 4 } )$ is the irrelevant ideal of $\\\\mathbb { C } [ u , x _ { 0 } , \\\\ldots , x _ { 4 } ]$ and $A$ is the action of ⎝ $\\\\mathbb { C } ^ { * } \\\\times \\\\mathbb { C } ^ { * }$ with weights:⎠ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[375,560,621,621]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0112/51a94e6c18c3cc4bfec4c4aee0f8319a6a66c1645622f525d68e706585275154.jpg\",\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left( \\\\begin{array} { c c c c c c } { { u \\\\ x _ { 0 } \\\\ s _ { 1 } } } & { { s _ { 2 } } } & { { s _ { 3 } } } & { { s _ { 4 } } } \\\\\\\\ { { 1 \\\\ 0 \\\\ - 1 \\\\ - 1 \\\\ - 1 \\\\ - 1 \\\\ - 1 } } \\\\\\\\ { { 0 \\\\ 1 } } & { { 1 } } & { { 1 } } & { { 1 } } & { { 1 } } \\\\end{array} \\\\right) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[175,638,378,654]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The equation of $X _ { p }$ is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[207,671,785,695]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0298/44c29223727eb4e66aa090544dd14d46cb94d9802f2c2e3b6f550df87e5dc57e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nX _ { p } = \\\\{ u ^ { 2 } \\\\big ( u ( s _ { 1 } ^ { 4 } + s _ { 2 } ^ { 4 } + s _ { 3 } ^ { 4 } ) + x _ { 0 } s _ { 1 } s _ { 2 } s _ { 3 } \\\\big ) + s _ { 4 } ( x _ { 0 } ^ { 3 } + u ^ { 3 } s _ { 4 } ^ { 3 } ) = 0 \\\\} ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,710,851,785]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"while $E ~ = ~ \\\\{ u ~ = ~ 0 \\\\}$ and $D = \\\\{ u ( s _ { 1 } ^ { 4 } + s _ { 2 } ^ { 4 } + s _ { 3 } ^ { 4 } ) + x _ { 0 } s _ { 1 } s _ { 2 } s _ { 3 } = 0 \\\\} .$ . By construction, $E$ is the projective plane with coordinates $s _ { 1 } , s _ { 2 } , s _ { 3 }$ . Note that $( X _ { p } , D + E )$ is not dlt because $D \\\\cap E \\\\ = \\\\ \\\\{ x _ { 0 } s _ { 1 } s _ { 2 } s _ { 3 } \\\\ = \\\\ 0 \\\\}$ consists of 3 concurrent lines $C _ { 1 } , C _ { 2 } , C _ { 3 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[170,785,733,803]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Consider $g _ { 1 } \\\\colon X _ { 1 } \\\\to X _ { p }$ the blowup of the nonsingular curve \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[350,819,646,840]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0964/0117b1c9a90f5d10a019d4b47fe616c8675f18df885ca0d33ab285663c60a29b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nC _ { 1 } = \\\\{ u = s _ { 1 } = s _ { 4 } = 0 \\\\} \\\\subset X _ { p } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[769,860,879,875]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(continued) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[742,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A.-S. Kaloghiros \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"70\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"64 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[148,109,364,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 2 (continued) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,126,851,163]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The exceptional divisor of $g _ { 1 }$ is a surface $E _ { 1 } \\\\simeq \\\\mathbb { P } ( N _ { C _ { 1 } / X _ { p } } )$ , and since $C _ { 1 } \\\\simeq$ $\\\\mathbb { P } ^ { 1 }$ , the restriction sequence of normal bundles gives \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[227,181,765,202]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0312/30ae614d69127b32fb96e9fcfc2743e8e4c33c50f277b529017838d35228edc3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nN _ { C _ { 1 } / X _ { p } } \\\\simeq N _ { C _ { 1 } / E } \\\\oplus ( N _ { E / X _ { p } } ) _ { | C _ { 1 } } \\\\simeq O _ { C _ { 1 } } ( 1 ) \\\\oplus O _ { E } ( - 1 ) _ { | C _ { 1 } } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[148,217,380,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"so that $E _ { 1 } = \\\\mathbb { F } _ { 2 }$ . Further, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[300,252,697,274]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0894/99b97ce164b72e04cebd88174923cd77299e262b45484ac0b8fd8cc875957ec6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { X _ { 1 } } + D + E + E _ { 1 } = g _ { 1 } ^ { * } ( K _ { X _ { p } } + D + E )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,288,854,379]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where, abusing notation, I denote by $D$ and $E$ the proper transforms of the divisors $D$ and $E$ . The “restricted pair” on $E _ { 1 }$ is a surface CY pair $( E _ { 1 } , ( D +$ $E ) _ { | E _ { 1 } } )$ by adjunction. By construction, $E \\\\cap E _ { 1 }$ is the negative section $\\\\sigma$ . The curve $\\\\Gamma = D \\\\cap E _ { 1 }$ is irreducible, and since $( D + E ) _ { | E _ { 1 } }$ is anticanonical, we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,432,851,506]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The divisors $D , E , E _ { 1 }$ meet in two points, the dual complex $\\\\mathcal { D } ( X _ { 1 } , D + E +$ $E _ { 1 }$ ) is not simplicial because it is a sphere $S ^ { 2 }$ whose triangulation is given by 3 vertices on an equator. While not strictly necessary, we consider a further blowup to obtain a (t, dlt) pair with simplicial dual complex. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[175,507,610,524]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Denote by $C _ { 2 }$ the proper transform of the curve \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[407,542,589,560]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0920/5ba46d063c71fa2b7d364697e757e87caa4c44c7587d4f798d76e95dff7143a5.jpg\",\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\{ u = s _ { 2 } = s _ { 4 } = 0 \\\\} .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[148,578,546,596]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Then $C _ { 2 } \\\\subset E \\\\cap D$ is rational, and as above \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[241,612,756,633]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0005/383c8c8809693750ae425b78614b085992f854f83f371ef381efb887c4e45210.jpg\",\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nN _ { C _ { 2 } / X _ { 1 } } \\\\simeq N _ { C _ { 2 } / E } \\\\oplus ( N _ { E / X _ { 2 } } ) _ { | C _ { 2 } } = O _ { C _ { 2 } } ( 1 ) \\\\oplus O _ { C _ { 2 } } ( - 2 ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[143,650,851,684]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $g _ { 2 } \\\\colon X _ { 2 } \\\\to X _ { 1 }$ be the blowup of $C _ { 2 }$ , then the exceptional divisor of $g _ { 2 }$ is a Hirzebruch surface \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[378,702,621,723]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0344/5c26de794ef924895665ae03730af0614863adbc044995ddcbcf55dd58c9f716.jpg\",\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nE _ { 2 } \\\\simeq \\\\mathbb { P } _ { \\\\mathbb { P } ^ { 1 } } ( N _ { C _ { 2 } / X _ { 1 } } ) \\\\simeq \\\\mathbb { F } _ { 3 } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,738,788,758]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Still denoting by $D , E , E _ { 1 }$ the strict transforms of $D , E , E _ { 1 }$ , we have: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[246,774,751,795]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0339/701ad4e8408e838ed7785ef1117721b50213cc77c96d143eab0b53394548c861.jpg\",\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { X _ { 2 } } + D + E + E _ { 1 } + E _ { 2 } = g _ { 2 } ^ { * } ( K _ { X _ { 1 } } + D + E + E _ { 1 } ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,810,669,830]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The pair $( X _ { 2 } , D + E + E _ { 1 } + E _ { 2 } )$ is dlt; the composition \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[209,843,788,867]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0053/a1bad00ec56cdc30be8b9001711b935340c47fcdc7325535206cd445f15c1369.jpg\",\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\ng _ { 2 } \\\\circ g _ { 1 } \\\\circ f \\\\colon ( \\\\widetilde { X } , D _ { \\\\widetilde { X } } ) = ( X _ { 2 } , D + E + E _ { 1 } + E _ { 2 } ) \\\\to ( X , D _ { X } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[769,879,879,896]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(continued) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[116,51,858,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[854,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"71\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"65 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[145,109,421,139]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 2 (continued) is a good (t, dlt) modification. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,142,851,177]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The “restrictions” of $( \\\\widetilde { X } , D _ { \\\\widetilde { X } } )$ to the component of the boundary are the following surface anticanonical pairs: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,186,854,351]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$- ~ \\\\mathrm { O n } D \\\\colon ( E + E _ { 1 } + E _ { 2 } ) _ { | D }$ is a cycle of $( - 3 )$ -curves, the morphism $D \\\\to D _ { X }$ is the familiar resolution of the $T _ { 4 , 4 , 4 }$ cusp singularity; On $E$ : $( D + E _ { 1 } + E _ { 2 } ) _ { | E }$ is the triangle of coordinate lines with selfintersections (1, 1, 1); On $E _ { 1 }$ $\\\\vec { \\\\iota } _ { 1 } \\\\colon ( D + E + E _ { 2 } ) _ { | E _ { 1 } }$ is an anticanonical cycle with self-intersections $( 5 , - 3 , - 1 )$ ; On $E _ { 2 }$ : $( D + E + E _ { 1 } ) _ { | E _ { 2 } }$ is an anticanonical cycle with self-intersections $( 5 , - 3 , 0 )$ (as above, $E _ { | E _ { 2 } } \\\\sim \\\\sigma$ is a negative section, ${ \\\\cal E } _ { 1 | E _ { 2 } } \\\\sim f$ a fibre of $\\\\mathbb { F } _ { 3 } \\\\to \\\\mathbb { P } ^ { 1 }$ , and $D _ { | E _ { 2 } } \\\\sim 4 f + \\\\sigma )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,360,854,430]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It follows that the dual complex ${ \\\\mathcal { D } } ( X , D _ { X } )$ is $\\\\mathrm { P L }$ homeomorphic to a tetrahedron and $( X , D _ { X } )$ has maximal intersection. Note that $( 0 ~ \\\\in ~ D _ { X } )$ ) is a maximal intersection lc point, and since $D _ { X }$ is a rational surface, it has a toric model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,487,498,506]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example $^ 3$ Let $X$ be the hypersurface \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[232,521,763,543]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0633/c506f8addaaae4368cdd8be5c2869e94ac650f29037386907f2b3c2f82da959b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nX = \\\\{ x _ { 3 } ( x _ { 0 } ^ { 3 } + x _ { 1 } ^ { 3 } ) + x _ { 2 } ^ { 4 } + x _ { 0 } x _ { 1 } x _ { 2 } x _ { 3 } + x _ { 4 } ( x _ { 3 } ^ { 3 } + x _ { 4 } ^ { 3 } ) = 0 \\\\} ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,560,558,578]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and $D _ { X }$ its hyperplane section $X \\\\cap \\\\{ x _ { 4 } = 0 \\\\}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[170,578,799,596]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The quartic $X$ has 3 ordinary double points at the intersection points \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[405,609,589,635]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0441/15e945254390b90ceb01847104f15bbd5289c5ad2bf65ec26af7012910df57b8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nL \\\\cap \\\\{ x _ { 0 } ^ { 3 } + x _ { 1 } ^ { 3 } = 0 \\\\} ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,650,851,704]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $L$ is the line $\\\\{ x _ { 2 } ~ = ~ x _ { 3 } ~ = ~ x _ { 4 } ~ = ~ 0 \\\\}$ . The singular locus of $D _ { X }$ is $\\\\operatorname { S i n g } ( X ) \\\\cup \\\\{ p \\\\}$ , where $p = ( 0 { : } 0 { : } 0 { : } 1 { : } 0 )$ is a $T _ { 3 , 3 , 4 }$ cusp, i.e. locally analytically equivalent to \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[350,717,644,740]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0865/138d4b0bae52c1ad10d7794280a656b4abf35a9cfa3aec1fe086f0ba7e1f0049.jpg\",\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n0 \\\\in \\\\{ x ^ { 3 } + y ^ { 3 } + z ^ { 4 } + x y z = 0 \\\\} .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[148,758,751,776]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The quartic surface $D _ { X }$ is rational; the projection of $D _ { X }$ from $p$ is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[414,789,580,815]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0761/cdbe416a30b891190e06650011274519b96d00d89749a3d23a0f9d44833f9411.jpg\",\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nD _ { X } \\\\ { - } { \\\\to } \\\\ \\\\mathbb { P } _ { x _ { 0 } , x _ { 1 } , x _ { 2 } } ^ { 2 } ;\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,828,854,867]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"this map is defined outside of the 12 points (counted with multiplicity) defined by $\\\\{ x _ { 2 } ^ { 4 } \\\\overset { \\\\cdot } { = } x _ { 0 } ^ { 3 } + x _ { 1 } ^ { 3 } + x _ { 0 } x _ { 1 } x _ { 2 } = 0 \\\\}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[744,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A.-S. Kaloghiros \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"66 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[769,882,879,899]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"72\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(continued) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[148,109,364,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"73\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 3 (continued) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,132,854,186]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"73\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $\\\\tilde { X } \\\\stackrel { f } { \\\\to } X$ is the composition of the blowups at the ordinary double points and at $p , \\\\widetilde { X }$ is smooth and $D _ { \\\\tilde { X } }$ is non-singular, so that $f$ is a good (t, dlt) modification. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,187,854,330]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"73\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The minimal resolution of $p \\\\in D _ { X }$ is a rational curve with self intersection-$C ^ { 2 } = - 3$ . Explicitly, taking the blowup of $X$ at $p$ , the proper transform is a rational surface $D$ . The exceptional curve is the preimage of a nodal cubic in $\\\\mathbb { P } ^ { 2 }$ blown up at 12 points counted with multiplicities. Note that $( \\\\widetilde { X } , D + E )$ is not dlt, but in order to obtain a (t, dlt) modification, we just need to blowup the node of $D \\\\cap E$ which is a nonsingular point of $\\\\widetilde { X } , \\\\bar { D ( \\\\mathbf { \\\\Phi } ) }$ and $E$ . The (t, dlt) modification of $( X , D _ { X } )$ in a neighbourhood of $p$ is good and the associated dual complex is 2-dimensional. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,331,851,367]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"73\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The pair $( X , D _ { X } )$ has maximal intersection; but as in the previous examples, $X$ is rigid, so that $( X , D _ { X } )$ can have no toric model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,423,681,441]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"73\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example $^ { 4 }$ Let $X$ be the nonsingular quartic hypersurface \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[218,456,781,480]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0918/4a562063a0074e5e55a6d6305169e71add7b6f6251999a1a8968f2608f7c5e8d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"73\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nX = \\\\{ x _ { 0 } ^ { 3 } x _ { 3 } + x _ { 1 } ^ { 4 } + x _ { 2 } ^ { 4 } + x _ { 0 } x _ { 1 } x _ { 2 } x _ { 3 } + x _ { 4 } ( x _ { 3 } ^ { 3 } + x _ { 4 } ^ { 3 } ) = 0 \\\\} \\\\subset \\\\mathbb { P } ^ { 4 }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,495,558,513]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"73\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and $D _ { X }$ its hyperplane section $X \\\\cap \\\\{ x _ { 4 } = 0 \\\\}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,515,854,549]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"73\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The surface $D _ { X }$ has a unique singular point $p = ( 0 { : } 0 { : } 0 { : } 1 { : } 0 )$ , which is a cusp $T _ { 3 , 4 , 4 }$ , i.e. is locally analytically equivalent to \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[350,564,644,587]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0594/db663062178195ec0a970bcc6780137c32adbbc31beb7c975ce78166a2865d47.jpg\",\n \"list_items\": null,\n \"page_idx\": \"73\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n0 \\\\in \\\\{ x ^ { 3 } + y ^ { 4 } + z ^ { 4 } + x y z = 0 \\\\} .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,603,854,732]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"73\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As in Example 2, $X$ is non-singular, and finding a good (t, dlt) modification of $( X , D _ { X } )$ will amount to taking a minimal resolution of the singular point of $D _ { X }$ . Let $X _ { p } X$ be the blowup of $X$ at $p ; X _ { p }$ is non-singular and if $D$ denotes the proper transform of $D _ { X }$ , and $E$ the exceptional divisor, $D \\\\cap E$ consists of 2 rational curves of self intersection $^ { - 3 }$ and $^ { - 4 }$ . These curves are the proper transforms of $\\\\{ x _ { 0 } = 0 \\\\}$ and of $\\\\{ x _ { 0 } ^ { 2 } + x _ { 1 } x _ { 2 } = 0 \\\\}$ under the blow up of P2 x ,x ,x at the points \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[346,749,651,773]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0795/731dfece77a55abca32d78604fced71cb3d51e689a4a6198d4dcd118203e21c2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"73\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\{ x _ { 1 } ^ { 4 } + x _ { 2 } ^ { 4 } = x _ { 0 } ( x _ { 1 } x _ { 2 } + x _ { 0 } ^ { 2 } ) = 0 \\\\} .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,788,854,860]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"73\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The dual complex consists of 3 vertices that are joined by edges and span 2 distinct faces: ${ \\\\mathcal { D } } ( X , D _ { X } )$ is PL homeomorphic to a sphere $S ^ { 2 }$ whose triangulation is given by 3 vertices on an equator. The CY pair $( X , D _ { X } )$ has maximal intersection but no toric model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,822,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"73\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"73\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,615,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.2 Examples with Non-normal Boundary \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,151,826,171]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 5 This example is due to R. Svaldi. Consider the cubic threefold \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[268,184,731,210]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0035/1006d3a7c27ecf1f37f1439e8058fa1151a6b4ece51d0d3ba7a9ddd23c45ef5f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nX = \\\\{ x _ { 0 } x _ { 1 } x _ { 2 } + x _ { 1 } ^ { 3 } + x _ { 2 } ^ { 3 } + x _ { 3 } q + x _ { 4 } q ^ { \\\\prime } = 0 \\\\} \\\\subset \\\\mathbb { P } ^ { 4 }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,225,851,261]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $q , q ^ { \\\\prime }$ are homogeneous polynomials of degree 2 in $x _ { 0 } , \\\\cdots , x _ { 4 }$ . If the quadrics $q$ and $q ^ { \\\\prime }$ are general and if \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[302,277,694,298]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0630/132e170903f0011cc983f0181d16a86b16a03fd5890d712e93ca4232ae8d4416.jpg\",\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n( q ( 1 , 0 , 0 , 0 , 0 ) , q ^ { \\\\prime } ( 1 , 0 , 0 , 0 , 0 ) ) \\\\neq ( 0 , 0 ) ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,313,790,333]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"then $X$ and $S = \\\\{ x _ { 3 } = 0 \\\\} \\\\cap X$ and $T = \\\\{ x _ { 4 } = 0 \\\\} \\\\cap X$ are nonsingular. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,334,851,385]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $D _ { X }$ be the anticanonical divisor $S + T$ . The curve $C = S \\\\cap T = \\\\Pi \\\\cap X$ for $\\\\Pi = \\\\{ x _ { 3 } = x _ { 4 } = 0 \\\\}$ is a nodal cubic. It follows that both $( S , C )$ and $( T , C )$ are log canonical, and therefore so is \\u0014 $( X , D _ { X } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[141,387,851,421]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since $s$ and $T$ are smooth, $\\\\mathrm { S i n g } ( D _ { X } ) = S \\\\cap T = C$ , and if $p$ is the node of $C$ , we have:\\u0014 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[195,435,788,490]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0449/a475fd5adc6bd29ae7062a496a5742c0ed0f7fdcad6cd3d2103379a19e5b6079.jpg\",\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { ( p \\\\in D _ { X } ) \\\\sim 0 \\\\in \\\\big \\\\{ ( x y + x ^ { 3 } + y ^ { 3 } + z ) ( x y + x ^ { 3 } + y ^ { 3 } + t ) = 0 \\\\big \\\\} } \\\\\\\\ & { \\\\sim 0 \\\\in \\\\big \\\\{ ( x y + z ) ( x y + t ) = 0 \\\\big \\\\} \\\\sim 0 \\\\in \\\\big \\\\{ ( x y + z ) ( x y - z ) = 0 \\\\big \\\\} , } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,504,851,540]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\sim$ denotes analytic equivalence. Thus, $p \\\\in D _ { X }$ is a double pinch point, i.e. $p$ is locally analytically equivalent to $0 \\\\in \\\\{ x ^ { 2 } y ^ { 2 } - z ^ { 2 } = 0 \\\\}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,540,849,576]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We now construct a good (t, dlt) modification of $( X , D _ { X } )$ . Let $f \\\\colon X _ { C } \\\\to X$ be the blowup of $X$ along $C$ ; ${ \\\\mathrm { ; ~ } } \\\\mathrm { S i n g } ( X _ { C } )$ is an ordinary double point. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,578,851,648]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Indeed, let $\\\\Pi = \\\\{ x _ { 3 } = x _ { 4 } = 0 \\\\}$ , then $f$ is the restriction to $X$ of the blowup $\\\\mathcal { F } \\\\to \\\\mathbb { P } ^ { 4 }$ of $\\\\mathbb { P } ^ { 4 }$ along . Note that ⎛ $\\\\mathcal { F }$ is the rank 2 toric variety⎞ $\\\\mathrm { T V } ( I , A )$ , where $I = ( u , x _ { 0 } , x _ { 1 } , x _ { 2 } ) \\\\cap ( x _ { 3 } , x _ { 4 } )$ is the irrelevant ideal of $\\\\mathbb { C } [ u , x _ { 0 } , \\\\ldots , x _ { 4 } ]$ and $A$ is the action of $\\\\mathbb { C } ^ { * } \\\\times \\\\mathbb { C } ^ { * }$ with weights:⎝ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[384,665,615,725]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0175/3a0b1b16bf18169ce219d2705048b94746410682bb89c64d764649660e2708ca.jpg\",\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left( { \\\\begin{array} { l l l l l } { u \\\\ x _ { 0 } \\\\ x _ { 1 } \\\\ x _ { 2 } \\\\ x _ { 3 } \\\\ x _ { 4 } } \\\\\\\\ { 1 \\\\ 0 \\\\ \\\\ 0 \\\\ 0 \\\\ - 1 \\\\ - 1 } \\\\\\\\ { 0 \\\\ 1 \\\\ 1 \\\\ 1 \\\\ 1 \\\\ 1 } \\\\end{array} } \\\\right) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[148,741,353,758]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The equation of $X _ { C }$ is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[300,774,692,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0874/9dc62af1ffecd0ab8b062a965dbaa6356428f58a3525327af25c4d362c1b3c44.jpg\",\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\{ x _ { 0 } x _ { 1 } x _ { 2 } + x _ { 1 } ^ { 3 } + x _ { 2 } ^ { 3 } + u ( x _ { 3 } q + x _ { 4 } q ^ { \\\\prime } ) = 0 \\\\} ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[742,52,876,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A.-S. Kaloghiros \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[769,816,879,833]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(continued) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"74\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"68 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[145,109,521,141]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"75\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 5 (continued) so that $X _ { C }$ has a unique singular point at \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[182,157,813,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0813/2176defa2b9e7dd03dba3f4616500402dcb8ce18f50a999629c525880f1541f3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"75\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nx _ { 0 } - 1 = u = x _ { 1 } = x _ { 2 } = x _ { 3 } q ( 1 , 0 , 0 , 0 , 0 ) + x _ { 4 } q ^ { \\\\prime } ( 1 , 0 , 0 , 0 , 0 ) = 0 ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[148,195,854,231]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"75\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and this is a threefold ordinary double point. In addition, denoting by $E _ { f } =$ $\\\\{ u = 0 \\\\} \\\\cap X _ { C }$ the exceptional divisor, we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,246,671,270]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0185/5f9f825085eb0d74cac8cd09d8629c24eed54196477241231f113141f20b0f63.jpg\",\n \"list_items\": null,\n \"page_idx\": \"75\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { X _ { C } } + \\\\tilde { S } + \\\\tilde { T } + E _ { f } = K _ { X } + S + T ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,285,640,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"75\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"so that the pair $( X _ { C } , \\\\tilde { S } + \\\\tilde { T } + E _ { f } )$ is a (t, lc) CY pair. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,307,854,451]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"75\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The pair $( X _ { C } , \\\\tilde { S } + \\\\tilde { T } + E _ { f } )$ is not dlt as the boundary has multiplicity 3 along the fibre $F$ over the node of $S \\\\cap T$ . The blowup of $F$ is not $\\\\mathbb { Q }$ - factorial, therefore in order to obtain a good (t, dlt) modification, we consider the divisorial contraction $g \\\\colon \\\\tilde { X } X _ { C }$ centred along $F$ . This is obtained by (a) blowing up the node, (b) then blowing up the proper transform of $F$ , (c) flopping a pair of lines with normal bundle $( - 1 , - 1 )$ and (d) contracting the proper transform of the $\\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 }$ above the node to a point $\\\\textstyle { \\\\frac { 1 } { 2 } } ( 1 , 1 , 1 )$ . The exceptional divisor of $g$ is denoted by $E _ { g }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,453,851,507]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"75\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The pair $( \\\\tilde { X } , \\\\tilde { S } + \\\\tilde { T } + \\\\tilde { E } _ { f } + E _ { g } )$ is the desired (t, dlt) modification of $( X , D _ { X } )$ , and it has maximal intersection. The dual complex is PL homeomorphic to a tetrahedron. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,564,567,582]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"75\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example $6$ Let $X$ be the quartic hypersurface \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,596,722,620]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0094/1d4fe569bb971885ddfb5fbea8b2009cde0b0682dec2dec2127ac1f1b8a791e5.jpg\",\n \"list_items\": null,\n \"page_idx\": \"75\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nX = \\\\{ x _ { 1 } ^ { 2 } x _ { 2 } ^ { 2 } + x _ { 1 } x _ { 2 } x _ { 3 } l + x _ { 3 } ^ { 2 } q + x _ { 4 } f _ { 3 } = 0 \\\\} \\\\subset \\\\mathbb { P } ^ { 4 } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[148,636,851,689]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"75\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $l$ (resp. $q$ ) is a general linear (resp. quadratic) form in $x _ { 0 } , \\\\cdots , x _ { 3 }$ , and $f _ { 3 }$ a general homogeneous form of degree 3 in $x _ { 0 } , \\\\cdots , x _ { 4 }$ . Let $D _ { X }$ be the hyperplane section\\u0014 $X \\\\cap \\\\{ x _ { 4 } = 0 \\\\}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,690,851,726]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"75\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As $l , q$ and $f _ { 3 }$ are general, $X$ has 6 ordinary double points. Indeed, denote by $L = \\\\{ x _ { 1 } = x _ { 3 } = x _ { 4 } = 0 \\\\}$ and $L ^ { \\\\prime } = \\\\{ x _ { 2 } = x _ { 3 } = x _ { 4 } = 0 \\\\}$ , then \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[154,741,845,764]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0503/4fecc342b72742e893dc59ad574982ee30fb998c0bb7bb7e6ec40a287e95e46a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"75\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathrm { S i n g } ( X ) = \\\\left\\\\{ L \\\\cap \\\\{ f _ { 3 } = 0 \\\\} \\\\right\\\\} \\\\cup \\\\left\\\\{ L ^ { \\\\prime } \\\\cap \\\\{ f _ { 3 } = 0 \\\\} \\\\right\\\\} = \\\\{ q _ { 1 } , q _ { 2 } , q _ { 3 } \\\\} \\\\cup \\\\{ q _ { 1 } ^ { \\\\prime } , q _ { 2 } ^ { \\\\prime } , q _ { 3 } ^ { \\\\prime } \\\\}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[769,780,879,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"75\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(continued) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,51,804,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"75\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[854,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"75\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"69 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[148,109,364,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"76\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example $6$ (continued) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,124,851,159]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"76\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"which consists of 3 points on each of the lines. In the neighbourhood of each point $q _ { i }$ (resp. $q _ { i } ^ { \\\\prime } ,$ ) for $i = 1 , 2 , 3$ , the equation of $X$ is of the form \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[410,177,585,196]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0682/5799c199bded8fe3bbc71ebe70dbeab68dc2fed4eda8f1690b9a334296a039b3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"76\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n0 \\\\in \\\\{ x y + z t = 0 \\\\}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[143,213,851,249]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"76\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(and $D _ { X } = \\\\{ t = 0 \\\\} ,$ ) so that all singular points of $X$ are ordinary double points. The quartic hypersurface $X$ is birationally rigid by Proposition 1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[141,250,851,285]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"76\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The surface $D _ { X }$ is non-normal as it has multiplicity 2 along $L$ and $L ^ { \\\\prime }$ . The point $p = L \\\\cap L ^ { \\\\prime }$ is locally analytically equivalent to \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[396,298,599,322]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0532/3d2120e1dd3b64de3101c9fb3efee3db69e5d451bd75b277a48fa6031bfc1d01.jpg\",\n \"list_items\": null,\n \"page_idx\": \"76\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n0 \\\\in \\\\{ x ^ { 2 } y ^ { 2 } + z ^ { 2 } = 0 \\\\} ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,339,851,375]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"76\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"so that $p \\\\in D _ { X }$ is a double pinch point. We conclude that the surface $D _ { X }$ has slc singularities, and hence $( X , D _ { X } )$ is a (t, lc) CY pair. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[170,376,653,393]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"76\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We construct a good (t, dlt) modification as follows. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,394,851,593]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"76\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"First, since $\\\\operatorname { S i n g } ( X ) \\\\cap L$ (resp. $\\\\operatorname { S i n g } ( X ) \\\\cap L ^ { \\\\prime } )$ is non-empty, the blowup of $X$ along $L$ (resp. along $L ^ { \\\\prime }$ ) is not $\\\\mathbb { Q }$ -factorial. In order to remain in the (t, dlt) category, we consider the divisorial extraction $f \\\\colon X _ { L } \\\\ \\\\to \\\\ X$ centered on $L$ (resp. $L ^ { \\\\prime }$ ).This is obtained by (a) blowing up the 3 nodes lying on $L$ , (b) blowing up the proper transform of $L$ , (c) flopping 3 pairs of lineswith normal bundle $( - 1 , - 1 )$ and (d) contracting the proper transforms of the three exceptional divisors $\\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 }$ lying above the nodes to points $\\\\textstyle { \\\\frac { 1 } { 2 } } ( 1 , 1 , 1 )$ . The exceptional divisor of $f$ is denoted by $E$ . Let $p \\\\colon { \\\\widetilde { X } } \\\\to X$ denote the morphism obtained by composing the divisorial extraction centered on $L$ with that centered on $L ^ { \\\\prime }$ (in any order), and let - $E , E ^ { \\\\prime }$ denote the exceptional divisors of the divisorial extractions. Then-\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[332,606,665,630]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0962/e2ab4adecb5b3e51fe085770d6eed7d68835f8810e560b0f9c5d078eb2b0024f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"76\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { \\\\widetilde { X } } + \\\\widetilde { D } + E + E ^ { \\\\prime } = p ^ { * } ( K _ { X } + D )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,647,854,702]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"76\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is a (t, dlt) modification of $( X , D _ { X } )$ and it has maximal intersection. The dual complex ${ \\\\mathcal { D } } ( X , D _ { X } )$ is PL homeomorphic to a sphere $S ^ { 2 }$ whose triangulation is given by 3 vertices on an equator. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,764,817,807]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"76\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 Further Results on Quartic Threefold CY Pairs: Beyond Maximal Intersection \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,830,879,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"76\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This section concentrates on $( t , l c )$ CY pairs $( X , D _ { X } )$ , where $X$ is a factorial quartic hypersurface in $\\\\mathbb { P } ^ { 4 }$ and $D$ is an irreducible hyperplane section of $X , ~ \\\\mathrm { I }$ give some more detail on the possible dual complexes of such pairs. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[744,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"76\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A.-S. Kaloghiros \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,54,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"76\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"77\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As explained in Sect. 2.2, $D _ { X }$ is slc because $( X , D _ { X } )$ is lc. In order to completely study the dual complexes of such (t, lc) CY pairs, one needs a good understanding of the normal forms of slc singularities that can lie on $D _ { X }$ . In the case of a general Fano $X$ , this step would require additional work, but here, $D _ { X }$ is a quartic surface in $\\\\mathbb { P } ^ { 3 }$ and the study of singularities of such surfaces has a rich history. I recall some results directly relevant to the construction of degenerate CY pairs $( X , D _ { X } )$ . The classification of singular quartic surfaces in $\\\\mathbb { P } ^ { 3 }$ can be broken in three independent cases. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,243,883,421]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"77\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(a) Quartic surfaces with no worse than rational double points: the minimal resolution is a $K 3$ surface. Possible configurations of canonical singularities were studied by several authors using the moduli theory of $K 3$ surfaces; there are several thousands possible configurations. The pair $( X , D _ { X } )$ is (t, dlt) and the dual complex of $( X , D _ { X } )$ is reduced to a point. \\n(b) Non-normal quartic surfaces were classified by Urabe [19]; there are a handful of cases recalled in Theorem 2. \\n(c) Non-canonical quartic surfaces with isolated singularities. These are studied by Wall [21] and Degtyarev [6] among others; their results are recalled in Theorem 3. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,430,740,450]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"77\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2 ([19]) A non-normal quartic surface $D \\\\subset \\\\mathbb { P } ^ { 3 }$ is one of: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,459,876,493]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"77\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. the cone over an irreducible plane quartic curve with a singular point of type $A _ { 1 }$ or $A _ { 2 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,495,765,513]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"77\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. a ruled surface over a smooth elliptic curve $G$ , $D = \\\\varphi _ { \\\\mathcal { L } } ( Z )$ , where: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,521,881,629]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"77\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(a) $M$ is a line bundle of degree 2 over $G , ~ N$ is a non-trivial line bundle of degree 0 over $G$ , $\\\\pi \\\\colon Z = \\\\mathbb { P } ( O _ { G } \\\\oplus N ) \\\\to G$ is the $\\\\mathbb { P } ^ { 1 }$ -bundle associated to $O _ { G } \\\\oplus N$ , $C _ { 1 }$ and $C _ { 2 }$ are the sections of $\\\\pi$ associated to $O _ { G } \\\\oplus N \\\\to O _ { G }$ and to $O _ { G } \\\\oplus N \\\\to N$ , and $\\\\mathcal { L } = O _ { Z } ( C _ { 1 } ) \\\\otimes \\\\pi ^ { * } M$ , $\\\\varphi \\\\underline { { c } }$ is the map associated to the linear system $| { \\\\mathcal { L } } |$ . Denoting by $L _ { i }$ the image by $\\\\varphi \\\\mathcal { L }$ of $C _ { i }$ , we have $\\\\mathrm { S i n g } ( D ) = L _ { 1 } \\\\cup L _ { 2 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[143,630,874,666]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"77\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(b) $M$ is a line bundle of rank 2 over $G , E$ is a rank 2 vector bundle over $G$ which fits in a non-splitting exact sequence \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[396,683,662,702]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0370/efa623d708bc826a55266989dcaada46b6b9d41fb6741167ab5dda584f73fc09.jpg\",\n \"list_items\": null,\n \"page_idx\": \"77\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n0 O _ { G } E O _ { G } 0 ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[182,719,881,792]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"77\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\pi \\\\colon Z = \\\\mathbb { P } ( E ) \\\\to G$ is the $\\\\mathbb { P } ^ { 1 }$ -bundle associated to $E , C$ is the section of $\\\\pi$ associated to $E O _ { G }$ ${ \\\\bf \\\\varepsilon } , { \\\\mathcal { L } } = O _ { Z } ( C ) \\\\otimes \\\\pi ^ { * } M , \\\\varphi$ $\\\\varphi \\\\underline { { c } }$ is the map associated to the linear system $| \\\\mathcal { L } |$ . Denoting by $L$ the image by $\\\\varphi \\\\mathcal { L }$ of $C$ , we have $\\\\mathrm { S i n g } ( D ) = L$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,800,473,819]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"77\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3. a rational surface $D \\\\subset \\\\mathbb { P } ^ { 3 }$ which is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[138,827,881,909]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"77\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(a) the image of a smooth $S \\\\subset \\\\mathbb { P } ^ { 5 }$ under the projection from a line disjoint from S; $D$ has no isolated singular point and – $S = v _ { 2 } ( \\\\mathbb { P } ^ { 2 } )$ , where $v _ { 2 }$ is the Veronese embedding; $D$ is the Steiner Roman surface; \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,783,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"77\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"77\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[177,90,879,165]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"78\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$- \\\\ S = \\\\varphi ( \\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 } )$ , where $\\\\varphi$ is the embedding defined by $| l _ { 1 } + 2 l _ { 2 } | f o r l _ { 1 , 2 }$ the rulings of $\\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 }$ ; \\n$- \\\\ S \\\\ = \\\\ \\\\varphi ( \\\\mathbb { F } _ { 2 } )$ , where $\\\\varphi$ is the embedding defined by $\\\\vert \\\\sigma + 3 f \\\\vert$ for $\\\\sigma$ the negative section and $f$ the fibre of $\\\\mathbb { F } _ { 2 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,171,881,226]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"78\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(b) the image of a surface $\\\\hat { D } \\\\subset \\\\mathbb { P } ^ { 4 }$ with canonical singularities under the projection from a point not lying on it; $\\\\hat { D }$ is a degenerate dP4 surface which is the blowup of $\\\\mathbb { P } ^ { 2 }$ in 5 points in almost general position. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,228,881,334]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"78\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(c) a rational surface embedded by a complete linear system on its normalisation $\\\\hat { D }$ ; the non-normal locus of $D$ is a line $L$ and $D$ may have isolated singularities outside $L$ . The minimal resolution of the normalisation of $D$ is a blowup of $\\\\mathbb { P } ^ { 2 }$ in 9 points. The normalisation of $D$ has at most two rational triple points lying on the inverse image of the non-normal locus; their images on $D$ are also triple points. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,343,419,361]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"78\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 5 $D$ is not slc in case 1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,370,881,426]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"78\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 1 Let $( X , D _ { X } )$ be a $( t , \\\\ l c )$ quartic CY pair with non-normal boundary. Then, $( X , D _ { X } )$ has maximal intersection except in the cases described in 2.(a) and $( b )$ of Theorem 2. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,462,567,481]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"78\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 7 Consider the pair $( X , D _ { X } )$ where: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[168,495,824,519]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0955/b533738050f72bc3daecd4a17f3f169de6c8d759454747f4215e020b83070d2d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"78\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nX = \\\\{ x _ { 0 } ^ { 2 } x _ { 3 } ^ { 2 } + x _ { 1 } ^ { 2 } x _ { 2 } x _ { 3 } + x _ { 2 } ^ { 2 } q ( x _ { 0 } , x _ { 1 } ) + x _ { 4 } f _ { 3 } = 0 \\\\} , D _ { X } = X \\\\cap \\\\{ x _ { 4 } = 0 \\\\} ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[148,534,851,570]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"78\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $q$ is a general quadratic form in $( x _ { 1 } , x _ { 2 } )$ and $f _ { 3 }$ a general cubic in $x _ { 0 } , \\\\cdots , x _ { 4 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,572,854,680]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"78\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since $q$ and $f _ { 3 }$ are general, the quartic hypersurface $X$ has 6 ordinary double points. Indeed, denote by $L ~ = ~ \\\\{ x _ { 0 } ~ = ~ x _ { 1 } ~ = ~ x _ { 4 } ~ = ~ 0 \\\\}$ , and by $L ^ { \\\\prime } = \\\\{ x _ { 2 } = x _ { 3 } = x _ { 4 } = 0 \\\\}$ , then ${ \\\\mathrm { S i n g } } ( X )$ consists of points of intersection of $\\\\{ f _ { 3 } = 0 \\\\}$ with $L \\\\cup L ^ { \\\\prime }$ ; there are 3 such points $\\\\{ q _ { 1 } , q _ { 2 } , q _ { 3 } \\\\}$ on $L$ and 3 points $\\\\{ q _ { 1 } ^ { \\\\prime } , q _ { 2 } ^ { \\\\prime } , q _ { 3 } ^ { \\\\prime } \\\\}$ on $L ^ { \\\\prime }$ because $f _ { 3 }$ is general. In the neighbourhood of each point $q _ { i }$ (resp. $q _ { i } ^ { \\\\prime } )$ for $i = 1 , 2 , 3$ , the equation of $X$ is of the form \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[410,696,585,716]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0399/ae44ea1fa0cba127f9f565bd503bf5e11bca49107c1652f730a1970f199ecabd.jpg\",\n \"list_items\": null,\n \"page_idx\": \"78\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n0 \\\\in \\\\{ x y + z t = 0 \\\\}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[148,732,851,786]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"78\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(and $D _ { X } = \\\\{ t = 0 \\\\} )$ ) so that all singular points of $X$ are ordinary double points. The nodal quartic - $X$ is terminal and $\\\\mathbb { Q }$ -factorial because it has less than 9 ordinary double points; $X$ is birationally rigid by [2, 18]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,788,854,876]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"78\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Taking the divisorial extraction of the lines $L$ and $L ^ { \\\\prime }$ is enough to produce a dlt modification $( \\\\widetilde { X } , D _ { \\\\widetilde { X } } + E + E ^ { \\\\prime } )$ of $( X , D _ { X } )$ . As the lines $L$ , $L ^ { \\\\prime }$ are skew, $E$ and $E ^ { \\\\prime }$ are disjoint and $( X , D _ { X } )$ does not have maximal intersection. The dual complex has two 1-strata, The quartic surface $D _ { X }$ is an example of case 2.(a) in Theorem 2. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[744,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"78\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A.-S. Kaloghiros \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"78\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"72 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,876,129]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"79\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3 ([21]) A normal quartic surface $D \\\\subset \\\\mathbb { P } ^ { 3 }$ with at least one noncanonical singular point is one of: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,136,651,190]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"79\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. D has a single elliptic singularity and $D$ is rational, or \\n2. D is a cone, or \\n3. D is elliptically ruled and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,198,881,307]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"79\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(a) $D$ has a double point $p$ with tangent cone $z ^ { 2 }$ , the projection away from $p$ is the double cover of $\\\\mathbb { P } ^ { 2 }$ branched over a sextic curve $\\\\Gamma$ . The curve $\\\\Gamma$ is the union of 3 conics in a pencil that also contains a double line. When this line is a common chord, $D$ has two $T _ { 2 , 3 , 6 }$ singularities; when this line is $a$ common tangent, $D$ has one singularity of type $E _ { 4 , 0 }$ . In the first case, $D$ may have an additional $A _ { 1 }$ singular point. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,307,881,399]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"79\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(b) $D$ is $\\\\{ ( x _ { 0 } x _ { 3 } + q ( x _ { 1 } , x _ { 2 } ) ) ^ { 2 } + f _ { 4 } ( x _ { 1 } , x _ { 2 } , x _ { 3 } ) = 0 \\\\}$ and $\\\\{ f _ { 4 } ~ = ~ 0 \\\\}$ is four concurrent lines. Depending on whether $L = \\\\{ x _ { 3 } = 0 \\\\}$ is one of these lines or not and on whether the point of concurrence lies on $L , D$ has either two $T _ { 2 , 4 , 4 }$ singular points or one trimodal elliptic singularity. The surface may have additional canonical points $A _ { n }$ for $n = 1 , 2 , 3$ or $2 A _ { 1 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,435,681,454]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"79\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 8 Let $X$ be the nonsingular quartic hypersurface \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[198,468,801,492]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0849/500a21be1e3f9a828c1460cd808377d10714fbed25184d62e0044dabfd42f252.jpg\",\n \"list_items\": null,\n \"page_idx\": \"79\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nX = \\\\{ x _ { 0 } ^ { 2 } x _ { 3 } ^ { 2 } + x _ { 0 } x _ { 1 } ^ { 3 } + x _ { 3 } x _ { 2 } ^ { 3 } + x _ { 0 } x _ { 1 } x _ { 2 } x _ { 3 } + x _ { 4 } ( x _ { 0 } ^ { 3 } + x _ { 3 } ^ { 3 } + x _ { 4 } ^ { 3 } ) = 0 \\\\}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[143,507,806,527]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"79\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and $D _ { X }$ its hyperplane section $X \\\\cap \\\\{ x _ { 4 } = 0 \\\\}$ . The surface $D _ { X }$ is normal, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[271,542,722,564]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0014/cdfc160e95ff0ab2b5a453f512a75d88a2d4ab7fd21bb16b0df9014e5a218b72.jpg\",\n \"list_items\": null,\n \"page_idx\": \"79\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname { S i n g } ( D _ { X } ) = \\\\{ p , p ^ { \\\\prime } \\\\} = \\\\{ ( 1 { : } 0 { : } 0 { : } 0 { : } 0 ) , ( 0 { : } 0 { : } 0 { : } 1 { : } 0 ) \\\\} ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[148,579,849,615]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"79\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and each singular point is simple elliptic $J _ { 2 , 0 } ~ = ~ T _ { 2 , 3 , 6 }$ , i.e. is locally analytically equivalent to- $0 \\\\in \\\\{ x ^ { \\\\bar { 2 } } + y ^ { 3 } + z ^ { 6 } + x y z = 0 \\\\}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,617,854,743]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"79\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Here $X$ is nonsingular and $D _ { X }$ is irreducible and normal, and as I explain below, finding a good (t, dlt) modification amounts to constructing a minimal resolution of $D _ { X }$ . Let - ${ \\\\widetilde { X } } \\\\to X$ be the composition of the weighted blowups at $p = ( 1 { : } 0 { : } 0 { : } 0 { : } 0 )$ with weights $( 0 , 2 , 1 , 3 , 1 )$ and at $p ^ { \\\\prime } = ( 0 { : } 0 { : } 0 { : } 1 { : } 0 )$ with weights $( 3 , 1 , 2 , 0 , 1 )$ , and denote by $E$ and $E ^ { \\\\prime }$ the corresponding exceptional divisors. Note that $\\\\widetilde { X }$ is terminal and - $\\\\mathbb { Q }$ -factorial by [13, Theorem 3.5] and has no worse than cyclic quotient singularities. The morphism \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[364,758,635,785]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0901/54404990ccfb008908285a3461a5cd14cf4dcecaa2271a63fb9ab70a73c602c9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"79\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n( \\\\widetilde { X } , D + E + E ^ { \\\\prime } ) \\\\stackrel { f } { \\\\to } ( X , D )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,801,854,857]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"79\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is volume preserving and the intersection of $D$ with each exceptional divisor is a smooth elliptic curve $C _ { 6 } \\\\subset \\\\mathbb { P } ( 1 , 1 , 2 , 3 )$ not passing through the singular points of $E$ and $E ^ { \\\\prime }$ ; $f$ is a good (t, dlt) modification. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,51,858,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"79\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[769,873,879,888]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"79\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(continued) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"79\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[148,109,364,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"80\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 8 (continued) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,126,854,177]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"80\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The dual complex ${ \\\\mathcal { D } } ( X , D _ { X } )$ is 1-dimensional, it has 3 vertices and 2 edges; $( X , D _ { X } )$ does not have maximal intersection. The quartic surface $D _ { X }$ is an example of case 3(a) in Theorem 3. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,213,879,267]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"80\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 2 Let $( X , D _ { X } )$ be a $( t , \\\\ l c )$ quartic CY pair. Assume that $D _ { X }$ is normal, has non-canonical singularities but is not a cone. Then $( X , D _ { X } )$ has maximal intersection except in cases 3.(a) and (b) of Theorem 3. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,277,879,367]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"80\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark $6$ When $\\\\dim { \\\\mathcal { D } } ( X , D _ { X } ) \\\\ = \\\\ 1$ , $D _ { X }$ either has two $T _ { 2 , 3 , 6 }$ or two $T _ { 2 , 4 , 4 }$ singularities. Indeed, as is explained in Sect. 2.2, singular points $p \\\\in D$ are Kodaira singularities, and in particular are at worst bimodal. The description of cases 3.(a) and (b) of Theorem 3 immediately implies the result, because a surface singularity of type $E _ { 4 , 0 }$ is trimodal. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,414,248,433]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"80\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[125,454,881,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"80\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Brown, M., McKernan, J., Svaldi, R., Zong, H.: A geometric characterisation of toric varieties. Duke Math. J. 167, 923–968 (2018) \\n2. Cheltsov, I.: Factorial threefold hypersurfaces. J. Algebraic Geom. 19, 781–791 (2010) \\n3. Corti, A., Kaloghiros, A.-S.: The Sarkisov program for Mori fibred Calabi–Yau pairs. Algebr. Geom. 3, 370–384 (2016) \\n4. Cox, D., Little, J., Schenck, H., Toric Varieties. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011) \\n5. de Fernex, T., Kollár, J., Xu, C.: The dual complex of singularities. Adv. Stud. Pure Math. 74, 103–130 (2017) \\n6. Degtyarev, A.: Classification of surfaces of degree four having a nonsimple singular point, Math. USSR Izv. 35, 607–627 (1990) \\n7. Gross, M., Hacking, P., Keel, S.: Mirror symmetry for log Calabi–Yau surfaces I. Inst. Hautes Etudes Sci. Publ. Math. 122, 65–168 (2015) \\n8. Gross, M., Hacking, P., Keel, S.: Birational geometry of cluster algebras. Algebr. Geom. 2, 137–175 (2015) \\n9. Gross, M., Hacking, P., Keel, S., Kontsevich, M.: Canonical bases for cluster algebras. J. Am. Math. Soc. 31, 497–608 (2018) \\n10. Gross, M., Siebert, B.: Intrinsic mirror symmetry and punctured Gromov–Witten invariants. Proc. Sympos. Pure Math. 97(2), 199–230 (2018) \\n11. Iskovskih, V., Manin, Ju.: Three-dimensional quartics and counterexamples to the Lüroth problem. Mat. Sb. 86, 140–166 (1971) \\n12. Karras, U.: On pencils of curves and deformations of minimally elliptic singularities. Math. Ann. 47, 43–65 (1980) \\n13. Kawakita, M.: Divisorial contractions in dimension three which contract divisors to smooth points. Invent. Math. 145, 105–119 (2001) \\n14. Kollár, J., Xu, C.: The dual complex of Calabi–Yau pairs. Invent. Math. 205, 527–557 (2016) \\n15. Kollár, J., et al.: Flips and abundance. Astérisque 211, 249–254 (1992) \\n16. Laufer, H.: On minimally elliptic singularities. Am. J. Math. 99, 1257–1295 (1977) \\n17. Liu, W., Rollenske, S.: Two-dimensional semi-log-canonical hypersurfaces. Le Matematiche 67, 185–202 (2012) \\n18. Mella, M.: On the birational geometry of quartic 3-folds II: the importance of being $\\\\mathbb { Q }$ -factorial. Math. Ann. 330, 107–126 (2004) \\n19. Urabe, T.: Classification of non-normal quartic surfaces. Tokyo J. Math. 9, 265–295 (1986) \\n20. Wall, C.: Classification of unimodal isolated singularities of complete intersections. Proc. Sympos. Pure Math. 40, 625–640 (1983) \\n21. Wall, C.: Sextic curves and quartic surfaces with higher singularities (1999). Preprint. https:// www.liverpool.ac.uk/\\\\~ctcw/hsscqs.ps \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[744,54,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"80\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A.-S. Kaloghiros \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,54,141,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"80\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,91,881,226]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"81\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[113,52,865,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"81\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"81\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,589,141]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"82\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,175,439,193]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"82\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Emma Lepri and Marco Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,286,881,394]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"82\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract In this paper we study classical deformations of diagrams of commutative algebras over a field of characteristic 0. In particular we determine several homotopy classes of DG-Lie algebras, each one of them controlling this above deformation problem: the first homotopy type is described in terms of the projective model structure on the category of diagrams of differential graded algebras, the others in terms of the Reedy model structure on truncated Bousfield-Kan approximations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,396,879,448]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"82\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The first half of the paper contains an elementary introduction to the projective model structure on the category of commutative differential graded algebras, while the second half is devoted to the main results. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,468,870,486]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"82\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords Model categories $\\\\cdot$ Deformation theory $\\\\cdot$ Differential graded algebras \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,522,309,543]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"82\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,567,881,728]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"82\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $\\\\mathbb { K }$ be a fixed field, $s$ a Noetherian commutative $\\\\mathbb { K }$ -algebra and $X = { \\\\mathrm { S p e c } } ( S )$ the associated affine scheme. It is well known that every deformation of $X$ , in the category of schemes over $\\\\mathbb { K }$ , is affine, hence the deformation theory of $X$ is the same of the deformation theory of $s$ inside the category $\\\\mathbf { A l g } _ { \\\\mathbb { K } }$ of unitary commutative $\\\\mathbb { K }$ - algebras. Similarly, the deformation theory of a separated Noetherian scheme $X$ over $\\\\mathbb { K }$ is the same as the deformation theory of a diagram in $\\\\mathbf { A l g } _ { \\\\mathbb { K } }$ . More precisely, if $N$ is the nerve of an affine open cover $\\\\{ U _ { i } \\\\}$ of $X$ , it is not difficult to prove that the deformations of $X$ (up to isomorphism) are the same as the deformations (up to isomorphism) of the diagram \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[234,746,758,767]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0140/a83cf2affd42a316cda131d9e7f9d1d0421ee086f7f610f8bec45002b6434948.jpg\",\n \"list_items\": null,\n \"page_idx\": \"82\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nS _ { \\\\bullet } \\\\colon N \\\\to \\\\mathbf { A l g } _ { \\\\mathbb { K } } , \\\\qquad S _ { i _ { 0 } , \\\\dots , i _ { n } } = \\\\Gamma ( U _ { i _ { 0 } } \\\\cap \\\\cdots \\\\cap U _ { i _ { n } } , O _ { X } ) ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,897,692,941]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"82\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_6 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,818,865,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"82\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Lepri $\\\\cdot$ M. Manetti $( \\\\boxtimes$ ) Università degli Studi di Roma La Sapienza, Dipartimento di Matematica “Guido Castelnuovo”, Roma, Italy e-mail: manetti@mat.uniroma1.it \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[804,91,876,139]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"82\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,899,876,909]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"82\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,90,876,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"83\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $N$ is considered as a poset and as a small category in the obvious way, inside the category of diagrams $\\\\operatorname { F u n } ( N , \\\\mathbf { A } \\\\mathbf { l g } _ { \\\\mathbb { K } } ) = \\\\{ N \\\\to \\\\mathbf { A } \\\\mathbf { l g } _ { \\\\mathbb { K } } \\\\}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,126,881,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"83\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is well known (see e.g. [5, 13, 15]) that, if $\\\\mathbb { K }$ has characteristic 0, then every (commutative) deformation of an algebra $S \\\\in \\\\mathbf { A l g } _ { \\\\mathbb { K } }$ is isomorphic to $H ^ { 0 } ( R ^ { \\\\prime } )$ , where $R ^ { \\\\prime }$ is obtained by perturbing the differential of a fixed Tate resolution $R S$ [20]. This easily implies that the deformations of $s$ are controlled, in the sense of [12], by the differential graded Lie algebra of derivations of $R$ . A short introduction to differential graded algebras is given here in Sect. 2. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,234,881,322]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"83\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is possible to prove that the above strategy generalises to arbitrary Noetherian separated schemes, where Tate resolution is replaced by the algebraic analogue of Palamodov’s resolvent [16, 17]. This is possible because the nerve $N$ of a covering is a direct Reedy category, i.e., there exists a degree function deg $N \\\\to \\\\mathbb { N }$ such that every non identity arrow increases degree. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,324,879,412]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"83\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The aim of this paper is to study deformations of diagrams ${ \\\\mathcal { D } } \\\\to \\\\mathbf { A l g } _ { \\\\mathbb { K } }$ for a general small category $\\\\mathcal { D }$ . The first result is to extend the above strategy by detecting what is the correct notion of Tate resolution of a diagram $\\\\left[ = \\\\right.$ the correct notion of Palamodov’s resolvent for a diagram). In doing this it is extremely convenient to work in the framework of model structures, briefly recalled in Sect. 3. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,414,881,469]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"83\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The category $\\\\mathbf { A l g } _ { \\\\mathbb { K } }$ can be considered in an obvious way as a full subcategory of $\\\\mathbf { C D G A } _ { \\\\mathbb { K } }$ (resp.: $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ ), the category of commutative differential graded algebras (resp.: in non-positive degrees). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,469,879,524]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"83\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By a classical result of Bousfield and Gugenheim [1] the category $\\\\mathbf { C D G A } _ { \\\\mathbb { K } }$ admits a model structure where weak equivalences are the quasi-isomorphisms and fibrations are the surjective maps, cf. [4]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,524,879,612]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"83\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The category $\\\\mathbf { C D } \\\\mathbf { \\\\bar { G } A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ carries a similar model structure, where weak equivalences are the quasi-isomorphisms and fibrations are the surjective maps in negative degrees. Due to the lack of appropriate references, in Sect. 4 we provide an elementary proof of this fact, based on the properties of free and semifree extensions. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,615,881,720]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"83\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Section 5 is devoted to some technical lemmas that are probably well known to experts. In Sect. 6 we prove the main result of this paper (Theorem 4), namely that the deformation theory of a diagram $S _ { \\\\bullet } \\\\colon { \\\\mathcal { D } } \\\\ \\\\to \\\\ \\\\mathbf { A l g } _ { \\\\mathbb { K } }$ is controlled by the differential graded Lie algebra of derivations of a cofibrant replacement of $S _ { \\\\bullet }$ in the model category of diagrams $\\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } )$ ), equipped with the projective model structure. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,723,879,794]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"83\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Unfortunately, for general index categories $\\\\mathcal { D }$ , cofibrant replacements in the projective model structure are difficult to describe from the constructive point of view. For this reason, in the last sections we propose a different approach by describing a countable family of functors between small categories (Definition 15) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[300,812,692,831]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0430/ed3658eddca4cd81894bcfd932989c3b8d328ed11bff178b5cff3926e24207fa.jpg\",\n \"list_items\": null,\n \"page_idx\": \"83\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\epsilon _ { k } \\\\colon N ( \\\\mathcal { D } ) _ { \\\\leq k } \\\\to \\\\mathcal { D } , \\\\qquad k = 2 , 3 , \\\\dots , \\\\infty ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[687,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"83\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Lepri and M. Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,54,141,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"83\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,487,106]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"84\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"such that for every $k$ in the above range: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,115,879,151]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"84\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. every diagram $S _ { \\\\bullet } \\\\colon { \\\\mathcal { D } } \\\\to \\\\mathbf { A l g } _ { \\\\mathbb { K } }$ has the same isomorphism classes of deformations as $S _ { \\\\bullet } \\\\circ \\\\epsilon _ { k }$ ; \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,153,879,228]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"84\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. $N ( \\\\mathcal { D } ) _ { \\\\leq k }$ is a Reedy category (see Sect. 8) and the projective model structure on the category $\\\\mathrm { F u n } ( N ( \\\\mathcal { D } ) _ { \\\\leq k } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } )$ is the same as the Reedy model structure, hence with cofibrations described constructively in terms of latching objects and cofibrations in $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,235,879,289]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"84\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In our construction the functor $\\\\epsilon _ { \\\\infty }$ is the forgetful functor from the simplex category of $\\\\mathcal { D }$ (see Sect. 7), and $\\\\epsilon _ { k }$ is its restriction to the full subcategory of $p$ - simplexes, with $p \\\\leq k$ . The composition map \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[248,306,749,333]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0308/87e12017b2fa82feaa4f7e04ee21639f08f98a31665504af3d714015be6f0a45.jpg\",\n \"list_items\": null,\n \"page_idx\": \"84\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } ) \\\\xrightarrow { - \\\\circ \\\\epsilon _ { \\\\infty } } \\\\mathrm { F u n } ( N ( \\\\mathcal { D } ) _ { \\\\leq \\\\infty } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,348,876,384]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"84\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is called Bousfield-Kan approximation and plays an important role in the homotopy theory of diagrams [3]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[138,385,760,402]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"84\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Putting together all the above facts, the main result of this paper is: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,411,876,447]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"84\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 1 $\\\\ Y = \\\\ \\\"$ Theorem $^ { 4 + }$ Corollary 1) Let $\\\\mathcal { D }$ be a small category, $S _ { \\\\bullet } \\\\colon { \\\\mathcal { D } } $ $\\\\mathbf { A l g } _ { \\\\mathbb { K } }$ a diagram of unitary commutative algebras. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,456,881,510]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"84\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Let $R _ { \\\\bullet } S _ { \\\\bullet }$ be a cofibrant replacement in $\\\\begin{array} { r } { \\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } , } \\\\end{array}$ ) with respect to the projective model structure. Then the DG-Lie algebra $L = \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { \\\\bullet } , R _ { \\\\bullet } )$ controls the deformations of $S _ { \\\\bullet }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,513,881,584]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"84\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. For every $k ~ = ~ 2 , \\\\ldots , \\\\infty$ , let $\\\\epsilon _ { k } \\\\colon N ( \\\\mathcal { D } ) _ { \\\\leq k } \\\\ \\\\to \\\\ \\\\mathcal { D }$ be the functor defined in Definition 15 and let $R _ { \\\\bullet k } ~ ~ S _ { \\\\bullet } \\\\circ \\\\epsilon _ { k }$ be a Reedy cofibrant replacement in $\\\\mathrm { F u n } ( N ( \\\\mathcal { D } ) _ { \\\\leq k } , { \\\\bf C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } ,$ • • ). Then the DG-Lie algebra $L _ { k } \\\\ = \\\\ \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { \\\\bullet k } , R _ { \\\\bullet k } )$ controls the deformations of $S _ { \\\\bullet }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,630,400,650]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"84\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1.1 Notation and Setup \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,672,881,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"84\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Throughout this paper we will work over a fixed field $\\\\mathbb { K }$ of characteristic 0. Unless otherwise specified, every (graded) vector space is assumed over $\\\\mathbb { K }$ and the symbol $\\\\otimes$ denotes the tensor product over $\\\\mathbb { K }$ . If $V = \\\\oplus _ { n \\\\in \\\\mathbb { Z } } V ^ { n }$ is a graded vector space, we denote by $\\\\overline { { a } }$ the degree of a non-zero homogeneous element $a$ : in other words ${ \\\\overline { { a } } } = n$ whenever $a \\\\neq 0$ and $a \\\\in V ^ { n }$ . It is implicitly assumed that if a mathematical formula contains the degree symbols $\\\\overline { { a } }$ , $\\\\overline { { b } }$ , . . . then all the elements $a , b , \\\\ldots$ involved are homogeneous and different from 0. As usual, for every complex of vector spaces $V$ , we shall denote by $Z ^ { n } ( V )$ , $B ^ { n } ( V )$ and $H ^ { n } ( V )$ the space of $n$ -cocycles, the space of $n$ -coboundaries and the nth cohomology group, respectively. We denote by Set the category of sets, by Grp the category of groups, by $\\\\mathbf { A l g } _ { \\\\mathbb { K } }$ the category of unitary commutative $\\\\mathbb { K }$ -algebras and by $\\\\mathbf { A r t _ { \\\\mathbb { K } } } \\\\subset \\\\mathbf { A l g _ { \\\\mathbb { K } } }$ the full subcategory of local Artin algebras with residue field $\\\\mathbb { K }$ . Finally, in order to avoid an excessive length we assume that the reader has a basic knowledge of differential graded Lie algebras and of the associated deformation functors: for instance, the papers [12, 14] contain everything needed for the comprehension of this paper. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,562,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"84\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,52,879,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"84\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"79 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,879,144]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"85\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,174,669,196]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"85\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 Commutative Differential Graded Algebras \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,219,881,292]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"85\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the first four sections of this paper we shall give a short survey, addressed to a wide mathematical audience, of some homotopical algebra that we use in the second part of the paper. We begin by recalling the definition and the first properties of unitary commutative differential graded algebras (DG-algebras for short) over $\\\\mathbb { K }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,300,881,376]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"85\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 1 A unitary commutative graded algebra is a graded vector space $A =$ $\\\\textstyle \\\\bigoplus _ { n \\\\in \\\\mathbb { Z } } A ^ { n }$ with a product $A ^ { i } \\\\times A ^ { j } \\\\longrightarrow A ^ { i + j }$ which is $\\\\mathbb { K }$ -linear, associative and graded commutative, i.e., such that $a b = ( - 1 ) ^ { \\\\overline { { { a } } } \\\\overline { { { b } } } } b a$ for every $a , b \\\\in A$ . Moreover there exists a unit $1 \\\\in A ^ { 0 }$ such that $1 a = a 1 = a$ for every $a \\\\in A$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,385,879,457]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"85\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A morphism of unitary commutative graded algebras is a morphism of graded vector spaces that commutes with products and preserves the units. We denote by $\\\\mathbf { C G A } _ { \\\\mathbb { K } }$ the category of unitary commutative graded algebras. In the above definition it is allowed that $1 = 0$ , and this happens if and only if $A = 0$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,457,881,584]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"85\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The usual construction of polynomials extends without difficulties to the graded case. Given a unitary commutative graded algebra $A$ and a set $\\\\{ x _ { i } \\\\} , ~ i ~ \\\\in ~ I$ , of indeterminates, each one equipped with a degree ${ \\\\overline { { x _ { i } } } } \\\\ \\\\in \\\\ \\\\mathbb { Z }$ , the polynomial algebra $A [ \\\\{ x _ { i } \\\\} ]$ is defined as the graded vector space generated by the monomials in $x _ { i }$ with coefficients in $A$ , subject to the relations $\\\\overline { { { x _ { i } } { x _ { j } } } } ~ = ~ ( - 1 ) ^ { \\\\overline { { { x _ { i } } } } \\\\overline { { { x _ { j } } } } } x _ { j } x _ { i }$ and $\\\\begin{array} { r l } { a x _ { i } } & { { } = } \\\\end{array}$ $( - 1 ) ^ { \\\\overline { { x _ { i } } } \\\\overline { { a } } } x _ { i } a$ , $a \\\\in A$ . For instance, if $\\\\overline { { x } } = 0$ and $\\\\overline { { y } } = 1$ , then $x y = y x$ , $y ^ { 2 } = 0$ and therefore $\\\\mathbb { K } [ x , y ] = \\\\mathbb { K } [ x ] \\\\oplus \\\\mathbb { K } [ x ] y$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,584,876,620]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"85\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Given $A \\\\in \\\\mathbf { C G A } _ { \\\\mathbb { K } }$ , a derivation of degree $k$ of $A$ is a linear map $\\\\alpha \\\\colon A \\\\to A$ such that $\\\\alpha ( A ^ { n } ) \\\\subset A ^ { n + k }$ for every $n$ , satisfying the (graded) Leibniz identity: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[346,635,651,656]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0283/9befbcd1f13b4468aa2c6a7b18c0c2f151b20db3ac8408aff206b10f11fcd4f4.jpg\",\n \"list_items\": null,\n \"page_idx\": \"85\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\alpha ( a b ) = \\\\alpha ( a ) b + ( - 1 ) ^ { k \\\\overline { { { a } } } } a \\\\alpha ( b ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,672,733,692]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"85\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The vector space of derivations of degree $k$ is denoted $\\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { k } ( A , A )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,693,881,731]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"85\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $A \\\\ = \\\\ \\\\mathbb { K } [ \\\\{ x _ { i } \\\\} ]$ , by the Leibniz identity every derivation $\\\\alpha \\\\in { \\\\mathrm { D e r } } _ { \\\\mathbb { K } } ^ { k } ( A , A )$ is uniquely defined by the values $\\\\alpha ( x _ { i } ) \\\\in A ^ { \\\\overline { { x _ { i } } } + k }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,740,879,794]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"85\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 2 A commutative differential graded algebra (DG-algebra for short) is a graded commutative algebra $A$ equipped with a derivation $d \\\\in \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { 1 } ( A , A )$ , called differential, such that $d ^ { 2 } = 0$ . In other words: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,801,669,858]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"85\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. $d ( A ^ { n } ) \\\\subseteq A ^ { n + 1 }$ , \\n2. $d ^ { 2 } = 0$ , \\n3. (Graded Leibniz identity) $d ( a b ) = d ( a ) b + ( - 1 ) ^ { \\\\overline { { { a } } } } a d ( b ) .$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,866,881,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"85\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A morphism of commutative differential graded algebras is a morphism of commutative graded algebras that commutes with differentials. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"85\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"80 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[687,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"85\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Lepri and M. Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We denote by $\\\\mathbf { C D G A } _ { \\\\mathbb { K } }$ the category of commutative differential graded algebras. Notice that $d ( 1 ) = 0$ , and that $\\\\mathbb { K }$ and 0 are respectively the initial and the final object in the category $\\\\mathbf { C D G A } _ { \\\\mathbb { K } }$ . It is easy to see that this category is complete and cocomplete. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,169,881,261]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3 (Free Extensions) Let $\\\\begin{array} { r } { A \\\\in \\\\begin{array} { r } { \\\\mathbf { C D G A } _ { \\\\mathbb { K } } } \\\\end{array} } \\\\end{array}$ and $x _ { i }$ , $\\\\textit { i } \\\\in \\\\textit { I }$ , a set of indeterminates of degree ${ \\\\overline { { x _ { i } } } } ~ \\\\in ~ \\\\mathbb { Z }$ . Consider a parallel set of indeterminates $d x _ { i }$ , with $\\\\overline { { d x _ { i } } } = \\\\overline { { x _ { i } } } + 1$ and the polynomial extension $A \\\\to A [ \\\\{ x _ { i } , d x _ { i } \\\\} ]$ . The differential $d$ on $A$ can be extended to a differential on $A [ \\\\{ x _ { i } , d x _ { i } \\\\} ]$ by setting $d ( x _ { i } ) = d x _ { i }$ and $d ( d x _ { i } ) = 0$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,270,881,342]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The name free extension is motivated by the following property: for every morphism $f \\\\colon A \\\\ \\\\to \\\\ B$ in $\\\\mathbf { C D G A } _ { \\\\mathbb { K } }$ and every subset $\\\\{ b _ { i } \\\\} \\\\subset B$ with $b _ { i } ~ \\\\in ~ B ^ { \\\\overline { { x _ { i } } } }$ for every $i$ , there exists a unique morphism of DG-algebras $g \\\\colon A [ \\\\{ x _ { i } , d x _ { i } \\\\} ] B$ extending $f$ and such that $g ( x _ { i } ) = b _ { i }$ for every $i$ . Clearly $g ( d x _ { i } ) = d ( b _ { i } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,351,881,387]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 1 Every free extension of $_ { D G }$ -algebras is a quasi-isomorphism, i.e., the inclusion $A \\\\to A [ \\\\{ x _ { i } , d x _ { i } \\\\} ]$ induces an isomorphism in cohomology. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,394,881,466]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Since every element of $A [ \\\\{ x _ { i } , d x _ { i } \\\\} ]$ is a polynomial in a finite number of indeterminates, we can assume the set of variables finite, say $x _ { 1 } , \\\\ldots , x _ { n }$ , and proceed by induction on $n$ . Therefore it is sufficient to show that the inclusion $A \\\\to A [ x , d x ]$ is a quasi-isomorphism. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,468,876,504]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $x$ has even degree, then $( d x ) ^ { 2 } = 0$ and every homogeneous element of the quotient $A [ x , d x ] / A$ is of type \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[298,515,699,567]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"1016/33408f49e3b6513cf57157a77ffe3fa7a90f7b6fdb133156c4e4b0fae65f645e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nv = \\\\sum _ { i = 0 } ^ { n } x ^ { i + 1 } a _ { i } + x ^ { i } d x b _ { i } , \\\\qquad a _ { i } , b _ { i } \\\\in A .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,578,881,596]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $d v = 0$ then $d ( b _ { i } ) = ( i + 1 ) a _ { i }$ and therefore (notice the assumption char $( \\\\mathbb { K } ) = 0$ ) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[387,608,610,659]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"1008/428784783f2c8d85ca28ad12e30622708d3b042e825f084c769491b75a940fd9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nv = d \\\\left( \\\\sum _ { i = 0 } ^ { n } \\\\frac { x ^ { i + 1 } } { i + 1 } b _ { i } \\\\right) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[123,672,881,710]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $x$ has odd degree, then $x ^ { 2 } = 0$ and every homogeneous element of the quotient $A [ x , d x ] / A$ is of type \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[275,722,722,773]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0356/ffa50fea34f3505cae393ca117998e3c68a7b86d33a1fb75eb1fd50c3215bcc1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nu = \\\\sum _ { i = 0 } ^ { n } ( d x ) ^ { i + 1 } a _ { i } + x ( d x ) ^ { i } b _ { i } , \\\\qquad a _ { i } , b _ { i } \\\\in A .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,783,649,801]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $d u = 0$ then $d a _ { i } + b _ { i } = 0$ for every $i$ , and we can write \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[382,813,615,866]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0701/ea04aeb7b83ff1b5681e35d7ad403866f5d7ce311ae2cc940b0f9bd5d8c83e3c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nu = d \\\\left( \\\\sum _ { i = 0 } ^ { m } x ( d x ) ^ { i } a _ { i } \\\\right) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[113,879,826,899]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Thus we have proved that $A [ x , d x ] / A$ is an acyclic complex of vector spaces. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,562,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[861,882,879,896]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"86\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"□ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,91,879,147]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"87\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 4 Let $f \\\\colon A \\\\to B$ be a morphism in $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ , with $A = A ^ { 0 } \\\\in \\\\mathbf { A l g } _ { \\\\mathbb { K } }$ We shall say that $f$ is flat, or that $B$ is a flat $A$ -algebra if $B$ is a complex of flat $A$ -modules. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,156,879,211]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"87\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Clearly the above definition extends the usual notion of flat morphism of algebras. It is worth pointing out that there also exists a good notion of flatness for every morphism in $\\\\mathbf { C D G } \\\\mathbf { \\\\bar { A } } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ [15]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,211,881,271]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"87\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For every $A \\\\in \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ we shall denote by $\\\\mathbf { C D G A } _ { A } ^ { \\\\leq 0 }$ the undercategory of maps $A B$ : the morphisms in $\\\\mathbf { C D G A } _ { A } ^ { \\\\leq 0 }$ are the commutative triangles. The following lemma is completely standard, see e.g. [19, Lemma A.4 and Theorem A.10]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[113,279,799,300]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"87\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 2 Let $A \\\\in \\\\mathbf { A r t } _ { \\\\mathbb { K } }$ and let $f \\\\colon B \\\\to C$ be a morphism in $\\\\mathbf { C D G A } _ { A } ^ { \\\\leq 0 }$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,307,881,435]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"87\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. if $C$ is flat over A and the induced map $B \\\\otimes _ { A } \\\\mathbb { K } \\\\to C \\\\otimes _ { A } \\\\mathbb { K }$ is an isomorphism, then $f$ is also an isomorphism; \\n2. if $B$ , $C$ are flat $A$ -algebras and the induced map $B \\\\otimes _ { A } \\\\mathbb { K } \\\\to C \\\\otimes _ { A } \\\\mathbb { K }$ is a quasiisomorphism, then $f$ is also a quasi-isomorphism; \\n3. if $B$ is flat over $A$ and $H ^ { i } ( B \\\\otimes _ { A } \\\\mathbb { K } ) = 0$ for every $i \\\\ < 0$ , then $H ^ { 0 } ( B )$ is a flat $A$ -algebra and the natural map $H ^ { 0 } ( B ) \\\\to H ^ { 0 } ( B \\\\otimes _ { A } \\\\mathbb { K } )$ induces an isomorphism $H ^ { 0 } ( B ) \\\\otimes _ { A } \\\\mathbb { K } = H ^ { 0 } ( B \\\\otimes _ { A } \\\\mathbb { K } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,478,710,500]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"87\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 A Very Short Introduction to Model Structures \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,522,881,576]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"87\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We briefly recall the definition of model category and some few basic results about them; the reader may consult [6, 7] for a deeper and more complete exposition of the subject. Throughout this section M will denote a fixed category. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,585,876,623]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"87\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 5 (Lifting Properties) Consider two morphisms $i \\\\colon A \\\\to B , f \\\\colon C \\\\to$ $D$ in M. If for every solid commutative diagram \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[432,638,562,707]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0901/d6c590c3afdb16ac4843d49db5aaec57e6ac985e149065895cf8c44695948ecf.jpg\",\n \"list_items\": null,\n \"page_idx\": \"87\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[118,726,879,782]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"87\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"there exists the dotted arrow that makes both triangles commute, we shall say that the map $i$ has the left lifting property with respect to $f$ , and the map $f$ has the right lifting property with respect to $i$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,791,881,845]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"87\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For instance, in the category of sets, every injective map $i$ has the left lifting property with respect to any surjective map $f$ . The same holds in the category of vector spaces. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[687,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"87\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Lepri and M. Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"87\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"82 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,881,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"88\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 6 (Retracts) A morphism $f$ in $\\\\mathbf { M }$ is called a retract of a morphism $g$ in $\\\\mathbf { M }$ if there exists a commutative diagram: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[389,142,608,265]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0604/35b67bdee57017e081f39e3bae7cc5b317786d8c7b09d7960285b3262a2a2cdb.jpg\",\n \"list_items\": null,\n \"page_idx\": \"88\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[120,288,881,324]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"88\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 7 A model structure on $\\\\mathbf { M }$ is the data of three classes of maps: weak equivalences, fibrations and cofibrations, which satisfy the following axioms: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,333,881,384]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"88\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(M1) (2-out-of-3) If $f$ and $g$ are morphisms in $\\\\mathbf { M }$ such that the composition $g f$ is defined, and two out of the three $f , g$ and $g f$ are weak equivalences, so is the third. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,387,876,421]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"88\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(M2) (Retracts) If $f$ and $g$ are maps in $\\\\mathbf { M }$ such that $f$ is a retract of $g$ , and $g$ is a weak equivalence, a cofibration or a fibration, then so is $f$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,423,881,475]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"88\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(M3) (Lifting) A trivial fibration is map which is both a fibration and a weak equivalence; a trivial cofibration is map which is both a cofibration and a weak equivalence. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[157,486,879,540]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"88\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(a) Trivial fibrations have the right lifting property with respect to cofibrations. (b) Trivial cofibrations have the left lifting property with respect to fibrations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[113,549,783,567]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"88\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(M4) (Factorisation) Every morphism $g$ in $\\\\mathbf { M }$ admits two factorisations: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[177,576,820,612]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"88\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(CW, F): $g = q j$ , where $j$ is a trivial cofibration and $q$ is a fibration, (C, FW): $g = p i$ , where $i$ is a cofibration and $p$ is a trivial fibration. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,621,879,656]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"88\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 8 A model category is a complete and cocomplete category equipped\\u0018 with a model structure. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,665,881,755]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"88\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In particular every model category has an initial object $\\\\amalg _ { \\\\varnothing }$ and a final object $\\\\Pi _ { \\\\emptyset }$ ; an object $X$ is called cofibrant in the morphism $\\\\operatorname { U } _ { \\\\varnothing } \\\\to X$ is a cofibration; it is called fibrant if the morphism $X \\\\to \\\\Pi _ { \\\\emptyset }$ is a fibration. A cofibrant replacement of an object $Y$ is a trivial fibration $X Y$ with $X$ cofibrant. The factorisation axiom guarantees that cofibrant replacements always exist. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,756,881,827]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"88\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For notational simplicity we shall denote by $\\\\mathcal { W }$ , $\\\\mathcal { F }$ and $c$ the classes of weak equivalences, fibrations and cofibrations, respectively. We shall denote by ${ \\\\mathcal { F W } } =$ $\\\\mathcal { F } \\\\cap \\\\mathcal { W }$ the class of trivial fibrations and by $C W = C \\\\cap \\\\mathcal { W }$ the class of trivial cofibrations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,837,879,873]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"88\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 3 If $j$ has the left (right) lifting property with respect to $f$ , and $i$ is a retract of $j$ , then i has the left (right) lifting property with respect to $f$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,882,387,900]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"88\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For a proof, see [6, 7.2.8]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,562,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"88\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"88\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,881,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"89\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 1 (Retract Argument) Let g be a map which can be factored as $g =$ pi \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,135,806,171]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"89\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. If g has the left lifting property with respect to p then g is a retract of i. \\n2. If g has the right lifting property with respect to i then g is a retract of $p$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,178,387,198]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"89\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For a proof, see [7, 1.1.9]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,205,480,223]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"89\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 4 Let M be a model category: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,232,881,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"89\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. A map in M that has the left lifting property with respect to all trivial fibrations is a cofibration. \\n2. A map in M that has the right lifting property with respect to all trivial cofibrations is a fibration. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,313,398,331]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"89\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For a proof, see [7, 1.1.10]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,342,881,412]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"89\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 1 It follows from the previous lemma that isomorphisms belong to all three classes of maps. Furthermore, two of the three classes $\\\\mathcal { W }$ , $c$ , $\\\\mathcal { F }$ determine the third. Pay attention to the fact that, for example, if $\\\\mathcal { W }$ and $\\\\mathcal { F }$ are two classes satisfying (M1) and (M2), in general they do not extend to a model structure. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,423,419,439]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"89\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The following lemma is clear. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,448,879,486]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"89\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 5 If $h = g f$ and both $f , g$ have the left (right) lifting property with respect to $p$ , then h has the left (right) lifting property with respect to $p$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,495,874,530]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"89\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 2 The previous lemmas show that the three classes of fibrations, cofibrations and weak equivalences are closed by composition. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,539,881,647]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"89\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Model categories were introduced by Quillen [18] under the name of (complete and cocomplete) closed model categories. Nowadays many authors (e.g. [6, 7]) assume that the (C,FW) and (CW,F) factorisations are functorial. Since in algebraic geometry it is often important to resolve minimally algebraic structures, we prefer here to adopt the original Quillen’s assumption, and require only the existence of factorisations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,693,421,713]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"89\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.1 Pre-model Structures \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,737,881,881]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"89\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In several concrete cases, a convenient way to describe model structures is in terms of pre-model structures. Since two out of the three classes ${ \\\\mathcal { W } } , C , { \\\\mathcal { F } }$ determine the third, it is typical to try to construct a model structure by establishing two out of the three classes and seeing whether they extend to a model structure. Often it happens that two classes have an easy description and the third is more complicated, but it has a nice subclass sufficiently large to ensure axiom (M4). The notion of left premodel structure applies when one has fixed the weak equivalences, fibrations and two more classes, as in the next definition. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[687,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"89\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Lepri and M. Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"89\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"84 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,879,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"90\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 9 A left pre-model structure on $\\\\mathbf { M }$ is the data of four classes of maps: $\\\\mathcal { W } , \\\\mathcal { F } , C ^ { \\\\prime } , C \\\\mathcal { W } ^ { \\\\prime }$ such that: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,135,881,244]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"90\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. (2-out-of-3) The maps in $\\\\mathcal { W }$ satisfy the 2-out-of-3 property; \\n2. (Retracts) The classes $\\\\mathcal { W }$ and $\\\\mathcal { F }$ are closed under retracts; \\n3. $C W ^ { \\\\prime } \\\\subseteq \\\\mathcal { W }$ ; \\n4. (Lifting) The maps in $C ^ { \\\\prime }$ have the left lifting property with respect to the maps in $\\\\mathcal { F } \\\\cap \\\\mathcal { W }$ ; the maps in $c w ^ { \\\\prime }$ have the left lifting property with respect to the maps in $\\\\mathcal { F }$ ; \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,243,667,261]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"90\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5. (Factorisation) Every map $g$ in $\\\\mathbf { M }$ has two factorisations: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,270,589,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"90\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(a) $g = q j$ , where $j$ is in $C ^ { \\\\prime }$ and $q$ is in $\\\\mathcal { F } \\\\cap \\\\mathcal { W }$ , (b) $g = p i$ , where $i$ is in $c w ^ { \\\\prime }$ and $p$ is in $\\\\mathcal { F }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,315,881,385]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"90\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2 Given a left pre-model structure ${ \\\\mathcal { W } } , { \\\\mathcal { F } } , C ^ { \\\\prime }$ , $c w ^ { \\\\prime }$ there exists a unique model structure where the weak equivalences are the maps in $\\\\mathcal { W }$ and the fibrations are the maps in $\\\\mathcal { F } .$ Notably, the cofibrations are the retracts of $C ^ { \\\\prime }$ , and the trivial cofibrations are the retracts of $C \\\\mathcal W ^ { \\\\prime }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,394,881,466]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"90\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof We set $c$ the retracts of $C ^ { \\\\prime }$ and check that $c , \\\\mathcal { F } , \\\\mathcal { W }$ satisfy the model category axioms (Definition 7). Axioms (M1) and (M2) follow immediately from the definition of pre-model structure and of $c$ . As above, for notational simplicity we shall denote by $C W = C \\\\cap \\\\mathcal { W }$ and $\\\\mathcal { F W } = \\\\mathcal { F } \\\\cap \\\\mathcal { W }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,468,881,546]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"90\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We first show that $c w ^ { \\\\prime } \\\\subset C$ . Let $g \\\\colon A \\\\ \\\\to \\\\ B$ be a morphism in $c w ^ { \\\\prime }$ and consider a factorisation $g \\\\colon A \\\\ { \\\\overset { C ^ { \\\\prime } } { \\\\to } } \\\\ X \\\\ { \\\\overset { \\\\mathcal { F W } } { \\\\to } } \\\\ B$ . Since $g$ has the left lifting property with respect to maps in $\\\\mathcal { F W }$ it follows by the retract argument that $g$ is a retract of an element of $C ^ { \\\\prime }$ . Since $C W ^ { \\\\prime } \\\\subset \\\\mathcal { W }$ by assumption, we have $C W ^ { \\\\prime } \\\\subset C W$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,548,879,624]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"90\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We now show that every map in $c w$ is a retract of a map in $c w ^ { \\\\prime }$ . Let $f \\\\colon A \\\\longrightarrow$ $B$ be a map in $c w$ , using the factorisation axiom $f \\\\colon A \\\\ { \\\\xrightarrow { C { \\\\mathcal { W } } ^ { \\\\prime } } } \\\\ X \\\\ { \\\\xrightarrow { \\\\mathcal { F } } } \\\\ B$ , and by the 2-out-of-3 axiom $X \\\\longrightarrow B$ is in $\\\\mathcal { F W } .$ . Therefore by the lifting axiom and the retract argument (1), $A \\\\longrightarrow B$ is a retract of $A \\\\longrightarrow X$ , so it is the retract of a map in $c w ^ { \\\\prime }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,626,879,716]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"90\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By definition of pre-model structure, maps in $C ^ { \\\\prime }$ have the left lifting property with respect to maps in $\\\\mathcal { F W }$ ; then by Lemma 3 maps in $c$ have the left lifting property with respect to maps in $\\\\mathcal { F W }$ . Similarly, maps in $c w ^ { \\\\prime }$ have the left lifting property with respect to maps in $\\\\mathcal { F } _ { : }$ , so we have that maps in $c w$ have the left lifting property with respect to maps in $\\\\mathcal { F }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,716,874,752]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"90\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The factorisation axiom (M4) is clear since we have already proved that $C \\\\subset C ^ { \\\\prime }$ and $C W ^ { \\\\prime } \\\\subseteq C W$ . □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,761,881,833]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"90\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is plain that one can also give the analogous notion of right pre-model structure, simply working in the opposite category and exchanging the role of $c$ and $\\\\mathcal { F } .$ . Finally, the reader should be aware that some authors use the name of pre-model structure for a completely different concept. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,562,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"90\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"90\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,553,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"91\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 Model Structure on DG-Algebras \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,132,879,205]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"91\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is well known that the category $\\\\mathbf { C D G A } _ { \\\\mathbb { K } }$ admits a model structure where weak equivalences are the quasi-isomorphisms and fibrations are the surjective maps [1, 4]. Consequently, by Lemma 4 the cofibrations are the morphisms that have the left lifting property with respect to the class of surjective quasi-isomorphisms. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,205,881,349]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"91\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Similarly, the category $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ of DG-algebras concentrated in non-positive degree admits a model structure where weak equivalences are the quasiisomorphisms and fibrations are the surjective maps in negative degree. It is worth noticing that the existence of the model structure on $\\\\mathbf { C } \\\\mathbf { \\\\bar { D } G } \\\\mathbf { A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ is an immediate consequence of [11, Proposition 4.5.4.6] applied to the standard (cofibrantly generated) model structure on the category of non-positively graded DG-vector spaces. Moreover, by Lurie’s result also follows that the model structure on $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ is combinatorial and cofibrantly generated. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,351,881,403]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"91\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section, following the ideas of [8], we give an elementary proof of the above mentioned model structure on $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ A≤0K , which relies on the notion of semifree extension. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,409,881,466]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"91\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 10 (Semifree Extension) Let $A \\\\in \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ , $I$ be a set, and let $x _ { i } , i \\\\in I$ be indeterminates of non-positive degree $\\\\overline { { x _ { i } } } \\\\in \\\\mathbb { Z } _ { \\\\leq 0 }$ . Any inclusion of DG-algebras of type \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[423,480,571,500]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0022/6d255f8454464f4dfca207a5d9434b30c26879217c01d9c49183dba423285c2b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"91\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nA \\\\longrightarrow A [ \\\\{ x _ { i } \\\\} ] ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,513,760,531]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"91\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"regardless of the differential on $A [ \\\\{ x _ { i } \\\\} ]$ , is called a semifree extension. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,540,876,576]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"91\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recall that a differential on $A [ \\\\{ x _ { i } \\\\} ]$ is determined by the differential on $A$ and by the values $d ( x _ { i } )$ . Every free extension is also semifree. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,584,876,638]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"91\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3 There exists a model structure on the category $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ , where weak equivalences are the quasi-isomorphisms and fibrations are the maps surjective in negative degree. Moreover: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,648,653,702]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"91\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. cofibrations are the retracts of semifree extensions, \\n2. trivial cofibrations are the retracts of free extensions, \\n3. trivial fibrations are the surjective quasi-isomorphisms. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,711,881,819]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"91\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The proof that every trivial fibration is surjective is a simple argument in basic homological algebra. In fact, if $f \\\\colon A \\\\ \\\\longrightarrow \\\\ B$ is a quasi-isomorphism which is surjective in negative degree, for every $x \\\\in B ^ { 0 }$ , since $d x = 0$ and $f \\\\colon H ^ { 0 } ( A ) \\\\to$ $H ^ { 0 } ( B )$ is bijective, there exist $y \\\\in A ^ { 0 }$ and $z \\\\in B ^ { - 1 }$ such that $f ( y ) = x + d z $ . Since $f \\\\colon A ^ { \\\\ - 1 } \\\\longrightarrow B ^ { - 1 }$ is surjective there exists $u \\\\in A ^ { - 1 }$ such that $f ( u ) = z$ and therefore $x = f ( y - d u )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,819,879,854]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"91\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now the proof of Theorem 3 follows, according to Theorem 2, from the fact that the four classes: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,864,489,900]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"91\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. quasi-isomorphisms, \\n2. $\\\\mathcal { F }$ maps surjective in negative degree, \\n3. $C ^ { \\\\prime }$ semifree extensions, \\n4. $c w ^ { \\\\prime }$ free extensions. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"91\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"86 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[687,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"91\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Lepri and M. Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,362,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,135,881,189]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"form a left pre-model structure. We have already proved in Lemma 1 that $c w ^ { \\\\prime } \\\\subset$ $\\\\mathcal { W }$ . The 2-out-of-3 axiom for $\\\\mathcal { W }$ is clear, and the retract axiom for $\\\\mathcal { W }$ , $\\\\mathcal { F }$ is also obvious: the retract of a injective (surjective) map is also injective (surjective). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,198,879,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 2 The maps in $C ^ { \\\\prime }$ , i.e., the semifree extensions, have the left lifting property with respect to all trivial fibrations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,243,651,262]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Consider the following solid commutative diagram \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[410,280,587,352]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0720/3040397b5c7fc224650f7bf519e1f0d653502ebb9350059b16d6afd4af642dd6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n{ \\\\underset { A \\\\left[ \\\\left\\\\{ X _ { i } \\\\right\\\\} \\\\right] } { \\\\overset { A } { \\\\longrightarrow } } } { \\\\underset { \\\\substack { \\\\overset { . . . . } { \\\\longrightarrow } } } { \\\\overset { \\\\alpha } { \\\\longrightarrow } } } C \\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,372,876,409]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $g$ is a trivial fibration and $f$ a semifree extension. For every integer $n \\\\geq - 1$ consider the DG-subalgebra of $A [ \\\\{ x _ { i } \\\\} ]$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[380,426,619,445]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0121/3e070fbdd1357698a346bf4f1bc201ff4ae9fe088de68ef20dcd1919be56fac5.jpg\",\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nA _ { n } = A [ \\\\{ x _ { i } \\\\mid { \\\\overline { { x _ { i } } } } \\\\geq - n \\\\} ] .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,462,881,500]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We have $A _ { - 1 } = A$ , $\\\\cup _ { n } A _ { n } = A [ \\\\{ x _ { i } \\\\} ]$ and therefore, setting $\\\\gamma _ { - 1 } = \\\\alpha$ it is sufficient to prove by induction that for every $n \\\\geq 0$ we have a commutative diagram \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[421,515,576,590]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0570/8e224e1d7af8586ff240bcf6cddc67042aa189b4e4e4609ffaaeb35061fdb8b9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{c} \\\\begin{array} { c } { { A _ { n - 1 } \\\\xrightarrow { \\\\gamma _ { n - 1 } } C } } \\\\\\\\ { { \\\\downarrow \\\\underbrace { \\\\gamma _ { n } } \\\\end{array} \\\\downarrow g } _ { \\\\beta } } \\\\\\\\ { { A _ { n } \\\\xrightarrow { \\\\beta _ { n } } D } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,609,881,663]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where the left vertical arrow is the inclusion $A _ { n - 1 } \\\\subset A _ { n }$ . If $n = 0$ , then for every $x _ { i }$ with $\\\\overline { { x _ { i } } } = 0$ we have $d x _ { i } = 0$ . Since $g$ is surjective there exists $c _ { i } \\\\in C$ such that $g ( c _ { i } ) = \\\\beta ( x _ { i } )$ . We define $\\\\gamma _ { 0 }$ by setting $\\\\gamma _ { 0 } ( x _ { i } ) = c _ { i }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,665,876,699]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Assume now $n > 0$ and $\\\\gamma _ { n - 1 }$ already defined. If ${ \\\\overline { { x _ { i } } } } = - n$ , then ${ \\\\overline { { d ( x _ { i } ) } } } = - n + 1$ and therefore $d x _ { i } \\\\in A _ { n - 1 }$ . We have that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[179,714,817,738]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0126/6d85afc4794f03c6287f198c78be4429351e2e2eb12ba5ad624ed305b433eaa2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nd \\\\gamma _ { n - 1 } ( d x _ { i } ) = \\\\gamma _ { n - 1 } ( d ^ { 2 } x _ { i } ) = 0 , \\\\qquad g \\\\gamma _ { n - 1 } ( d x _ { i } ) = \\\\beta ( d x _ { i } ) = d \\\\beta ( x _ { i } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,753,879,791]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and since $g$ is injective in cohomology we have $\\\\gamma _ { n - 1 } ( d x _ { i } ) = d y _ { i }$ with $y _ { i } ~ \\\\in ~ C$ . Setting $z _ { i } = \\\\beta ( x _ { i } ) - g ( y _ { i } )$ we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[250,807,749,828]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0303/e46d5271feb501f725b0b9bf9d88053b37082638834fdfdaef43f67d414c8c86.jpg\",\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nd z _ { i } = \\\\beta ( d x _ { i } ) - g ( d y _ { i } ) = \\\\beta ( d x _ { i } ) - g \\\\gamma _ { n - 1 } ( d x _ { i } ) = 0 .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[116,843,881,881]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since $g$ is a surjective quasi-isomorphism there exists $c _ { i } \\\\in C$ such that $d c _ { i } = 0$ and $g ( c _ { i } ) = z _ { i }$ . We can now define $\\\\gamma _ { n } ( x _ { i } ) = y _ { i } + c _ { i }$ . □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,562,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"92\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,879,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 3 Maps in $c w ^ { \\\\prime }$ have the left lifting property with respect to all fibrations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,135,651,154]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Consider the following solid commutative diagram \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[387,168,610,240]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0757/b474cc5e2158b36d4595f3378cce96508a1c044a586ba8c1534505a05ca698ca.jpg\",\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { A \\\\xrightarrow { d } C \\\\qquad } \\\\\\\\ { i \\\\xrightarrow { i } \\\\qquad _ { i \\\\cdots \\\\cdots } \\\\qquad \\\\downarrow g } \\\\\\\\ { A [ \\\\{ x _ { i } , d x _ { i } \\\\} ] \\\\xrightarrow { d } \\\\qquad \\\\beta \\\\xrightarrow { d } D } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,259,881,315]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"with $g$ surjective in negative degrees. If $A = 0$ there is nothing to prove; otherwise, since $A [ \\\\{ x _ { i } , d x _ { i } \\\\} ] \\\\in \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\le 0 }$ , every $x _ { i }$ has negative degree and there exists $c _ { i } \\\\in C$ such that $g ( c _ { i } ) = \\\\beta ( x _ { i } )$ . We set $h ( x _ { i } ) = c _ { i }$ , $h ( d x _ { i } ) = d c _ { i }$ , and $h | _ { A } = \\\\alpha$ . □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,321,881,360]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 4 Every map in $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ can be factored as a free extension followed by a fibration. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,366,881,423]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Let $f : A B$ be a map in $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ . For every homogeneous element $b \\\\in B$ of strictly negative degree we add two indeterminates $x _ { b }$ and $d x _ { b }$ to $A$ , with $x _ { b }$ of degree $\\\\overline { { b } }$ and $d x _ { b }$ of degree $\\\\overline { { b } } + 1$ , obtaining the free extension \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[403,439,594,465]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0849/48a557bb97573fd53bc715929257a907ad70e7bccc8d5e56ba6881d38b58be21.jpg\",\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nA \\\\xrightarrow { i } A [ \\\\{ x _ { b } , d x _ { b } \\\\} ]\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,481,626,501]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We define $\\\\pi : A [ \\\\{ x _ { b } , d x _ { b } \\\\} ] B$ in the following way: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,509,343,563]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. $\\\\pi$ is equal to $f$ on $A$ , \\n2. $\\\\pi ( x _ { b } ) = b$ , \\n3. $\\\\pi ( d x _ { b } ) = d b$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,572,658,590]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The map $\\\\pi$ is obviously a fibration and the composition \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[369,606,628,632]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0847/00ed1198a12a6f874a42c70d69622bd1e8cbe969025fb71d3a35de0ae413e731.jpg\",\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nA \\\\ { \\\\overset { i } { \\\\longrightarrow } } \\\\ A [ \\\\{ x _ { b } , d x _ { b } \\\\} ] \\\\ { \\\\overset { \\\\pi } { \\\\longrightarrow } } \\\\ B\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,650,241,668]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is equal to $f$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,674,879,713]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 5 Every map in $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ can be factored as a semifree extension followed by a trivial fibration. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,720,881,774]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Since semifree extensions are closed by composition, according to Proposition 4 it is sufficient to prove that every morphism $f \\\\colon A \\\\ \\\\to \\\\ B$ factors as a composition of a semifree extension and a quasi-isomorphism. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,776,879,828]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We use the differential graded analog of the classical argument about the existence of Tate-Tyurina resolutions: we construct recursively a countable sequence of semifree extensions \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[343,848,653,866]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0502/33c18d0ad1c6488713dcd5a581ffee8514979d6f10ded8dc41dd6afa7a4274b6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nA = A _ { 0 } \\\\subset A _ { 1 } \\\\subset \\\\cdots \\\\subset A _ { n } \\\\subset \\\\cdots\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[687,52,876,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Lepri and M. Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"88 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[858,651,879,666]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"93\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"□ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[116,88,831,108]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"94\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"together with morphisms of DG-algebras $f _ { n } \\\\colon A _ { n } \\\\to B$ such that $f _ { 0 } = f$ and: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,115,669,189]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"94\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. $A _ { n + 1 } = A _ { n } [ \\\\{ x _ { i } \\\\} ]$ , with ${ \\\\overline { { x _ { i } } } } = - n$ ; \\n2. $f _ { n + 1 }$ extends $f _ { n }$ ; \\n3. $f _ { n } \\\\colon Z ^ { i } ( A _ { n } ) \\\\to Z ^ { i } ( B )$ is surjective for every $i > - n$ ; \\n4. $f _ { n } \\\\colon H ^ { i } ( A _ { n } ) \\\\to H ^ { i } ( B )$ is bijective for every $i > - n + 1$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[141,195,876,216]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"94\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $\\\\{ b _ { i } \\\\} , i \\\\in I$ , be a set of generators of $B ^ { 0 }$ as a $A ^ { 0 }$ -algebra; then we may define \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[246,232,751,252]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0686/c3087e339e50c919ecabbfb4d7048df602af50093becd8cbacda51e79b7c1d38.jpg\",\n \"list_items\": null,\n \"page_idx\": \"94\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nA _ { 1 } = A [ \\\\{ x _ { i } \\\\} ] , \\\\qquad { \\\\overline { { x _ { i } } } } = 0 , \\\\quad d x _ { i } = 0 , \\\\quad f _ { 1 } ( x _ { i } ) = b _ { i } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,268,881,328]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"94\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Assume now $n > 0$ and $f _ { n } \\\\colon A _ { n } \\\\to B$ defined. By choosing a suitable set of generators of $Z ^ { - n } ( B )$ as $A _ { n } ^ { 0 }$ -module we can first consider a factorisation $f _ { n } \\\\colon A _ { n } \\\\subset$ $C { \\\\stackrel { g } { \\\\to } } B$ such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[209,345,785,366]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0614/36d217f64b352757a472fd8594b75d999c8bd9dd520546fee5b11b227e19204b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"94\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nC = A _ { n } [ \\\\{ x _ { i } \\\\} ] , \\\\qquad { \\\\overline { { x _ { i } } } } = - n , \\\\quad d x _ { i } = 0 , \\\\quad f _ { n } ( x _ { i } ) \\\\in Z ^ { - n } ( B ) ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,379,881,472]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"94\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and such that $g \\\\colon Z ^ { - n } ( C ) \\\\to Z ^ { - n } ( B )$ is surjective. If $g \\\\colon H ^ { - n + 1 } ( C ) \\\\to H ^ { - n + 1 } ( B )$ is bijective we can define $A _ { n + 1 } \\\\ = \\\\ C$ and $f _ { n + 1 } ~ = ~ g$ . Otherwise let $\\\\left\\\\{ c _ { i } \\\\right\\\\}$ be a set of elements in $Z ^ { - n + 1 } ( C )$ whose cohomology classes generate the kernel of $g \\\\colon H ^ { - n + 1 } ( C ) \\\\to H ^ { - n + 1 } ( B )$ , choose elements $b _ { i } ~ \\\\in ~ B ^ { - n }$ such that $d b _ { i } = g ( c _ { i } )$ and consider the factorisation \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[211,489,785,510]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0610/404290579577058c0a289e42838b10236d2a1bea34410806c268d06b45e81787.jpg\",\n \"list_items\": null,\n \"page_idx\": \"94\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nA _ { n + 1 } = C [ \\\\{ x _ { i } \\\\} ] , \\\\qquad { \\\\overline { { x _ { i } } } } = - n , \\\\quad d x _ { i } = c _ { i } , \\\\quad f _ { n + 1 } ( x _ { i } ) = b _ { i } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,527,881,581]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"94\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is easy to verify that the map $f _ { n + 1 } \\\\colon A _ { n + 1 } \\\\ \\\\to \\\\ B$ has the required properties. Finally, since $H ^ { i } ( A _ { n } ) ~ = ~ H ^ { i } ( A _ { n + 1 } )$ for every $i \\\\ > \\\\ - n + 1$ , the colimit of the sequence $f _ { n + 1 } \\\\colon A _ { n + 1 } \\\\to B$ gives the required factorisation. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,585,881,647]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"94\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 3 Let $f$ be a morphism in $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ . We have already proved that ⎧ $f$ is a trivial fibration in $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ if and only if it is a trivial fibration in ⎪⎪⎨ $\\\\mathbf { C D G A } _ { \\\\mathbb { K } }$ . Since the truncation functor \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[209,657,785,737]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0897/9192c16da09704e76c167dafe714ceeba85699ecae5af773a29f3b02f49edcba.jpg\",\n \"list_items\": null,\n \"page_idx\": \"94\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\tau \\\\colon \\\\mathbf { C D G A } _ { \\\\mathbb { K } } \\\\to \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } , \\\\qquad ( \\\\tau A ) ^ { n } = \\\\left\\\\{ \\\\begin{array} { l l } { A ^ { n } } & { \\\\quad n < 0 , } \\\\\\\\ { Z ^ { 0 } ( A ) } & { \\\\quad n = 0 , } \\\\\\\\ { 0 } & { \\\\quad n > 0 , } \\\\end{array} \\\\right.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,750,881,812]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"94\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is right adjoint to the faithful natural inclusion $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } \\\\subset \\\\mathbf { C D G A } _ { \\\\mathbb { K } }$ and preserves trivial fibrations, by Lemma 4 it follows that $f$ is cofibration in $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ if and only if it is a cofibration in $\\\\mathbf { C D G A } _ { \\\\mathbb { K } }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,819,881,893]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"94\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The notions of semifree extension and left pre-model structure apply to many other contexts, for instance cochain complexes over a commutative ring, DGalgebras, DG-Lie algebras etc.: full details will appear in the forthcoming thesis of the first author. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,51,562,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"94\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[854,52,879,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"94\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"89 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,455,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 Modules and Derivations \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,132,879,187]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $( A , d _ { A } )$ be in $\\\\mathbf { C D G A } _ { \\\\mathbb { K } }$ , an $A$ -module is a differential graded vector space $( M , d _ { M } )$ together with an associative and distributive $\\\\mathbb { K }$ -linear left multiplication map $A \\\\times M \\\\to M$ , with the properties: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,193,731,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. $A ^ { i } M ^ { j } \\\\subset M ^ { i + j }$ , \\n2. $d _ { M } ( a m ) = d _ { A } ( a ) m + ( - 1 ) ^ { \\\\overline { { { a } } } } a d _ { M } ( m ) \\\\mathrm { f o r e v e r y } a \\\\in A , m \\\\in M .$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,240,881,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A morphism of $A$ -modules is a morphism of differential graded vector spaces commuting with multiplications. Since $A$ is graded commutative, we can also define an associative right multiplication map $M \\\\times A \\\\to M$ by setting $m a = ( - 1 ) ^ { \\\\overline { { { a } } } \\\\overline { { { m } } } } a m$ , $a \\\\in A$ , $m \\\\in M$ . Notice that $a ( m b ) = ( a m ) b$ for every $a , b \\\\in A$ , $m \\\\in M$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,313,879,349]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The trivial extension of a DG-algebra $A$ by the $A$ -module $M$ is the direct sum of complexes $A \\\\oplus M$ equipped with the product: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[355,367,642,385]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0412/023b6f1bba25a92ad5376148cc8445e237ce99bd3d22c863145ed21505e771d8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n( a , m ) ( b , n ) = ( a b , m b + a n ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,402,876,439]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is immediate to see that $A \\\\oplus M \\\\in \\\\mathbf { C D G A } _ { \\\\mathbb { K } }$ , the projection $A \\\\oplus M \\\\to A$ is a morphism of DG-algebras and $M$ is a square-zero ideal of $A \\\\oplus M$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,439,881,512]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For a given graded vector space $M$ and an integer $n$ we shall denote by $M [ - n ]$ the same space with the degrees shifted by $- n$ , namely $M [ - n ] ^ { i } = M ^ { i - n }$ , and by $s ^ { n } \\\\colon M \\\\to M [ - n ]$ the tautological (bijective) map of degree $n$ . In other words, $s ^ { n }$ is the essentially the identity and its only effect is changing the degree: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[318,525,678,548]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"1012/99f1d4f829db06b194bea7a1806c93e94c8fa53197376c73c450ce5ada3555df.jpg\",\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\ns ^ { n } \\\\colon M ^ { i } \\\\to M [ - n ] ^ { i + n } , \\\\qquad x \\\\mapsto s ^ { n } x .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,564,879,602]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $M$ is an $A$ -module, then $M [ - n ]$ is also an $A$ -module, where the differential and the product are defined accordingly to the Koszul sign rule: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[136,615,858,638]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0941/cd03942fde595c6d17f909bfa61bd94a4a4fa96abddd4bf846b779a0807116c9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nd ( s ^ { n } x ) = ( - 1 ) ^ { n } s ^ { n } d ( x ) , \\\\qquad a ( s ^ { n } x ) = ( - 1 ) ^ { n { \\\\overline { { a } } } } s ^ { n } ( a x ) , \\\\qquad ( s ^ { n } x ) a = s ^ { n } ( x a ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,656,876,692]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 11 Let $M$ be an $A$ -module. A $\\\\mathbb { K }$ -linear map $\\\\alpha \\\\colon A \\\\to M$ is a derivation of degree $j \\\\in \\\\mathbb Z$ if $\\\\alpha ( A ^ { n } ) \\\\subset M ^ { n + j }$ and it satisfies Leibniz’s law: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[350,705,646,728]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0173/74ce2498ffc6dc8b618f8366c2bd3ca4615ebc3fdcd90cfae910226cc1dcda38.jpg\",\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\alpha ( a b ) = \\\\alpha ( a ) b + ( - 1 ) ^ { \\\\overline { { { a } } } j } a \\\\alpha ( b )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,746,881,840]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The vector space of derivations of degree $j$ from $A$ to $M$ is denoted $\\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { j } ( A , M )$ . The graded vector space $\\\\begin{array} { r } { \\\\mathrm { D e r } _ { \\\\mathbb K } ^ { * } ( A , M ) = \\\\bigoplus _ { j \\\\in \\\\mathbb Z } \\\\mathrm { D e r } _ { \\\\mathbb K } ^ { j } ( A , M ) } \\\\end{array}$ has a natural structure of $A$ -module, with multiplication $( a \\\\alpha ) ( x ) = a ( \\\\alpha ( x ) )$ and differential $( d \\\\alpha ) ( x ) =$ $d ( \\\\alpha ( x ) ) ~ - ~ ( - 1 ) ^ { \\\\overline { { { \\\\alpha } } } } \\\\alpha ( d x )$ . Observe that for every integer $n$ there is a natural isomorphism of $A$ -modules \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[309,858,687,879]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0567/7ae88f4da2f76a42d4cb7dace0f333051b2ce8b633df565986d7bf0717b0f689.jpg\",\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathrm { D e r } _ { \\\\mathbb K } ^ { * } ( A , M [ - n ] ) \\\\to \\\\mathrm { D e r } _ { \\\\mathbb K } ^ { * } ( A , M ) [ - n ] .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[687,52,879,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Lepri and M. Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"95\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"90 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,163]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Every morphism of DG-algebras $f \\\\colon A \\\\ \\\\to \\\\ B$ induces in the natural way an $A$ -module structure on $B$ . In this case the module of derivations will be denoted $\\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( A , B ; f )$ : a $\\\\mathbb { K }$ -linear map $\\\\alpha \\\\colon A \\\\ \\\\to \\\\ B$ is an $f$ -derivation of degree $k$ if $\\\\alpha ( A ^ { n } ) \\\\subset B ^ { n + k }$ and $\\\\alpha ( a b ) = \\\\alpha ( a ) f ( b ) + ( - 1 ) ^ { k \\\\overline { { { a } } } } f ( a ) \\\\alpha ( b )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,171,881,283]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 4 Let $f \\\\colon A \\\\to B$ be a morphism of DG-algebras, $I \\\\subset B$ a square-zero ideal and $\\\\pi \\\\colon B \\\\to B / I$ the quotient map. Then $I$ is a $B / I$ -module and then also an $A$ -module via the morphism $\\\\pi f$ . It is immediate to check that if $g \\\\colon A B$ is a morphism of graded algebras such that $\\\\pi g = \\\\pi f$ then $g - f \\\\colon A \\\\to I$ is a derivation of degree 0. Conversely, if $\\\\alpha \\\\in { \\\\mathrm { D e r } } _ { \\\\mathbb { K } } ^ { 0 } ( A , I )$ , then $f + \\\\alpha$ is a morphism of graded algebras, and it is a morphism of DG-algebras if and only if $\\\\alpha \\\\in Z ^ { 0 } ( \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( A , I ) )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,291,876,327]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 6 Let $A \\\\in \\\\mathbf { C D G A } _ { \\\\mathbb { K } }$ be a cofibrant algebra and $f \\\\colon M \\\\to N$ a surjective quasi-isomorphism of $A$ -modules. Then the map \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[266,346,728,367]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0182/12317d3df4c1441fa73d72f8ecdbcba8f778012c253d66a85336466e4accf78a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nf _ { * } \\\\colon \\\\operatorname { D e r } _ { \\\\mathbb { K } } ^ { * } ( A , M ) \\\\to \\\\operatorname { D e r } _ { \\\\mathbb { K } } ^ { * } ( A , N ) , \\\\qquad \\\\alpha \\\\mapsto f \\\\alpha ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,387,432,405]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is a surjective quasi-isomorphism. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,412,879,448]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Since $f \\\\colon M [ - n ] \\\\ \\\\to \\\\ N [ - n ]$ is a surjective quasi-isomorphism for every integer $n$ it is sufficient to prove that: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,456,660,498]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. $f _ { * } \\\\colon \\\\operatorname { D e r } _ { \\\\mathbb { K } } ^ { 0 } ( A , M ) \\\\to \\\\operatorname { D e r } _ { \\\\mathbb { K } } ^ { 0 } ( A , N )$ is surjective; \\n2. $f _ { * } \\\\colon H ^ { 0 } ( \\\\mathrm { D e r } _ { \\\\mathbb K } ^ { * } ( A , M ) ) \\\\to H ^ { 0 } ( \\\\mathrm { D e r } _ { \\\\mathbb K } ^ { * } ( A , N ) )$ is bijective. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,504,881,576]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let’s denote by $C ( P ) = P \\\\oplus P [ 1 ]$ the mapping cone of the identity of an $A$ - module $P$ , with the differential defined by the formula $d ( x + s ^ { - 1 } y ) = d x + y -$ $s ^ { - 1 } d y$ ; notice that $C ( P )$ is acyclic and the natural projection $C ( P ) P [ 1 ]$ is a morphism of- $A$ -modules. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[143,578,703,596]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For every linear map $\\\\alpha \\\\colon A P$ of degree 0 we shall denote \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[182,612,815,635]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0250/73fa32d07f0a51d4d5888ae03d3e3a0746b68945b49da44286489fb856655875.jpg\",\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n{ \\\\widetilde \\\\alpha } \\\\colon A \\\\to A \\\\oplus C ( P ) , \\\\qquad { \\\\widetilde \\\\alpha } ( a ) = a + \\\\alpha ( a ) + s ^ { - 1 } ( \\\\alpha ( d a ) - d \\\\alpha ( a ) ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,654,879,708]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is straightforward to check that $\\\\widetilde { \\\\alpha }$ is a morphism of complexes and that every morphism of complexes $A \\\\to A \\\\oplus C ( P )$ lifting the identity on $A$ is obtained this way. Moreover, $\\\\alpha$ is a derivation if and only if - $\\\\widetilde { \\\\alpha }$ is a morphism in $\\\\mathbf { C D G A } _ { \\\\mathbb { K } }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,710,881,780]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since A ⊕ C(M) → A ⊕ C(N), $a + x + s ^ { - 1 } y \\\\mapsto a + f ( x ) + s ^ { - 1 } f ( y )$ , is a trivial fibration, the lifting of a derivation $\\\\alpha \\\\in \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { 0 } ( A , N )$ is obtained by taking the lifting of the morphism of DG-algebras ${ \\\\widetilde { \\\\alpha } } \\\\colon A \\\\to A \\\\oplus C ( N )$ . This proves the first item. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,782,781,800]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $K$ is the kernel of $f$ , then we have an exact sequence of complexes \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[236,818,763,840]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0802/6e6ed6e857e990dcffb035fa8c0b751ed0ae93136fba22d749c0d283aac6b3cc.jpg\",\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n0 \\\\to \\\\operatorname { D e r } _ { \\\\mathbb { K } } ^ { * } ( A , K ) \\\\to \\\\operatorname { D e r } _ { \\\\mathbb { K } } ^ { * } ( A , M ) \\\\to \\\\operatorname { D e r } _ { \\\\mathbb { K } } ^ { * } ( A , N ) \\\\to 0\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,858,879,896]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and in order to prove the second item it is sufficient to show that $\\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( A , K )$ is acyclic. By the shifting degree argument it is sufficient to prove that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,51,562,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[854,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"96\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"91 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,681,111]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"97\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$H ^ { 1 } ( \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( A , K ) ) = 0 .$ . Given $\\\\beta \\\\in Z ^ { 1 } ( \\\\operatorname { D e r } _ { \\\\mathbb { K } } ^ { * } ( A , K ) )$ , the map \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[280,129,717,151]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0504/c1121281114b36bb118763e5e265df1056611a8afbb25af94b0b553d9ce43dde.jpg\",\n \"list_items\": null,\n \"page_idx\": \"97\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n{ \\\\widehat { \\\\beta } } \\\\colon A \\\\to A \\\\oplus K [ 1 ] , \\\\qquad { \\\\widehat { \\\\beta } } ( a ) = a + s ^ { - 1 } \\\\beta ( a ) ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,171,881,226]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"97\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is a morphism of DG-algebras and the proof that $\\\\beta \\\\in B ^ { 1 } ( \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( A , K ) )$ follows immediately by considering a lifting of $\\\\hat { \\\\boldsymbol { \\\\beta } }$ along the trivial fibration $A \\\\oplus C ( K ) \\\\to$ $A \\\\oplus K [ 1 ]$ . □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,256,760,298]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"97\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 Deformations of Diagrams via Projective Cofibrant Resolutions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,321,881,394]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"97\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Throughout this section we shall denote by $\\\\mathcal { D }$ a fixed small category. For every category $\\\\mathbf { M }$ we shall denote by $\\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { M } )$ the category of diagrams $\\\\mathcal { D } \\\\mathbf { M }$ . For every local Artin $\\\\mathbb { K }$ -algebra $A$ with residue field $\\\\mathbb { K }$ we shall denote by $\\\\mathbf { A } \\\\mathbf { g } _ { A }$ the category of unitary commutative $A$ -algebras. For simplicity of notation, if \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[312,408,683,429]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0341/bae1f2833d697a579eacca58072060af5c1972b6c120104c4423857db3dbbd04.jpg\",\n \"list_items\": null,\n \"page_idx\": \"97\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nP _ { \\\\bullet } \\\\in \\\\operatorname { F u n } ( { \\\\mathcal { D } } , \\\\mathbf { A l g } _ { A } ) , \\\\qquad { \\\\mathcal { D } } \\\\ni a \\\\mapsto P _ { a } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,441,876,478]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"97\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is a diagram of $A$ -algebras and $A B$ is a morphism of algebras, we shall denote $P , \\\\otimes _ { A } B$ the diagram $( P _ { \\\\bullet } \\\\otimes _ { A } B ) _ { a } = P _ { a } \\\\otimes _ { A } B , a \\\\in { \\\\mathcal { D } }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,478,874,515]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"97\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Here we are interested in studying the deformation theory of a diagram $S _ { \\\\bullet } \\\\colon { \\\\mathcal { D } } $ $\\\\mathbf { A l g } _ { \\\\mathbb { K } }$ of unitary commutative algebras. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,522,881,578]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"97\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 12 A deformation over $A \\\\in \\\\mathbf { A r t } _ { \\\\mathbb { K } }$ of a diagram $S _ { \\\\bullet } \\\\colon { \\\\mathcal { D } } \\\\to \\\\mathbf { A l g } _ { \\\\mathbb { K } }$ is the data of a diagram $S _ { \\\\bullet A } : { \\\\mathcal { D } } \\\\to \\\\mathbf { A l g } _ { A }$ of flat $A$ -algebras and a morphism of diagrams of algebras $\\\\phi \\\\colon S _ { \\\\bullet A } S _ { \\\\bullet }$ inducing an isomorphism $S _ { \\\\bullet A } \\\\otimes _ { A } \\\\mathbb { K } \\\\simeq S _ { \\\\bullet }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,578,874,615]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"97\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Two deformations $\\\\phi \\\\colon S _ { \\\\bullet A } S _ { \\\\bullet }$ and $\\\\psi \\\\colon S _ { \\\\bullet A } ^ { \\\\prime } \\\\to S _ { \\\\bullet }$ are isomorphic if there exists an isomorphism of diagrams of $A$ -algebras $\\\\eta \\\\colon S _ { \\\\bullet A } \\\\to S _ { \\\\bullet A } ^ { \\\\prime }$ such that $\\\\phi = \\\\psi \\\\eta$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,621,881,770]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"97\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is possible to prove, see e.g. [10, A.2], that for every small category $\\\\mathcal { D }$ there exist model structures on $\\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } )$ and $\\\\mathrm { F u n } ( { \\\\mathcal { D } } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } )$ , called projective model structures, such that a morphism of diagrams $F G$ is a weak equivalence (resp.: fibration) if and only if $F _ { a } \\\\ \\\\to \\\\ G _ { a }$ is a weak equivalence (resp.: fibration) for every $a \\\\in \\\\mathcal { D }$ . The same argument used in Remark 3 shows that a morphism $f$ in $\\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } )$ ) is a weak equivalence, cofibration, trivial fibration in $\\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } )$ if and only if it is a weak equivalence, cofibration, trivial fibration in $\\\\mathrm { F u n } ( { \\\\mathcal { D } } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } )$ , respectively. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,770,867,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"97\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The notions of module and derivation extend naturally to the context of diagrams For every diagram $R _ { \\\\bullet } \\\\colon \\\\mathcal { D } { \\\\bf C D G A } _ { \\\\mathbb { K } } ^ { \\\\le 0 }$ the DG-Lie algebra of derivations is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[150,818,847,872]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0526/93baa3aed7de1749f3a1b61593cdb6abf295499cdbc75827450bbc5107868af6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"97\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { \\\\bullet } , R _ { \\\\bullet } ) = \\\\{ \\\\{ \\\\alpha _ { a } \\\\} \\\\in \\\\prod _ { a \\\\in \\\\mathcal { D } } \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { a } , R _ { a } ) \\\\mid \\\\alpha _ { b } R _ { f } = R _ { f } \\\\alpha _ { a } , \\\\forall a \\\\stackrel { f } { } b \\\\} .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[111,882,822,905]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"97\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is plain that $\\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { \\\\bullet } , R _ { \\\\bullet } )$ is a DG-Lie subalgebra of $\\\\begin{array} { r } { \\\\prod _ { a \\\\in \\\\mathcal { D } } \\\\operatorname { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { a } , R _ { a } ) } \\\\end{array}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[685,52,879,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"97\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Lepri and M. Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"97\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"92 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,204]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"98\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An $R _ { \\\\bullet }$ -module $M$ is a diagram of differential graded vector spaces over $\\\\mathcal { D }$ such that $M _ { a }$ is an $R _ { a }$ -module for every $a \\\\in \\\\mathcal { D }$ and, for every arrow $a \\\\ { \\\\overset { f } { \\\\to } } \\\\ b$ in $\\\\mathcal { D }$ , the map $M _ { f } \\\\colon M _ { a } \\\\to M _ { b }$ is a morphism of $R _ { a }$ -modules, where $M _ { b }$ is considered as a $R _ { a }$ -module via the morphism of DG-algebras $R _ { f } \\\\colon R _ { a } \\\\ \\\\ R _ { b }$ . A morphism $g \\\\colon M _ { \\\\bullet } \\\\to N _ { \\\\bullet }$ of $R _ { \\\\bullet }$ -modules is a morphism of diagrams of DG-vector spaces such\\u0005 that $g _ { a } \\\\colon M _ { a } \\\\to N _ { a }$ is a morphism of $R _ { a }$ -modules for every $a \\\\in \\\\mathcal { D }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,204,628,222]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"98\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The differential graded vector space of derivations is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[143,234,851,288]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0192/82f41a0d5dc9969a95898df7ee3531e1921578179bee6e59bbf6d8e235505136.jpg\",\n \"list_items\": null,\n \"page_idx\": \"98\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { \\\\bullet } , M _ { \\\\bullet } ) = \\\\{ \\\\{ \\\\alpha _ { a } \\\\} \\\\in \\\\prod _ { a \\\\in \\\\mathcal { D } } \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { a } , M _ { a } ) \\\\mid \\\\alpha _ { b } R _ { f } = M _ { f } \\\\alpha _ { a } , \\\\forall a \\\\stackrel { f } { } b \\\\} .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,303,876,339]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"98\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The same argument used in the proof of Lemma 6 works, mutatis mutandis, also for diagrams and gives the following result. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,346,881,402]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"98\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 7 Let $R _ { \\\\bullet } ~ \\\\in ~ \\\\operatorname { F u n } ( { \\\\mathcal { D } } , \\\\mathbf { C D G A _ { \\\\mathbb { K } } } )$ be a projective cofibrant diagram and $f \\\\colon M _ { \\\\bullet } \\\\to N _ { \\\\bullet }$ a morphism of $R _ { \\\\bullet }$ -modules such that $f _ { a } \\\\colon M _ { a } \\\\to N _ { a }$ is a surjective quasi-isomorphism for every $a \\\\in \\\\mathcal { D }$ . Then the map \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[252,418,742,439]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0704/e66ed0b2c253743269f6ea03bab27303fa61b4b27fc5795815f53208ef115a78.jpg\",\n \"list_items\": null,\n \"page_idx\": \"98\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nf _ { * } \\\\colon \\\\operatorname { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { \\\\bullet } , M _ { \\\\bullet } ) \\\\to \\\\operatorname { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { \\\\bullet } , N _ { \\\\bullet } ) , \\\\qquad \\\\alpha \\\\mapsto f \\\\alpha ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,456,432,474]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"98\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is a surjective quasi-isomorphism. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,481,726,501]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"98\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The main goal of this section is to prove the following theorem. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,509,881,582]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"98\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 4 Let $\\\\mathcal { D }$ be a small category and $S _ { \\\\bullet } \\\\colon { \\\\mathcal { D } } \\\\ \\\\to \\\\ \\\\mathbf { A l g } _ { \\\\mathbb { K } } \\\\ a$ diagram of unitary commutative algebras. Let $R _ { \\\\bullet } S _ { \\\\bullet }$ be a cofibrant replacement in $\\\\begin{array} { r } { \\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } , } \\\\end{array}$ • → • ) with respect to the projective model structure. Then the DG-Lie algebra $\\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { \\\\bullet } , R _ { \\\\bullet } )$ controls the deformations of $S _ { \\\\bullet }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,590,881,717]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"98\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In other words, the functor of isomorphism classes of deformations of $S _ { \\\\bullet }$ is isomorphic to the functor of Maurer-Cartan solutions in $\\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { \\\\bullet } , R _ { \\\\bullet } )$ modulus gauge equivalence. We shall prove Theorem 4 after a certain number of preliminary results. Unless otherwise specified we always equip the categories $\\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } )$ ) and $\\\\mathrm { F u n } ( { \\\\mathcal { D } } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } )$ with the projective model structure. Therefore $R _ { \\\\bullet } ~ ~ S _ { \\\\bullet }$ is a cofibrant resolution also in the model category $\\\\mathrm { F u n } ( { \\\\mathcal { D } } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } )$ and we can apply Lemma 7 to the diagram $R _ { \\\\bullet }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,725,658,744]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"98\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 8 Consider a commutative square of solid arrows \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[432,764,560,852]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0843/8ccbca62ea9b3d6b7b1928e0910aa2da2748785568fb890d2cde88b2b72e8bc0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"98\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { l } { P _ { \\\\bullet } \\\\xrightarrow { g } E _ { \\\\bullet } } \\\\\\\\ { i \\\\Bigg \\\\downarrow _ { \\\\begin{array} { c } { \\\\diagdown ^ { \\\\prime } } \\\\\\\\ { ⨏ _ { \\\\ell } ^ { \\\\prime } } \\\\end{array} } ^ { \\\\ell } \\\\Bigg \\\\downarrow ^ { \\\\prime } } \\\\\\\\ { C _ { \\\\bullet } \\\\xrightarrow [ f ^ { \\\\prime } ] { g } D _ { \\\\bullet } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,52,562,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"98\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[854,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"98\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"93 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,881,147]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"99\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"in $\\\\begin{array} { r } { \\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } , } \\\\end{array}$ ). If $i$ is a cofibration and $p _ { a } \\\\colon E _ { a } \\\\to D _ { a }$ is surjective for every $a \\\\in \\\\mathcal { D }$ , then there exists a lifting $\\\\gamma \\\\colon C _ { \\\\bullet } \\\\to E .$ in the category of diagrams of graded algebras. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,156,881,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"99\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Consider the contractible polynomial algebra $\\\\mathbb { K } [ d ^ { - 1 } ] \\\\ \\\\in \\\\ \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\le 0 }$ , where $\\\\overline { { d ^ { - 1 } } } = - 1$ and $d ( d ^ { - 1 } ) = 1$ , and notice that the natural inclusion $\\\\alpha \\\\colon \\\\mathbb { K } \\\\to \\\\mathbb { K } [ d ^ { - 1 } ]$ is a morphism of DG-algebras, while the natural projection $\\\\beta \\\\colon \\\\mathbb { K } [ d ^ { - 1 } ] \\\\mathbb { K }$ is a morphism of graded algebras; moreover $\\\\beta \\\\alpha$ is the identity on $\\\\mathbb { K }$ . Now, the morphism \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[328,247,674,276]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0194/df85b0f16bcc4974284012cb122694d8f0640576ea7aab89bc1e15fd0858ac55.jpg\",\n \"list_items\": null,\n \"page_idx\": \"99\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nE _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } { \\\\mathbb { K } } [ d ^ { - 1 } ] \\\\xrightarrow { p \\\\otimes \\\\mathrm { I d } } D _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } { \\\\mathbb { K } } [ d ^ { - 1 } ]\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[116,292,678,312]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"99\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is a trivial fibration, and so there exists a commutative square \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[343,325,656,423]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0695/e5c2b70951e3ab59e732f5f5bc599ede97748dd8fd95ca5c98b4b3d4715b3f62.jpg\",\n \"list_items\": null,\n \"page_idx\": \"99\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { P _ { \\\\bullet } \\\\xrightarrow { ( \\\\mathrm { I d } \\\\otimes \\\\alpha ) g } \\\\quad \\\\ J _ { \\\\bullet } \\\\otimes _ { \\\\mathbb K } \\\\mathbb K [ d ^ { - 1 } ] } \\\\\\\\ & { i \\\\underbrace { \\\\quad \\\\quad \\\\quad \\\\varphi \\\\quad \\\\quad } _ { ( \\\\mathrm { I d } \\\\otimes \\\\alpha ) f } \\\\quad \\\\quad \\\\ J \\\\otimes \\\\mathrm { I d } } \\\\\\\\ & { C _ { \\\\bullet } \\\\xrightarrow { ( \\\\mathrm { I d } \\\\otimes \\\\alpha ) g } \\\\quad \\\\ J _ { \\\\bullet } \\\\otimes _ { \\\\mathbb K } \\\\mathbb K [ d ^ { - 1 } ] } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,444,708,466]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"99\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"in $\\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } )$ ). It is now sufficient to take $\\\\gamma = ( \\\\operatorname { I d } \\\\otimes \\\\beta ) \\\\varphi$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,474,883,572]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"99\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 9 Let $\\\\begin{array} { r } { A \\\\in \\\\mathrm { ~ \\\\bf ~ A r t { } ~ } _ { \\\\mathbb { K } } } \\\\end{array}$ and let $N _ { \\\\bullet A } \\\\colon \\\\mathcal { D } \\\\ \\\\to \\\\ \\\\mathbf { C D G A } _ { A } ^ { \\\\leq 0 }$ be a diagram of flat $A$ -algebras. Then every cofibrant replacement $f \\\\colon P _ { \\\\bullet } \\\\ \\\\quad N _ { \\\\bullet A } \\\\otimes _ { A } \\\\mathbb { K }$ in $\\\\begin{array} { r } { \\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } ) } \\\\end{array}$ ) lifts to an A-linear differential on $P _ { \\\\bullet } \\\\otimes A$ and to a trivial fibration $P _ { \\\\bullet } \\\\otimes A N _ { \\\\bullet A }$ in the category $\\\\mathrm { F u n } ( \\\\mathcal { D } , { \\\\bf C D G A } _ { A } ^ { \\\\le 0 } )$ ). The above lifting is unique up to $A$ -linear algebra isomorphisms of $P \\\\cdot \\\\otimes A$ lifting the identity on $P _ { \\\\bullet }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,578,881,651]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"99\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof (Existence) We proceed by induction on the length of the Artin ring. Since for $A = \\\\mathbb { K }$ there is nothing to prove, we may assume $A \\\\in \\\\mathbf { A r t } _ { \\\\mathbb { K } }$ of length $l ( A ) > 1$ and then there exists a non-trivial element $t \\\\in A$ annihilated by the maximal ideal ${ \\\\mathfrak { m } } _ { A }$ , giving a small extension \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[268,666,728,692]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0144/57c3846e58c1fa472dd0c3c56b4c4baca4ffbdb473ba3b295208230332865ad1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"99\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n0 \\\\to \\\\mathbb { K } { \\\\stackrel { \\\\cdot t } { \\\\longrightarrow } } A \\\\to B \\\\to 0 , \\\\qquad l ( B ) = l ( A ) - 1 .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,710,879,747]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"99\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By induction there exist a $B$ -linear differential on $P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } B$ and a commutative square in $\\\\mathrm { F u n } ( \\\\mathcal { D } , { \\\\bf C D G A } _ { B } ^ { \\\\le 0 } )$ ): \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[371,762,624,846]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0522/0bfd901b0855df75b9f62a82db725a26cf9ed54f3b8092eddffb76dfc68318a0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"99\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { c c } { P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } B \\\\xrightarrow { q } \\\\sim N _ { \\\\bullet A } \\\\otimes _ { A } B } \\\\\\\\ { \\\\downarrow } \\\\\\\\ { P _ { \\\\bullet } \\\\xrightarrow { \\\\qquad f \\\\qquad \\\\ } N _ { \\\\bullet A } \\\\otimes _ { A } \\\\mathbb { K } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[687,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"99\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Lepri and M. Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"99\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"94 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[858,448,879,463]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"99\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"□ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,883,181]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"100\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In view of the embedding $\\\\mathbb { K } \\\\subset B$ , the $B$ -linear morphism of diagrams $q$ is uniquely determined by its restriction $q _ { \\\\vert } P _ { \\\\bullet } \\\\colon P _ { \\\\bullet } \\\\to N _ { \\\\bullet A } \\\\otimes _ { A } B$ , which is a morphism in $\\\\begin{array} { r } { \\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } , } \\\\end{array}$ ). By Lemma 8 we can lift $q _ { \\\\lvert P _ { \\\\bullet } }$ to a morphism of diagrams of graded algebras $P _ { \\\\bullet } N _ { \\\\bullet A }$ and then we get a commutative diagram with (pointwise) exact rows \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[218,190,781,276]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0769/4b2e11a02a0188918b700d8fb3d62d9915bc01592cf292840fd88c7ed37289db.jpg\",\n \"list_items\": null,\n \"page_idx\": \"100\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } { 0 \\\\xrightarrow { \\\\quad \\\\quad } } & { { } \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad t \\\\xrightarrow { t } { } P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } A \\\\xrightarrow { \\\\quad \\\\pi \\\\quad \\\\quad } { } P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } B \\\\xrightarrow { \\\\quad \\\\quad \\\\quad } { } 0 } \\\\\\\\ { \\\\quad \\\\quad } & { { } \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad } \\\\\\\\ { 0 \\\\xrightarrow { \\\\quad \\\\quad \\\\quad } } & { { } \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad } \\\\\\\\ { \\\\quad \\\\quad \\\\quad } & { { } \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad } \\\\\\\\ { 0 \\\\xrightarrow { \\\\quad \\\\quad \\\\quad \\\\quad } } & { { } \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad } \\\\\\\\ { \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad } \\\\\\\\ { \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad } \\\\\\\\ { \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad } \\\\\\\\ { \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad } \\\\\\\\ { \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad } \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,294,881,420]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"100\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since $f$ and $q$ are trivial fibrations, to conclude the proof it is sufficient to show that there exists a lifting of the differential of $P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } B$ to an $A$ -linear differential of $P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } A$ making $p$ a morphism of diagrams of DG-algebras. Let $d = \\\\{ d _ { a } \\\\colon P _ { a } $ $P _ { a } \\\\mid a \\\\in { \\\\mathcal { D } } \\\\}$ be the differential of $P _ { \\\\bullet }$ , since the $A$ -linear derivations of $P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } A$ of degree 1 lifting $d$ are of type $d + \\\\eta$ with $\\\\eta \\\\in \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { 1 } ( P _ { \\\\bullet } , P _ { \\\\bullet } ) \\\\otimes \\\\mathfrak { m } _ { A }$ , we can lift the differential of $P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } B$ to an $A$ -linear derivation $\\\\delta \\\\colon P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } A \\\\to P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } A$ of degree 1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,421,876,456]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"100\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now it is sufficient to prove that there exists a derivation in $\\\\xi \\\\in \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { 1 } ( P _ { \\\\bullet } , P _ { \\\\bullet } )$ such that: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,463,330,503]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"100\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. $( \\\\delta + t \\\\xi ) ^ { 2 } = 0$ , \\n2. $p ( \\\\delta + t \\\\xi ) = d _ { N _ { \\\\bullet } } p$ , \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,510,881,638]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"100\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and consider $\\\\delta + t \\\\xi$ as the differential of $P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } A$ . Since $t$ is annihilated by the maximal ideal ${ \\\\mathfrak { m } } _ { A }$ , the condition $( \\\\delta + t \\\\xi ) ^ { 2 } = 0$ is equivalent to $d \\\\xi + \\\\xi d = 0$ , i.e., the above condition (1) holds if and only if $\\\\xi \\\\in Z ^ { 1 } ( \\\\operatorname { D e r } _ { \\\\mathbb { K } } ^ { * } ( P _ { \\\\bullet } , P _ { \\\\bullet } ) )$ . The map $\\\\psi = d _ { N _ { \\\\bullet } } p - p \\\\delta$ is $A$ -linear and its image is contained in $t ( N _ { \\\\bullet A } \\\\otimes _ { A } \\\\mathbb { K } )$ , hence it factors to a derivation $\\\\phi \\\\in Z ^ { 1 } ( \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( P _ { \\\\bullet } , N _ { \\\\bullet A } \\\\otimes _ { A } \\\\mathbb { K } ; f ) )$ and the above condition (2) is equivalent to $\\\\phi = f \\\\xi$ . It is now sufficient to observe that since $f$ is a trivial fibration, by Lemma 7 the morphism \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[287,653,712,675]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0950/3ab1506876d8bd758d4875ec02fefd31194026deb41f79711cefc9efab65b56c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"100\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nf \\\\colon \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( P _ { \\\\bullet } , P _ { \\\\bullet } ) \\\\to \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( P _ { \\\\bullet } , N _ { \\\\bullet A } \\\\otimes _ { A } \\\\mathbb { K } ; f )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,690,555,708]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"100\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is a surjective quasi-isomorphism and therefore \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[246,723,751,747]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0010/83e23c77eb95ca6ebe196f35b47e5b0943e7fdcf177caa841f1219dfc577905d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"100\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nf \\\\colon Z ^ { 1 } ( \\\\mathrm { D e r } _ { \\\\mathbb K } ^ { * } ( P _ { \\\\bullet } , P _ { \\\\bullet } ) ) \\\\to Z ^ { 1 } ( \\\\mathrm { D e r } _ { \\\\mathbb K } ^ { * } ( P _ { \\\\bullet } , N _ { \\\\bullet A } \\\\otimes _ { A } \\\\mathbb K ; f ) )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,764,298,780]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"100\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is a surjective map. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,789,879,825]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"100\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Unicity Let $\\\\delta , \\\\delta ^ { \\\\prime }$ be two $A$ linear differentials on $P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } A$ lifting the differential $d$ on $P _ { \\\\bullet }$ and let \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[252,842,742,863]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0801/eb1e7e17492d1008937dd190b9c0898fae65cb2e2371603d32452e72cb6b7142.jpg\",\n \"list_items\": null,\n \"page_idx\": \"100\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\np \\\\colon ( P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } A , \\\\delta ) \\\\to N , \\\\qquad q \\\\colon ( P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } A , \\\\delta ^ { \\\\prime } ) \\\\to N ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,51,562,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"100\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[854,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"100\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"95 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,879,148]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"101\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"be two morphisms in $\\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { C D G A } _ { A } ^ { \\\\leq 0 } )$ ) lifting the trivial fibration $f \\\\colon P . \\\\ \\\\to$ $N _ { \\\\bullet A } \\\\otimes _ { A } \\\\mathbb { K }$ . We need to prove that there exists an isomorphism of diagrams of differential graded $A$ -algebras $\\\\phi \\\\colon ( P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } A , \\\\delta ) ( P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } A , \\\\delta ^ { \\\\prime } )$ such that $p = q \\\\phi$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,157,881,373]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"101\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By induction on the length we can assume that there exists an isomorphism of diagrams of differential graded $B$ -algebras $\\\\phi ^ { \\\\prime } \\\\colon ( P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } B , \\\\delta ) \\\\ \\\\ ( P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } B , \\\\delta ^ { \\\\prime } )$ such that $p = q \\\\phi ^ { \\\\prime }$ . By Lemma 8 we can lift $\\\\phi ^ { \\\\prime }$ to an isomorphism of diagrams of graded $A$ -algebras $\\\\psi ^ { \\\\prime } \\\\colon ( P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } A , \\\\delta ) \\\\to ( P _ { \\\\bullet } \\\\otimes _ { \\\\mathbb { K } } A , \\\\delta ^ { \\\\prime } )$ ; therefore, replacing $\\\\delta ^ { \\\\prime }$ with $( \\\\psi ^ { \\\\prime } ) ^ { - 1 } \\\\delta ^ { \\\\prime } ( \\\\psi )$ and $q$ with $q \\\\psi ^ { \\\\prime }$ if necessary, it is not restrictive to assume $\\\\phi ^ { \\\\prime }$ equal to the identity. The derivation $p - q \\\\colon P _ { \\\\bullet } \\\\to t ( N _ { \\\\bullet A } \\\\otimes _ { A } \\\\mathbb { K } )$ can be lifted to a derivation $\\\\alpha \\\\in \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { 0 } ( P _ { \\\\bullet } , P _ { \\\\bullet } )$ and then, replacing $q$ with $q ( \\\\mathrm { I d } + t \\\\alpha )$ and $\\\\delta ^ { \\\\prime }$ with $( \\\\mathrm { I d } - t \\\\alpha ) \\\\delta ^ { \\\\prime } ( \\\\mathrm { I d } + t \\\\alpha )$ if necessary, it is not restrictive to assume $p = q$ . This implies in particular that $\\\\delta ^ { \\\\prime } = \\\\delta + t \\\\xi$ , for some $\\\\xi \\\\in Z ^ { 1 } ( \\\\operatorname { D e r } _ { \\\\mathbb { K } } ^ { * } ( P _ { \\\\bullet } , P _ { \\\\bullet } ) )$ . Since the kernel of $f \\\\colon \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( P _ { \\\\bullet } , P _ { \\\\bullet } ) \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( P _ { \\\\bullet } , N _ { \\\\bullet A } \\\\otimes _ { A } \\\\mathbb { K } ; f ) )$ is acyclic and $p \\\\delta = p \\\\delta ^ { \\\\prime }$ , we have $f \\\\xi = 0$ and therefore $\\\\xi = [ d , \\\\alpha ]$ for some $\\\\alpha \\\\in \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { 0 } ( P _ { \\\\bullet } , P _ { \\\\bullet } )$ such that $f \\\\alpha = 0$ . Now $e ^ { t \\\\alpha }$ is the required isomorphism. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,382,879,436]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"101\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof of Theorem 4 We assume that the reader has a certain familiarity with the theory of deformation functors associated to DG-Lie algebras; the basic facts exposed in [12, 14] are sufficient for our needs. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,438,879,512]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"101\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $\\\\mathcal { D }$ be a small category and $S _ { \\\\bullet } \\\\colon { \\\\mathcal { D } } \\\\to \\\\mathbf { A l g } _ { \\\\mathbb { K } }$ a diagram of unitary commutative algebras. Let $R _ { \\\\bullet } ~ ~ S _ { \\\\bullet }$ be a cofibrant replacement in $\\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } )$ with respect to the projective model structure and consider the DG-Lie algebra $L \\\\ =$ $\\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { \\\\bullet } , R _ { \\\\bullet } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,512,876,548]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"101\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Denoting by $ { \\\\mathrm { D e f } _ { S _ { \\\\bullet } } } \\\\colon { \\\\mathbf { A r t } } _ { \\\\mathbb { K } } \\\\ \\\\to$ Set the functor of isomorphism classes of deformations we want to describe an isomorphism \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[400,564,594,585]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0543/0b441880be78bf300cdbc5843f54a3e9b2a725e65ad3ff008d2db90f582eaa24.jpg\",\n \"list_items\": null,\n \"page_idx\": \"101\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\phi \\\\colon \\\\mathrm { D e f } _ { L } \\\\mathrm { D e f } _ { S _ { \\\\bullet } } \\\\ .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,600,881,636]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"101\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Denoting by $d \\\\in \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { 1 } ( R _ { \\\\bullet } , R _ { \\\\bullet } )$ the differential of $R _ { \\\\bullet }$ , for every $A \\\\in \\\\mathbf { A r t } _ { \\\\mathbb { K } }$ with maximal ideal ${ \\\\mathfrak { m } } _ { A }$ , a Maurer-Cartan element \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[246,650,751,693]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0286/e7b685f52c76054c18e2b20e81c92a50401c26139fe2f3d10930cd8d11b1362e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"101\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\xi \\\\in \\\\mathbf { M C } _ { L } ( A ) = \\\\left\\\\{ x \\\\in L ^ { 1 } \\\\otimes \\\\mathfrak { m } _ { A } \\\\ \\\\bigg | \\\\ d _ { L } x + \\\\frac { 1 } { 2 } [ x , x ] = 0 \\\\right\\\\}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,708,881,786]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"101\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is exactly a derivation $\\\\xi \\\\in \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { 1 } ( R _ { \\\\bullet } , R _ { \\\\bullet } \\\\otimes \\\\mathfrak { m } _ { A } )$ such that $( R _ { \\\\bullet } \\\\otimes A , d + \\\\xi )$ is a flat diagram in $\\\\mathrm { F u n } ( \\\\mathcal { D } , { \\\\bf C D G A } _ { A } ^ { \\\\le 0 } )$ ). Moreover $\\\\boldsymbol { \\\\xi } , \\\\eta \\\\in \\\\mathbf { M C } _ { L } ( A )$ are gauge equivalent if and only if there exists an isomorphism of diagrams of DG-algebras $( R _ { \\\\bullet } \\\\otimes A , d +$ $\\\\xi ) \\\\simeq ( R _ { \\\\bullet } \\\\otimes A , d + \\\\eta )$ , lifting the identity over $R _ { \\\\bullet }$ . According to Lemma 2 the map \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[243,798,749,822]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0681/e6392e53308d2b51c69da4e5698e9e866394a81fc1dce8e6183233b899abfcc5.jpg\",\n \"list_items\": null,\n \"page_idx\": \"101\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n{ \\\\bf M C } _ { L } ( A ) \\\\mathrm { D e f } _ { S _ { \\\\bullet } } ( A ) , \\\\qquad \\\\xi \\\\mapsto H ^ { 0 } ( R _ { \\\\bullet } \\\\otimes A , d + \\\\xi ) ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,839,879,875]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"101\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is properly defined and factors to a natural transformation $\\\\phi \\\\colon \\\\mathrm { D e f } _ { L } \\\\ \\\\ \\\\mathrm { D e f } _ { S _ { \\\\bullet } }$ . Finally Lemma 9 implies immediately that $\\\\phi$ is an isomorphism. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[687,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"101\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Lepri and M. Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,54,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"101\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"102\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the situation of Theorem 4, according to Lemma 7, the natural map $L \\\\ =$ $\\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { \\\\bullet } , R _ { \\\\bullet } ) \\\\ \\\\to \\\\ \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { \\\\bullet } , S _ { \\\\bullet } )$ is a quasi-isomorphism of complexes, hence $H ^ { i } ( L ) ~ = ~ 0$ for every $\\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ : \\\\ c \\\\ c \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F \\\\ F$ . The $S _ { \\\\bullet }$ -module $\\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { \\\\bullet } , S _ { \\\\bullet } )$ , defined up to quasi-isomorphism, is called the tangent complex of $S _ { \\\\bullet }$ . Its cohomology groups are denoted by $T ^ { i } ( S _ { \\\\bullet } )$ . According to [12, 14], an immediate consequence of Theorem 4 is that the space of first order deformations of the diagram $S _ { \\\\bullet }$ is $H ^ { 1 } ( \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { \\\\bullet } , R _ { \\\\bullet } ) ) \\\\ : = \\\\ : T ^ { \\\\bar { 1 } } ( S _ { \\\\bullet } )$ , and obstructions to deformations are contained in the space $H ^ { 2 } ( \\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { \\\\bullet } , R _ { \\\\bullet } ) ) = T ^ { 2 } ( S _ { \\\\bullet } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,234,881,324]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"102\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Although in principle Theorem 4 gives a complete answer to our initial problem, for diagrams over a general small category $\\\\mathcal { D }$ it may be very difficult to concretely describe a cofibrant replacement, since projective cofibrations are described either as maps satisfying the left lifting property with respect to trivial fibrations, or as transfinite compositions of certain elementary cofibrations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,324,876,358]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"102\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A possible strategy to overcome this difficulty is to give an explicit functor of small categories $\\\\epsilon : N \\\\to \\\\mathcal { D }$ such that: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,367,879,426]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"102\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. for every diagram $S _ { \\\\bullet } \\\\in \\\\operatorname { F u n } ( \\\\mathcal { D } , \\\\mathbf { A l g } _ { \\\\mathbb { K } } )$ , the deformation theory of $S _ { \\\\bullet }$ is the same as the deformation theory of $S _ { \\\\bullet } \\\\circ \\\\epsilon \\\\in \\\\mathrm { F u n } ( N , \\\\mathbf { A l g _ { \\\\mathbb { K } } } )$ ; 2. cofibrations in $\\\\mathrm { F u n } ( N , { \\\\bf C D G A } _ { \\\\mathbb { K } } ^ { \\\\le 0 } )$ ) admit a constructive description. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,433,881,522]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"102\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the next sections we follow this strategy by setting as $\\\\epsilon$ a simplified version of the Bousfield-Kan approximation [3]. In our construction the category $N$ will be in particular a Reedy category (see Sect. 8) and the projective model structure in $\\\\mathrm { F u n } ( N , { \\\\bf C D G A } _ { \\\\mathbb { K } } ^ { \\\\le 0 } ,$ ) will be the same as the Reedy model structure, hence with a simpler description of cofibrations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,569,387,590]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"102\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7 Simplex Categories \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,612,881,686]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"102\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $\\\\pmb { \\\\Delta }$ be the category with objects the finite ordinals $[ n ] ~ = ~ \\\\{ 0 , 1 , \\\\cdots , n \\\\}$ and morphisms non-decreasing maps, also known as the simplex category. We denote by $\\\\delta _ { k } \\\\colon [ n - 1 ] \\\\to [ n ]$ , and by $\\\\sigma _ { k } \\\\colon [ n + 1 ] \\\\to [ n ]$ , $k = 0 , \\\\ldots , n$ , the usual face and degeneracy maps: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[159,704,833,756]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0061/1d65b4169e0d05893f821f21446670af8202efb12989f5d5244b93f8cfe4aaf1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"102\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{c} \\\\delta _ { k } : [ n - 1 ] \\\\to [ n ] , \\\\qquad \\\\delta _ { k } ( p ) = \\\\left\\\\{ p \\\\qquad { \\\\mathrm { i f ~ } } p < k \\\\\\\\ { p + 1 \\\\quad { \\\\mathrm { i f ~ } } p \\\\geq k } \\\\end{array} , \\\\right. \\\\qquad k = 0 , \\\\ldots , n ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[159,770,835,821]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0318/5342bde39db3c8245a0b5b3575eab11ba8ca77227c6f882bce6dd680c1c143c9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"102\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\sigma _ { k } \\\\colon [ n + 1 ] \\\\to [ n ] , \\\\qquad \\\\sigma _ { k } ( p ) = { \\\\left\\\\{ \\\\begin{array} { l l } { p } & { { \\\\mathrm { i f ~ } } p \\\\leq k } \\\\\\\\ { p - 1 } & { { \\\\mathrm { i f ~ } } p > k } \\\\end{array} \\\\right. } , \\\\qquad k = 0 , \\\\ldots , n ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,52,562,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"102\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[856,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"102\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,478,108]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"103\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"They satisfy the cosimplicial identities: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[337,123,658,256]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0686/0b60f318b4a31ab6f4af699cd3713dceb86d6bd1d4afc23dc177891e54d5c027.jpg\",\n \"list_items\": null,\n \"page_idx\": \"103\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\sigma _ { i } \\\\sigma _ { j } = \\\\sigma _ { j } \\\\sigma _ { i + 1 } \\\\quad \\\\mathrm { ~ f o r ~ } i \\\\geq j } \\\\\\\\ & { \\\\delta _ { i } \\\\delta _ { j } = \\\\delta _ { j + 1 } \\\\delta _ { i } \\\\quad \\\\mathrm { ~ f o r ~ } i \\\\leq j } \\\\\\\\ & { \\\\sigma _ { i } \\\\delta _ { j } = \\\\left\\\\{ \\\\begin{array} { l l } { \\\\delta _ { j - 1 } \\\\sigma _ { i } , } & { \\\\mathrm { ~ i f ~ } j > i + 1 } \\\\\\\\ { \\\\mathrm { I d } , } & { \\\\mathrm { ~ i f ~ } j = i , i + 1 } \\\\\\\\ { \\\\delta _ { j } \\\\sigma _ { i - 1 } , } & { \\\\mathrm { ~ i f ~ } j < i . } \\\\end{array} \\\\right. } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,270,881,324]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"103\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We recall that a cosimplicial group is a functor $G \\\\colon \\\\Delta \\\\to \\\\mathbf { G r p } , [ n ] \\\\mapsto G _ { n }$ ; in the sequel we shall need the following proposition, which is an easy generalisation of a well known result about cosimplicial groups, cf. [2, Prop. X.4.9]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,331,881,387]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"103\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 6 Let $G$ be a cosimplicial group, let $n \\\\geq 1$ , and $I \\\\subseteq [ n ]$ . Assume there are given elements $x _ { i } ~ \\\\in ~ G _ { n }$ , $i \\\\in I$ , such that $\\\\sigma _ { i - 1 } x _ { j } = \\\\sigma _ { j } x _ { i }$ for all $i \\\\ > \\\\ j$ and $i , j \\\\in I$ . Then there exists $x \\\\in G _ { n + 1 }$ such that $\\\\sigma _ { i } x = x _ { i }$ for all $i \\\\in I$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,394,876,432]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"103\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Writing $I = \\\\{ i _ { 0 } < i _ { 1 } < \\\\cdots < i _ { k } \\\\} \\\\subseteq [ n ]$ , consider the sequence $z _ { i _ { 0 } } , \\\\ldots , z _ { i _ { k } }$ defined recursively by the formula: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[191,445,806,471]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0959/099cdb066c0b7fd39c9749fc2d0ed69b5d9be8c280024267331425b11e1619b9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"103\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nz _ { i _ { k } } = \\\\delta _ { i _ { k } } x _ { i _ { k } } , \\\\qquad z _ { i _ { p } } = z _ { i _ { p + 1 } } \\\\cdot ( \\\\delta _ { i _ { p } } \\\\sigma _ { i _ { p } } z _ { i _ { p + 1 } } ) ^ { - 1 } \\\\cdot ( \\\\delta _ { i _ { p } } x _ { i _ { p } } ) , \\\\quad p < k .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[116,486,879,522]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"103\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For later use we point out that in the construction of this sequence we have only used the group homomorphisms \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[257,539,740,558]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0868/9fe23cb1471409321f71bec21e3d374efe2ef8daa2b5318de8c57a3bd1c914b4.jpg\",\n \"list_items\": null,\n \"page_idx\": \"103\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\sigma _ { i } \\\\colon G _ { n + 1 } \\\\to G _ { n } , \\\\qquad \\\\delta _ { i } \\\\colon G _ { n } \\\\to G _ { n + 1 } , \\\\qquad i \\\\in I .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,576,881,612]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"103\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We claim that $x = z _ { i _ { 0 } }$ is the required element: we show by induction on $k - p$ that that $\\\\sigma _ { i _ { m } } z _ { i _ { p } } = x _ { i _ { m } }$ for all $m \\\\geq p$ . For $m > p$ , \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[252,629,742,747]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0830/d37b4277c3e515d20fb104993cd5d1609ad26e0eee86b67558e917d07c3ea463.jpg\",\n \"list_items\": null,\n \"page_idx\": \"103\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\sigma _ { i _ { m } } z _ { i _ { p } } = ( \\\\sigma _ { i _ { m } } z _ { i _ { p + 1 } } ) \\\\cdot ( \\\\sigma _ { i _ { m } } \\\\delta _ { i _ { p } } \\\\sigma _ { i _ { p } } z _ { i _ { p + 1 } } ) ^ { - 1 } \\\\cdot ( \\\\sigma _ { i _ { m } } \\\\delta _ { i _ { p } } x _ { i _ { p } } ) } \\\\\\\\ & { \\\\qquad = x _ { i _ { m } } \\\\cdot ( \\\\delta _ { i _ { p } } \\\\sigma _ { i _ { m } - 1 } \\\\sigma _ { i _ { p } } z _ { i _ { p + 1 } } ) ^ { - 1 } \\\\cdot ( \\\\delta _ { i _ { p } } \\\\sigma _ { i _ { m } - 1 } x _ { i _ { p } } ) } \\\\\\\\ & { \\\\qquad = x _ { i _ { m } } \\\\cdot ( \\\\delta _ { i _ { p } } \\\\sigma _ { i _ { p } } \\\\sigma _ { i _ { m } } z _ { i _ { p + 1 } } ) ^ { - 1 } \\\\cdot ( \\\\delta _ { i _ { p } } \\\\sigma _ { i _ { p } } x _ { i _ { m } } ) } \\\\\\\\ & { \\\\qquad = x _ { i _ { m } } \\\\cdot ( \\\\delta _ { i _ { p } } \\\\sigma _ { i _ { p } } x _ { i _ { m } } ) ^ { - 1 } \\\\cdot ( \\\\delta _ { i _ { p } } \\\\sigma _ { i _ { p } } x _ { i _ { m } } ) = x _ { i _ { m } } , } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,759,257,777]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"103\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and for $m = p$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[259,792,740,849]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"1004/458608a5d23cc5cc299102e0b2710e55cd7b6347c533fab143a918a69ee5ae75.jpg\",\n \"list_items\": null,\n \"page_idx\": \"103\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\sigma _ { i _ { p } } z _ { i _ { p } } = ( \\\\sigma _ { i _ { p } } z _ { i _ { p + 1 } } ) \\\\cdot ( \\\\sigma _ { i _ { p } } \\\\delta _ { i _ { p } } \\\\sigma _ { i _ { p } } z _ { i _ { p + 1 } } ) ^ { - 1 } \\\\cdot ( \\\\sigma _ { i _ { p } } \\\\delta _ { i _ { p } } x _ { i _ { p } } ) } \\\\\\\\ & { \\\\qquad = ( \\\\sigma _ { i _ { p } } z _ { i _ { p + 1 } } ) \\\\cdot ( \\\\sigma _ { i _ { p } } z _ { i _ { p + 1 } } ) ^ { - 1 } \\\\cdot ( x _ { i _ { p } } ) = x _ { i _ { p } } . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[685,52,879,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"103\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Lepri and M. Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,52,143,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"103\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"98 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[858,861,881,876]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"103\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"□ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,144]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"104\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The simplex category $\\\\pmb { \\\\Delta }$ admits the following useful generalisation. Let $\\\\mathcal { B }$ be a small category and consider, for every $n \\\\geq 0$ , the set $N ( { \\\\mathcal { B } } ) _ { n }$ of $n$ -simplexes of the nerve of $\\\\mathcal { B }$ : every element of $N ( { \\\\mathcal { B } } ) _ { n }$ is a string \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[350,159,649,183]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0482/6edc782ee05a8c1e3f782b9df3316333fb5551bef05cf5441fa9fc6485ef1c1b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"104\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nx = [ x _ { 0 } { \\\\overset { \\\\alpha _ { 1 } } { \\\\longrightarrow } } x _ { 1 } \\\\cdots x _ { n - 1 } { \\\\overset { \\\\alpha _ { n } } { \\\\longrightarrow } } x _ { n } ]\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,199,881,237]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"104\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"of $n$ morphisms of $\\\\mathcal { B }$ . The simplex category $N ( { \\\\mathcal { B } } )$ of $\\\\mathcal { B }$ is defined in the following way: the set of objects is the disjoint union of $N ( { \\\\mathcal { B } } ) _ { n } , n \\\\geq 0$ . Given two objects \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[168,252,826,280]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0063/75ee1a58b7b816b4807c0cbd2410ec4ddc9e5ec79827603f60086b4a0f6794bd.jpg\",\n \"list_items\": null,\n \"page_idx\": \"104\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { x = [ x _ { 0 } \\\\xrightarrow { \\\\alpha _ { 1 } } x _ { 1 } \\\\cdots x _ { n - 1 } \\\\xrightarrow { \\\\alpha _ { n } } x _ { n } ] , \\\\qquad y = [ y _ { 0 } \\\\xrightarrow { \\\\beta _ { 1 } } y _ { 1 } \\\\cdots y _ { m - 1 } \\\\xrightarrow { \\\\beta _ { m } } y _ { m } ] , } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,297,881,352]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"104\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"a morphism $f \\\\colon x y$ is a monotone map $f \\\\colon [ n ] [ m ]$ such that $y _ { f ( i ) } = x _ { i }$ for every $i \\\\in [ n ]$ , and for every $0 \\\\leq i \\\\leq n$ the morphism $\\\\alpha _ { i }$ is the composition of $\\\\beta _ { j }$ , for $f ( i - 1 ) < j \\\\leq f ( i )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[193,366,804,397]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0423/11c905fd89535b4557a74443aaf364cfac55fdc53901f3591c4135109c691e4c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"104\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\alpha _ { i } \\\\colon x _ { i - 1 } = y _ { f ( i - 1 ) } { \\\\xrightarrow { { \\\\beta _ { f ( i - 1 ) + 1 } } } } \\\\cdots { \\\\xrightarrow { } } y _ { f ( i ) - 1 } { \\\\xrightarrow { { \\\\beta _ { f ( i ) } } } } y _ { f ( i ) } = x _ { i } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,411,881,472]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"104\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Notice that the equality $f ( i - 1 ) = f ( i )$ implies $x _ { i } ~ = ~ x _ { i - 1 }$ and $\\\\alpha _ { i } ~ = \\\\mathrm { I d }$ . For example, if $[ x \\\\stackrel { \\\\alpha } { } y \\\\stackrel { \\\\beta } { } z ] \\\\in N ( \\\\mathcal { B } ) _ { 2 }$ , then we have in the category $N ( { \\\\mathcal { B } } )$ the following morphisms: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[223,487,772,654]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0220/100a6031f004d7260c979251644d59c6ab3ffc315433c030e78696a7fbf0fc65.jpg\",\n \"list_items\": null,\n \"page_idx\": \"104\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { [ z ] \\\\xrightarrow { \\\\delta _ { 0 } } \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad } \\\\\\\\ & { \\\\downarrow \\\\quad \\\\delta _ { 0 } \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\delta _ { 1 } } \\\\\\\\ & { [ y \\\\xrightarrow { \\\\beta } z ] \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad } \\\\\\\\ & { \\\\quad \\\\quad \\\\quad \\\\quad \\\\sigma _ { 0 } ^ { ( 1 ) } \\\\Bigg | } \\\\\\\\ & { \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\sigma _ { 1 } \\\\xrightarrow { \\\\beta } \\\\ y \\\\ ] { \\\\beta } \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\sigma _ { 1 } \\\\xrightarrow { \\\\alpha } \\\\ y ] } \\\\\\\\ & { [ y \\\\xrightarrow { \\\\mathrm { I d } } \\\\ y \\\\xrightarrow { \\\\beta } z ] { \\\\beta } \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad } \\\\\\\\ & { \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad } \\\\\\\\ & { \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\sigma _ { 1 } } \\\\\\\\ & { \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\sigma _ { 2 } ^ { ( 1 ) } \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\sigma _ { 1 } } \\\\\\\\ & \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\quad \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[143,671,792,690]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"104\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Notice that the simplex category of the singleton $\\\\mathcal { B } = \\\\{ * \\\\}$ is exactly $\\\\pmb { \\\\Delta }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,698,564,717]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"104\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 13 A morphism $f \\\\colon x y$ in $N ( { \\\\mathcal { B } } )$ : \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[287,734,708,753]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0657/abd35a29b1c17f0faf13b66ac93de985ac7d76c6d5b66f682397ae2d9adda9ae.jpg\",\n \"list_items\": null,\n \"page_idx\": \"104\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { x \\\\in N ( \\\\mathcal { B } ) _ { n } , \\\\quad y \\\\in N ( \\\\mathcal { B } ) _ { m } , \\\\quad f \\\\colon [ n ] \\\\to [ m ] , } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,770,421,788]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"104\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is called an anchor if $f ( n ) = m$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,797,881,852]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"104\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 14 For every $k ~ \\\\in ~ \\\\mathbb { N } \\\\cup \\\\{ + \\\\infty \\\\}$ we shall denote by $N ( { \\\\mathcal { B } } ) _ { \\\\leq k }$ the full subcategory of $N ( { \\\\mathcal { B } } )$ with objects the (disjoint) union of $N ( { \\\\mathcal { B } } ) _ { i }$ for $i ~ \\\\leq k$ , and by \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[309,866,687,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0550/c31e396d5b813b9322bc4cd2c835c21dd3386ed3b30fd623f633871aa9f42882.jpg\",\n \"list_items\": null,\n \"page_idx\": \"104\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathrm { F u n } ^ { \\\\star } ( N ( \\\\mathcal { B } ) _ { \\\\leq k } , \\\\mathbf { M } ) \\\\subseteq \\\\mathrm { F u n } ( N ( \\\\mathcal { B } ) _ { \\\\leq k } , \\\\mathbf { M } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,51,562,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"104\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[854,52,879,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"104\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"99 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"the full subcategory of diagrams $F \\\\colon N ( \\\\mathcal { B } ) _ { \\\\leq k } \\\\ \\\\ \\\\textbf { M }$ such that $F ( f )$ is an isomorphism for every anchor map $f$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,133,781,153]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 15 The forgetful functor $\\\\epsilon \\\\colon N ( \\\\mathcal { B } ) \\\\mathcal { B }$ is defined by setting \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[328,168,671,192]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0787/9e4b160aff8141a849d62264f34b4a584d1683261f0388fc25c6571c49f645f0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\epsilon ( [ x _ { 0 } \\\\xrightarrow { \\\\alpha _ { 1 } } x _ { 1 } \\\\cdots x _ { n - 1 } \\\\xrightarrow { \\\\alpha _ { n } } x _ { n } ] ) = x _ { n }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,210,430,228]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"on the objects. For any morphism \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[198,243,797,271]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0841/0d4fe98c248f9592bddf57c2dcf8fda0f192b45ae38477457930c27220cc1323.jpg\",\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nf \\\\colon [ x _ { 0 } \\\\ { \\\\xrightarrow { \\\\alpha _ { 1 } } } \\\\ x _ { 1 } \\\\ \\\\cdots \\\\ x _ { n - 1 } \\\\ { \\\\xrightarrow { \\\\alpha _ { n } } } \\\\ x _ { n } ] \\\\to [ y _ { 0 } \\\\ { \\\\xrightarrow { \\\\beta _ { 1 } } } \\\\ y _ { 1 } \\\\ \\\\cdots \\\\ y _ { m - 1 } \\\\ { \\\\xrightarrow { \\\\beta _ { m } } } \\\\ y _ { m } ] ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,288,198,304]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[277,324,722,343]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0786/e4499e59c286d0c79795f4be903afe477698ffc4a3a083778574db53f5e5f708.jpg\",\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\epsilon ( f ) = \\\\beta _ { m } \\\\circ \\\\cdot \\\\cdot \\\\cdot \\\\circ \\\\beta _ { f ( n ) + 1 } \\\\colon x _ { n } = y _ { f ( n ) } y _ { m } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[116,358,879,396]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In particular, $\\\\epsilon ( f ) = \\\\operatorname { I d }$ for any anchor $f$ . It is clear that the composition with the functor $\\\\epsilon$ gives, for every $k$ , a natural transformation: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[157,409,790,433]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0477/b0023dffab1bec40a4b61e84883d446794f49102a2505854399e9a12905ea44f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\epsilon ^ { * } \\\\colon \\\\mathrm { F u n } ( \\\\mathcal { B } , \\\\mathbf { M } ) \\\\to \\\\mathrm { F u n } ^ { \\\\star } ( N ( \\\\mathcal { B } ) _ { \\\\leq k } , \\\\mathbf { M } ) , \\\\qquad \\\\epsilon ^ { * } ( F ) = F \\\\circ \\\\epsilon _ { | N ( \\\\mathcal { B } ) _ { \\\\leq k } } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,448,571,468]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $k \\\\geq 2$ we also have a natural transformation \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,481,678,504]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"1005/b5d227f309b35887422b8feeab1051dad334146a3b0c989ab48d742610473118.jpg\",\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\boldsymbol { \\\\tau } : \\\\mathrm { F u n } ^ { \\\\star } ( N ( \\\\mathcal { B } ) _ { \\\\leq k } , \\\\mathbf { M } ) \\\\to \\\\mathrm { F u n } ( \\\\mathcal { B } , \\\\mathbf { M } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,521,881,557]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"defined in the following way: given $G \\\\in \\\\mathrm { F u n } ^ { \\\\star } ( N ( \\\\mathcal { B } ) _ { \\\\leq k } , \\\\mathbf { M } )$ and an object $x \\\\in { \\\\mathcal { B } }$ we set \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[405,575,594,594]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0561/62340f08abc59f80ea1bb56a5124ac1588d2d65ea5b9e7e109f5435dc4bc68ca.jpg\",\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\tau ( G ) ( x ) = G ( [ x ] ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,611,514,632]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Given a morphism $x { \\\\stackrel { \\\\alpha } { \\\\to } } y$ in $\\\\mathcal { B }$ we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[307,647,692,675]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0814/69cd02caef053897930fbaad011c26c477df83383f234541aa56e0e4edabfa3b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nG ( [ x ] ) \\\\xrightarrow { G ( \\\\delta _ { 1 } ) } G ( [ x \\\\xrightarrow { \\\\alpha } y ] ) \\\\xrightarrow { G ( \\\\delta _ { 0 } ) } G ( [ y ] )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,692,578,710]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and, since $G ( \\\\delta _ { 0 } )$ is an isomorphism we can define \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[280,723,719,746]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0406/97fd8332613682491530deb5bfe319fec9ea1c8380e24a8f1083b4e8c238d7be.jpg\",\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\tau ( G ) ( \\\\alpha ) = G ( \\\\delta _ { 0 } ) ^ { - 1 } G ( \\\\delta _ { 1 } ) \\\\colon G ( [ x ] ) \\\\to G ( [ y ] ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[143,762,876,783]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We need to prove that $\\\\tau ( G )$ is a functor: applying $G$ to the commutative diagram \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[357,798,640,888]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0265/b558999db3772922efecc42e24d402697d11c1676e776b0e6a9ff9e2a5e65c53.jpg\",\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n[ x ] \\\\xrightarrow [ \\\\mathrm { I d } ] { \\\\delta _ { \\\\begin{array} { l } { \\\\delta _ { 1 } } \\\\\\\\ { \\\\qquad \\\\mathrm { I d } } \\\\end{array} } [ x \\\\xrightarrow [ \\\\mathrm { I d } ] { \\\\mathrm { I d } } x ] \\\\mathord { \\\\left. \\\\right. } \\\\mathord { \\\\left[ x \\\\right] } } [ x ]\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[685,51,879,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Lepri and M. Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,152,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"105\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"100 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,93,881,135]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"106\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"we prove that $\\\\tau ( G )$ preserves the identities. Given $[ x \\\\stackrel { \\\\alpha } { } y \\\\stackrel { \\\\beta } { } z ] \\\\in N ( { \\\\mathcal { B } } ) _ { 2 }$ , applying $G$ to the commutative diagram \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[289,151,710,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0501/db22096c72c8ad6537c2d5e51723845ab7f00f27446046f12c252991c80414d0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"106\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\left[ z \\\\right] \\\\xrightarrow { \\\\delta _ { 0 } } \\\\left[ x \\\\xrightarrow { \\\\beta \\\\alpha } z \\\\right] { - \\\\delta _ { 1 } } \\\\left[ x \\\\right] } \\\\\\\\ & { \\\\left[ \\\\begin{array} { l } { \\\\delta _ { 0 } } \\\\\\\\ { \\\\delta _ { 0 } } \\\\end{array} \\\\right] \\\\xrightarrow [ { \\\\delta _ { 1 } } ] { \\\\left[ \\\\begin{array} { l l l } { \\\\delta _ { 1 } } & & { \\\\left\\\\lfloor x \\\\right\\\\rfloor } \\\\\\\\ { \\\\delta _ { 2 } } \\\\end{array} \\\\right] { - \\\\delta _ { 2 } } } \\\\left[ x \\\\xrightarrow [ { \\\\delta _ { 2 } ] { \\\\delta _ { 2 } } } y \\\\right] } \\\\\\\\ & { \\\\left[ y \\\\xrightarrow [ { \\\\delta _ { 1 } } ] { \\\\delta _ { 1 } } \\\\bigoplus _ { \\\\substack { \\\\left[ \\\\underline { { y } } \\\\right] } } \\\\right] { - \\\\delta _ { 1 } } \\\\left[ x \\\\xrightarrow [ { \\\\delta _ { 2 } ] { \\\\delta _ { 3 } } } z \\\\right] { - \\\\delta _ { 2 } } \\\\left[ x \\\\xrightarrow [ { \\\\delta _ { 3 } ] { \\\\alpha } } y \\\\right] } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,328,881,366]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"106\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"we obtain $\\\\tau ( G ) ( \\\\beta \\\\alpha ) = \\\\tau ( G ) ( \\\\beta ) \\\\circ \\\\tau ( G ) ( \\\\alpha )$ . Therefore $\\\\tau$ is properly defined and its functoriality is clear. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[113,373,808,393]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"106\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 7 The above functors $\\\\epsilon ^ { * }$ and $\\\\tau$ are equivalences of categories. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,400,881,438]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"106\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof It is immediate from the definition that $\\\\tau \\\\circ \\\\epsilon ^ { * }$ is the identity. On the other hand every anchor map of type \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[375,454,621,474]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0706/b4980cdffe8b54c63d1b86823091e2fb0f794397306514883d998451130d6170.jpg\",\n \"list_items\": null,\n \"page_idx\": \"106\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n[ x _ { n } ] \\\\to [ x _ { 0 } \\\\to \\\\cdot \\\\cdot \\\\cdot \\\\to x _ { n } ]\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,490,756,510]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"106\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"induces, for every $G \\\\in \\\\mathrm { F u n } ^ { \\\\star } ( N ( \\\\mathcal { B } ) _ { \\\\leq k } , \\\\mathbf { M } )$ , a canonical isomorphism \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,525,863,551]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0070/91076d9e75ee0fb7bbfc41fd7633dce6c9bc62a9b461bc4ac0bfab006c828d46.jpg\",\n \"list_items\": null,\n \"page_idx\": \"106\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\epsilon ^ { * } \\\\tau ( G ) ( [ x _ { 0 } \\\\cdot \\\\cdot \\\\cdot x _ { n } ] ) = \\\\tau ( G ) ( x _ { n } ) = G ( [ x _ { n } ] ) \\\\xrightarrow { \\\\simeq } G ( [ x _ { 0 } \\\\cdot \\\\cdot \\\\cdot x _ { n } ] ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,594,881,650]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"106\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 8 Let $S _ { \\\\bullet } \\\\in \\\\mathrm { F u n } ( \\\\mathcal { B } , \\\\mathbf { A l g } _ { \\\\mathbb { K } } )$ be a diagram of commutative algebras and $k \\\\geq 2$ . Then the isomorphism classes of deformations of S are the same as the isomorphism classes of deformations of $\\\\epsilon ^ { * } S _ { \\\\bullet } \\\\in \\\\mathrm { F u n } ( N ( \\\\mathcal { B } ) _ { \\\\leq k } , \\\\mathbf { A l g } _ { \\\\mathbb { K } } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,657,876,693]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"106\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof It is immediate from the definition that for every $A \\\\in \\\\mathbf { A r t } _ { \\\\mathbb { K } }$ the equivalences of categories \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[291,710,706,744]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0628/4eaeff36674d51e43561662ffdee785e58da5551ffe9b72b143fe587d51f3f40.jpg\",\n \"list_items\": null,\n \"page_idx\": \"106\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathrm { F u n } ( \\\\mathbf { \\\\mathcal { B } } , \\\\mathbf { A } \\\\mathbf { l g } _ { A } ) \\\\xrightarrow [ \\\\tau ] { \\\\epsilon ^ { * } } \\\\mathrm { F u n } ^ { \\\\star } ( N ( \\\\mathbf { \\\\mathcal { B } } ) _ { \\\\leq k } , \\\\mathbf { A } \\\\mathbf { l g } _ { A } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,765,876,801]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"106\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"preserve flatness. Therefore, for every deformation $\\\\phi \\\\colon S _ { A , \\\\bullet } \\\\to S .$ of $S _ { \\\\bullet }$ the map $\\\\epsilon ^ { * } \\\\phi \\\\colon \\\\epsilon ^ { * } S _ { A , \\\\bullet } \\\\to \\\\epsilon ^ { * } S _ { \\\\bullet }$ is a deformation of the diagram $\\\\epsilon ^ { * } S _ { \\\\bullet }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,803,879,858]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"106\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In order to conclude the proof we only need to show that, if $R _ { \\\\bullet } \\\\colon N ( \\\\mathcal { B } ) _ { \\\\leq k } \\\\ $ $\\\\mathbf { A } \\\\mathbf { g } _ { A }$ is a deformation of $\\\\epsilon ^ { * } S _ { \\\\bullet }$ , and $f \\\\colon \\\\alpha \\\\beta$ is an anchor in $N ( { \\\\mathcal { B } } ) _ { \\\\leq k }$ , then $R _ { f }$ is an isomorphism. This implies that $R _ { \\\\bullet } \\\\in \\\\mathrm { F u n } ^ { \\\\star } ( N ( \\\\mathcal { B } ) _ { \\\\leq k } , \\\\mathbf { A l g } _ { A } )$ . We have by definition \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,51,564,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"106\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"106\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"101 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[861,570,879,584]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"106\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"□ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,876,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"107\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"that $\\\\phi \\\\colon R _ { \\\\bullet } \\\\to \\\\epsilon ^ { * } S _ { \\\\bullet }$ induces an isomorphism $R _ { \\\\bullet } \\\\otimes _ { A } \\\\mathbb { K } \\\\to \\\\epsilon ^ { * } S _ { \\\\bullet }$ and, since $\\\\epsilon ^ { * } S _ { \\\\alpha } $ $\\\\epsilon ^ { * } S _ { \\\\beta }$ is the identity, by the commutativity of the diagram \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[371,142,628,223]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0929/602f0df01959acdab795da4e6e46a0ead9e360d5384c89e581e82657a7e8763c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"107\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l r } { { R _ { \\\\alpha } \\\\otimes _ { A } \\\\mathbb { K } \\\\longrightarrow R _ { \\\\beta } \\\\otimes _ { A } \\\\mathbb { K } } } \\\\\\\\ & { } & { \\\\cong \\\\downarrow \\\\phi } \\\\\\\\ & { } & { \\\\epsilon ^ { * } S _ { \\\\alpha } \\\\longrightarrow \\\\epsilon ^ { * } S _ { \\\\beta } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,244,881,282]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"107\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"we obtain that $R _ { \\\\alpha } \\\\otimes _ { A } \\\\mathbb { K } \\\\to R _ { \\\\beta } \\\\otimes _ { A } \\\\mathbb { K }$ is also an isomorphism. By Lemma 2, $R _ { \\\\alpha } $ $R _ { \\\\beta }$ is an isomorphism too. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,327,446,348]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"107\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"8 Reedy Model Structures \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,372,881,426]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"107\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We briefly recall the notion of Reedy category, for more details see [6]. We do so in view of Theorem 5, which yields a model structure on the category $\\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { M } )$ of diagrams on a model category $\\\\mathbf { M }$ indexed by a Reedy category $\\\\mathcal { D }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,433,876,472]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"107\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 16 A Reedy category is a small category $\\\\mathcal { D }$ together with two subcategories $\\\\vec { \\\\mathcal { D } } , \\\\overleftarrow { \\\\mathcal { D } }$ , such that: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,480,881,584]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"107\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. $\\\\mathrm { O b } ( \\\\mathcal { D } ) = \\\\mathrm { O b } ( \\\\mathcal { \\\\vec { D } } ) = \\\\mathrm { O b } ( \\\\mathcal { \\\\overleftarrow { D } } )$ \\n2. Every morphism $f$ in $\\\\mathcal { D }$ has a unique factorisation $f = g h$ , where $g$ is in $\\\\vec { \\\\mathcal { D } }$ and $h$ is in . \\n3. There exists a function deg $\\\\mathrm { O b } ( \\\\mathcal { D } ) \\\\mathbb { N }$ such that every non-identity morphism in $\\\\vec { \\\\mathcal { D } }$ raises degree and every non-identity morphism in lowers degree. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,591,881,669]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"107\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is easy to see that in a Reedy category every isomorphism is an identity: if $f$ is an isomorphism and $f = g _ { 1 } h _ { 1 }$ $g _ { 1 } h _ { 1 } , f ^ { - 1 } = g _ { 2 } h _ { 2 } , h _ { 1 } g _ { 2 } = g _ { 3 } h _ { 3 }$ are factorisations as in (2), then $g _ { 1 } g _ { 3 }$ must be the identity. A Reedy category is called direct if $\\\\mathcal { D } = \\\\overline { { \\\\mathcal { D } } }$ , or equivalently if contains only the identities. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,671,758,689]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"107\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For instance, we have the following examples of Reedy categories: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,696,883,855]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"107\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. A category whose only morphisms are the identities is called discrete. Every discrete category is trivially a Reedy category. \\n2. Let $I$ be a finite poset such that there exists a function deg $I \\\\mathbb { N }$ such that $\\\\deg ( x ) > \\\\deg ( y )$ for every $x > y$ . Then $I$ is a direct Reedy category. \\n3. The simplex category $\\\\pmb { \\\\Delta }$ is a Reedy category, with de ${ \\\\mathfrak { g } } ( [ n ] ) = n$ , $\\\\vec { \\\\Delta }$ the injective maps and $\\\\overleftarrow { \\\\Delta }$ the surjective maps. \\n4. If $c$ and $\\\\mathcal { D }$ are Reedy categories then so is the product $\\\\boldsymbol { C } \\\\times \\\\boldsymbol { \\\\mathcal { D } }$ , with ${ \\\\overrightarrow { C \\\\times { \\\\mathcal { D } } } } =$ −→ × −→, ←−−− × = ←− × ←− and deg(c, d) = deg(c) + deg(d). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,861,881,899]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"107\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $a$ is an object of a category $\\\\mathcal { D }$ we denote by $a \\\\downarrow \\\\mathcal { D }$ the undercategory of maps $a b$ in $\\\\mathcal { D }$ and by $\\\\mathcal { D } \\\\downarrow a$ the overcategory of maps $b \\\\to a$ in $\\\\mathcal { D }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[687,52,879,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"107\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Lepri and M. Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,152,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"107\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"102 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,706,108]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"108\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 17 Let $\\\\mathcal { D }$ be a Reedy category and $a$ an object in $\\\\mathcal { D }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,117,881,193]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"108\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. The matching category $M _ { a } \\\\mathcal { D }$ of $\\\\mathcal { D }$ at $a$ is the full subcategory of $a \\\\downarrow \\\\overleftarrow { \\\\mathcal { D } }$ containing all objects except the identity map of $a$ . \\n2. The latching category $L _ { a } \\\\mathcal { D }$ of $\\\\mathcal { D }$ at $a$ is the full subcategory of $\\\\vec { \\\\mathcal { D } } \\\\downarrow a$ containing all objects except the identity map of $a$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,202,881,276]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"108\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 18 Let $\\\\mathcal { D }$ be a Reedy category, let $\\\\mathbf { M }$ be a complete and cocomplete category, let $X$ be a $\\\\mathcal { D }$ -diagram in $\\\\mathbf { M }$ , and $a$ be an object in $\\\\mathcal { D }$ . For notational simplicity $X$ also denotes the induced $M _ { a } \\\\mathcal { D }$ -diagram, with $X _ { a b } = X _ { b }$ , and the induced $L _ { a } \\\\mathcal { D }$ -diagram, with $X _ { b a } = X _ { b }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,283,630,321]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"108\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. The matching object of $X$ at $a$ is $M _ { a } X = \\\\operatorname* { l i m } _ { M _ { a } \\\\mathcal { D } } X$ . \\n2. The latching object of $X$ at $a$ is $L _ { a } X = \\\\operatorname { c o l i m } _ { L _ { a } \\\\mathcal { D } } X$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,328,699,348]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"108\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There are natural morphisms $L _ { a } X \\\\longrightarrow X$ and $X \\\\longrightarrow M _ { a } X$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,355,881,409]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"108\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The main use of Reedy categories originates from the following theorem: the category of diagrams in a model category indexed by a Reedy category has a model category structure. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,417,881,459]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"108\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 5 (Reedy-Kan) Let $\\\\mathcal { D } = ( \\\\overline { { \\\\mathcal { D } } } , \\\\overleftarrow { \\\\mathcal { D } } )$ be a Reedy category, and M a model category. There is a model structure on \\\" $\\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { M } )$ where a map $f \\\\colon X \\\\longrightarrow Y$ is: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,466,788,522]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"108\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. a weak equivalence iff $X _ { i } \\\\longrightarrow Y _ { i }$ is a weak equivalence for all $i \\\\in \\\\mathcal { D }$ ; \\n2. a fibration iff $X _ { i } \\\\longrightarrow M _ { i } X \\\\times _ { { M _ { i } } Y } Y _ { i }$ is a fibration for all $i \\\\in \\\\mathcal { D }$ ; \\n3. a cofibration iff $X _ { i } \\\\coprod _ { L _ { i } X } L _ { i } Y \\\\longrightarrow Y _ { i }$ is a cofibration for all $i \\\\in \\\\mathcal { D }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,530,382,548]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"108\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For a proof, see [6, 15.3]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,549,881,620]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"108\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We call Reedy weak equivalences, Reedy fibrations and Reedy cofibrations the weak equivalences, fibrations and cofibrations of this model structure, to avoid confusion with other model structures on the same category. For example, the commutative square \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[441,638,558,719]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0009/b119f35d10a0556fdd3323de4232f8f55865a94f142599f82b6e560942972d12.jpg\",\n \"list_items\": null,\n \"page_idx\": \"108\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left. \\\\begin{array} { c } { { A \\\\longrightarrow B } } \\\\\\\\ { { g \\\\left\\\\downarrow \\\\begin{array} { c } { { } } \\\\\\\\ { { } } \\\\end{array} \\\\right\\\\downarrow } } \\\\\\\\ { { C \\\\longrightarrow D } } \\\\end{array} \\\\right.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,740,881,812]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"108\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"may be considered as a diagram over the Reedy poset of subsets of $\\\\{ 0 , 1 \\\\}$ . Then it is a Reedy fibrant diagram if and only if $A , B , C , D$ are fibrant objects; it is a Reedy cofibrant diagram if and only if $A$ is a cofibrant object and the three maps $f , g$ and $B \\\\ U _ { A } \\\\ C \\\\to D$ are cofibrations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,813,881,866]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"108\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We say that a map $f \\\\colon X \\\\to Y$ in $\\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { M } )$ is a pointwise weak equivalence (cofibration, fibration) if $f _ { i } : X _ { i } Y _ { i }$ is a weak equivalence (cofibration, fibration) for all $i \\\\in \\\\mathcal { D }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,562,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"108\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"108\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"103 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,879,142]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"109\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 10 Let $\\\\mathcal { D }$ be a Reedy category. Then the Reedy model structure on $\\\\begin{array} { r } { \\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } ) } \\\\end{array}$ ) coincides with the projective model structure if and only if every object is Reedy fibrant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,151,881,223]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"109\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof By definition the two model structures have the same weak equivalences. Let $\\\\varphi \\\\colon X Y$ be a morphism in $\\\\begin{array} { r } { \\\\mathrm { F u n } ( \\\\mathcal { D } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } ) } \\\\end{array}$ ). If $\\\\varphi$ is a Reedy fibration, then it is not difficult to prove it is a pointwise fibration ([6], 15.3.11); therefore if every pointwise fibration is a Reedy fibration then the two model structures coincide. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,225,881,372]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"109\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Assume that every object is Reedy fibrant and that $\\\\varphi _ { i } : X _ { i } \\\\longrightarrow Y _ { i }$ is a fibration in $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ for all $i \\\\in \\\\mathcal { D }$ ; we want to prove that $\\\\varphi$ is a Reedy fibration. We denote by $c ( \\\\mathbb { K } ) \\\\colon \\\\mathcal { D } \\\\to \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\le 0 }$ the constant diagram $i \\\\mapsto \\\\mathbb { K }$ , and by $K = c ( \\\\mathbb { K } ) \\\\times _ { Y } X$ the fibre product of $\\\\varphi$ and the initial morphism $c ( \\\\mathbb { K } ) \\\\to Y$ . Note that $M _ { i } c ( \\\\mathbb { K } )$ is concentrated in degree 0. Since the fibre product and the matching objects are both limits, they commute by Fubini’s theorem [9, Prop. 6.2.8], and we have $M _ { i } ( K ) =$ $M _ { i } ( c ( \\\\mathbb { K } ) \\\\times _ { Y } X ) \\\\cong M _ { i } c ( \\\\mathbb { K } ) \\\\times _ { M _ { i } Y } M _ { i } X$ . Thus we have a morphism of cartesian squares: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[339,388,656,561]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0967/0a233222d76221dd8756a48c257e25d3403127fe7ac6992cf0fefb4dc4c9581a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"109\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[118,584,881,711]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"109\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The map $\\\\varphi _ { i } \\\\colon X _ { i } \\\\to Y _ { i }$ induces $\\\\widehat { \\\\varphi _ { i } } \\\\colon M _ { i } X \\\\to M _ { i } Y$ ; let $Y _ { i } \\\\times _ { M _ { i } Y } M _ { i } X$ be the fibre product of $\\\\widehat { \\\\varphi _ { i } }$ and the natural map $Y _ { i } M _ { i } Y$ . We have to show that the map $X _ { i } $ $Y _ { i } \\\\times _ { M _ { i } Y } M _ { i } X$ is surjective in strictly negative degree. Let $( \\\\alpha , \\\\beta ) \\\\in Y _ { i } \\\\times _ { M _ { i } Y } M _ { i } X$ , with $\\\\overline { { \\\\alpha } } = \\\\overline { { \\\\beta } } < 0$ . Without loss of generality, because of the surjectivity of $\\\\varphi _ { i }$ , we can assume $\\\\alpha = 0$ , so $\\\\widehat { \\\\varphi _ { i } } ( \\\\beta ) = 0$ , which means $\\\\beta$ lifts to $( 0 , \\\\beta ) \\\\in M _ { i } K$ , and then to $K _ { i }$ , since the map $K _ { i } M _ { i } K$ is by hypothesis a fibration. By the commutativity of the above diagram, we have the thesis. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,711,881,767]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"109\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conversely, if the Reedy and projective model structures coincide, an object is Reedy fibrant if and only if it is point-wisely fibrant, and that is clearly true in CDGA≤0K . □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,774,410,792]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"109\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The following result is clear. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,801,879,858]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"109\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 11 The simplex category $N ( { \\\\mathcal { B } } )$ is a Reedy category, where the direct subcategory $\\\\overrightarrow { N ( \\\\mathcal { B } ) }$ is defined by injective maps and the inverse subcategory $\\\\overleftarrow { \\\\Lambda } ( \\\\mathcal { B } )$ by surjective maps. The same applies to its subcategories $N ( { \\\\mathcal { B } } ) _ { \\\\leq k }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[687,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"109\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Lepri and M. Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,54,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"109\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"110\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In particular, when $\\\\mathcal { B } = \\\\{ * \\\\}$ we recover the usual Reedy structure on $\\\\pmb { \\\\Delta }$ . By Theorem 5 we have the Reedy model structure on the category $\\\\mathrm { F u n } ( N ( \\\\mathcal { B } ) , { \\\\bf C D G A } _ { \\\\mathbb { K } } ^ { \\\\le 0 } )$ ; we show that the Reedy and projective model structures on this category coincide, using Lemma 10. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,168,876,207]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"110\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 6 Every object in $\\\\mathrm { F u n } ( N ( \\\\mathbf { \\\\mathcal { B } } ) _ { \\\\leq k } , \\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 } )$ is Reedy fibrant, and so the Reedy model structure coincides with the projective model structure. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,213,881,270]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"110\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof In view of the definition of fibrations in $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ , it is sufficient to show that for every $X \\\\in \\\\mathrm { F u n } ( N ( \\\\mathcal { B } ) _ { \\\\leq k } , \\\\mathbf { G r p } )$ the map $X _ { \\\\alpha } M _ { \\\\alpha } X$ is surjective for all $\\\\alpha \\\\in N ( \\\\mathcal { B } ) _ { \\\\leq k }$ . Fix $n \\\\leq k$ and let $\\\\alpha ^ { ( m ) } \\\\in N ( { \\\\mathcal { B } } ) _ { n }$ , \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[337,286,658,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0420/1c84b992339e1bc40f24b741ee8ffaa8593f7d33ddbac5c5466a2b84f6dd713f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"110\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\alpha ^ { ( m ) } = [ x _ { 0 } \\\\xrightarrow { h _ { 1 } } x _ { 1 } \\\\xrightarrow { h _ { 2 } } \\\\cdot \\\\cdot \\\\cdot \\\\xrightarrow { h _ { n } } x _ { n } ] ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,328,881,382]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"110\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $m$ is the number of morphisms $h _ { i }$ equal to the identity, $0 \\\\leq m \\\\leq n$ . If $m = 0$ there is nothing to prove, because the matching category of $\\\\alpha ^ { ( 0 ) }$ is empty, so every $X _ { \\\\alpha ^ { ( 0 ) } }$ is automatically fibrant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,385,410,400]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"110\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In case $n = m = 1$ , we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[432,418,562,490]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0643/67dc1dfa0a85b73e7a58f8d8de6305dd1ff338e06d88bc40573a9078fa320524.jpg\",\n \"list_items\": null,\n \"page_idx\": \"110\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { c } { { [ a a ] } } \\\\\\\\ { { \\\\nonumber } } \\\\\\\\ { { \\\\delta _ { 0 } } } \\\\end{array} \\\\overset { \\\\triangledown } { ( \\\\begin{array} { l } { { \\\\overline { { { \\\\sigma _ { 0 } } } } } } \\\\\\\\ { { \\\\overline { { { a } } } } } \\\\end{array} ) } _ { \\\\delta _ { 1 } }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,510,876,546]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"110\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"with $\\\\sigma _ { 0 } \\\\delta _ { 0 } = \\\\sigma _ { 0 } \\\\delta _ { 1 } = \\\\mathrm { I d }$ by the cosimplicial identities, so ${ \\\\cal X } ( \\\\sigma _ { 0 } ) \\\\colon X _ { \\\\alpha ^ { ( 1 ) } } \\\\to M _ { \\\\alpha ^ { ( 1 ) } } X$ has a section and hence is surjective. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,546,881,647]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"110\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In general, assume $n \\\\geq 2$ and $0 < m \\\\le n$ ; let ${ \\\\cal I } = \\\\{ i \\\\in \\\\{ 0 , \\\\cdot \\\\cdot \\\\cdot n - 1 \\\\} \\\\mid h _ { i + 1 } = \\\\mathrm { I d } \\\\}$ , $| I | = m$ . For every $i \\\\in I$ we have a degeneracy map $\\\\sigma _ { i } \\\\colon \\\\alpha ^ { ( m ) } \\\\alpha _ { i } ^ { ( m - 1 ) }$ , where the α(m−1)i a re suitable objects in $N ( { \\\\mathcal { B } } ) _ { n - 1 }$ . From each $\\\\alpha _ { i } ^ { ( m - 1 ) }$ there are $m - 1$ degeneracy maps to other objects $\\\\alpha _ { i , j } ^ { ( m - 2 ) } \\\\in N ( \\\\mathcal { B } ) _ { n - 2 }$ , and so on. For example, for $n = 3 , m = 2 , I = \\\\{ 0 , 2 \\\\}$ : \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[236,662,763,789]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0199/cf2a827d638e4153098586550bf73920b82f0d8c38ac90862bee8e089819ddfd.jpg\",\n \"list_items\": null,\n \"page_idx\": \"110\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n[ a b \\\\underbrace { b } _ { \\\\sigma _ { 1 } } ^ { [ a a b b ] } \\\\underbrace { \\\\vphantom { \\\\int } _ { [ a b ] } ^ { \\\\sigma _ { 2 } } } _ { [ a b ] } \\\\underbrace { \\\\vphantom { \\\\int } _ { \\\\sigma _ { 1 } } ^ { \\\\sigma _ { 2 } } } _ { [ a b ] } \\\\nonumber\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,801,881,863]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"110\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For every map $\\\\sigma _ { i } \\\\colon \\\\alpha ^ { ( m ) } \\\\to \\\\alpha _ { i } ^ { ( m - 1 ) }$ we also have two sections $\\\\delta _ { i } , \\\\delta _ { i + 1 } \\\\colon \\\\alpha _ { i } ^ { ( m - 1 ) } \\\\to$ $\\\\alpha ^ { ( m ) }$ , so using an identical computation to Lemma 6, we have that $X _ { \\\\alpha ^ { ( m ) } }$ maps surjectively into \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[177,878,815,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0628/df863ab338a6fcdb179f9473a35b8b64dd69cd6787cbbea4b8a5fa34db5f3cb0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"110\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nV : = \\\\{ x _ { l } \\\\in \\\\alpha _ { l } ^ { ( m - 1 ) } , \\\\ l \\\\in I \\\\ | \\\\ \\\\sigma _ { i - 1 } x _ { j } = \\\\sigma _ { j } x _ { i } \\\\ \\\\mathrm { ~ f o r ~ e v e r y } \\\\ i , j \\\\in I , \\\\ i > j \\\\} .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,51,562,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"110\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"110\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"105 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,881,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"111\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is clear that the map $X _ { \\\\alpha ^ { ( m ) } } V$ factors through the inclusion $M _ { \\\\alpha ^ { ( m ) } } X V$ , so $X _ { \\\\alpha ^ { ( m ) } }$ also maps surjectively into $M _ { \\\\alpha ^ { ( m ) } } X$ , and we have the thesis. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[136,135,876,153]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"111\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally, the following corollary is an immediate consequence of the above results. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,160,881,252]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"111\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 1 Let $\\\\mathcal { B }$ be a small category and $S _ { \\\\bullet } \\\\colon { \\\\mathcal { B } } \\\\to \\\\mathbf { A l g } _ { \\\\mathbb { K } }$ a diagram of unitary commutative algebras. Let $\\\\epsilon \\\\colon N ( \\\\mathcal { B } ) _ { \\\\leq k } \\\\to \\\\mathcal { B }$ be the functor defined in Definition 15 for some $k \\\\_ \\\\geq \\\\ 2$ and let $R _ { \\\\bullet } ~ ~ S _ { \\\\bullet } ~ \\\\circ ~ \\\\epsilon$ be a Reedy cofibrant replacement in $\\\\mathrm { F u n } ( N ( { \\\\mathcal { B } } ) _ { \\\\leq k }$ , $\\\\mathbf { C D G A } _ { \\\\mathbb { K } } ^ { \\\\leq 0 }$ ). Then the DG-Lie algebra $\\\\mathrm { D e r } _ { \\\\mathbb { K } } ^ { * } ( R _ { \\\\bullet } , R _ { \\\\bullet } )$ controls the deformations of $S _ { \\\\bullet }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,259,881,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"111\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An example of deformation problem which is naturally encoded by a diagram over a non-Reedy category is the case of deformations of pairs (algebra, idempotent), cf. [16, Example 5.1]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,315,876,351]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"111\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $e \\\\colon R R$ be an idempotent morphism of an algebra $R \\\\in \\\\mathbf { A l g } _ { \\\\mathbb { K } }$ , then the deformations of the pair $( R , e )$ can be interpreted as the deformations of the diagram \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[428,366,569,397]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0508/412cb5667aaef80639436b46a455bd54d1bc901e121c5b903b143351828e7a63.jpg\",\n \"list_items\": null,\n \"page_idx\": \"111\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nR _ { \\\\bullet } \\\\colon \\\\textit { \\\\textbf { R } } ^ { \\\\sim } \\\\bigcup \\\\mathfrak { e }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,415,881,507]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"111\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"over the (non-Reedy) category $\\\\mathcal { B }$ that has one object $\\\\bullet$ and two morphisms $\\\\{ \\\\mathrm { I d } , \\\\alpha \\\\}$ , with $\\\\alpha ^ { 2 } = \\\\alpha$ . By Proposition 8, the diagrams $R$ and $\\\\epsilon ^ { * } R _ { \\\\bullet } \\\\in \\\\mathrm { F u n } ( N ( \\\\mathcal { B } ) _ { \\\\leq 2 }$ , $\\\\mathbf { A l g } _ { \\\\mathbb { K } } )$ have the same deformation theory. Moreover, since $\\\\epsilon ^ { * } R _ { \\\\bullet } \\\\in \\\\mathrm { F u n } ^ { \\\\star } ( N ( \\\\mathcal { B } ) _ { \\\\leq 2 } ^ { - } , \\\\mathbf { A l g _ { \\\\mathbb { K } } } )$ , it is easy to see that the diagram $\\\\epsilon ^ { * } R$ has the same deformation theory of the diagram \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[225,525,772,612]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0807/4f2d4fbcea69a6ce67589dd1f7d079212a7edd9f9a1a96e6cf4f2bee8a48dd26.jpg\",\n \"list_items\": null,\n \"page_idx\": \"111\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[118,635,760,653]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"111\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"of algebras over the following (direct Reedy) subcategory of $N ( { \\\\mathcal { B } } ) _ { \\\\leq 2 }$ : \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[164,669,831,762]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0225/78887379985eb22ec2d9948ac4d858029d19c6b94468e8e1e5e59faa08576c95.jpg\",\n \"list_items\": null,\n \"page_idx\": \"111\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { \\\\overbrace { \\\\phantom { \\\\left( \\\\frac { \\\\partial \\\\theta } { \\\\partial \\\\theta } \\\\right) ^ { 2 } } \\\\left[ \\\\bullet \\\\xrightarrow { \\\\alpha } \\\\bullet \\\\right] \\\\xrightarrow [ \\\\delta _ { 0 } ] { \\\\delta _ { 1 } } \\\\left[ \\\\bullet \\\\xrightarrow [ \\\\delta ] { \\\\delta _ { 2 } } \\\\bullet \\\\xrightarrow [ \\\\delta ] { \\\\alpha } \\\\bullet \\\\right] \\\\delta _ { 0 } ^ { 2 } \\\\left[ \\\\delta _ { 1 } = \\\\delta _ { 2 } \\\\delta _ { 0 } \\\\right] , } ^ { \\\\delta _ { 1 } } \\\\qquad \\\\delta _ { 0 } ^ { 2 } = \\\\delta _ { 1 } \\\\delta _ { 0 } , } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,785,879,831]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"111\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acknowledgements Both authors thank Francesco Meazzini for useful discussions about the topics of this paper, and the anonymous referee for useful comments and remarks. M.M. is partially supported by Italian MIUR under PRIN project 2017YRA3LK “Moduli and Lie theory”. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[687,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"111\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E. Lepri and M. Manetti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,52,152,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"111\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"106 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,248,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"112\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,121,879,737]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"112\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Bousfield, A., Gugenheim, V.: On PL De Rham Theory and Rational Homotopy Type. Memoirs of the American Mathematical Society, vol. 8, no. 179. American Mathematical Society, Providence (1976) \\n2. Bousfield, A.K., Kan, D.M.: Homotopy Limits, Completions and Localizations. Springer Lecture Notes in Mathematics, vol. 304. Springer, New York (1972) \\n3. Chachólski, W., Scherer, J.: Homotopy Theory of Diagrams. Memoirs of the American Mathematical Society, no. 736. American Mathematical Society, Providence (2002) \\n4. Gelfand, S.I., Manin, Y.I.: Methods of Homological Algebra. Springer Monographs in Mathematics. Springer, Berlin (2003) \\n5. Hinich, V.: Deformations of homotopy algebras. Commun. Algebra 32(2), 473–494 (2004) \\n6. Hirschhorn, P.S.: Model Categories and their Localizations. Mathematical Surveys and Monographs, vol. 99. American Mathematical Society, Providence (2003) \\n7. Hovey, M.: Model Categories. Mathematical Surveys and Monographs, vol. 63. American Mathematical Society, Providence (1999) \\n8. Jardine, J.F.: A closed model structure for differential graded algebras. In: Cyclic Cohomology and Noncommutative Geometry (Waterloo, ON, 1995). Fields Institute Communications, vol. 17, pp. 55–58. American Mathematical Society, Providence (1997) \\n9. Leinster, T.: Basic Category Theory. Cambridge Studies in Advanced Mathematics, vol. 143. Cambridge University Press, Cambridge (2014) \\n10. Lurie, J.: Higher Topos Theory. Annals of Mathematical Studies, vol. 170. Princeton University Press, Princeton (2009) \\n11. Lurie, J.: Higher Algebra. Available on J. Lurie’s web page (2014). Preprint \\n12. Manetti, M.: Deformation theory via differential graded Lie algebras. In: Seminari di Geometria Algebrica 1998–1999. Scuola Normale Superiore, Pisa (1999). arXiv: math.AG/0507284 \\n13. Manetti, M.: Deformation of singularities via differential graded Lie algebras. In: Seminar Notes, Roma (2001). www1.mat.uniroma1.it/people/manetti/dispense/defosing.pdf \\n14. Manetti, M.: Differential graded Lie algebras and formal deformation theory. In: Algebraic Geometry: Seattle 2005. Proceedings of Symposia in Pure Mathematics, vol. 80, pp. 785–810 (2009) \\n15. Manetti, M., Meazzini, F.: Formal deformation theory in left-proper model categories. New York J. Math. 25, 1259–1311 (2019) \\n16. Manetti, M., Meazzini, F.: Deformations of algebraic schemes via Reedy-Palamodov cofibrant resolutions. Indag. Math. (N.S.) (2019). https://doi.org/10.1016/j.indag.2019.08.007 \\n17. Palamodov, V.P.: Deformations of complex spaces. Usp. Mat. Nauk 31(3), 129–194 (1976). English translation: Russian Math. Surveys 31, 129–197 (1976) \\n18. Quillen, D.: Homotopical Algebra. Lecture Notes in Mathematics, vol. 43. Springer, New York (1967) \\n19. Sernesi, E.: Deformations of Algebraic Schemes. Grundlehren der mathematischen Wissenschaften, vol. 334. Springer, New York (2006) \\n20. Tate, J.: Homology of Noetherian rings and local rings. Illinois J. Math. 1, 14–27 (1957) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,562,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"112\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On Deformations of Diagrams of Commutative Algebras \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"112\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,694,141]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"113\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,175,537,192]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"113\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"César Lozano Huerta and Alex Massarenti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,286,881,412]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"113\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract Inspired by the Weak Lefschetz Principle, we study when a smooth projective variety fully determines the birational geometry of some of its subvarieties. In particular, we consider the natural embedding of the space of complete quadrics into the space of complete collineations and we observe that their birational geometry, from the point of view of Mori theory, fully determines each other. When two varieties are related in this way, we call them birational twins. We explore this notion and its various flavors for other embeddings between Mori dream spaces. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,432,822,466]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"113\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords Mori dream spaces $\\\\cdot$ Cox rings $\\\\cdot$ Spherical varieties $\\\\cdot$ Complete collineations and quadrics \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,486,840,521]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"113\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2010 Mathematics Subject Classification Primary 14E30; Secondary 14J45, 14N05, 14E07, 14M27 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,558,309,579]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"113\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,603,881,710]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"113\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The celebrated Lefschetz hyperplane theorem asserts that if there is a smooth complex projective variety $Y$ with the inclusion of a smooth ample hypersurface i $X Y$ , then the ambient variety $Y$ often fully determines many of the topological invariants of $X$ . Such invariants include homology, cohomology (compatible with Hodge structures) and homotopy groups among others. This phenomenon leads to the question: what are some algebro-geometric invariants to which a result of this type applies? In other words, we are asking if the Weak Lefschetz Principle holds beyond topological invariants. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,897,692,941]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"113\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_7 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[804,91,876,139]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"113\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,899,876,909]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"113\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[116,743,867,879]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"113\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. L. Huerta Universidad Nacional Autónoma de México, Instituto de Matemáticas, Oaxaca, Mexico e-mail: lozano@im.unam.mx A. Massarenti $( \\\\boxtimes )$ ) Dipartimento di Matematica e Informatica, Università di Ferrara, Ferrara, Italy Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Rio de Janeiro, Brazil e-mail: alex.massarenti@unife.it; alexmassarenti@id.uff.br \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,88,876,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"114\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,126,879,214]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"114\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The purpose of this paper is to investigate the Weak Lefschetz Principle in the context of birational geometry. In other words, if we consider two varieties $X$ , Y as well as an embedding between them $i : X \\\\to Y$ , then we ask about the birational invariants $X$ which are determined by those of $Y$ . Such invariants may include effective and ample cones, or finer information as we now explain. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,216,881,466]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"114\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let us recall that a normal $\\\\mathbb { Q }$ -factorial projective variety $X$ with finitely generated Picard group is a Mori dream space if and only if its Cox ring is finitely generated [11, Proposition 2.9]. The birational geometry of these varieties is encoded in this ring and we may ask whether the Weak Lefschetz Principle applies to it. This question has recently been studied in the case of hypersurfaces [10, 12, 20, 22], and for Mori dream spaces of Picard rank two [13]. In addition to considering the Cox ring, let us recall that the pseudo-effective cone ${ \\\\overline { { \\\\operatorname { E f f } } } } ( X )$ of a projective variety $X$ with irregularity zero, such as a Mori dream space, can be decomposed into chambers depending on the stable base locus of the corresponding linear series. This decomposition is called stable base locus decomposition and in general it is coarser than the so-called Mori chamber decomposition; which encodes the isomorphism type of the birational models of $X$ . Mori dream spaces and these decompositions have been widely studied by many authors recently, for instance see [1, 7, 17, 18] and the references therein. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,468,879,540]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"114\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We may also ask whether the previous two decompositions satisfy the Weak Lefschetz Principle. We then summarize the previous discussion in the following definition. Let us denote by SBLD $( X )$ the stable base locus decomposition of ${ \\\\overline { { \\\\operatorname { E f f } } } } ( X )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,549,881,602]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"114\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 1.1 Let $X , Y$ be $\\\\mathbb { Q }$ -factorial projective varieties and $i : X \\\\to Y$ be an embedding. These varieties are said Lefschetz divisorially equivalent if the pull-back $i ^ { * } : \\\\operatorname { P i c } ( Y ) \\\\to \\\\operatorname { P i c } ( X )$ induces an isomorphism such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[173,618,822,639]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0048/dfcee5d0e3152c3baf93195dbf27a5663215d2ba2d002dc66aa66ab67070daf3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"114\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\ni ^ { * } \\\\operatorname { E f f } ( Y ) = \\\\operatorname { E f f } ( X ) , \\\\quad i ^ { * } \\\\operatorname { M o v } ( Y ) = \\\\operatorname { M o v } ( X ) , \\\\quad i ^ { * } \\\\operatorname { N e f } ( Y ) = \\\\operatorname { N e f } ( X )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,656,879,711]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"114\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Furthermore, $X$ and $Y$ are said strongly Lefschetz divisorially equivalent if they are Lefschetz divisorially equivalent and in addition $i ^ { * } \\\\operatorname { S B L D } ( Y ) \\\\ : = \\\\ : \\\\operatorname { S B L D } ( X )$ , with $h ^ { 1 } ( X , { \\\\mathcal { O } } _ { X } ) = h ^ { 1 } ( Y , { \\\\mathcal { O } } _ { Y } ) { = } 0 .$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,719,881,845]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"114\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is an immediate consequence of the Lefschetz hyperplane theorem that a smooth variety $Y$ , of Picard rank one and dimension at least four, is Lefschetz divisorially equivalent to any of its smooth ample divisors. If the assumption that the Picard rank is one is removed in this case, B. Hassett et al. [10] exhibited an example of a variety and an ample divisor $D \\\\subset Y$ whose Picard groups are isomorphic by the Lefschetz hyperplane theorem, but such that the nef cone is not preserved; hence $D$ and $Y$ are not Lefschetz divisorially equivalent. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,846,876,881]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"114\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The previous definition is not restricted to subvarieties of codimension one. In fact, the results in this paper involve classical spaces and subvarieties of high codimension, such that the birational geometry of both objects can now be linked using Definition 1.1. Let us recall those spaces so we can state our results precisely. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[633,54,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"114\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. L. Huerta and A. Massarenti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,152,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"114\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"110 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,90,876,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"115\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,127,881,484]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"115\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $V$ , W be two $K$ -vector spaces of dimensions $n + 1$ and $m + 1$ , respectively, with $n ~ \\\\leqslant m$ . The field $K$ is algebraically closed of characteristic zero. We will denote by ${ \\\\mathcal { X } } ( n , m )$ the space of complete collineations $V W$ and by $\\\\boldsymbol { \\\\mathcal { Q } } ( \\\\boldsymbol { n } )$ the space of complete $( n - 1 )$ -dimensional quadrics of $V$ . In [23, 24], I. Vainsencher showed that these spaces can be understood as sequences of blow-ups along the subvariety parametrizing rank one matrices and the strict transforms of its secant varieties. Such blowups are central characters of this paper, so let us be more precise about them. Given an irreducible and reduced non-degenerate variety $X \\\\subset \\\\mathbb { P } ^ { N }$ , and a positive integer $h \\\\leqslant N$ , the $h$ -secant variety $\\\\mathbb { S } e c _ { h } ( X )$ of $X$ is the subvariety of $\\\\mathbb { P } ^ { N }$ obtained as the closure of the union of all $( h - 1 )$ -planes spanned by $h$ general points of $X$ . Spaces of matrices and symmetric matrices admit a natural stratification dictated by the rank and observe that a general point of the $h$ -secant variety of a Segre, or a Veronese, corresponds to a matrix of rank $h$ . More precisely, let $\\\\mathbb { P } ^ { N }$ be the projective space parametrizing $( n + 1 ) \\\\times ( n + 1 )$ matrices modulo scalars, $\\\\mathbb { P } ^ { N _ { + } }$ the subspace of symmetric matrices, $\\\\mathcal { S } \\\\subset \\\\mathbb { P } ^ { N }$ the Segre variety, and $\\\\mathcal { V } \\\\subset \\\\mathbb { P } ^ { N _ { + } }$ the Veronese variety. Set $\\\\mathcal { X } ( n ) : = \\\\mathcal { X } ( n , n )$ and denote by $\\\\mathcal { X } ( n ) _ { i } , \\\\mathcal { Q } ( n ) _ { i }$ respectively the varieties obtained at the step $i$ in Vainsencher’s constructions 3.4; which roughly says that $\\\\mathcal { X } ( n ) _ { i }$ , ${ \\\\mathcal { Q } } ( n ) _ { i }$ are the blow-ups respectively of $\\\\mathbb P ^ { N }$ and $\\\\mathbb { P } ^ { N _ { + } }$ along all the $j$ -secant varieties of $s$ and $\\\\nu$ for $j \\\\ \\\\leq \\\\ i$ . Since $\\\\mathbb { S } e c _ { h } ( \\\\mathcal { V } ) = \\\\mathbb { S } e c _ { h } ( \\\\mathcal { S } ) \\\\cap \\\\mathbb { P } ^ { N _ { + } }$ , the natural inclusion $\\\\mathbb { P } ^ { N _ { + } } \\\\hookrightarrow \\\\mathbb { P } ^ { N }$ lifts to an embedding ${ \\\\mathcal { Q } } ( n ) _ { i } \\\\hookrightarrow { \\\\mathcal { X } } ( n ) _ { i }$ for any $i$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,486,879,557]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"115\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"These families of spaces $\\\\mathcal { X } ( n ) _ { i }$ and $\\\\mathcal { Q } ( n ) _ { i }$ are very different in may aspects, for instance their dimensions are different. However, when it comes to certain birational invariants they are indistinguishable. Indeed, by Proposition 3.12 and Lemma 3.18, we have our first result. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,567,879,638]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"115\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem A The varieties $\\\\mathcal { X } ( n ) _ { i }$ and ${ \\\\mathcal { Q } } ( n ) _ { i }$ are Lefschetz divisorially equivalent via the usual embedding for any i. In particular, the spaces of complete collineations ${ \\\\mathcal { X } } ( n )$ and quadrics $\\\\mathcal { Q } ( n )$ are Lefschetz divisorially equivalent via the usual embedding. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,648,881,773]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"115\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A more thorough version of the Weak Lefschetz Principle in our context would be the following: a small modification of the ambient variety $Y$ restricts to a small modification, possibly an isomorphism, of the subvariety $X$ , and all the small modifications of $X$ can be obtained in this way. This is stronger than determining invariants and it is the content of the following definition. We will test this definition on some Mori dream spaces. The Mori chamber decomposition of Eff $( X )$ , of a Mori dream space $X$ , will be denoted by $\\\\operatorname { M C D } ( X )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,783,879,855]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"115\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 1.2 Let $X , Y$ be as in the previous definition, we say that they are birational twins if they are Lefschetz divisorially equivalent Mori dream spaces and in addition $i ^ { * } \\\\mathsf { M C D } ( Y ) \\\\ : = \\\\ : \\\\mathbf { M C D } ( X )$ . Also, they are said to be strong birational twins if they are birational twins and in addition $i ^ { * } \\\\operatorname { S B L D } ( Y ) = \\\\operatorname { S B L D } ( X )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,694,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"115\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,874,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"115\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,879,142]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"116\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our next result shows that the relationship between the birational geometry of complete collineations and complete quadrics, in some cases, goes deeper than Theorem A. The following result follows from Theorem 3.20 and Corollary 3.22. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,153,874,187]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"116\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem B The varieties $\\\\mathcal { X } ( n ) _ { 3 }$ and $\\\\mathcal { Q } ( n ) _ { 3 }$ are birational twins for any $n \\\\geqslant 1$ Furthermore, $\\\\mathcal { X } ( 3 )$ and $\\\\mathcal { Q } ( 3 )$ are strong birational twins. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,198,881,286]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"116\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In general, the spaces $\\\\mathcal { X } ( n ) _ { i } , \\\\mathcal { Q } ( n ) _ { i }$ are Mori dream spaces for any $i$ ; we can actually count the number of chambers in the Mori chamber decomposition in various cases. Furthermore, the ample, movable and effective cones are preserved by Theorem A. However, we do not know if the structure of the Mori chambers is also preserved. In other words, if Theorem B holds in general. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,297,829,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"116\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Question 1.3 Are $\\\\mathcal { X } ( n ) _ { i }$ and ${ \\\\mathcal { Q } } ( n ) _ { i }$ birational twins for $n \\\\geqslant 4$ and any $i \\\\geq 4 2$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,324,881,484]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"116\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We organize this paper as follows. In Sect. 2, we discuss Definitions 1.1 and 1.2 and the relations among the various flavors of these definitions. In particular, Theorem 2.6 provides examples of Mori dream spaces that are strongly Lefschetz divisorially equivalent but not birational twins, and of Mori dream spaces that are birational twins but not strong birational twins. In Sect. 3, we introduce the spaces of complete collineations and quadrics, and we study their birational geometry from a Mori theoretic viewpoint. Finally, in Sect. 4 we recall the modular description of the unique flip of $\\\\mathcal { Q } ( 3 )$ given in [14, Section 5.2], and based on it, we give a conjectural description of the unique flip of $\\\\mathcal { X } ( 3 )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,530,658,572]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"116\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 Lefschetz Divisorially Equivalent Varieties and Birational Twins \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,596,879,650]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"116\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This section contains preliminaries and explores various aspects of Definitions 1.1 and 1.2. In particular, Theorem 2.6 and Proposition 2.9 are the main results of this section. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,651,881,794]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"116\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $X$ be a normal projective variety over an algebraically closed field of characteristic zero. We denote by $N ^ { 1 } ( X )$ the real vector space of $\\\\mathbb { R }$ -Cartier divisors modulo numerical equivalence. The nef cone of $X$ is the closed convex cone $\\\\mathsf { N e f } ( X ) \\\\subset N ^ { 1 } ( X )$ generated by classes of nef divisors. The movable cone of $X$ is the convex cone $\\\\mathsf { M o v } ( X ) \\\\subset N ^ { 1 } ( X )$ generated by classes of movable divisors; these are Cartier divisors whose stable base locus has codimension at least two in $X$ . The effective cone of $X$ is the convex cone $\\\\operatorname { E f f } ( X ) \\\\subset N ^ { 1 } ( X )$ generated by classes of effective divisors. We have inclusions $\\\\operatorname { N e f } ( X ) \\\\subset { \\\\overline { { \\\\operatorname { M o v } ( X ) } } } \\\\subset { \\\\overline { { \\\\operatorname { E f f } ( X ) } } }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,795,881,848]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"116\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We will denote by $N _ { 1 } ( X )$ the real vector space of numerical equivalence classes of 1-cycles on $X$ . The closure of the cone in $N _ { 1 } ( X )$ generated by the classes of irreducible curves in $X$ is called the Mori cone of $X$ , we will denote it by $\\\\operatorname { N E } ( X )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,849,879,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"116\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A class $[ C ] \\\\in N _ { 1 } ( X )$ is called moving if the curves in $X$ of class $[ C ]$ cover a dense open subset of $X$ . The closure of the cone in $N _ { 1 } ( X )$ generated by classes of \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[633,54,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"116\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. L. Huerta and A. Massarenti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,54,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"116\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"112 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,88,879,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"117\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"moving curves in $X$ is called the moving cone of $X$ and we will denote it by mov $( X )$ . \\nWe refer to [9, Chapter 1] for a comprehensive treatment of these topics. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,126,881,252]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"117\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We say that a birational map $f : X \\\\to X ^ { \\\\prime }$ to a normal projective variety $X ^ { \\\\prime }$ is a birational contraction if its inverse does not contract any divisor. We say that it is a small $\\\\mathbb { Q }$ -factorial modification if $X ^ { \\\\prime }$ is $\\\\mathbb { Q }$ -factorial and $f$ is an isomorphism in codimension one. If $f \\\\ : \\\\ X \\\\quad - \\\\quad X ^ { \\\\prime }$ is a small $\\\\mathbb { Q }$ -factorial modification, then the natural pull-back map $f ^ { * } ~ : ~ N ^ { 1 } ( X ^ { \\\\prime } ) ~ ~ N ^ { 1 } ( X )$ sends $\\\\operatorname { M o v } ( X ^ { \\\\prime } )$ and $\\\\operatorname { E f f } ( X ^ { \\\\prime } )$ isomorphically onto $\\\\operatorname { M o v } ( X )$ and $\\\\operatorname { E f f } ( X )$ , respectively. In particular, we have $f ^ { * } ( \\\\operatorname { N e f } ( { \\\\bar { X ^ { \\\\prime } } } ) ) \\\\subset { \\\\bar { \\\\operatorname { M o v } } } ( X )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,259,879,297]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"117\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 2.1 A normal projective $\\\\mathbb { Q }$ -factorial variety $X$ is called a Mori dream space if the following conditions hold: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,304,881,396]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"117\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"– Pic# $( X )$ is finitely generated, or equivalently $h ^ { 1 } ( X , { \\\\mathcal { O } } _ { X } ) = 0$ , $- \\\\mathrm { ~ \\\\ N e f ~ } ( X )$ is generated by the classes of finitely many semi-ample divisors, there is a finite collection of small $\\\\mathbb { Q }$ -factorial modifications $f _ { i } : X \\\\to X _ { i }$ , such that each $X _ { i }$ satisfies the second condition above, and Mov $\\\\begin{array} { r l } { ( X ) } & { { } = } \\\\end{array}$ $\\\\textstyle \\\\bigcup _ { i } \\\\ f _ { i } ^ { * } ( { \\\\mathrm { N e f } } ( X _ { i } ) )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,405,881,493]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"117\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By [3, Corollary 1.3.2] smooth Fano varieties are Mori dream spaces. In fact, there is a larger class of varieties called log Fano varieties which are Mori dream spaces as well. By work of M. Brion [5] we have that $\\\\mathbb { Q }$ -factorial spherical varieties are Mori dream spaces. An alternative proof of this result can be found in [21, Section 4]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,495,881,603]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"117\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The collection of all faces of all cones $f _ { i } ^ { * } ( \\\\mathrm { N e f } \\\\left( X _ { i } \\\\right) )$ in Definition 2.1 forms a fan which is supported on $\\\\operatorname { M o v } ( X )$ . If two maximal cones of this fan, say $f _ { i } ^ { * } ( \\\\mathrm { N e f } \\\\left( X _ { i } \\\\right) )$ and $f _ { j } ^ { * } ( \\\\mathrm { N e f } ( X _ { j } ) )$ , meet along a facet, then there exist a normal projective variety $Y$ , a small modification $\\\\varphi : X _ { i } \\\\ \\\\to \\\\ Y _ { j }$ , and $h _ { i } : X _ { i } \\\\to Y$ and $h _ { j } : X _ { j } \\\\to Y$ small birational morphism of relative Picard number one such that $h _ { j } \\\\circ \\\\varphi = h _ { i }$ . The fan structure on $\\\\operatorname { M o v } ( X )$ can be extended to a fan supported on $\\\\operatorname { E f f } ( X )$ as follows. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,611,881,755]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"117\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 2.2 Let $X$ be a Mori dream space. We describe a fan structure on the effective cone $\\\\operatorname { E f f } ( X )$ , called the Mori chamber decomposition. We refer to [11, Proposition 1.11] and [19, Section 2.2] for details. There are finitely many birational\\\\$ \\u0003 \\u0004 % contractions from $X$ to Mori dream spaces, denoted by $g _ { i } \\\\ : \\\\ X \\\\to \\\\ Y _ { i }$ . The set $\\\\operatorname { E x c } ( g _ { i } )$ of exceptional prime divisors of $g _ { i }$ has cardinality $\\\\rho ( X / Y _ { i } ) = \\\\rho ( X ) -$ $\\\\rho ( Y _ { i } )$ . The maximal cones $\\\\mathcal { C }$ of the Mori chamber decomposition of $\\\\operatorname { E f f } ( X )$ are of the form: $\\\\begin{array} { r l } { \\\\mathcal { C } _ { i } } & { = \\\\Big \\\\langle g _ { i } ^ { * } \\\\big ( \\\\mathrm { N e f } ( Y _ { i } ) \\\\big ) , \\\\mathrm { E x c } ( g _ { i } ) \\\\Big \\\\rangle } \\\\end{array}$ . We call $\\\\mathcal { C } _ { i }$ or its interior $\\\\boldsymbol { \\\\mathcal { C } } _ { i } ^ { \\\\circ }$ a maximal chamber of $\\\\operatorname { E f f } ( X )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,764,881,819]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"117\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 2.3 Let $X$ be a normal projective variety with finitely generated divisor class group $\\\\mathrm { C l } ( X ) : = \\\\mathrm { W D i v } ( X ) / \\\\mathrm { P D i v } ( X )$ , in particular $h ^ { 1 } ( X , { \\\\mathcal { O } } _ { X } ) = 0$ The $C o x$ sheaf and Cox ring of $X$ are defined as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[248,833,749,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0798/561115f984aaf982d26fe361a71e991727dcf985a59c5484724f14e8b5197554.jpg\",\n \"list_items\": null,\n \"page_idx\": \"117\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n{ \\\\mathcal { R } } : = \\\\bigoplus _ { [ D ] \\\\in \\\\operatorname { C l } ( X ) } { \\\\mathcal { O } } _ { X } ( D ) \\\\qquad \\\\operatorname { C o x } ( X ) : = \\\\Gamma ( X , { \\\\mathcal { R } } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,52,694,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"117\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"117\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"113 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"118\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recall that $\\\\mathcal { R }$ is a sheaf of $\\\\operatorname { C l } ( X )$ -graded ${ \\\\mathcal { O } } _ { X }$ -algebras, whose multiplication maps are discussed in [2, Section 1.4]. In case the divisor class group is torsionfree, one can just take the direct sum over a subgroup of $\\\\mathrm { W D i v } ( X )$ , isomorphic to $\\\\operatorname { C l } ( X )$ via the quotient map, getting immediately a sheaf of ${ \\\\mathcal { O } } _ { X }$ -algebras. Denote by $\\\\widehat { X }$ the relative spectrum of $\\\\mathcal { R }$ and by $\\\\overline { { X } }$ the spectrum of $\\\\operatorname { C o x } ( X )$ . The $\\\\operatorname { C l } ( X )$ - grading induces an action of the quasitorus $H _ { X } : = \\\\operatorname { S p e c } \\\\mathbb { C } [ \\\\operatorname { C l } ( X ) ]$ on both spaces. The inclusion $\\\\mathcal { O } _ { X } \\\\mathcal { R }$ induces a good quotient $p _ { X } \\\\colon { \\\\widehat { X } } \\\\to X$ with respect to this action. Summarizing, we have the following diagram \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[448,247,548,321]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0180/d08fd5d27f24c3b014cdebeae8dd36ac6afc3495c5ff101038110b7bd951d57a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"118\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n{ \\\\underset { X } { \\\\widehat { X } } } \\\\subseteq { \\\\overline { { X } } }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,340,881,414]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"118\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and we will refer to it as the Cox construction of $X$ . In case $\\\\operatorname { C o x } ( X )$ is a finitely generated algebra, the complement of $\\\\widehat { X }$ in the affine variety $\\\\overline { { X } }$ has codimension $\\\\geqslant 2$ . This subvariety is the irrelevant locus and its defining ideal is the irrelevant ideal $\\\\mathcal { I } _ { \\\\mathrm { i r r } } ( X ) \\\\subseteq \\\\mathbf { C o x } ( X )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,414,881,486]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"118\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $X$ be a normal $\\\\mathbb { Q }$ -factorial projective variety, and let $D$ be an effective $\\\\mathbb { Q }$ - divisor on $X$ . The stable base locus $\\\\mathbf { B } ( D )$ of $D$ is the set-theoretic intersection of the base loci of the complete linear systems $| s D |$ for all positive integers $s$ such that $s D$ is integral. In other words, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[405,501,589,542]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0976/f94fd44579a0426f2054ac1c966a7ebfecad80fe9d65401c0a01e96343769600.jpg\",\n \"list_items\": null,\n \"page_idx\": \"118\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf { B } ( D ) = \\\\bigcap _ { s > 0 } B ( s D )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,555,881,609]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"118\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since stable base loci do not behave well with respect to numerical equivalence, we will assume that $h ^ { 1 } ( X , { \\\\mathcal { O } } _ { X } ) = 0$ . This implies that linear and numerical equivalence of $\\\\mathbb { Q }$ -divisors coincide. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,611,881,681]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"118\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since numerically equivalent $\\\\mathbb { Q }$ -divisors on $X$ have the same stable base locus, then ${ \\\\overline { { \\\\operatorname { E f f } } } } ( X )$ the pseudo-effective cone of $X$ can be decomposed into chambers depending on the stable base locus of the corresponding linear series. This decomposition is called stable base locus decomposition. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,683,881,737]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"118\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $X$ is a Mori dream space, satisfying then the condition $h ^ { 1 } ( X , { \\\\mathcal { O } } _ { X } ) = 0$ , determining the stable base locus decomposition of $\\\\operatorname { E f f } ( X )$ is a first step in order to compute its Mori chamber decomposition. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,744,881,782]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"118\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 2.4 Recall that two divisors $D _ { 1 } , D _ { 2 }$ are said to be Mori equivalent if $\\\\mathbf { B } ( D _ { 1 } ) = \\\\mathbf { B } ( D _ { 2 } )$ and the following diagram of rational maps is commutative \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[364,798,635,869]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0579/0a89dc884b4dce3b83c7b11c8b4ab4d5d76749b8a41f89bc32b4c04e772940b8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"118\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { c } { { \\\\phi _ { D _ { 1 } } \\\\_ { \\\\stackrel { \\\\displaystyle \\\\bigwedge } \\\\gamma } , \\\\ ~ \\\\stackrel { \\\\displaystyle } { \\\\bigwedge } \\\\ _ { { { { \\\\mathrm { \\\\Large ~ \\\\ddots ~ } } \\\\atop { { \\\\mathrm { \\\\Large ~ \\\\ddots ~ } } \\\\atop { { \\\\mathrm { \\\\Large ~ \\\\ddots ~ } } \\\\atop { { \\\\mathrm { \\\\Large ~ \\\\ddots ~ } } \\\\atop { { \\\\mathrm { \\\\Large ~ \\\\ddots ~ } } \\\\atop { { \\\\mathrm { \\\\Large ~ \\\\ddots ~ } } \\\\atop { { \\\\mathrm { \\\\scriptsize ~ \\\\ddots ~ } } \\\\atop { { \\\\mathrm { \\\\scriptsize ~ \\\\ddots ~ } } \\\\scriptstyle { { \\\\mathrm { \\\\scriptsize ~ \\\\ � } } } \\\\left( D _ { 2 } \\\\right) } } } } } } } } } } } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[633,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"118\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. L. Huerta and A. Massarenti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"118\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,88,879,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"119\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where the horizontal arrow is an isomorphism. Therefore, the Mori chamber decomposition is a refinement of the stable base locus decomposition. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,133,876,171]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"119\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $X$ be a Mori dream space with Cox ring $\\\\operatorname { C o x } ( X )$ and grading matrix $Q$ . The matrix $Q$ defines a surjection \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[423,187,576,207]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0836/743d7d472d5fc4df516f9b22d11de9b2a65858daae45c1a0c3ee39ee68bca007.jpg\",\n \"list_items\": null,\n \"page_idx\": \"119\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nQ \\\\colon E \\\\to \\\\mathbf { C l } ( X )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,223,881,403]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"119\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"from a free, finitely generated, abelian group $E$ to the divisor class group of $X$ . Denote by $\\\\gamma$ the positive quadrant of $E _ { \\\\mathbb { Q } } : = E \\\\otimes _ { \\\\mathbb { Z } } \\\\mathbb { Q }$ . Let $e _ { 1 } , \\\\ldots , e _ { r }$ be the canonical basis of $E _ { \\\\mathbb { Q } }$ . Given a face $\\\\gamma _ { 0 } \\\\preceq \\\\gamma$ , we say that $i \\\\in \\\\{ 1 , \\\\ldots , r \\\\}$ is a cone index of $\\\\gamma _ { 0 }$ if $e _ { i } \\\\in \\\\gamma _ { 0 }$ . The face $\\\\gamma _ { 0 }$ is an $\\\\mathfrak { F }$ -face if there exists a point of ${ \\\\overline { { X } } } = \\\\operatorname { S p e c } ( \\\\operatorname { C o x } ( X ) )$ whose $i$ -th coordinate is non-zero exactly when $i$ is a cone index of $\\\\gamma _ { 0 }$ [2, Construction 3.3.1.1]. The set of these points is denoted by $\\\\overline { { X } } ( \\\\gamma _ { 0 } )$ . Given the Cox construction of $X$ we denote by $X ( \\\\gamma _ { 0 } ) \\\\subseteq X$ the image of $\\\\overline { { X } } ( \\\\gamma _ { 0 } )$ , and given an $\\\\mathfrak { F }$ -face $\\\\gamma _ { 0 }$ its image $Q ( \\\\gamma _ { 0 } ) \\\\subseteq \\\\mathbf { C l } ( X ) _ { \\\\mathbb { Q } }$ is an orbit cone of $X$ . The set of all orbit cones of& $X$ is denoted by $\\\\Omega$ . Accordingly to [2, Definition 3.1.2.6] an effective class $w \\\\in \\\\operatorname { C l } ( X )$ defines the GIT chamber \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[396,417,601,460]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0187/a449d6d9146d9be7f65dd9ab9d76ae7e5898f59df4fb5b09a8ae252441241ba9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"119\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\lambda ( w ) : = \\\\bigcap _ { \\\\{ \\\\omega \\\\in \\\\Omega : w \\\\in \\\\omega \\\\} } \\\\omega\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,474,881,545]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"119\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $w$ is an ample class of $X$ , then the corresponding GIT chamber is the semi-ample cone of $X$ . The variety $X$ can be reconstructed from the pair $( \\\\operatorname { C o x } ( X ) , \\\\Phi )$ formed by the Cox ring together with a bunch of cones, consisting of certain subsets of the orbit cones [2, Definition 3.1.3.2]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,548,881,618]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"119\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We now explore different aspects of Definitions 1.1 and 1.2. We exhibit two varieties which are birational twins but fail to be strong birational twins. We do this inspired by [13, Theorem 1.1.]. We also exhibit two Mori dream spaces which are strongly Lefschetz divisorially equivalent but fail to be birational twins. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,627,879,663]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"119\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2.6 Let $Z$ be the toric variety with Cox ring $K [ T _ { 1 } , \\\\dots , T _ { 1 1 } ]$ whose grading matrix and irrelevant ideal are the following \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[177,678,820,722]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0217/92f05396ad6a75b1cc07ef615de96342d6636f4ad754e27010c8e0749ecd5a88.jpg\",\n \"list_items\": null,\n \"page_idx\": \"119\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nQ = \\\\left[ \\\\begin{array} { c c } { 1 \\\\mathrm { ~ 1 ~ 2 ~ 2 ~ 2 ~ 2 ~ 1 ~ 0 ~ 0 ~ 0 ~ 0 ~ } } \\\\\\\\ { 0 \\\\mathrm { ~ 0 ~ 1 ~ 1 ~ 1 ~ 2 ~ 1 ~ 1 ~ 1 ~ 1 ~ } } \\\\end{array} \\\\right] \\\\qquad \\\\mathcal { J } _ { \\\\mathrm { i r r } } ( Z ) = \\\\langle T _ { 1 } , T _ { 2 } \\\\rangle \\\\cap \\\\langle T _ { 3 } , \\\\ldots , T _ { 1 1 } \\\\rangle\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,735,881,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"119\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and let $F _ { \\\\mathrm { ~ } } = T _ { 3 } T _ { 8 } + T _ { 4 } T _ { 9 } + T _ { 1 } T _ { 7 } , G _ { \\\\mathrm { ~ } } = T _ { 5 } T _ { 9 } + T _ { 6 } T _ { 1 1 } + T _ { 2 } T _ { 7 }$ . Then the ring K[T1,...,T11](F,G) is the Cox ring of a projective normal Q-factorial Mori dream space $X \\\\subset Z$ of Picard rank two. The varieties $X$ and $Z$ are birational twins but fail to be strong birational twins. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,806,881,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"119\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Furthermore, the ring $\\\\frac { K [ T _ { 1 } , . . . , T _ { 1 1 } ] } { ( F , G , T _ { 7 } + T _ { 1 } T _ { 1 1 } ^ { 2 } + T _ { 2 } T _ { 8 } T _ { 1 1 } + T _ { 2 } T _ { 1 0 } ^ { 2 } + T _ { 2 } T _ { 8 } T _ { 9 } ) }$ is the Cox ring of a projective normal $\\\\mathbb { Q }$ -factorial Mori dream space $Y ~ \\\\subset ~ X$ of Picard rank two. The varieties $Y$ and $X$ are strongly Lefschetz divisorially equivalent but fail to be birational twins. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,51,697,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"119\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"119\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"115 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,879,147]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"120\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Let us denote by $w _ { i } \\\\in \\\\mathbb { Z } ^ { 2 }$ the degree of $T _ { i }$ . We will denote by $V ( T _ { i _ { 1 } } , \\\\dots , T _ { i _ { r } } ) : = \\\\{ T _ { i _ { 1 } } = \\\\dots = T _ { i _ { r } } = 0 \\\\}$ . The following picture represents the degrees of the generators of the Cox ring \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[312,166,683,286]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0649/8bcbdabe131d1e8122aab3314e3bb2f0ee59012581777e5ff9b54aae9ea3369c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"120\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[118,307,881,363]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"120\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\operatorname { E f f } ( Z ) = \\\\operatorname { M o v } ( Z ) = \\\\lambda _ { A } \\\\cup \\\\lambda \\\\cup \\\\lambda ^ { \\\\prime }$ and $\\\\mathbf { N e f } ( Z ) = \\\\lambda _ { A }$ . Let ${ \\\\overline { { Z } } } = K ^ { 1 1 }$ be the spectrum of the Cox ring of $Z$ and let $\\\\overline { { X } }$ be the affine subvariety defined by $\\\\overline { { X } } = \\\\overset { \\\\cdot } { \\\\{ F = G = 0 \\\\} } \\\\subset \\\\overline { { Z } }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,363,881,456]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"120\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A standard computation shows that $\\\\overline { { X } }$ is irreducible and $\\\\mathrm { c o d i m } _ { \\\\overline { { X } } } ( \\\\mathrm { S i n g } ( \\\\overline { { X } } ) ) \\\\geqslant 2$ , and then Serre’s criterion on normality yields that $\\\\overline { { X } }$ is normal. Let $p _ { Z } \\\\colon { \\\\widehat { Z } } \\\\to Z$ be the characteristic space morphism of $Z$ , and let ${ \\\\widehat { X } } : = { \\\\overline { { X } } } \\\\cap { \\\\widehat { Z } }$ . Note that the irrelevant locus with respect to the chamber $\\\\lambda _ { A }$ has two components given by $\\\\Gamma _ { 1 } = \\\\{ T _ { 1 } =$ $T _ { 2 } = 0 \\\\}$ and $\\\\Gamma _ { 2 } = \\\\{ T _ { 3 } = \\\\cdot \\\\cdot \\\\cdot = T _ { 1 1 } = 0 \\\\}$ . Hence $\\\\widehat { Z } = K ^ { 1 1 } \\\\setminus \\\\overline { { \\\\{ \\\\Gamma _ { 1 } \\\\cup \\\\bar { \\\\Gamma _ { 2 } } \\\\} } }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,456,879,509]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"120\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The image of $\\\\widehat { X }$ via $p _ { Z }$ is a subvariety $X$ of $Z$ , let $i : X \\\\to Z$ be the inclusion. Since $\\\\overline { { X } }$ is irreducible and normal, and $X$ is a GIT quotient of $\\\\overline { { X } }$ by a reductive group [6, Theorem 1.24 (vi)] yields that $X$ is irreducible and normal as well. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,510,879,584]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"120\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $Z ^ { \\\\prime }$ be the smooth locus of $Z$ . Note that $Z ^ { \\\\prime }$ contains the open subset $Z ^ { \\\\prime \\\\prime }$ of $Z$ obtained by removing the union of all the toric subvarieties of the form $p z ( V ( T _ { i } , T _ { j } ) )$ . Since the Zariski closure of $p _ { Z } ^ { - 1 } ( X \\\\cap Z ^ { \\\\prime \\\\prime } )$ in $\\\\widehat { Z }$ is $\\\\widehat { X }$ we conclude that $\\\\widehat { X }$ is the Zariski closure of $p _ { Z } ^ { - 1 } ( X \\\\cap Z ^ { \\\\prime } )$ in $\\\\widehat { Z }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[113,585,876,623]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"120\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note that, according to the grading matrix $Q$ , the $K ^ { * } \\\\times K ^ { * }$ action on $\\\\overline { { X } }$ is given by \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,636,847,680]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0810/703f8ccb4d358800d43439bac6b635b57fc31006625183e2df148fb6a441780e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"120\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { c c } { { ( K ^ { * } \\\\times K ^ { * } ) \\\\times \\\\overline { { { X } } } } } & { { \\\\longrightarrow } } \\\\\\\\ { { ( ( \\\\lambda , \\\\mu ) , ( T _ { 1 } , \\\\dots , T _ { 1 1 } ) ) \\\\longmapsto ( \\\\lambda T _ { 1 } , \\\\lambda T _ { 2 } , \\\\lambda ^ { 2 } \\\\mu T _ { 3 } , \\\\dots , \\\\lambda \\\\mu ^ { 2 } T _ { 7 } , \\\\mu T _ { 8 } , \\\\dots , \\\\mu T _ { 1 1 } ) } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,695,881,894]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"120\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Therefore, if $T _ { 2 } ~ \\\\ne ~ 0 , T _ { 8 } ~ \\\\ne ~ 0$ we may set $\\\\lambda ~ = ~ \\\\mu ~ = ~ 1$ , and write $\\\\begin{array} { r l } { T _ { 3 } } & { { } = } \\\\end{array}$ \\u0012=T4T9+T1T7 , T $\\\\begin{array} { r } { T _ { 7 } ~ { \\\\stackrel { - } { = } } ~ - \\\\frac { T _ { 5 } T _ { 9 } + T _ { 6 } T _ { 1 1 } } { T _ { 2 } } } \\\\end{array}$ − T5T9+T6T11 . Therefore, if we remove the images of V (T2) ∪ $V ( T _ { 8 } )$ 8 from $X$ 2 , the resulting variety is isomorphic to an affine space. Therefore, $\\\\operatorname { C l } ( X )$ is generated by the classes of the images of the two irreducible divisors $V ( T _ { i } ) \\\\cap \\\\overline { { X } }$ , with $i \\\\in \\\\{ 2 , 8 \\\\}$ , and $\\\\rho ( X ) \\\\quad \\\\leqslant \\\\quad 2$ . Moreover, crossing the wall corresponding to $w _ { 1 } , w _ { 2 }$ we get a morphism $f : Z \\\\to \\\\mathbb { P } ^ { 1 }$ . Furthermore, $X \\\\subset Z$ is not contained in any fiber of $f$ , and hence $f$ restricts to a surjective morphism $f _ { | X } : X \\\\to \\\\mathbb { P } ^ { 1 }$ . This forces $\\\\rho ( X ) \\\\ \\\\geqslant \\\\ 2$ . Finally, we conclude that the images of $V ( T _ { 2 } ) , V ( T _ { 8 } )$ form a basis of $\\\\operatorname { C l } ( X )$ . So $\\\\rho ( X ) = 2$ and the pull-back $i ^ { * } : \\\\mathbf { C l } ( Z ) \\\\to$ $\\\\operatorname { C l } ( X )$ , induced by the inclusion, is an isomorphism. Now, note that $\\\\Gamma _ { 2 } \\\\subset { \\\\overline { { X } } }$ has codimension greater that two in $\\\\overline { { X } }$ , and $\\\\Gamma _ { 1 } \\\\cap { \\\\overline { { X } } } = \\\\{ T _ { 1 } = T _ { 2 } = T _ { 3 } T _ { 8 } + T _ { 4 } T _ { 9 } =$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[633,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"120\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. L. Huerta and A. Massarenti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,152,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"120\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"116 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,876,129]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"121\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$T _ { 5 } T _ { 9 } + T _ { 6 } T _ { 1 1 } = 0 \\\\}$ has codimension two in $\\\\overline { { X } }$ . Hence codi $\\\\operatorname { r } _ { \\\\overline { { X } } } ( { \\\\overline { { X } } } \\\\setminus { \\\\widehat { X } } ) \\\\geqslant 2$ and [2, Corollary 4.1.1.5] yields that the Cox ring of $X$ is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[371,142,628,184]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0070/4a3d9c9156c39e25ebc454a4d31fb0f75f8605bd0dc4242e6f5602b808974357.jpg\",\n \"list_items\": null,\n \"page_idx\": \"121\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname { C o x } ( X ) = { \\\\frac { K [ T _ { 1 } , \\\\dots , T _ { 1 1 } ] } { I ( { \\\\overline { { X } } } ) } }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,196,881,270]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"121\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In general, if $X$ is a Mori dream space such that $\\\\operatorname { C l } ( X )$ has rank two we can fix a total order on the classes in the effective cone $w \\\\leqslant w ^ { \\\\prime }$ if $w$ is on the left of $w ^ { \\\\prime }$ . Given two convex cones $\\\\lambda , \\\\lambda ^ { \\\\prime }$ contained in the effective cone we will write- - $\\\\lambda \\\\leqslant \\\\lambda ^ { \\\\prime }$ if $w \\\\leqslant w ^ { \\\\prime }$ for any $w \\\\in \\\\lambda$ and $w ^ { \\\\prime } \\\\in \\\\lambda$ . Note that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[284,285,712,307]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0793/045e62935d08cc53fcbee6ca860aec11434092bb954919b0d3407400b1ed4dc6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"121\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nV ( f _ { i } : w _ { i } \\\\geqslant \\\\lambda _ { A } ) = \\\\{ T _ { 1 } = T _ { 2 } = \\\\widetilde { F } = \\\\widetilde { G } = 0 \\\\}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,327,660,348]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"121\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\widetilde { F } = T _ { 3 } T _ { 8 } + T _ { 4 } T _ { 9 }$ , $\\\\widetilde { G } = T _ { 5 } T _ { 9 } + T _ { 6 } T _ { 1 1 }$ . Furthermore \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,364,874,408]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0273/8817a17b7160d9c7377aabe65bb9cee2854ee8258e0b2046ad6cb9f0a6c81702.jpg\",\n \"list_items\": null,\n \"page_idx\": \"121\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { V ( f _ { i } : w _ { i } \\\\leqslant \\\\lambda ) = \\\\{ T _ { 7 } = T _ { 8 } = T _ { 9 } = T _ { 1 0 } = T _ { 1 1 } = 0 \\\\} ; } \\\\\\\\ & { V ( f _ { i } : w _ { i } \\\\leqslant \\\\lambda ^ { \\\\prime } ) = \\\\{ T _ { 1 } = T _ { 2 } = T _ { 8 } = T _ { 9 } = T _ { 1 0 } = T _ { 1 1 } = 0 \\\\} \\\\cup \\\\{ T _ { 7 } = T _ { 8 } = T _ { 9 } = T _ { 1 0 } = T _ { 1 1 } = 0 \\\\} . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,421,881,495]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"121\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"So $V ( f _ { i } ~ : ~ w _ { i } ~ \\\\geqslant ~ \\\\lambda _ { A } ) ~ \\\\supset ~ \\\\{ T _ { 1 } ~ = ~ T _ { 2 } ~ = ~ T _ { 8 } ~ = ~ T _ { 9 } ~ = { \\\\cal T } _ { 1 0 } ~ = ~ T _ { 1 1 } ~ = ~ 0 \\\\}$ contains a component of the set $V ( f _ { i } : w _ { i } \\\\leqslant \\\\lambda ^ { \\\\prime } )$ . Finally, [13, Lemmas 4.3, 4.4, 4.5] yields that $\\\\mathsf { M C D } ( X ) \\\\ = \\\\ i ^ { * } \\\\mathbf { M C D } ( Z )$ , and that $\\\\operatorname { S B L D } ( X )$ has just two chambers while $\\\\mathrm { S B L D } ( Z ) = \\\\mathbf { M C D } ( Z )$ has three chambers. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,495,576,513]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"121\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let ${ \\\\overline { { Y } } } \\\\subset { \\\\overline { { Z } } } = K ^ { 1 1 }$ be the subvariety cut out by \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[205,528,792,551]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0935/8edfc9a44eda826f29d2410d875be0f9c24b278345517be0c7c58da59962be93.jpg\",\n \"list_items\": null,\n \"page_idx\": \"121\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n{ \\\\overline { { Y } } } = \\\\{ F = G = T _ { 7 } + T _ { 1 } T _ { 1 1 } ^ { 2 } + T _ { 2 } T _ { 8 } T _ { 1 1 } + T _ { 2 } T _ { 1 0 } ^ { 2 } + T _ { 2 } T _ { 8 } T _ { 9 } = 0 \\\\}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,564,881,621]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"121\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We have that $\\\\overline { { Y } }$ is irreducible of dimension eight and $\\\\mathrm { c o d i m } _ { \\\\overline { { { Y } } } } ( \\\\mathrm { S i n g } ( { \\\\overline { { { Y } } } } ) ) ~ \\\\geqslant ~ 2$ , so Serre’s criterion on normality yields that $\\\\overline { { Y } }$ is normal, and [6, Theorem 1.24 (vi)] yields that the GIT quotient $Y$ is irreducible and normal as well. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,623,881,831]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"121\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $T _ { 2 } \\\\neq 0 , T _ { 8 } \\\\neq 0$ we may write $\\\\begin{array} { r } { T _ { 3 } ~ = ~ - \\\\frac { T _ { 4 } T _ { 9 } + T _ { 1 } T _ { 7 } } { T _ { 8 } } } \\\\end{array}$ T4T9+T1T7 , T $\\\\begin{array} { r } { T _ { 7 } ~ = ~ - \\\\frac { T _ { 5 } T _ { 9 } + T _ { 6 } T _ { 1 1 } } { T _ { 2 } } } \\\\end{array}$ and − T2T 210+T2T8T11+T1T 211+T7 . Therefore, Y \\\\ (V (T2) ∪ V (T8)) is isomorphic to an affine space and $\\\\operatorname { \\\\mathbb { C } l } ( Y )$ is generated by the classes of the images of the two irreducible divisors $V ( T _ { i } ) \\\\cap \\\\overline { { Y } }$ , with $i \\\\in \\\\{ 2 , 8 \\\\}$ , and $\\\\rho ( Y ) \\\\ \\\\leqslant \\\\ 2$ . Again crossing the wall corresponding to $w _ { 1 }$ , $w _ { 2 }$ we get a morphism $f : Z \\\\to \\\\mathbb { P } ^ { 1 }$ and since $Y \\\\subset Z$ is not contained in any fiber of $f$ this morphism restricts to a surjective morphism $f _ { | Y } : Y \\\\to \\\\mathbb { P } ^ { 1 }$ . This forces $\\\\rho ( Y ) \\\\geqslant 2$ . We conclude that the images of $V ( T _ { 2 } ) , V ( T _ { 8 } )$ form a basis of $\\\\operatorname { C l } ( Y )$ . So $\\\\rho ( Y ) = 2$ and the pull-back $j ^ { * } : \\\\operatorname { C l } ( X ) \\\\to \\\\operatorname { C l } ( Y )$ , induced by the inclusion $j : Y X$ , is an isomorphism. As before $\\\\Gamma _ { 2 } \\\\subset \\\\overline { { Y } }$ has codimension greater that two in $\\\\overline { { Y } }$ , and $\\\\Gamma _ { 1 } \\\\cap \\\\overline { { Y } }$ has codimension two in $\\\\overline { { Y } }$ . Hence [2, Corollary 4.1.1.5] yields that the Cox ring of $Y$ is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[371,845,626,887]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0068/0c8f52042474e60817ce67d007517fa902f491a5393b838c4ea372edbe5dee27.jpg\",\n \"list_items\": null,\n \"page_idx\": \"121\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf { C o x } ( Y ) = \\\\frac { K [ T _ { 1 } , \\\\hdots , T _ { 1 1 } ] } { I ( \\\\overline { { Y } } ) }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,51,697,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"121\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"121\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"117 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,507,108]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"122\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now, for the Mori dream space $Y$ we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,124,874,166]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0043/df8f48cb0e01dc454535fb538d1584d2444f3b76af7da34ab6c5ad166d4c2197.jpg\",\n \"list_items\": null,\n \"page_idx\": \"122\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { V ( f _ { i } : w _ { i } \\\\leqslant \\\\lambda ) = V ( f _ { i } : w _ { i } \\\\leqslant \\\\lambda ^ { \\\\prime } ) = \\\\{ T _ { 7 } = T _ { 8 } = T _ { 9 } = T _ { 1 0 } = T _ { 1 1 } = 0 \\\\} ; } \\\\\\\\ & { V ( f _ { i } : w _ { i } \\\\geqslant \\\\lambda ) = V ( f _ { i } : w _ { i } \\\\geqslant \\\\lambda ^ { \\\\prime } ) = \\\\{ T _ { 1 } = T _ { 2 } = T _ { 3 } = T _ { 4 } = T _ { 5 } = T _ { 6 } = T _ { 7 } = 0 \\\\} . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,180,876,216]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"122\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hence [13, Lemma 4.3] yields that $\\\\operatorname { M C D } ( Y )$ has two chambers while $\\\\mathbf { M C D } ( X ) =$ $i ^ { * } \\\\mathrm { M C D } ( Z )$ has three chambers. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,225,881,315]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"122\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 2.7 Note that any pair of varieties among the Mori dream spaces $Y \\\\subset X \\\\subset$ $Z$ in Theorem 2.6 gives a pair of Lefschetz divisorially equivalent varieties. On the other hand, $i ^ { * } \\\\operatorname { S B L D } ( Z ) \\\\neq \\\\operatorname { S B L D } ( X )$ and $( i \\\\circ j ) ^ { * } \\\\mathsf { M C D } ( Z ) \\\\ne \\\\mathsf { M C D } ( Y )$ . Therefore, Lefschetz divisorially equivalent does not imply strongly Lefschetz divisorially equivalent nor birational twins. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,325,881,360]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"122\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"However, when a Mori dream space inherits the stable base locus decomposition from an ambient toric variety something more can be said. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,369,881,496]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"122\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 2.8 Given a Mori dream space $X$ , then there is an embedding $i : X \\\\to \\\\mathcal { T } _ { X }$ into a simplicial projective toric variety $\\\\mathcal { T } _ { X }$ such that $i ^ { * } : \\\\operatorname { P i c } ( \\\\mathcal T _ { X } ) \\\\to \\\\operatorname { P i c } ( X )$ is an isomorphism which induces an isomorphism $\\\\begin{array} { l l l } { \\\\operatorname { E f f } ( { \\\\mathcal { T } } _ { X } ) } & { \\\\to } & { \\\\operatorname { E f f } ( X ) } \\\\end{array}$ , [11, Proposition 2.11]. Furthermore, the Mori chamber decomposition of $\\\\operatorname { E f f } ( { \\\\mathcal { T } } _ { X } )$ is a refinement of the Mori chamber decomposition of $\\\\operatorname { E f f } ( X )$ . Indeed, if $\\\\operatorname { C o x } ( X ) \\\\ \\\\cong$ $\\\\textstyle \\\\frac { K [ T _ { 1 } , \\\\ldots , T _ { s } ] } { I }$ where the $T _ { i }$ = are homogeneous generators with non-trivial effective $\\\\operatorname { P i c } ( X )$ -degrees then $\\\\operatorname { C o x } ( \\\\mathcal { T } _ { X } ) \\\\cong K [ T _ { 1 } , \\\\dots , T _ { s } ]$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,504,879,558]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"122\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 2.9 Let $X \\\\mathcal { T } _ { X }$ be a Mori dream space embedded in its canonical toric embedding. Assume that both $X$ and $\\\\mathcal { T } _ { X }$ have Picard rank two. If $X$ and $\\\\mathcal { T } _ { X }$ are strongly Lefschetz divisorially equivalent then they are strong birational twins. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,567,881,677]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"122\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof By Remark $2 . 8 \\\\mathrm { M C D } ( \\\\mathcal { T } _ { X } )$ refines $\\\\operatorname { M C D } ( X )$ , and moreover $\\\\operatorname { M C D } ( X )$ refines $\\\\mathrm { S B L D } ( X )$ . By our hypothesis $\\\\mathrm { S B L D } ( X ) = i ^ { * } \\\\mathrm { S B L D } ( \\\\mathcal { T } _ { X } )$ , where $i : X \\\\to \\\\mathcal { T } _ { X }$ is the inclusion. Now, since $\\\\mathcal { T } _ { X }$ is a toric projective variety of Picard rank two [13, Theorem 1.2] yields $\\\\mathrm { S B L D } ( \\\\mathcal { T } _ { X } ) = \\\\mathbf { M C D } ( \\\\mathcal { T } _ { X } )$ . Summing up, $\\\\mathrm { M C D } ( \\\\mathcal { T } _ { X } ) = \\\\mathrm { S B L D } ( \\\\mathcal { T } _ { X } )$ refines $\\\\operatorname { M C D } ( X )$ which in turn refines $\\\\Re \\\\mathrm { L D } ( X ) = \\\\Im \\\\mathrm { B L D } ( \\\\mathcal { T } _ { X } ) = \\\\Re \\\\mathrm { C D } ( \\\\mathcal { T } _ { X } )$ , and this forces $\\\\mathsf { M C D } ( X ) = i ^ { * } \\\\mathsf { M C D } ( \\\\mathcal { T } _ { X } )$ . □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,720,744,764]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"122\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 Birational Twin Varieties: Complete Collineations and Quadrics \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,786,876,840]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"122\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This section is the core of the paper. It contains the definitions and crucial properties of the spaces of complete collineations and complete quadrics. It also contains the details of the proofs of Theorems A and B listed in the introduction. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,842,881,912]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"122\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $V$ and $W$ be $K$ -vector spaces of dimension $n + 1$ and $m + 1$ , respectively with $n \\\\leqslant m$ . Let $\\\\mathbb { P } ^ { N }$ denote the projective space whose points parametrize non-zero linear maps $W V$ up to a scalar multiple. We recall that a point in this spaces is called a collineation from $W$ to $V$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[633,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"122\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. L. Huerta and A. Massarenti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"122\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"118 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,144]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"123\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Observe that a full-rank collineation $p \\\\in \\\\mathsf { H o m } ( W , V )$ induces another collineation ${ \\\\textstyle \\\\bigwedge } ^ { k } W \\\\ \\\\to \\\\ { \\\\textstyle \\\\bigwedge } ^ { k } V$ , for any $\\\\textit { k } \\\\leq n + 1$ , by taking the wedge product. Consider the rational map \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,157,879,202]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0262/c7dd9085129764287dbb18d31dc5faff338da4a3453dfee8b32fb007048557a2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"123\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\phi : \\\\mathbb { P } ( \\\\mathbf { H o m } ( W , V ) ) \\\\ \\\\xrightarrow { } \\\\ \\\\mathbb { P } ( \\\\mathbf { H o m } ( { \\\\textstyle \\\\bigwedge } ^ { 2 } W , { \\\\textstyle \\\\bigwedge } ^ { 2 } V ) ) \\\\times \\\\cdots \\\\times \\\\mathbb { P } ( \\\\mathbf { H o m } ( { \\\\textstyle \\\\bigwedge } ^ { n + 1 } W , { \\\\textstyle \\\\bigwedge } ^ { n + 1 } V ) )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,235,879,273]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"123\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3.2 The space of complete collineations ${ \\\\mathcal { X } } ( n , m )$ from $W$ to $V$ is defined as the closure of the graph of the rational map (3.1). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,282,876,316]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"123\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In order to understand the closure of the image of $\\\\phi$ , we now recall an alternative description of the space ${ \\\\mathcal { X } } ( n , m )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,318,881,477]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"123\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Given an irreducible, reduced and non-degenerate variety $X \\\\subset \\\\mathbb { P } ^ { N }$ , for any $h <$ $N$ , we can consider the $h$ -secant variety of $X$ , which is defined as the closure of the union of all $( h - 1 )$ -planes spanned by $h$ general points in $X$ . We denote this variety by $\\\\mathbb { S } e c _ { h } ( X )$ . Observe that any point $p \\\\in \\\\mathbb { P } ^ { N } = \\\\mathbb { P } ( \\\\operatorname { H o m } ( W , V ) )$ can be represented by a $( n + 1 ) { \\\\times } ( m + 1 )$ matrix $Z$ , and that the locus of matrices of rank one is the Segre variety $\\\\mathcal { S } \\\\subset \\\\mathbb { P } ^ { N }$ . More generally, any $p \\\\in \\\\mathbb { S } e c _ { h } ( S )$ can be represented by a matrix $Z$ which is a linear combination of $h$ matrices of rank one, and conversely. In other words, a point $p$ is contained in $\\\\mathbb { S } e c _ { h } ( S )$ if and only if the rank of its representing matrix is at most $h$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,480,876,515]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"123\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Setting up notation, we will think of the point $p = [ z _ { 0 , 0 } : \\\\cdot \\\\cdot \\\\cdot : z _ { n , m } ] \\\\in \\\\mathbb { P } ^ { N }$ as the matrix \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[391,528,605,599]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0376/050266ccd7dbe7273bb1fe29a7753fbf5dbc30a66cefdd91b8ea1e6d9fb4dcdf.jpg\",\n \"list_items\": null,\n \"page_idx\": \"123\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nZ = \\\\left( \\\\begin{array} { c } { { z _ { 0 , 0 } \\\\ldots z _ { 0 , m } } } \\\\\\\\ { { \\\\vdots \\\\ddots \\\\vdots } } \\\\\\\\ { { z _ { n , 0 } \\\\ldots z _ { n , m } } } \\\\end{array} \\\\right)\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[136,614,863,632]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"123\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We can now describe the space ${ \\\\mathcal { X } } ( n , m )$ as a sequence of blow-ups as follows. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,641,774,659]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"123\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Construction 3.4 Let us consider the following sequence of blow-ups: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,666,856,795]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"123\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\begin{array} { r l } { - } & { { } \\\\mathcal { X } ( n , m ) _ { 1 } } \\\\end{array}$ is the blow-up of $\\\\mathcal { X } ( n , m ) _ { 0 } : = \\\\mathbb { P } ^ { N }$ along the Segre variety $s$ ; \\n– $\\\\mathbf { \\\\chi } - \\\\mathcal { X } ( n , m ) _ { 2 }$ is the blow-up of $\\\\mathcal { X } ( n , m ) _ { 1 }$ along the strict transform of $\\\\mathbb { S } e c _ { 2 } ( S )$ ; . \\n$\\\\mathbf { \\\\chi } - \\\\mathbf { \\\\chi } \\\\mathcal { X } ( n , m ) _ { i }$ is the blow-up of $\\\\mathcal { X } ( n , m ) _ { i - 1 }$ along the strict transform of $\\\\mathbb { S } e c _ { i } ( \\\\boldsymbol { S } )$ ; .. \\n$\\\\vdash \\\\mathcal { X } ( n , m ) _ { n }$ is the blow-up of $\\\\mathcal { X } ( n , m ) _ { n - 1 }$ along the strict transform of $\\\\mathbb { S } e c _ { n } ( S )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,803,881,894]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"123\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It follows from [24] that the variety $\\\\mathcal { X } ( n , m ) _ { n }$ is isomorphic to ${ \\\\mathcal { X } } ( n , m )$ , the space of complete collineations from $W$ to $V$ . Furthermore, we have that for any $i = 1 , \\\\ldots , n$ the variety $\\\\mathcal { X } ( n , m ) _ { i }$ is smooth, the strict transform of $\\\\mathbb { S } e c _ { i } ( \\\\boldsymbol { S } )$ in $\\\\mathcal { X } ( n , m ) _ { i - 1 }$ is smooth, and the divisor $E _ { 1 } \\\\cup E _ { 2 } \\\\cup \\\\dots \\\\cup E _ { i - 1 }$ in $\\\\mathcal { X } ( n , m ) _ { i - 1 }$ is simple normal crossing. When $n = m$ we will write ${ \\\\mathcal { X } } ( n )$ for $\\\\mathcal { X } ( n , n )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,51,697,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"123\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"123\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"119 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,881,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"124\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Setting up notation, let $f _ { i } : \\\\mathcal { X } ( n , m ) _ { i } \\\\to \\\\mathcal { X } ( n , m ) _ { i - 1 }$ be the blow-up morphism and $E _ { i } = \\\\operatorname { E x c } ( f _ { i } )$ its exceptional divisor. By abusing notation, we will also denote by $E _ { i }$ the strict transform in the subsequent blow-ups. Let $H$ denote the pull-back to $\\\\mathcal { X } ( n , m ) _ { i }$ of the hyperplane section of $\\\\mathbb { P } ^ { N }$ . We will denote by $f : \\\\mathcal { X } ( n , m ) \\\\to \\\\mathbb { P } ^ { N }$ the composition of the $f _ { i }$ ’s. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,180,881,325]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"124\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $n = m$ , then the last blow-up in Construction 3.4 is the blow-up of a Cartier divisor and hence ${ \\\\mathcal { X } } ( n ) _ { n } \\\\cong { \\\\mathcal { X } } ( n ) _ { n - 1 }$ . Furthermore, we have a distinguished linear subspace in $\\\\mathbb { P } ^ { N } = \\\\mathbb { P } ( \\\\mathrm { H o m } ( V , V ) )$ . Indeed, the space of symmetric matrices is defined by $\\\\mathbb { P } ^ { N _ { + } } : = \\\\mathbb { P } ( \\\\operatorname { S y m } ^ { 2 } V ) = \\\\{ z _ { i , j } = z _ { j , i } | i \\\\neq j \\\\}$ , where $N _ { + } = { \\\\binom { n + 2 } { 2 } } - 1$ . Furthermore, the space $\\\\mathbb { P } ^ { N _ { + } }$ cuts out scheme-theoretically on $s$ the Veronese variety $\\\\nu \\\\subseteq \\\\mathbb { P } ^ { N _ { + } }$ ; which parametrizes $( n + 1 ) { \\\\times } ( n + 1 )$ symmetric matrices of rank one. More generally, the space $\\\\mathbb { P } ^ { N _ { + } }$ cuts out scheme-theoretically on $\\\\mathbb { S } e c _ { h } ( \\\\cal S )$ the $h$ - secant variety $\\\\mathbb { S } e c _ { h } ( \\\\mathbb { \\\\nu } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,325,879,379]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"124\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Observe that a full-rank symmetric matrix ) $q \\\\in \\\\mathsf { S y m } ^ { 2 } ( V )$ induces another symmetric matrix ${ \\\\textstyle \\\\bigwedge } ^ { k } q$ , for any $k \\\\ \\\\leq \\\\ n + 1$ , by taking the wedge product. There is a rational map \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[186,393,744,438]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0154/9dba7ddf42fa877cdf813a9e7d7ce96f37e046475ec165ef2a393f19f707c473.jpg\",\n \"list_items\": null,\n \"page_idx\": \"124\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { \\\\rho : \\\\mathbb { P } ( \\\\mathbf { S y m } ^ { 2 } V ) \\\\xrightarrow { } \\\\mathbb { P } ( \\\\mathbf { S y m } ^ { 2 } \\\\bigwedge ^ { 2 } V ) \\\\times \\\\dots \\\\times \\\\mathbb { P } ( \\\\mathbf { S y m } ^ { 2 } \\\\bigwedge ^ { n } V ) } \\\\\\\\ { q \\\\longmapsto \\\\qquad ( \\\\bigwedge ^ { 2 } q , \\\\dots , \\\\bigwedge ^ { n } q ) } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,453,876,489]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"124\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3.6 The space of complete quadrics in $\\\\mathbb P ^ { n }$ is defined as the closure of the graph of the map (3.5). We denote this space by $\\\\boldsymbol { \\\\mathcal { Q } } ( \\\\boldsymbol { n } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,498,881,554]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"124\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"One can apply Construction 3.4 in the symmetric setting to the spaces $\\\\mathbb { P } ^ { N _ { + } }$ . In doing so, we obtain the following blow-up construction for the space of complete quadrics [23, Theorem 6.3]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,561,772,579]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"124\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Construction 3.7 Let us consider the following sequence of blow-ups: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,587,797,716]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"124\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\begin{array} { r l } { - } & { { } \\\\mathcal { Q } ( n ) _ { 1 } } \\\\end{array}$ is the blow-up of $\\\\mathcal { Q } ( n ) _ { 0 } : = \\\\mathbb { P } ^ { N _ { + } }$ along the Veronese variety $\\\\nu$ ; \\n– $\\\\mathcal { Q } ( n ) _ { 2 }$ is the blow-up of $\\\\mathcal { Q } ( n ) _ { 1 }$ along the strict transform of $\\\\mathbb { S } e c _ { 2 } ( \\\\mathcal { V } )$ ; . \\n$\\\\ c - \\\\mathcal { Q } ( n ) _ { i }$ is the blow-up of $\\\\mathcal { Q } ( n ) _ { i - 1 }$ along the strict transform of $\\\\mathbb { S } e c _ { i } ( \\\\gamma )$ ; . \\n$\\\\begin{array} { r l } { - } & { { } \\\\mathcal { Q } ( n ) _ { n } } \\\\end{array}$ is the blow-up of $\\\\scriptstyle { \\\\mathcal { Q } } ( n ) _ { n - 1 }$ along the strict transform of $\\\\mathbb { S } e c _ { n } ( \\\\mathbb { \\\\nu } )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,723,881,798]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"124\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It follows from [23] that the variety ${ \\\\mathcal { Q } } ( n ) _ { n - 1 } \\\\cong { \\\\mathcal { Q } } ( n ) _ { n }$ is isomorphic to the space of complete $( n - 1 )$ -dimensional quadrics $\\\\boldsymbol { \\\\mathcal { Q } } ( \\\\boldsymbol { n } )$ . Furthermore, the variety ${ \\\\mathcal { Q } } ( n ) _ { i }$ is smooth, the strict transform of $\\\\mathbb { S } e c _ { i } ( \\\\gamma )$ in $\\\\mathcal { Q } ( n ) _ { i - 1 }$ is smooth, and the divisor $E _ { 1 } ^ { + } \\\\cup E _ { 2 } ^ { + } \\\\cup \\\\cdots \\\\cup E _ { i - 1 } ^ { + }$ in $\\\\mathcal { Q } ( n ) _ { i - 1 }$ − is simple normal crossing for any $i = 1 , \\\\ldots , n$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,798,881,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"124\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Setting up notation, let $f _ { i } ^ { + } : \\\\mathcal { Q } ( n ) _ { i } \\\\ \\\\to \\\\ \\\\mathcal { Q } ( n ) _ { i - 1 }$ be the blow-up morphism and $E _ { i } ^ { + } = \\\\mathrm { E x c } ( f _ { i } ^ { + } )$ its exceptional divisor. We will also denote by $E _ { i } ^ { + }$ the strict transforms of this divisor in the subsequent blow-ups, if confusion does not arise. Let $H ^ { + }$ stand for the pull-back to ${ \\\\mathcal { Q } } ( n ) _ { i }$ of the hyperplane section of $\\\\mathbb { P } ^ { N _ { + } }$ . We will denote by $f ^ { + } : \\\\mathcal { Q } ( n ) \\\\to \\\\mathbb { P } ^ { N _ { + } }$ the composition of the $f _ { i } ^ { + }$ ’s. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[633,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"124\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. L. Huerta and A. Massarenti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,152,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"124\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"120 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,722,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"125\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.1 Birational Geometry of the Intermediate Spaces \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,132,879,187]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"125\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section we will investigate the birational geometry of the intermediate blowups $\\\\mathcal { X } ( n ) _ { i }$ , ${ \\\\mathcal { Q } } ( n ) _ { i }$ appearing in Constructions 3.4 and 3.7. We begin by recalling the notion of spherical variety. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,195,881,250]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"125\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3.8 A spherical variety is a normal variety $X$ together with an action of a connected reductive affine algebraic group $\\\\mathcal { G }$ , a Borel subgroup $\\\\mathcal { B } \\\\subset \\\\mathcal { G }$ , and a base point $x _ { 0 } \\\\in X$ such that the ${ \\\\mathcal { B } }$ -orbit of $x _ { 0 }$ in $X$ is a dense open subset of $X$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,250,881,322]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"125\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $( X , { \\\\mathcal { G } } , { \\\\mathcal { B } } , x _ { 0 } )$ be a spherical variety. We distinguish two types of $\\\\mathcal { B }$ -invariant prime divisors: a boundary divisor of $X$ is a $\\\\mathcal { G }$ -invariant prime divisor on $X$ , a color of $X$ is a $\\\\mathcal { B }$ -invariant prime divisor that is not $\\\\mathcal { G }$ -invariant. We will denote by $B ( X )$ and ${ \\\\mathcal { C } } ( X )$ respectively the set of boundary divisors and colors of $X$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,330,876,367]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"125\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Notation 3.9 For any $i = 1 , \\\\ldots , n + 1$ let us denote by $D _ { i }$ the strict transform of the divisor in $\\\\mathbb { P } ^ { N }$ given by⎜⎝ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[312,384,687,454]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0067/c92405c831209ae328b2b506d66793060dfcb9b000a0b1a1aa3814d0ee4ce96b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"125\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname* { d e t } \\\\left( \\\\begin{array} { c c c } { { z _ { n - i + 1 , m - i + 1 } } } & { { \\\\ldots } } & { { z _ { n - i + 1 , m } } } \\\\\\\\ { { \\\\vdots } } & { { \\\\ddots } } & { { \\\\vdots } } \\\\\\\\ { { z _ { n , m - i + 1 } } } & { { \\\\ldots } } & { { z _ { n , m } } } \\\\end{array} \\\\right) = 0\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,468,879,504]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"125\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"in the intermediate spaces of complete collineations. Similarly, we will denote by $D _ { i } ^ { + }$ the analogous divisors in the intermediate spaces of complete quadrics. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,513,879,569]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"125\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 3.11 Let $\\\\mathcal { X } ( n , m ) _ { i }$ be the intermediate space appearing at the step $1 \\\\leqslant$ $i ~ \\\\leqslant ~ n - 1$ of Construction 3.4. Then \\u0005 $\\\\mathcal { X } ( n , m ) _ { i }$ is spherical. Furthermore,\\u0005 $\\\\operatorname { P i c } ( \\\\mathcal X ( n , m ) _ { i } ) = \\\\mathbb Z [ H , E _ { 1 } , \\\\dots , E _ { i } ]$ and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,582,876,626]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0711/e8e4c53c2e6ade0c62a15dd43ea8d49cbd058ccd7f7c176d5803a0732b38f1d7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"125\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nB ( \\\\mathcal { X } ( n , m ) _ { i } ) = \\\\left\\\\{ \\\\begin{array} { l l } { \\\\{ E _ { 1 } , \\\\dots , E _ { i } \\\\} } & { i f n < m } \\\\\\\\ { \\\\{ E _ { 1 } , \\\\dots , E _ { i } , D _ { n + 1 } \\\\} } & { i f n = m } \\\\end{array} \\\\right. \\\\quad \\\\mathcal { C } ( \\\\mathcal { X } ( n , m ) _ { i } ) = \\\\left\\\\{ \\\\begin{array} { l l } { \\\\{ D _ { 1 } , \\\\dots , D _ { n + 1 } \\\\} } & { i f n < m } \\\\\\\\ { \\\\{ D _ { 1 } , \\\\dots , D _ { n } \\\\} } & { i f n = m } \\\\end{array} \\\\right.\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,639,769,659]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"125\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally, for the intermediate spaces $\\\\mathcal { Q } ( n ) _ { i }$ in Construction 3.7 we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[191,672,806,698]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0970/480167ce8f1e09357959a43dda9923e1c38c81b644b6809ab5a03658b07a1e32.jpg\",\n \"list_items\": null,\n \"page_idx\": \"125\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { \\\\mathcal { B } ( \\\\mathcal { Q } ( n ) _ { i } ) = \\\\{ E _ { 1 } ^ { + } , \\\\ldots , E _ { i } ^ { + } , D _ { n + 1 } ^ { + } \\\\} , \\\\quad \\\\mathcal { C } ( \\\\mathcal { Q } ( n ) _ { i } ) = \\\\{ D _ { 1 } ^ { + } , \\\\ldots , D _ { n } ^ { + } \\\\} . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,711,881,857]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"125\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Note that $\\\\mathcal { X } ( n , m ) _ { i }$ is spherical, even though it is not wonderful, with respect to the action of $\\\\begin{array} { l } { \\\\mathcal { G } ~ = ~ \\\\varsigma L ( n + 1 ) \\\\times S L ( m + 1 ) } \\\\end{array}$ , with the Borel subgroup $\\\\mathcal { B }$ given by pair of upper triangular matrices. Indeed, by [24, Theorem 1] ${ \\\\mathcal { X } } ( n , m )$ is a wonderful variety and $\\\\mathcal { B }$ acts on it with a dense open orbit, and since the actions of $\\\\mathcal { B }$ on ${ \\\\mathcal { X } } ( n , m )$ and $\\\\mathcal { X } ( n , m ) _ { i }$ are compatible with the blow-up maps in Construction 3.4 we conclude that $\\\\mathcal { B }$ acts on $\\\\mathcal { X } ( n , m ) _ { i }$ with a dense open orbit as well. For the definition and the basic properties of wonderful varieties we refer to [2, Section 4.5.5]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,51,697,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"125\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"125\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"121 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,198]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"126\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since the actions of $\\\\mathcal { G }$ on ${ \\\\mathcal { X } } ( n , m )$ and $\\\\mathcal { X } ( n , m ) _ { i }$ are compatible with the blow-up morphism ${ \\\\mathcal { X } } ( n , m ) \\\\ \\\\to \\\\ { \\\\mathcal { X } } ( n , m ) _ { i }$ , the intermediate space $\\\\mathcal { X } ( n , m ) _ { i }$ has at most the same number of boundary divisors and colors as ${ \\\\mathcal { X } } ( n , m )$ . Now, to compute $B ( \\\\mathcal { X } ( n , m ) _ { i } )$ and $\\\\mathcal { C } ( \\\\mathcal { X } ( n , m ) _ { i } )$ it is enough to note that the ones listed in the statement are clearly boundary divisors and colors, and to apply [15, Proposition 3.6]. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,205,574,225]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"126\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 3.12 Let $n < m$ . Then $D _ { 1 } \\\\sim H$ and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[373,238,626,292]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0519/d50ebee67001ab2b1a20f06917d57af0de5b4fba8e321ae610dd40f585842ae1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"126\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nD _ { k } \\\\sim k H - \\\\sum _ { h = 1 } ^ { k - 1 } ( k - h ) E _ { h }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[116,306,879,342]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"126\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for $k = 2 , \\\\ldots , n + 1$ . Furthermore, $\\\\operatorname { E f f } ( \\\\mathcal { X } ( n , m ) _ { i } ) \\\\ = \\\\ \\\\langle E _ { 1 } , \\\\dots , E _ { i } , D _ { n + 1 } \\\\rangle$ and $\\\\mathrm { N e f } ( \\\\mathcal { X } ( n , m ) _ { i } ) = \\\\langle D _ { 1 } , . . . , D _ { i + 1 } \\\\rangle$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,343,546,361]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"126\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $n = m$ we have $D _ { 1 } \\\\sim H$ , $D _ { 1 } ^ { + } \\\\sim H ^ { + }$ and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[179,378,740,432]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0644/7b7e7b795997405e56a3aecb00770d3ce551b215369ea9605fb0135b7e931d35.jpg\",\n \"list_items\": null,\n \"page_idx\": \"126\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nD _ { k } \\\\sim k H - \\\\sum _ { h = 1 } ^ { k - 1 } ( k - h ) E _ { h } , ~ D _ { k } ^ { + } \\\\sim k H ^ { + } - \\\\sum _ { h = 1 } ^ { k - 1 } ( k - h ) E _ { h } ^ { + }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[113,447,744,466]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"126\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for $k = 2 , \\\\ldots , n$ , and for the boundary divisors $E _ { n }$ and $E _ { n } ^ { + }$ we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,481,874,536]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0695/13ebbf8adb88cf947056f952313c8509f72c6dffa34e0bdb9146bb7fc5633cac.jpg\",\n \"list_items\": null,\n \"page_idx\": \"126\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nE _ { n } \\\\sim ( n + 1 ) H - \\\\sum _ { h = 1 } ^ { n - 1 } ( n - h + 1 ) E _ { h } , \\\\quad E _ { n } ^ { + } \\\\sim ( n + 1 ) H ^ { + } - \\\\sum _ { h = 1 } ^ { n - 1 } ( n - h + 1 ) E _ { h } ^ { + } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,567,881,623]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"126\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Furthermore, $\\\\begin{array} { r } { \\\\mathrm { E f f } ( \\\\mathcal { X } ( n ) _ { i } ) = \\\\langle E _ { 1 } , \\\\ldots , E _ { i } , D _ { n + 1 } \\\\rangle , \\\\mathrm { N e f } ( \\\\mathcal { X } ( n ) _ { i } ) = \\\\langle D _ { 1 } , \\\\ldots , D _ { i + 1 } \\\\rangle } \\\\end{array}$ and similarly $\\\\mathrm { E f f } ( \\\\boldsymbol { \\\\mathcal { Q } } ( n ) _ { i } ) = \\\\left. E _ { 1 } ^ { + } , \\\\ldots , E _ { i } ^ { + } , D _ { n + 1 } ^ { + } \\\\right. , \\\\mathrm { N e f } ( \\\\boldsymbol { \\\\mathcal { Q } } ( n ) _ { i } ) = \\\\left. D _ { 1 } ^ { + } , \\\\ldots , D _ { i + 1 } ^ { + } \\\\right.$ for $i = 1 , \\\\ldots , n - 1$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,632,881,669]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"126\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Consider the case $n < m$ , a similar argument works for the case $n = m$ . Let $f : \\\\mathcal { X } ( n , m ) \\\\to \\\\mathbb { P } ^ { N }$ be the blow-up morphism in Construction 3.4. We have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[275,683,722,737]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0327/7bf2f410e40b5942d4e388ee73965afbf16d47d03719156415f286db45742042.jpg\",\n \"list_items\": null,\n \"page_idx\": \"126\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nD _ { k } = \\\\deg ( f _ { * } D _ { k } ) H - \\\\sum _ { h = 1 } ^ { i } \\\\operatorname { m u l t s } _ { e c _ { h } ( \\\\mathcal { S } ) } ( f _ { * } D _ { k } ) E _ { h }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,750,881,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"126\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now, since $f _ { * } D _ { k }$ is the zero locus of a $k \\\\times k$ minor of the matrix $Z$ then $\\\\deg ( f _ { * } D _ { k } ) = k$ . Furthermore, [15, Lemma 3.10] yields $\\\\mathrm { m u l t } _ { \\\\mathbb { S } e c _ { h } ( S ) } ( f _ { * } D _ { k } ) = k - h$ if $h = 1 , \\\\ldots k - 1$ and $\\\\mathrm { m u l t } _ { \\\\mathbb { S } e c _ { h } ( { \\\\cal S } ) } ( f _ { * } D _ { k } ) = 0$ if $h \\\\geqslant k$ , and hence we get (3.13). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[633,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"126\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. L. Huerta and A. Massarenti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"126\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"122 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,881,142]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"127\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By [2, Proposition 4.5.4.4] and [4, Section 2.6] the generators of the effective and the nef cones are respectively generated by the boundary divisors and the colors. Lemma 3.11 then concludes the proof. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,178,478,198]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"127\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.1.1 Generators of $\\\\mathbf { C o x } ( \\\\mathcal { X } ( \\\\pmb { n } , \\\\pmb { m } ) _ { i } )$ \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,214,881,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"127\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $I = \\\\{ i _ { 0 } , \\\\ldots , i _ { k } \\\\}$ , $J = \\\\{ j _ { 0 } , \\\\ldots , j _ { k } \\\\}$ be two ordered sets of indexes with $0 \\\\leqslant$ $i _ { 0 } \\\\leqslant \\\\cdots \\\\leqslant i _ { k } \\\\leqslant n$ and $0 \\\\ \\\\leqslant \\\\ j _ { 0 } \\\\ \\\\leqslant \\\\ \\\\cdots \\\\ \\\\leqslant \\\\ j _ { k } \\\\ \\\\leqslant \\\\ m$ , and denote by $Z _ { I , J }$ the $( k + 1 ) \\\\times ( k + 1 )$ minor of $Z$ built with the rows indexed by $I$ and the columns indexed by $J$ . Similarly, let $Z _ { I , J } ^ { + }$ be the $( k + 1 ) \\\\times ( k + 1 )$ minor of the symmetrization of $Z$ built with the rows indexed by $I$ and the columns indexed by $J$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,315,879,387]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"127\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 3.16 Let $T _ { I , J }$ be the canonical section associated to the strict transform of the hypersurface $\\\\{ \\\\operatorname* { d e t } ( Z _ { I , J } ) = 0 \\\\} \\\\subset \\\\mathbb { P } ^ { N }$ , and let $S _ { j }$ be the canonical section associated to the exceptional divisor $E _ { j }$ in Construction 3.4. Then $\\\\mathbf { C o x } ( \\\\mathcal { X } ( n , m ) ) _ { i }$ is generated by the $T _ { I , J }$ for $1 \\\\leqslant | I | , | J | \\\\leqslant n + 1$ and the $S _ { j }$ for $j = 1 , \\\\dotsc , i$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,387,879,487]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"127\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now, consider $\\\\mathcal { Q } ( n ) _ { i }$ . Let $T _ { I , J } ^ { + }$ be the canonical section associated to the strict transform of the hypersurface $\\\\{ \\\\mathsf { d e t } ( Z _ { I , J } ^ { + } ) ~ = ~ 0 \\\\} ~ \\\\subset ~ \\\\mathbb { P } ^ { N _ { + } }$ , and let $S _ { j } ^ { + }$ be the canonical section associated to the exceptional divisor $E _ { j } ^ { + }$ in Construction 3.7. Then $\\\\mathrm { C o x } ( \\\\mathcal { Q } ( n ) _ { i } )$ is generated by the $T _ { I , J } ^ { + }$ for $1 \\\\leqslant | I | , | J | \\\\leqslant n + 1$ and the $S _ { j } ^ { + }$ for $j = 1 , \\\\dots , i$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,495,881,585]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"127\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof By [2, Theorem 4.5.4.6] if $\\\\mathcal { G }$ is a semi-simple and simply connected algebraic group and $( X , { \\\\mathcal { G } } , { \\\\mathcal { B } } , x _ { 0 } )$ is a spherical variety with boundary divisors $E _ { 1 } , \\\\ldots , E _ { r }$ and colors $D _ { 1 } , \\\\ldots , D _ { s }$ , then $\\\\operatorname { C o x } ( X )$ is generated as a $K$ -algebra by the canonical sections of the $E _ { i }$ ’s and the finite dimensional vector subspaces $\\\\operatorname* { l i n } _ { K } ( { \\\\mathcal { G } } \\\\cdot D _ { i } ) \\\\subseteq \\\\mathbf { C o x } ( X )$ for $1 \\\\leqslant i \\\\leqslant s$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,587,881,677]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"127\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Consider the case $n < m$ . For $\\\\mathcal { X } ( n ) _ { i }$ and $\\\\mathcal { Q } ( n ) _ { i }$ an analogous argument works. By Lemma 3.11 we have that $B ( \\\\mathcal { X } ( n , m ) _ { i } ) = \\\\{ E _ { 1 } , . . . , E _ { i } \\\\}$ and $\\\\mathcal { C } ( \\\\mathcal { X } ( n , m ) _ { i } ) =$ $\\\\{ D _ { 1 } , \\\\ldots , D _ { n + 1 } \\\\}$ . Recall that for any $k = 0 , \\\\ldots , n$ the divisor $D _ { k + 1 }$ is the strict transform of the hypersurface in $\\\\mathbb { P } ^ { N }$ defined by the determinant of the $( k + 1 ) \\\\times$ $( k + 1 )$ most right down minor of the matrix $Z$ . Let us denote by $Z _ { I , J } ^ { r d }$ such minor. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,678,881,758]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"127\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note that $\\\\operatorname* { d e t } ( Z _ { I , J } ^ { r d } ) \\\\ \\\\in \\\\ I ( \\\\mathbb { S } e c _ { k } ( { \\\\cal S } ) )$ . Therefore, considering the action of $\\\\mathcal { G } =$ $S L ( n + 1 ) \\\\times S L ( m + 1 )$ we have that $g \\\\cdot \\\\operatorname* { d e t } ( Z _ { I , J } ^ { r d } ) \\\\in I ( \\\\mathbb { S } e c _ { k } ( { \\\\cal S } ) )$ for any $g \\\\in { \\\\mathcal { G } }$ , and hence $\\\\begin{array} { r } { \\\\operatorname* { l i n } _ { K } ( \\\\mathscr { G } \\\\cdot \\\\mathrm { d e t } ( Z _ { I , J } ^ { r d } ) ) \\\\subseteq I ( \\\\mathbb { S } e c _ { k } ( S ) ) } \\\\end{array}$ . To conclude it is enough to recall that $I ( \\\\mathbb { S } e c _ { k } ( \\\\mathcal { S } ) )$ is generated by the $( k + 1 ) \\\\times ( k + 1 )$ minors of $Z$ . □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,694,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"127\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"127\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"123 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[861,129,879,141]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"127\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"□ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,799,130]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"128\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.2 The Mori Chamber Decomposition of the Intermediate Space of Picard Rank Three \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,153,876,207]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"128\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section, using the techniques introduced in [15, 16], we will study the Mori chamber and stable base locus decomposition for the spaces \\\\$ $\\\\mathcal { X } ( n ) _ { 3 }$ in Construction 3.4.\\\\$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,217,879,273]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"128\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Notation 3.17 We will denote by $\\\\langle v _ { 1 } , \\\\ldots , v _ { s } \\\\rangle$ the cone in $\\\\mathbb { R } ^ { n }$ generated by the vectors $v _ { 1 } , \\\\dots , v _ { s } \\\\in \\\\mathbb { R } ^ { n }$ . Given two vectors $v _ { i } , v _ { j }$ we set $( v _ { i } , v _ { j } ] : = \\\\left. v _ { i } , v _ { j } \\\\right. \\\\backslash \\\\mathbb { R } _ { > 0 } v _ { i }$ and $( v _ { i } , v _ { j } ) : = \\\\big \\\\langle v _ { i } , v _ { j } \\\\big \\\\rangle \\\\setminus ( \\\\mathbb { R } _ { > 0 } v _ { i } \\\\cup \\\\mathbb { R } _ { > 0 } v _ { j } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,280,876,316]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"128\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 3.18 Consider the natural inclusion $i \\\\quad : \\\\quad \\\\mathcal { Q } ( n ) _ { i } \\\\quad \\\\quad \\\\mathcal { X } ( n ) _ { i }$ . Then $i ^ { * } \\\\operatorname { M o v } ( \\\\mathcal { X } ( n ) _ { i } ) = \\\\operatorname { M o v } ( \\\\mathcal { Q } ( n ) _ { i } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,325,879,361]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"128\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof By Proposition $3 . 1 6 \\\\mathrm { C o x } ( \\\\mathcal { Q } ( n ) _ { i } )$ is generated by the pull-backs of the generators of $\\\\mathrm { C o x } ( \\\\mathcal { X } ( n ) _ { i } )$ . Hence the statement follows from [2, Proposition 3.3.2.3]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,369,881,423]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"128\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 3.19 The movable cone of $\\\\mathcal { X } ( n ) _ { 3 }$ is generated by $D _ { 1 } \\\\sim H , D _ { 2 } \\\\sim$ $2 H - E _ { 1 }$ , $D _ { n } \\\\sim n H + ( 1 - n ) E _ { 1 } + ( 2 - n ) E _ { 2 }$ , $P _ { n } \\\\sim n ( n - 1 ) H + n ( 2 - n ) E _ { 1 } +$ $( n - 1 ) ( 2 - n ) E _ { 2 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,424,876,462]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"128\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Similarly, $\\\\mathbf { M o v } ( \\\\mathcal { Q } ( n ) _ { 3 } )$ is generated by $D _ { 1 } ^ { + } \\\\sim H ^ { + }$ , $D _ { 2 } ^ { + } \\\\sim 2 H ^ { + } - E _ { 1 } ^ { + }$ , $D _ { n } ^ { + } \\\\sim$ $n H ^ { + } + ( 1 - n ) E _ { 1 } ^ { + } + ( 2 - n ) E _ { 2 } ^ { + }$ , $P _ { n } ^ { + } \\\\sim n ( n - 1 ) H ^ { + } + n ( 2 - \\\\bar { n } ) E _ { 1 } ^ { + } + ( n - 1 ) ( \\\\bar { 2 } - n ) E _ { 2 } ^ { + } ;$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,468,881,558]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"128\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof We use [2, Proposition 3.3.2.3] which gives a way of computing the movable cone starting from the generators of the Cox ring. More specifically, in order to compute the movable cone it is enough to intersect all the cones generated by the sets of vectors obtained by removing just one element from the set of generators of the Cox ring. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,560,881,666]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"128\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In our specific case, using the generators of $\\\\mathrm { C o x } ( \\\\mathcal { X } ( n ) _ { 3 } )$ in Proposition 3.16, it is immediate to see that this procedure give us the vectors $( 1 , 0 , 0 )$ , $( 2 , - 1 , 0 )$ , $( n , 1 -$ $n , 2 - n )$ and the vector $( n ( n - 1 ) , n ( 2 - n )$ , $( n - 1 ) ( 2 - n ) )$ that is the intersection of the plane generated by $( 1 , 0 , 0 )$ , $( n + 1 , - n , - n + 1 )$ and the plane generated by $( 0 , 1 , 0 )$ , $( n , 1 - n , 2 - n )$ . Finally, the statement on $\\\\mathbf { M o v } ( \\\\mathcal { Q } ( n ) _ { 3 } )$ follows from Lemma 3.18. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,677,879,731]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"128\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3.20 Let $f ( n )$ be the number of chambers in the Mori chamber decomposition of $\\\\mathcal { X } ( n ) _ { 3 }$ . Then $f ( n + 1 ) = f ( n ) + n + 1$ . Furthermore, the decomposition can be described by the following 2-dimensional cross-section of $\\\\mathrm { E f f } ( \\\\mathcal { X } ( n ) _ { 3 } )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[633,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"128\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. L. Huerta and A. Massarenti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"128\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"124 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,93,872,450]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0970/a12e8f6c7295dee86d37b566ab60c2902b398dd1eba16ab1a8ef755ce988c8ba.jpg\",\n \"list_items\": null,\n \"page_idx\": \"129\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[118,471,881,525]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"129\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Furthermore, the same statements hold, by replacing the relevant divisors with their pull-backs via the embedding $i : \\\\mathcal { Q } ( n ) _ { 3 } \\\\to \\\\mathcal { X } ( n ) _ { 3 }$ , for the intermediate space of quadrics $\\\\mathcal { Q } ( n ) _ { 3 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,534,881,587]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"129\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof By [15, Theorem 6.11] the statement holds for $n = 3$ . Consider the case $n = 4$ . Then the Cox ring of $\\\\mathcal { X } ( 4 ) _ { 3 }$ has additional generators belonging to the class of $D _ { 5 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,588,879,642]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"129\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now, let $\\\\tau _ { \\\\mathcal { X } ( 4 ) _ { 3 } }$ be a projective toric variety as in Remark 2.8. Then there is an embedding $i : \\\\mathcal { X } ( 4 ) _ { 3 } \\\\mathcal { T } _ { \\\\mathcal { X } ( 4 ) _ { 3 } }$ such that $i ^ { * } : \\\\mathrm { P i c } ( \\\\mathcal { T } _ { \\\\mathcal { X } ( 4 ) _ { 3 } } ) \\\\mathrm { P i c } ( \\\\mathcal { X } ( 4 ) _ { 3 } )$ is an isomorphism inducing an isomorphism $\\\\mathrm { E f f } ( \\\\mathcal T _ { \\\\mathcal X ( 4 ) _ { 3 } } ) \\\\to \\\\mathrm { E f f } ( \\\\mathcal X ( 4 ) _ { 3 } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,644,881,839]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"129\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since $\\\\tau _ { \\\\mathcal { X } ( 4 ) _ { 3 } }$ is toric, the Mori chamber decomposition of $\\\\mathrm { E f f } ( \\\\mathcal T _ { \\\\mathcal { X } ( 4 ) _ { 3 } } )$ can be computed by means of the Gelfand-Kapranov-Zelevinsky decomposition [2, Section 2.2.2]. Roughly speaking such decomposition is given by all the convex cones we can construct using all the generators of $\\\\cos ( \\\\mathcal { T } _ { \\\\mathcal { X } ( 4 ) _ { 3 } } )$ . Starting from the decomposition for $\\\\tau _ { \\\\mathcal { X } ( 4 ) _ { 3 } }$ we are allowed to add just two more segments, namely $[ E _ { 2 } , D _ { 5 } ] , [ E _ { 1 } , D _ { 5 } ]$ which introduce a new chamber each, and $[ D _ { 1 } , D _ { 5 } ]$ which gives rise two three new chambers by dividing $\\\\left. D _ { 1 } , D _ { 3 } , E _ { 1 } \\\\right. , \\\\left. E _ { 1 } , D _ { 3 } , D _ { 4 } \\\\right. , \\\\left. E _ { 1 } , D _ { 4 } , D _ { 5 } \\\\right.$ . Note that we get exactly $\\\\begin{array} { r l } { f ( 4 ) } & { { } = } \\\\end{array}$ $f ( 3 ) + 4 + 1 = 9 + 4 + 1 = 1 4$ chambers. On the other hand, a priori this is just a refinement of the Mori chamber decomposition of Ef $\\\\yen ( \\\\mathcal { X } (4 ) _ { 3 } )$ . We will prove that it is indeed the Mori chamber decomposition. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,840,879,894]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"129\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note that by considering a general $\\\\bar { \\\\mathbb { P } } ^ { 3 } \\\\times \\\\mathbb { P } ^ { 3 } \\\\subset \\\\mathbb { P } ^ { 4 } \\\\times \\\\mathbb { P } ^ { 4 }$ we get a natural embedding $j : \\\\mathcal { X } ( 3 ) _ { 3 } \\\\ \\\\ \\\\mathcal { X } ( 4 ) _ { 3 }$ . Furthermore, this embedding preserves the generators of the Picard groups in Lemma 3.11 and the generators of the Cox rings in Proposition 3.16. Therefore, all the chambers of $\\\\mathbf { M C D } ( \\\\mathcal { X } ( 3 ) _ { 3 } )$ appear in $\\\\mathbf { M C D } ( \\\\mathcal { X } ( 4 ) _ { 3 } )$ . Furthermore, the segments $[ E _ { 2 } , D _ { 5 } ] , [ E _ { 1 } , D _ { 5 } ]$ must appear in $\\\\mathbf { M C D } ( \\\\mathcal { X } ( 4 ) _ { 3 } )$ simply because by Proposition 3.12 they define two external walls of $\\\\operatorname { E f f } ( \\\\mathcal { X } ( 4 ) _ { 3 } )$ , and the segment $[ D _ { 1 } , D _ { 5 } ]$ must appear as well since it is needed to intercept the vector $P _ { 4 }$ which by Proposition 3.19 is an extremal ray of $\\\\mathbf { M o v } ( \\\\mathcal { X } ( 4 ) _ { 3 } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,694,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"129\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"129\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"130\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,180,881,304]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"130\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now, assuming that the statement holds for $\\\\mathcal { X } ( n \\\\mathrm { ~ - ~ } 1 ) _ { 3 }$ we can prove it for $\\\\mathcal { X } ( n ) _ { 3 }$ arguing exactly as we did in order to pass from $\\\\mathcal { X } ( 3 ) _ { 3 }$ to $\\\\mathcal { X } ( 4 ) _ { 3 }$ . Indeed, arguing exactly as in the previous part of the proof we see that all the chambers of $\\\\mathrm { M C D } ( \\\\mathcal { X } ( n \\\\mathrm { ~ - ~ } 1 ) _ { 3 } )$ must appear in $\\\\mathbf { M C D } ( \\\\mathcal { X } ( n ) _ { 3 } )$ . Moreover, we have to add the segments $[ E _ { 1 } , D _ { n + 1 } ] , [ E _ { 2 } , D _ { n + 1 } ]$ which by Proposition 3.12 give external walls of $\\\\mathrm { E f f } ( \\\\mathcal { X } ( n ) _ { 3 } )$ , and $[ D _ { 1 } , D _ { n + 1 } ]$ needed to intercept the vectors $P _ { n }$ which by Proposition 3.19 is an extremal ray of $\\\\mathbf { M o v } ( \\\\mathcal { X } ( n ) _ { 3 } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,306,879,394]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"130\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally, note that $n - 2$ of the chambers in $\\\\mathcal { X } ( n - 1 ) _ { 3 }$ are subdivided in two chambers each by $[ D _ { 1 } , D _ { n + 1 } ]$ , the new cone $\\\\langle E _ { 1 } , D _ { n } , D _ { n + 1 } \\\\rangle$ also is subdivided in two chambers each by $[ D _ { 1 } , D _ { n + 1 } ]$ , and we must add also the chamber $\\\\langle E _ { 2 } , D _ { n } , D _ { n + 1 } \\\\rangle$ . Hence the number of chambers of $\\\\mathrm { E f f } ( \\\\mathcal { X } ( n ) _ { 3 } )$ is given by $f ( n ) = f ( n - 1 ) + n -$ $2 + 2 + 1 = f ( n - 1 ) + n + 1$ . □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,403,879,495]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"130\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark $3 . 2 I$ By [15, Theorem 6.11] $\\\\mathcal { X } ( 3 ) _ { 3 }$ has nine Mori chambers but just eight stable base locus chambers. Indeed, within the non-convex chamber determined by the divisors $D _ { 1 } , D _ { 2 } , D _ { 3 } , E _ { 2 }$ the stable base locus is $E _ { 2 }$ . Note that this last fact holds true more generally for the space $\\\\mathcal { X } ( n ) _ { 3 }$ for $n \\\\geqslant 3$ . Similarly, this holds also for the spaces of quadrics $\\\\mathcal { Q } ( n ) _ { 3 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,503,879,558]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"130\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 3.22 The varieties $\\\\mathcal { X } ( n ) _ { i }$ and ${ \\\\mathcal { Q } } ( n ) _ { i }$ are Lefschetz divisorially equivalent, and moreover $\\\\mathcal { X } ( n ) _ { 3 }$ and $\\\\mathcal { Q } ( n ) _ { 3 }$ are birational twins for any $n \\\\geqslant 1$ . Furthermore, $\\\\mathcal { X } ( 3 ) _ { 3 }$ and $\\\\mathcal { Q } ( 3 ) _ { 3 }$ are strong birational twins. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,567,879,620]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"130\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof The first statement follows from Proposition 3.12 and Lemma 3.18. The second statement follows then from Theorem 3.20. Finally, the third statement is a consequence of Remark 3.21. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,665,781,687]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"130\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 The Flip of the Spaces of Complete Quadric Surfaces \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,710,879,818]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"130\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3.20 and Corollary 3.22 imply that each of the spaces $\\\\mathcal { X } ( 3 ) , \\\\mathcal { Q } ( 3 )$ of complete collineations of $\\\\mathbb { P } ^ { 3 }$ and complete quadric surfaces, admits a single flip. Moreover, such flips are strongly related to each other. Indeed, by computing the Mori chamber decomposition in both cases, we observe that $i ^ { * } \\\\mathbf { M C D } ( \\\\mathcal { X } ( 3 ) ) \\\\ =$ $\\\\mathbf { M C D } ( \\\\mathcal { Q } ( 3 ) )$ , thus these two spaces are birational twins and the unique flip on $\\\\mathcal { X } ( 3 )$ induces that of $\\\\mathcal { Q } ( 3 )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,819,881,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"130\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section we exhibit the flip of $\\\\mathcal { Q } ( 3 )$ , denoted by $\\\\mathcal { Q } ( 3 ) ^ { + }$ , the space of complete quadric surfaces following [14]. Towards the end of the section, we conjecture the geometry of the flip of $\\\\mathcal { X } ( 3 )$ . We construct $\\\\mathcal { Q } ( 3 ) ^ { + }$ by analyzing a $\\\\mathbb { Z } / 2$ -action on the following Hilbert scheme. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[633,54,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"130\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. L. Huerta and A. Massarenti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,54,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"130\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,881,147]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"131\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 4.1 Let Hilb $\\\\begin{array} { r l } { = } & { { } \\\\mathbf { H i l b } ^ { 2 x + 1 } ( \\\\mathbb { G } ( 1 , 3 ) ) } \\\\end{array}$ denote the Hilbert scheme parametrizing subschemes of $\\\\mathbb { G } ( 1 , 3 ) \\\\subset \\\\mathbb { P } ^ { 5 }$ whose Hilbert polynomial is $P ( x ) = 2 x + 1$ . This space is isomorphic to the following blow-up \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[396,163,599,183]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0111/7ae5ee027e68f225dfca7c9fc77a21c85b467ea2382c02dedcd92e292b7a3a06.jpg\",\n \"list_items\": null,\n \"page_idx\": \"131\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbf { H i l b } \\\\cong B l _ { \\\\mathbb { O G } } \\\\mathbb { G } ( 2 , 5 )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,199,879,235]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"131\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\mathbb { O } \\\\mathbb { G } \\\\subset \\\\mathbb { G } ( 2 , 5 )$ denotes the orthogonal Grassmannian inside the Grassmannian of 2-planes in $\\\\mathbb { P } ^ { 5 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,244,881,336]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"131\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Observe that a generic smooth curve with Hilbert polynomial $P ( x ) = 2 x + 1$ in $\\\\mathbb { P } ^ { 5 }$ is a plane conic $C$ . Thus, its ideal $I _ { C } \\\\subset k [ p _ { 0 } , \\\\dotsc , p _ { 5 } ]$ is generated by a quadric $F$ and three independent linear forms $L _ { 1 } , L _ { 2 } , L _ { 3 }$ . Since $C \\\\subset \\\\mathbb { G } = \\\\mathbb { G } ( 1 , 3 )$ , the equation $F$ is the quadric corresponding to the Grassmannian $\\\\mathbb { G } \\\\subset \\\\mathbb { P } ^ { 5 }$ under the Plücker embedding. This description gives rise to a rational map \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[400,352,599,373]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"1015/a0627f4570e7965d7811b967972613faacfe7fc059537cb6f027e5a6ede654af.jpg\",\n \"list_items\": null,\n \"page_idx\": \"131\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nf : \\\\mathbb { G } ( 2 , 5 ) \\\\xrightarrow { } \\\\xrightarrow { \\\\smile } \\\\mathbb { H i l b }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,388,881,498]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"131\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"by assigning the 2-plane $P$ defined by the independent linear forms $( L _ { 1 } , L _ { 2 } , L _ { 3 } )$ to the ideal $\\\\langle L _ { 1 } , L _ { 2 } , L _ { 3 } \\\\rangle + \\\\langle F \\\\rangle \\\\subset k [ p _ { 0 } , \\\\ldots , p _ { 5 } ] .$ . Observe that the exceptional locus of $f$ consists of planes in $\\\\mathbb { P } ^ { 5 }$ such that there is a containment of ideals $\\\\langle F \\\\rangle \\\\subset$ $\\\\langle L _ { 1 } , L _ { 2 } , L _ { 3 } \\\\rangle$ , that is planes $P$ which are contained in the quadric $\\\\mathbb { G } \\\\subset \\\\mathbb { P } ^ { 5 }$ . The locus parametrizing such planes is exactly the orthogonal Grassmannian $\\\\mathbb { O G }$ . Now, we resolve the rational map $f$ , \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[366,513,633,596]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0297/4acb7cfd4891dff40f397d8f6494e3b322dcda96129e3fa837bb2eb6bb856805.jpg\",\n \"list_items\": null,\n \"page_idx\": \"131\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { l } { { \\\\displaystyle B l _ { \\\\mathbb { O G } } \\\\mathbb { G } ( 2 , 5 ) } } \\\\\\\\ { { \\\\displaystyle \\\\qquad \\\\pi \\\\int _ { { \\\\bf \\\\Phi } } \\\\widetilde { { \\\\bf \\\\Phi } } } } \\\\\\\\ { { \\\\displaystyle \\\\mathbb { G } ( 2 , 5 ) \\\\mathrm { ~ . ~ . ~ . ~ } \\\\ f \\\\longrightarrow \\\\pmb { \\\\mathrm { H i l b } } } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,615,881,765]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"131\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The morphism $\\\\tilde { f }$ is an isomorphism. Indeed, the rational map $f$ is birational as it has an inverse morphism $g : \\\\mathbf { H i l b } \\\\mathbb { G } ( 2 , 5 )$ defined as follows: let $[ C ] \\\\in \\\\mathbf { H i l b }$ be a point, then the ideal $I ( C ) = ( F ) + ( \\\\mathrm { p l a n e } ) \\\\stackrel { \\\\xi } { \\\\mapsto }$ (plane) $\\\\in \\\\mathbb { G } ( 2 , 5 )$ . It is clear that $f \\\\circ g = I d$ , hence $f$ , and consequently $\\\\tilde { f }$ , is birational. Furthermore, $\\\\tilde { f }$ is a bijection. Indeed, since the exceptional divisor $E \\\\subset B l _ { \\\\mathbb { O G } } \\\\mathbb { G } ( 2 , 5 )$ is a $\\\\mathbb { P } ^ { 5 }$ -bundle over $\\\\mathbb { O G }$ , whose points can be thought of as pairs $( P , C )$ , where $P \\\\subset \\\\mathbb { P } ^ { 5 }$ is a 2- plane and $C \\\\subset P$ is a plane conic. Thus, Zariski’s Main Theorem implies that $\\\\tilde { f }$ is an isomorphism. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,773,879,833]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"131\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 4.2 Let Hilb be as above, then $\\\\mathrm { P i c } ( \\\\mathbf { H i l b } ) \\\\cong \\\\langle H ^ { + } , E _ { 2 } ^ { + } , E _ { 1 , 1 } ^ { + } \\\\rangle$ where $H ^ { + }$ is the pullback of the unique generator of the group $A ^ { 1 } ( \\\\mathbb { G } ( 2 , 5 ) )$ and the $E ^ { + } \\\\ '$ are the exceptional divisors of the blow-up. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,51,697,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"131\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"131\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"127 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"132\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof The orthogonal Grassmannian $\\\\mathbb { O G }$ has two components, hence the result follows. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,135,879,189]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"132\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If the base field $K$ is algebraically closed, then for a given smooth quadric $Q \\\\subset$ $\\\\mathbb { P } ^ { 3 }$ , the Fano variety of lines $F _ { 1 } ( Q ) \\\\subset \\\\mathbb { G } ( 1 , 3 )$ consists of two smooth conics. We get a $\\\\mathbb { Z } / 2$ -action on $\\\\mathbf { H i l b } ^ { 2 x + 1 } ( \\\\mathbb { G } ( 1 , 3 ) )$ by exchanging such conics. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,196,874,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"132\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 4.3 There is a nontrivial globally defined $\\\\mathbb { Z } / 2$ -action on $\\\\mathbf { H } \\\\mathbf { i } \\\\mathbf { l } \\\\mathbf { b } ^ { 2 x + 1 }$ $( \\\\mathbb { G } ( 1 , 3 ) )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,241,876,279]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"132\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Let $Q \\\\subset \\\\mathbb { P } ^ { 3 }$ be a smooth quadric hypersurface. The Fano variety of lines $F _ { 1 } ( Q )$ is the zero locus of a section of the following bundle, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[444,294,553,376]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0425/d8d0ac3738467b31146619af5a014c0bc6fa244e5889b2b20da7cb4ddd17bac3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"132\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { c } { { \\\\mathrm { S y m } ^ { 2 } ( S ^ { * } ) } } \\\\\\\\ { { Q _ { \\\\mid L } ~ { \\\\Bigg \\\\{ } ~ { \\\\Bigg \\\\{ } ~ \\\\pi ~ } } \\\\\\\\ { { \\\\mathbb { G } ( 1 , 3 ) } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,399,879,453]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"132\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $S ^ { * }$ is the dual of the tautological bundle $s$ over $\\\\mathbb { G } ( 1 , 3 )$ . A smooth conic in $\\\\mathbb { P } ^ { 5 }$ determines uniquely a 2-plane, thus in the Plücker embedding $\\\\mathbb { G } ( 1 , 3 ) \\\\subset \\\\mathbb { P } ^ { 5 }$ , we have that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,462,708,500]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"132\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"– $F _ { 1 } ( Q )$ determines two 2-planes if $\\\\operatorname { r a n k } ( Q )$ is either 2 or 4, – $F _ { 1 } ( Q )$ determines a single 2-plane if $\\\\operatorname { r a n k } ( Q )$ is either 1 or 3. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,507,881,581]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"132\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Exchanging such planes gives rise to a $\\\\mathbb { Z } / 2$ -action on $\\\\mathbb { G } ( 2 , 5 )$ , the Grassmannian of 2-planes in $\\\\mathbb { P } ^ { 5 }$ . The second condition above, says that such a $\\\\mathbb { Z } / 2$ -action on $\\\\mathbb { G } ( 2 , 5 )$ preserves the orthogonal Grassmannian $\\\\mathbb { O G }$ , hence inducing a $\\\\mathbb { Z } / 2$ -action on the blow-up $\\\\mathbf { H i l b } ^ { 2 x + 1 } ( \\\\bar { \\\\mathbb { G } } ( 1 , 3 ) )$ . □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,588,881,683]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"132\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Observe that there is an $S L _ { 4 } ( \\\\mathbb { C } )$ -action on Hilb induced by the action of $S L _ { 4 }$ on $\\\\mathbb { P } ^ { 3 }$ . This action stratifies Hilb into $S L _ { 4 }$ -orbits compatible with the exceptional divisors $E _ { 2 } ^ { + }$ , $E _ { 1 , 1 } ^ { + }$ . Notice that $\\\\mathbb { Z } / 2$ acts trivially on $S L _ { 4 }$ -orbits of codimension 2. In codimension 1, we have that $\\\\mathbb { Z } / 2$ acts as the identity on the exceptional divisors $E _ { 2 } ^ { + }$ and $E _ { 1 , 1 } ^ { + }$ . Consider the quotient \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[391,699,605,722]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0764/069044543fb0195871dd520d50d2e03948ae92d6b477220fddc265a5895852e1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"132\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathcal { Q } ( 3 ) ^ { + } : = \\\\mathbf { H i l b } / ( \\\\mathbb { Z } / 2 )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,738,881,774]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"132\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where the $\\\\mathbb { Z } / 2$ -action is defined in the previous Lemma 4.3.The main result of this section is the following. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,783,879,839]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"132\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 4.4 There is a flip $f ~ : ~ \\\\mathcal { Q } ( 3 ) \\\\quad \\\\xrightarrow { } - ~ \\\\mathcal { Q } ( 3 ) ^ { + }$ over the Chow variety $\\\\mathbf { C h o w } ^ { 2 x + 1 } ( \\\\mathbb { G } ( 1 , 3 ) )$ , and the flipping locus in $\\\\mathcal { Q } ( 3 )$ is the intersection the divisors $E _ { 1 } \\\\cap E _ { 3 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[633,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"132\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. L. Huerta and A. Massarenti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"132\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"128 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[145,88,710,108]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"133\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The previous result can summarized in the following diagram where $\\\\phi _ { 1 }$ and $\\\\phi _ { 2 }$ are small contractions and the other maps, except for the flip, are all divisorial contractions described in previous sections. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,123,792,208]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0869/2f483848b3007c149d4bee2b4e98581b57335906a62b3de92124e15eeea07209.jpg\",\n \"list_items\": null,\n \"page_idx\": \"133\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[120,237,876,271]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"133\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,273,881,363]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"133\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In order to show that $\\\\mathcal { Q } ( 3 )$ and $\\\\mathcal { Q } ( 3 ) ^ { + }$ are related through a flip, we lift the maps so we avoid the action of $\\\\mathbb { Z } / 2$ . In this scenario, we need to describe the flip for the Hilbert scheme $\\\\mathbf { H i l b } ^ { 2 x + 1 } ( \\\\mathbb { G } ( 1 , 3 ) )$ . In doing so, we are lead to consider the Kontsevich moduli space $\\\\overline { { \\\\mathcal { M } } } _ { 0 , 0 } ( \\\\mathbb { G } , 2 )$ of degree two stable maps into the Grassmannian $\\\\mathbb { G } = \\\\mathbb { G } ( 1 , 3 )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,372,876,408]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"133\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 4.5 There is a nontrivial globally defined $\\\\mathbb { Z } / 2$ -action on the Kontsevich space $\\\\overline { { \\\\mathcal { M } } } _ { 0 , 0 } ( \\\\mathbb { G } ( 1 , 3 ) , 2 )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,415,881,469]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"133\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof We have a generic 2 to 1 morphism from the Kontsevich moduli space\\u0016 \\u0017 $\\\\mathcal { M } \\\\ = \\\\ \\\\overline { { \\\\mathcal { M } } } _ { 0 , 0 } ( \\\\mathbb { G } ( 1 , 3 ) , 2 ) \\\\ = \\\\ \\\\{ ( C , C ^ { * } ) \\\\}$ to the space , $\\\\mathcal { Q } ( 3 )$ of complete quadric surfaces defined as follows \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[378,481,621,534]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0693/6f7a4ed64ed12ae1620a0654b4d401c04ca0462ec8c873cb72d00ee467758add.jpg\",\n \"list_items\": null,\n \"page_idx\": \"133\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n( C , C ^ { * } ) \\\\mapsto \\\\left( \\\\bigcup _ { L \\\\in C } L , C ^ { * } \\\\right)\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,548,881,675]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"133\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where the notation $( S , C ^ { * } )$ means a surface $s$ , and a curve $C ^ { * }$ as its marking. This map is 2 to 1 over the open subset parametrizing smooth quadric surfaces as well as over the divisor of complete quadrics of rank two. Indeed, if $\\\\cup _ { L \\\\in C } L$ sweeps out a smooth quadric $s$ , then $L$ is a ruling of $s$ . The other ruling induces another conic $C ^ { \\\\prime }$ which gets mapped to $s$ . The situation is similar over the locus of complete quadrics of rank two. Note that this map is 1 to 1 outside two such regions. We now define the $\\\\mathbb { Z } / 2$ -action on $\\\\overline { { \\\\mathcal { M } } }$ by identifying the two curves $C$ and $C ^ { \\\\prime }$ . □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,683,876,720]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"133\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corollary 4.6 The quotient of $\\\\overline { { \\\\mathcal { M } } } _ { 0 , 0 } ( \\\\mathbb { G } , 2 )$ by the $\\\\mathbb { Z } / 2$ -action is isomorphic to $\\\\mathcal { Q } ( 3 )$ . In particular, the quotient is smooth. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,728,881,783]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"133\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Let $Z$ denote the quotient of $\\\\overline { { \\\\mathcal { M } } }$ by the $\\\\mathbb { Z } / 2$ -action defined above. Observe that $X _ { 3 }$ and $Z$ are birational and there is a bijection between them. Zariski’s Main Theorem implies now the corollary. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,789,881,864]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"133\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It follows from [8] that $\\\\overline { { \\\\mathcal { M } } } _ { 0 , 0 } ( \\\\mathbb { G } , 2 )$ and $\\\\mathbf { H i l b } ^ { 2 x + 1 } ( \\\\mathbb { G } ( 1 , 3 ) )$ are related through a flip over the Chow variety. Furthermore, the maps in such a flip diagram between these two spaces are $\\\\mathbb { Z } / 2$ -equivariant [14, Section 5.2]. The result in Theorem 4.4 now follows from the previous lemmas. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,864,879,900]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"133\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Following the construction of the space $\\\\mathcal { Q } ( 3 ) ^ { + }$ above, we conjecture a modular interpretation for the flip of the space of complete collineations $\\\\mathcal { X } ( 3 )$ . Let $A \\\\in \\\\mathcal { X } ( 3 )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,51,697,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"133\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,879,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"133\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"129 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,91,881,199]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"134\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"be a generic complete collineation of $\\\\mathbb { P } ^ { 3 }$ . Then, we associate to it a hypersurface of $X = \\\\mathbf { \\\\bar { P } } ^ { 3 } \\\\times \\\\mathbf { \\\\mathbb { P } } ^ { 3 }$ of bidegree $( 1 , 1 )$ , denoted by $Y _ { A }$ ; this is an element of the complete linear system $| \\\\mathcal { O } _ { X } ( 1 , 1 ) | \\\\cong \\\\mathbb { P } ^ { 1 5 }$ . The singular locus of $Y _ { A }$ , when non-empty, is a product of projective spaces. Hence, the space $\\\\mathcal { X } ( 3 )$ parametrizes hypersurfaces $Y _ { A }$ with a marking: complete collineation of the type $Y _ { Q }$ , where $Q \\\\in \\\\mathcal { X } ( i )$ , for some $i < 3$ . The hypersurface $Y _ { A }$ is smooth for generic $A \\\\in \\\\mathcal { X } ( 3 )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,201,881,253]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"134\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On another hand, let $F ( Y _ { A } ) \\\\subset \\\\mathbb { G } ( 1 , 3 ) \\\\times \\\\mathbb { G } ( 1 , 3 )$ be the Fano scheme of ruled surfaces, isomorphic to $\\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 }$ , embedded in $Y _ { A }$ . This is the zero locus of a global section induced by $Y _ { A }$ , \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[412,270,587,351]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0422/300742c1e1a8269ba5e0516e410b805eee0e0696c4537766353891c1f4bdce21.jpg\",\n \"list_items\": null,\n \"page_idx\": \"134\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { c } { S ^ { * } \\\\times S ^ { * } } \\\\\\\\ { { } } \\\\\\\\ { Y _ { A } \\\\left( \\\\begin{array} { l } { { } } \\\\\\\\ { { } } \\\\\\\\ { { \\\\pi } } \\\\end{array} \\\\right) \\\\pi } \\\\\\\\ { \\\\mathbb { G } ( 1 , 3 ) \\\\times \\\\mathbb { G } ( 1 , 3 ) } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,369,669,387]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"134\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $S ^ { * }$ is the dual of the tautological bundle over $\\\\mathbb { G } ( 1 , 3 )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,388,881,477]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"134\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We claim that the subscheme $F ( Y _ { A } )$ fully determines the hypersurface $Y _ { A }$ . Hence, the Hilbert scheme $\\\\mathbf { H i l b } _ { 1 , 1 }$ parametrizing subschemes $F ( Y _ { A } )$ inside the product $\\\\mathbb { G } ( 1 , 3 ) \\\\times \\\\mathbb { G } ( 1 , 3 )$ is birational to $\\\\mathcal { X } ( 3 )$ . In other words, $\\\\mathbf { H i l b } _ { 1 , 1 } \\\\cong \\\\mathcal { X } ( 3 )$ . We conjecture that this rational map is a flip over the Chow variety that parametrizes the cycle classes of $F ( Y _ { A } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,496,881,633]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"134\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acknowledgements The first named author is currently a CONACYT-Research Fellow and during the preparation of this article he was partly supported by the CONACYT Grant CB-2015/253061. The second named author is a member of the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni of the Istituto Nazionale di Alta Matematica “F. Severi” (GNSAGA-INDAM). We thank the organizers of the conferences Birational Geometry and Moduli Spaces, held in Rome, Italy, and Moduli Spaces in Algebraic Geometry and Applications, held in Campinas, Brazil, where part of this work was carried out. We have benefited from discussion with G. Reyes-Ahumada. We thank the referee for the comments that helped us to improve the paper. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,680,248,699]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"134\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[127,722,881,887]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"134\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Araujo, C., Massarenti, A.: Explicit log Fano structures on blow-ups of projective spaces. Proc. Lond. Math. Soc. (3) 113(4), 445–473 (2016). MR 3556488 \\n2. Arzhantsev, I., Derenthal, U., Hausen, J., Laface, A.: Cox rings. Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge (2015). MR 3307753 \\n3. Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010). MR 2601039 \\n4. Brion, M.: Groupe de Picard et nombres caractéristiques des variétés sphériques. Duke Math. J. 58(2), 397–424 (1989). MR 1016427 \\n5. Brion, M.: Variétés sphériques et théorie de Mori. Duke Math. J. 72(2), 369–404 (1993). MR 1248677 \\n6. Brion, M.: Introduction to actions of algebraic groups. Les cours du CIRM, 1 no. 1: Hamiltonian Actions: invariants et classification (2010) \\n7. Castravet, A.-M., Tevelev, J.: Hilbert’s 14th problem and Cox rings. Compos. Math. 142(6), 1479–1498 (2006). MR 2278756 8. Chen, D., Coskun, I.: Stable base locus decompositions of Kontsevich moduli spaces. Mich. Math. J. 59(2), 435–466 (2010). MR 2677631 \\n9. Debarre, O.: Higher-Dimensional Algebraic Geometry. Universitext. Springer, New York (2001). MR 1841091 \\n10. Hassett, B., Lin, H.W., Wang, C.L.: The weak Lefschetz principle is false for ample cones. Asian J. Math. 6(1), 95–100 (2002) \\n11. Hu, Y., Keel, S.: Mori dream spaces and GIT. Mich. Math. J. 48, 331–348 (2000). Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786494 \\n12. Jow, S.-Y.: A Lefschetz hyperplane theorem for Mori dream spaces. Math. Z. 268(1–2), 197– 209 (2011). MR 2805430 \\n13. Laface, A., Massarenti, A., Rischter, R.: On Mori chamber and stable base locus decompositions. Trans. Am. Math. Soc. 373(3), 1667–1700 (2020) \\n14. Lozano Huerta, C.: Birational geometry of the space of complete quadrics. Int. Math. Res. Not. IMRN 2015(23), 12563–12589 (2015). MR 3431630 \\n15. Massarenti, A.: On the birational geometry of spaces of complete forms I: collineations and quadrics (2018). arXiv:1803.09161 \\n16. Massarenti, A.: On the birational geometry of spaces of complete forms II: skew-forms. J Algebra 546, 178–200 (2020) \\n17. Massarenti, A., Rischter, R.: Spherical blow-ups of Grassmannians and Mori dream spaces. J. Algebra 494, 188–219 (2018). MR 3723177 \\n18. Mukai, S.: Counterexample to hilbert’s fourteenth problem for the 3-dimensional additive group. Technical report, Kyoto University, Research Institute for Mathematical Sciences (2001) \\n19. Okawa, S.: On images of Mori dream spaces. Math. Ann. 364(3–4), 1315–1342 (2016). MR 3466868 \\n20. Ottem, J.C.: Birational geometry of hypersurfaces in products of projective spaces. Math. Z. 280(1–2), 135–148 (2015). MR 3343900 \\n21. Perrin, N.: On the geometry of spherical varieties. Transform. Groups 19(1), 171–223 (2014). MR 3177371 \\n22. Szendroi, B.: On the ample cone of an ample hypersurface. Asian J. Math. 7(1), 1–5 (2003). MR 2015238 \\n23. Vainsencher, I.: Schubert calculus for complete quadrics. In: Enumerative Geometry and Classical Algebraic Geometry (Nice, 1981). Progress in Mathematics, vol. 24, pp. 199–235. Birkhäuser, Boston (1982). MR 685770 \\n24. Vainsencher, I.: Complete collineations and blowing up determinantal ideals. Math. Ann. 267(3), 417–432 (1984). MR 738261 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[633,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"134\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. L. Huerta and A. Massarenti \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,52,152,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"134\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"130 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[116,90,881,680]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"135\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,694,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"135\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Lefschetz Principle in Birational Geometry: Birational Twin Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"135\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"131 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,649,166]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"136\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,202,277,220]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"136\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Luigi Lunardon \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,288,879,376]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"136\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract In this survey, we discuss the state of art about the monodromy property for Calabi-Yau varieties. We explain what is the monodromy property for Calabi-Yau varieties and then discuss some examples of Calabi-Yau varieties that satisfy this property. After this recap, we discuss a possible approach to future research in this area. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,396,874,430]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"136\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords Degenerations of Calabi-Yau varieties $\\\\cdot$ Motivic integration $\\\\cdot$ Motivic monodromy conjecture $\\\\cdot$ Motivic zeta function $\\\\cdot$ Triple-points-free models \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,453,861,486]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"136\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2010 Mathematics Subject Classification Primary: 14E18; Secondary: 14G10, \\n14D06 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,525,309,545]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"136\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,569,881,677]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"136\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our aim is the study of the monodromy of degenerations of Calabi-Yau varieties, and we are interested, in particular, in the so-called motivic monodromy conjecture. While degenerations and monodromy are intuitive concepts in complex geometry, translating these ideas in the setting of algebraic geometry is not trivial. In this section we try to give an intuitive description of our topic; hopefully, this picture will guide the reader through the understanding of this paper. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,678,876,713]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"136\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Denote by $D \\\\subset \\\\mathbb { C }$ the set $\\\\{ x \\\\ \\\\in \\\\mathbb { C } \\\\colon | x | \\\\ < \\\\ 1 \\\\}$ . For the moment, a model for a degeneration of a Calabi-Yau is the datum of: a Calabi-Yau variety $X$ over $\\\\mathbb { C }$ , a smooth complex variety $\\\\mathcal { X }$ , endowed with a proper holomorphic map $\\\\pi : \\\\mathcal { X } D$ which is smooth on $D ^ { * } \\\\ = \\\\ D \\\\ \\\\backslash \\\\ \\\\{ 0 \\\\}$ and satisfying the following additional properties: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,897,692,941]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"136\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_8 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,834,546,879]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"136\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"L. Lunardon $( \\\\boxtimes )$ Office 671, Kensington, London e-mail: l.lunardon16@imperial.ac.uk; ll3916@ic.ac.uk \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[804,91,876,139]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"136\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,899,876,909]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"136\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"133 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,881,142]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"137\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,151,879,205]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"137\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(1) there exists $z _ { 0 } \\\\in D , \\\\ z _ { 0 } \\\\not = 0$ such that $\\\\pi ^ { - 1 } ( z _ { 0 } ) = \\\\mathcal { X } _ { z _ { 0 } } \\\\cong X$ ; (2) $X _ { 0 }$ is a strict normal crossing divisor on $\\\\mathcal { X }$ —by this we mean that the irreducible components of $( X _ { 0 } ) _ { r e d }$ are smooth and with transverse intersection. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,216,881,268]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"137\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The divisor $\\\\begin{array} { r c l } { X _ { 0 } } & { = } & { \\\\sum _ { i \\\\in I } N _ { i } E _ { i } } \\\\end{array}$ contains some of the geometrical data of the degeneration, but it is not enough to determine the motivic zeta function. As we see in Sect. 3 some additional information is required. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,270,881,324]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"137\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Up to homotopy, we have a natural action of $\\\\mathbb { Z } = \\\\pi _ { 1 } ( D ^ { * } )$ on the underlying topological space of $X$ . This action induces an action on the cohomology groups $H ^ { i } ( X , \\\\mathbb { Q } )$ . We can think of this as the cohomological datum of the degeneration. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,325,881,521]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"137\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A natural question is how the geometrical and the cohomological data of a degeneration are related. The motivic monodromy conjecture suggests a possible answer to this question; this conjecture states that poles of the motivic zeta function of a degeneration of Calabi-Yau varieties are linked to monodromy eigenvalues. The motivic zeta function is a power series which depends on $\\\\mathcal { X } ^ { \\\\ast }$ —the restriction of the model $\\\\mathcal { X }$ over $D ^ { * }$ —and on a relative volume form $\\\\omega$ on $\\\\mathcal { X } ^ { \\\\ast }$ . The motivic zeta function encodes all these data and some more information about the geometry of the special fiber. A degeneration of $X$ satisfies the monodromy property if any pole of the motivic zeta function determines an eigenvalue in monodromy. The motivic monodromy conjecture states that Calabi-Yau varieties satisfy the monodromy property. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,522,881,701]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"137\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The motivic monodromy conjecture is the more recent of a series of similar conjectures in different settings. The first conjecture of this series is known as the $p$ -adic monodromy conjecture and it was suggested by Igusa. Given a polynomial $f \\\\in \\\\mathbb { Z } [ x _ { 1 } , \\\\ldots , x _ { n } ]$ , this conjecture suggests a connection between the associated $p$ -adic zeta function and the local monodromy eigenvalues of $f \\\\colon \\\\mathbb { C } ^ { n } \\\\to \\\\mathbb { C }$ . After Kontsevich’s work on motivic integration, Denef and Loeser proposed an upgrade of this conjecture in [6]. Given a polynomial $f \\\\in \\\\mathbb { C } [ x _ { 1 } , \\\\ldots , x _ { n } ]$ , they introduced the associated motivic zeta function and formulated the motivic monodromy conjecture for hypersurface singularities, which statement is analogue to the one of the $p$ -adic monodromy conjecture. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,704,881,900]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"137\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The main aim of this survey is to present this topic in a rigorous way. In Sect. 2 we give the definition of a degeneration of a Calabi-Yau varieties and we explain what a model is. In Sect. 3 we recall some basic definitions in motivic integration and state the monodromy property for Calabi-Yau varieties. The main technical issue is that the motivic zeta function is a power series on a certain localization of an equivariant Grothendieck ring of varieties, so we have to be careful when we talk about poles of the function. For the definition of the monodromy zeta function and the A’Campo formula we refer to [16]. The equivariant version of the Grothendieck ring of varieties was introduced in [10]. The definition of the motivic zeta function and the Denef and Loeser’s formula are the ones given in [9]. The definitions of poles of a power series with coefficients in the Grothendieck ring of varieties were given in [17]. In Sect. 4 we talk about abelian varieties, Calabi-Yau varieties which admit an equivariant Kulikov model and Kummer surfaces (we refer to [9] for the first two cases and to [18] for the last one). In Sect. 5 we discuss triple-points-free models of K3 surfaces (we refer to [11, 12] and [15]). Finally, in Sect. 6 we briefly discuss some directions of the research in this topic. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[779,54,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"137\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"L. Lunardon \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,54,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"137\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,180]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"138\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,225,624,246]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"138\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 Models for Calabi-Yau Varieties over $\\\\pmb { K }$ \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,268,881,340]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"138\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the introduction we talked about degenerations of Calabi-Yau varieties, the picture we described was intuitive and geometric, however, it was not rigorous. In this section we define rigorously what we mean by degenerations of Calabi-Yau varieties and what a model is. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,342,881,394]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"138\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We fix the following notations $k ~ = ~ \\\\mathbb { C }$ , $K \\\\ = \\\\ \\\\mathbb { C } ( ( t ) )$ and $R ~ = ~ \\\\mathbb { C } [ [ t ] ]$ . For every positive integer $d$ , we have ${ \\\\cal K } ( d ) = \\\\mathbb { C } ( ( \\\\sqrt [ d ] { t } ) ) , { \\\\cal R } ( d ) = \\\\mathbb { C } [ [ \\\\sqrt [ d ] { t } ] ]$ . We set $\\\\overline { { K } } = \\\\bar { \\\\mathbb { C } } \\\\{ \\\\{ t \\\\} \\\\}$ , the field of Puiseux series: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[403,409,589,460]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0953/17364bcff87f24ece907057e436793fab65fbd7d7db37ec1d78219be67e5ea89.jpg\",\n \"list_items\": null,\n \"page_idx\": \"138\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathbb { C } \\\\{ \\\\{ t \\\\} \\\\} = \\\\bigcup _ { d = 1 } ^ { \\\\infty } K ( d ) ;\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,474,881,533]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"138\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"this is an algebraic closure of $K$ . The group $\\\\operatorname { G a l } ( K ( d ) / K )$ is canonically isomorphic to $\\\\mu _ { d }$ , the group of $d$ -th roots of unity in $k$ . The group $\\\\widehat { \\\\mu }$ is the profinite group of roots of unity in $k$ , it is obtained as $\\\\varprojlim \\\\mu _ { d }$ . We have that ${ \\\\widehat { \\\\mu } } \\\\cong \\\\operatorname { G a l } ( { \\\\overline { { K } } } / K )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,539,879,575]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"138\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 2.1 (Calabi-Yau Varieties) A Calabi-Yau variety $X$ over $K$ is a smooth, proper, geometrically connected variety with trivial canonical sheaf. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,584,876,620]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"138\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 2.2 (Abelian Variety) An abelian variety $A$ over $K$ is a smooth, proper, geometrically connected commutative group-scheme over $K$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,627,879,683]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"138\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 2.3 All abelian varieties are Calabi-Yau varieties. In fact, since they are group varieties, their tangent bundle is trivial. Thus it follows that the top exterior power of the cotangent bundle is the trivial line bundle. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,692,876,728]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"138\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 2.4 (K3 Surfaces) A K3 surface over $K$ is a 2-dimensional Calabi-Yau variety $X$ over $K$ with $H ^ { 1 } ( X , { \\\\mathcal { O } } _ { X } ) = 0$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,735,881,791]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"138\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 2.5 If we restrict out attention to surfaces over $\\\\overline { { K } }$ , it follows by the Enriques-Kodaira classification, that a 2 dimensional Calabi-Yau variety is either an abelian or a K3 surface. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,798,879,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"138\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 2.6 (Model) Let $X$ be a proper and smooth $K$ -scheme; a model for $X$ over $R$ is a flat $R$ -algebraic space $\\\\mathcal { X }$ endowed with an isomorphism of $K$ -schemes: $\\\\mathcal { X } _ { K } = \\\\mathcal { X } \\\\times _ { R } K \\\\to X$ . We say that $\\\\mathcal { X }$ is a strict normal crossing model (snc model) for $X$ if it is regular and proper over $R$ , and $\\\\mathcal { X } _ { k } = \\\\mathcal { X } \\\\times _ { R } k$ is a strict normal crossing divisor on $\\\\mathcal { X }$ . The variety $\\\\mathcal { X } _ { k }$ is called special fiber, while $\\\\mathcal { X } _ { K }$ is called generic fiber. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,719,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"138\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"138\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,88,876,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"139\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The special fiber, under this definition, need not to be reduced. If it is reduced, then the model is called semistable. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,135,881,225]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"139\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 2.7 In this remark we compare this definition of a model to the one that we gave in the introduction. The space SpecR corresponds to the disk $D$ ; the generic fiber corresponds to the degeneration over the punctured disk, while the special fiber corresponds to the fiber over 0. The generic fiber over $\\\\overline { { K } }$ corresponds to the universal fiber of the degeneration, i.e. the base change to a universal covering space of $D ^ { * }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,232,881,340]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"139\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 2.8 Snc models always exist by Hironaka’s resolution of singularities, while semistable models do not, in general. However, by the semi-stable reduction Theorem [13], given any proper model $\\\\mathcal { X }$ of $X$ there exist a positive integer $d$ and a semistable model $\\\\mathcal { X } _ { d }$ of $X \\\\times _ { K } K ( d )$ that dominates $\\\\mathcal { X } \\\\times _ { R } R ( d )$ . If, moreover, $X$ is projective, then also $\\\\mathcal { X } _ { d }$ may be taken projective. It is important to notice that \\b $\\\\mathcal { X } _ { d }$ is not a model for $X$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,349,881,459]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"139\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Notation 2.9 Choose an snc model $\\\\mathcal { X }$ for the Calabi-Yau variety $X$ . Express the special fiber as the divisor # $\\\\mathcal { X } _ { k } ~ = ~ \\\\sum _ { i \\\\in I } N _ { i } E _ { i }$ . For the rest of the paper we fix the following notation: for any $\\\\textit { \\\\textbf { J } } \\\\subset \\\\textit { \\\\textbf { I } }$ , we define $E _ { J } ~ = ~ \\\\bigcap _ { j \\\\in J } E _ { j }$ and $E _ { J } ^ { \\\\circ } \\\\ =$ $E _ { J } \\\\setminus { \\\\left( \\\\bigcup _ { i \\\\not \\\\in J } { E _ { i } } \\\\right) } .$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,466,744,486]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"139\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 2.10 Let $X$ be the K3 surface in $\\\\mathbb { P } _ { K } ^ { 3 }$ given by the equation: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[261,501,733,525]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0962/9dac773462ac603338b6bf42432f91c5c29795294965d4bf06895f35c139a8ed.jpg\",\n \"list_items\": null,\n \"page_idx\": \"139\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nx ^ { 2 } w ^ { 2 } + y ^ { 2 } w ^ { 2 } + z ^ { 2 } w ^ { 2 } + x ^ { 4 } + y ^ { 4 } + z ^ { 4 } + t w ^ { 4 } = 0 .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,540,881,612]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"139\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $\\\\mathcal { X }$ be the closed subscheme of $\\\\mathbb { P } _ { R } ^ { 3 }$ given by the same equation. The scheme $\\\\mathcal { X }$ is regular, but the special fiber $\\\\mathcal { X } _ { k }$ is a singular surface. The only singular point is $P = [ 0 , 0 , 0 , 1 ]$ ; the singularity at that point is a canonical singularity of type $A _ { 1 }$ . If we blow up $\\\\mathcal { X }$ at $P$ , we obtain an snc model for $X$ , call it $\\\\mathcal { X } ^ { \\\\prime }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,614,879,687]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"139\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The special fiber of this model is non-reduced, so in particular it is not a semistable model. As a divisor the special fiber of this model is $\\\\mathcal { X } _ { k } ^ { \\\\prime } = D + 2 E$ , where $D$ is the proper transform of $\\\\mathcal { X } _ { k }$ , which is a smooth K3 surface, and $E \\\\cong \\\\mathbb { P } _ { k } ^ { 2 }$ . The intersection of $D$ and $E$ is transverse and it is a smooth conic in $E$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,732,708,753]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"139\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 Monodromy Property for Calabi-Yau Varieties \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,777,881,831]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"139\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section we explain what the motivic monodromy property for Calabi-Yau varieties is. First we explain what is the monodromy action; then we introduce the motivic zeta function and explain how they are, conjecturally, related. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[779,54,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"139\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"L. Lunardon \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"139\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,453,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"140\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.1 The Monodromy Action \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,130,881,199]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"140\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recall that we fixed ${ \\\\overline { { K } } } = \\\\mathbb { C } \\\\{ \\\\{ t \\\\} \\\\}$ , the field of Puiseux series; let $\\\\sigma$ be the canonical topological described as is groupand is cal $\\\\operatorname { G a l } ( { \\\\overline { { K } } } / K )$ . The generator odromy operator $\\\\sigma$ can be $\\\\begin{array} { r } { \\\\sigma = \\\\left( \\\\exp \\\\left( \\\\frac { 2 \\\\pi i } { d } \\\\right) \\\\right) _ { d > 0 } } \\\\end{array}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,204,881,279]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"140\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3.1 (Monodromy Eigenvalue) If $X$ is a smooth, proper variety over $K$ , then, for all $i$ , the monodromy operator $\\\\sigma$ acts on the $l$ -adic étale cohomology group $H _ { \\\\mathit { e t } } ^ { i } ( X \\\\times _ { K } { \\\\overline { { K } } } , \\\\mathbb { Q } _ { l } )$ . We say that $\\\\lambda$ is a monodromy eigenvalue if there exists an index $i$ such that $\\\\lambda$ is an eigenvalue for the action of $\\\\sigma$ on $H _ { { \\\\acute { e } } t } ^ { i } ( X \\\\times _ { K } { \\\\overline { { K } } } , \\\\mathbb { Q } _ { l } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,286,879,322]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"140\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3.2 (Monodromy Zeta Function) The monodromy zeta function of $X$ is defined as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[209,334,788,384]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0267/b720da0428fab0941a53e24cff7a8bfc6a2bd055ab1dcc3fee4af4badc30eb91.jpg\",\n \"list_items\": null,\n \"page_idx\": \"140\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\zeta _ { X } ( T ) = \\\\prod _ { i > 0 } \\\\left( { \\\\operatorname * { d e t } \\\\left( { T \\\\cdot \\\\operatorname { I d } - \\\\sigma | _ { H _ { \\\\mathit { e t } } ^ { i } ( X \\\\times _ { K } \\\\overline { { K } } , \\\\mathbb { Q } _ { l } ) } } \\\\right) } \\\\right) ^ { ( - 1 ) ^ { i + 1 } } \\\\in \\\\mathbb { Q } _ { l } ( T ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,396,881,543]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"140\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 3.3 The monodromy zeta function does not encode all the information about the monodromy eigenvalues, in fact, some cancelations may occur. Moreover, there is an even more natural function that encodes all the monodromy eigenvalue— namely, the product of all the characteristic polynomials. One of the main reason why the monodromy zeta function is such a useful tool to study the monodromy eigenvalues is that Theorem 3.5 gives an alternative, and often easier, way to compute this function, while computing all the characteristic polynomials of the monodromy action is usually more complicated. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,549,881,605]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"140\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 3.4 If $X$ is a K3 surface over $K$ , then $H _ { { \\\\acute { e } } t } ^ { i } ( X \\\\times _ { K } { \\\\overline { { K } } } \\\\mathbb { Q } _ { l } )$ is trivial in odd degrees; as a consequence, the monodromy zeta function of a K3 surface has all the\\b monodromy eigenvalues as poles. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,612,881,669]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"140\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3.5 (A’Campo’s Formula, [16] Theorem 2.6.2) Let \\u0006 \\u0007 $X$ be a smooth and proper $K$ -variety. Fix an snc model $\\\\mathcal { X }$ for $X$ with special fiber $\\\\begin{array} { r } { \\\\mathcal { X } _ { k } = \\\\sum _ { i \\\\in I } N _ { i } E _ { i } } \\\\end{array}$ . Then the monodromy zeta function is given by \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[339,681,656,729]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0894/3ac903e14405c12b72bcd605cbfbc3910afa03758192da86f3f8d0cd23cb05ad.jpg\",\n \"list_items\": null,\n \"page_idx\": \"140\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\zeta _ { X } ( T ) = \\\\prod _ { i \\\\in I } \\\\left( T ^ { N _ { i } } - 1 \\\\right) ^ { - \\\\chi _ { t o p } ( E _ { i } ^ { \\\\circ } ) } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[111,744,808,764]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"140\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where χt op is the topological Euler characteristic and $E _ { i } ^ { \\\\circ } = E _ { i } \\\\setminus \\\\bigcup _ { j \\\\neq i } E _ { j }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[113,770,820,791]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"140\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 3.6 If follows from A’Campo formula that $\\\\zeta _ { X } ( T ) \\\\in \\\\mathbb { Q } ( T ) \\\\subset \\\\mathbb { Q } _ { l } ( T )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,51,722,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"140\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"140\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"137 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,487,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"141\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.2 The Motivic Zeta Function \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,132,881,205]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"141\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Next we discuss the second main ingredient to understand the monodromy property: the motivic zeta function. First we need some background in motivic integration. Since we want to keep track of the Galois action of $\\\\operatorname { \\\\bar { G } a l } ( { \\\\overline { { K } } } / K )$ , instead of working with the usual Grothendieck ring of varieties, we use an equivariant version of it. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,213,879,250]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"141\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3.7 (Category of $G$ -schemes Over $k$ ) Given any group $G$ , we define the category of $G$ -schemes over $k$ , denote by $\\\\mathbf { S c h } _ { k , G }$ , as the category that has: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,259,760,295]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"141\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(1) as objects separated $k$ -schemes of finite type with good $G$ -action; \\n(2) as morphisms $G$ -equivariant morphisms of $k$ -schemes. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,304,874,340]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"141\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We say that the action of the group $G$ on the scheme $X$ is good if $X$ has a finite partition in $G$ -stable affine subschemes. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,348,881,439]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"141\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 3.8 The definition of good action of the group $G$ on the scheme $X$ that we use is the one given in [9, (2.2.1)], and it is weaker than the one that is commonly used, i.e. that $X$ can be covered by $G$ -stable affine open subschemes. The reason we preferred our definition is that it can be generalized to algebraic spaces, and it gives rise to the same Grothendieck ring as the usual one. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,447,881,539]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"141\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3.9 (Equivariant Grothendieck Ring, [10] Definition 4.1) Fix a finite group $G$ . The equivariant Grothendieck ring of $G$ -varieties over $k$ is the ring generated as an abelian group by the isomorphism classes of objects $X \\\\in \\\\mathbf { S c h } _ { k , G }$ , with the ring structure given by the fiber product over $k$ , with the additional relations: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,548,838,566]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"141\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(1) (Scissor relation) Given a $G$ -scheme $X$ and a closed $G$ -subscheme $Y$ , then \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[414,582,621,603]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0453/b0f88a161f5db8c0f178c137f2c49ee0e969d17b3c9f9e08467171cc3b6adb8b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"141\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n[ X ] = [ Y ] + [ X \\\\setminus Y ] .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,618,879,656]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"141\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(2) Given $A _ { 1 }$ and $A _ { 2 }$ two $G$ -equivariant affine bundles of rank $d$ over a scheme $S \\\\in \\\\mathbf { S c h } _ { k , G }$ we have $[ A _ { 1 } ] = [ A _ { 2 } ]$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,663,380,684]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"141\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$K _ { 0 , k } ^ { G }$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,690,879,764]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"141\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 3.10 An algebraic space $X$ with good $G$ -action defines a class in the equivariant Grothendieck ring. In fact, since $X$ admits a partition in $G$ -stable affine schemes, it is possible to use the scissor relation to construct the class in the equivariant Grothendieck ring. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,771,881,830]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"141\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For our purposes, the ring $K _ { 0 , k } ^ { G }$ is not enough, we have to invert an element, namely the class $\\\\mathbb { L } = [ \\\\mathbb { A } ^ { 1 } ]$ , with trivial action of the group $G$ . Thus we define the ring $\\\\mathcal { M } _ { k } ^ { G }$ to be the ring \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[410,846,589,872]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0401/0bf5cb21ddebb82f06532c6f58f0c26118e2e6aa19e7b9e13e62f580146e0804.jpg\",\n \"list_items\": null,\n \"page_idx\": \"141\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r } { \\\\mathcal { M } _ { k } ^ { G } = K _ { 0 , k } ^ { G } [ \\\\mathbb { L } ^ { - 1 } ] . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[779,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"141\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"L. Lunardon \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"141\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"138 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,879,127]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"142\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If we have a profinite group ${ \\\\widehat { G } } = \\\\varprojlim G _ { i }$ , with all the groups $G _ { i }$ finite, then we define: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[416,138,578,169]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0860/77a561fbda99e45b89a3813aa345f30075390715ded431d3957fa2f25702e29a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"142\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathcal { M } _ { k } ^ { \\\\widehat { G } } = \\\\varinjlim \\\\mathcal { M } _ { k } ^ { G _ { i } }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,180,881,291]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"142\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3.11 (Equivariant Weak Néron Model) Let $X$ be a smooth proper $K$ - scheme. For every $d > 0$ , set $X ( d ) = X \\\\times _ { K } K ( d )$ . There is an action of $\\\\mu _ { d }$ on $X ( d )$ . An equivariant weak Néron model for $X ( d )$ is a separated and smooth $R ( d )$ -scheme $\\\\mathcal { X }$ , with a good action of $\\\\mu _ { d }$ . Moreover, we require that there exists an isomorphism of $K ( d )$ -schemes ${ \\\\mathcal { X } } _ { K ( d ) } \\\\to X ( d )$ which is $\\\\mu _ { d }$ -equivariant and such that the natural map ${ \\\\mathcal { X } } ( R ( d ) ) \\\\to X ( K ( d ) )$ is a bijection. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,298,881,388]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"142\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 3.12 An equivariant weak Néron model always exists, as explained in [9, 2.2.3]. If we have $\\\\mathcal { X }$ an snc model for a Calabi-Yau variety $X$ , then it is possible to construct an equivariant weak Néron model for $X ( d )$ . We have just to normalize the pullback of $\\\\mathcal { X }$ on the new base, apply a $\\\\mu _ { d }$ -equivariant resolution of singularities and then restrict to the $R ( d )$ -smooth locus. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,397,881,543]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"142\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now we are ready to talk about motivic integration. We fix our variety $X$ over $K$ with trivial canonical bundle, and choose a volume form $\\\\omega$ on $X$ . Denote by $\\\\omega _ { d }$ the pullback of $\\\\omega$ to $X ( d )$ . Choose a weak equivariant Néron model $\\\\mathcal { X }$ for $X ( d )$ . For every connected component $C$ of the special fiber $\\\\mathcal { X } _ { k }$ , the order of $\\\\omega _ { d }$ along $C$ is the unique integer $n$ such that $t ^ { - n / d } \\\\omega _ { d }$ is a generator for the sheaf $\\\\omega _ { \\\\mathcal { X } / R ( d ) }$ locally at the generic point of $C$ . For every integer $i$ , let $C ( i )$ the union of the connected components of $\\\\mathcal { X } _ { k }$ of order $i$ . It is important to remark that $C ( i )$ is stable under the action of $\\\\mu _ { d }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,551,881,587]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"142\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3.13 (Motivic Integral) With the above notation the motivic integral of $\\\\omega _ { d }$ on $X ( d )$ is defined as: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[334,600,658,653]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0197/7a42795e2a595061233637fee611dce32691c4e9ea4b7a09bb5d009b65e43d0e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"142\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\int _ { X ( d ) } | \\\\omega _ { d } | = \\\\sum _ { i \\\\in \\\\mathbb { Z } } [ C ( i ) ] \\\\mathbb { L } ^ { - i } \\\\in \\\\mathcal { M } _ { k } ^ { \\\\widehat { \\\\mu } } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,668,879,704]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"142\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 3.14 ([9], Proposition 2.2.5) The motivic integral of $\\\\omega _ { d }$ on $X ( d )$ is independent of the choice of the weak Néron model $\\\\mathcal { X }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,713,881,767]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"142\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3.15 (Motivic Zeta Function) Let ⎛ ⎞ $X$ be a Calabi-Yau variety over $K$ , choose $\\\\omega$ a volume form on $X$ . The motivic zeta function of\\t - $X$ with respect to the volume form $\\\\omega$ is: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[289,777,710,848]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0978/eeeaaa24e623d7d60ad896ece68843d802e93e9601512227f5a02f4e40e0057b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"142\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nZ _ { X , \\\\omega } ( T ) = \\\\sum _ { d > 0 } \\\\left( \\\\intop _ { X \\\\left( d \\\\right) } \\\\left| \\\\omega _ { d } \\\\right| \\\\right) T ^ { d } \\\\in \\\\mathcal { M } _ { k } ^ { \\\\widehat { \\\\mu } } [ [ T ] ] .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,861,881,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"142\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As in the case of the monodromy zeta function, once we fix an snc model, there exists an alternative way to compute the motivic zeta function. Assume that $\\\\mathcal { X }$ is an snc model for $X$ , and let $\\\\begin{array} { r } { \\\\mathcal { X } _ { k } = \\\\sum _ { i \\\\in I } N _ { i } E _ { i } } \\\\end{array}$ be the special fiber, the reduced special fiber is $\\\\begin{array} { r } { \\\\mathcal { X } _ { k , r e d } = \\\\sum _ { i \\\\in I } E _ { i } } \\\\end{array}$ ∈. The volume form $\\\\omega$ on $X$ defines a rational section of the line bundle $\\\\omega \\\\chi _ { / R } ( \\\\mathcal { X } _ { k , r e d } - \\\\mathcal { X } _ { k } )$ on $\\\\mathcal { X }$ . To this section we can associate a divisor supported on the special fiber. This divisor is of the form . $\\\\textstyle \\\\sum _ { i \\\\in I } \\\\nu _ { i } E _ { i }$ . By the numerical data of $E _ { i }$ we mean the couple $( N _ { i } , \\\\nu _ { i } )$ . For any subset $J \\\\subset I$ consider the varieties $E _ { J }$ and $E _ { J } ^ { \\\\circ }$ , defined as in Notation 2.9, and set $N _ { J } = \\\\operatorname* { g c d } \\\\{ N _ { j } | j \\\\in J \\\\}$ . Let $\\\\mathcal { X } ( N _ { J } )$ be the normalization of $\\\\mathcal { X } \\\\times _ { R } R ( N _ { J } )$ . Then $\\\\widetilde { E _ { J } ^ { \\\\circ } } = \\\\mathcal { X } ( N _ { J } ) \\\\times _ { \\\\mathcal { X } } E _ { J } ^ { \\\\circ }$ is a Galois cover of $E _ { J } ^ { \\\\circ }$ (an explicit description of this cover can be found in Section 2.3 of [16]). Now we are ready to give an alternative description of the motivic zeta function. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,51,722,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"142\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"142\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"139 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,271]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"143\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,280,876,316]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"143\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3.16 (Denef and Loeser’s Formula, [2], Corollary 4.3.2) In the above\\t . setting we have: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[189,331,810,382]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0693/f2ca1d1ac6c5c4456609b7779ce60a276572d358912f27a431234b3216c06867.jpg\",\n \"list_items\": null,\n \"page_idx\": \"143\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nZ _ { X , \\\\omega } ( T ) = \\\\sum _ { \\\\theta \\\\neq J \\\\subset I } [ \\\\widetilde { E _ { J } ^ { \\\\circ } } ] ( \\\\mathbb { L } - 1 ) ^ { | J | - 1 } \\\\prod _ { j \\\\in J } \\\\frac { \\\\mathbb { L } ^ { - \\\\nu _ { j } } T ^ { N _ { j } } } { 1 - \\\\mathbb { L } ^ { - \\\\nu _ { j } } T ^ { N _ { j } } } \\\\in \\\\mathcal { M } _ { k } ^ { \\\\widehat { \\\\mu } } [ [ T ] ]\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,396,881,453]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"143\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is clear from Definition 3.15 that the motivic zeta function is a power series with coefficients in the ring $\\\\mathcal { M } _ { k } ^ { \\\\widehat { \\\\mu } }$ . However, the definition of poles of this function is not immediate. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,456,881,546]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"143\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3.17 (Poles, [17]) Let $\\\\begin{array} { r } { Z ( T ) \\\\ \\\\in \\\\ \\\\mathcal { M } _ { k } ^ { \\\\widehat { \\\\mu } } \\\\left[ T , \\\\frac { 1 } { 1 - \\\\mathbb { L } ^ { a } T ^ { b } } \\\\right] _ { ( a , b ) \\\\in S } } \\\\end{array}$ be a rational function over $\\\\mathcal { M } _ { k } ^ { \\\\widehat { \\\\mu } }$ , and choose $q \\\\in \\\\mathbb { Q }$ . We say that the rational number $q$ is a pole of order at most $m \\\\geq 1$ for the function $Z ( T )$ if there exists a set $\\\\mathcal { P }$ whose elements are multisets contained in $\\\\mathbb { Z } \\\\times \\\\mathbb { Z } _ { > 0 }$ such that: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,554,881,612]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"143\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(1) each multiset $P \\\\in \\\\mathcal { P }$ contains at most $m$ elements $( a , b )$ such that ${ \\\\frac { a } { b } } = q$ and (2) $Z ( T )$ is an element in the $\\\\mathcal { M } _ { k } ^ { \\\\widehat { \\\\mu } } [ T ]$ -submodule of $\\\\mathcal { M } _ { k } ^ { \\\\widehat { \\\\mu } } [ [ T ] ]$ generated by the elements of the form \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[419,624,612,671]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0856/a435a40866eca6781a068f25d802297fe31dc21f2821b509773c1f1e1a8c3aad.jpg\",\n \"list_items\": null,\n \"page_idx\": \"143\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\prod _ { ( a , b ) \\\\in P } { \\\\frac { 1 } { ( 1 - \\\\mathbb { L } ^ { a } T ^ { b } ) } } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[157,686,378,704]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"143\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for all multisets $P \\\\in \\\\mathcal { P }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,713,881,749]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"143\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The order of a pole at $q$ is the minimal $m$ such that $Z ( T )$ has a pole at $q$ of order at most $m$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,758,881,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"143\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 3.18 The reason why the definition of a pole is so involved is that the ring $\\\\mathcal { M } _ { k } ^ { \\\\widehat { \\\\mu } }$ is not a domain. Actually in [19] it is proven that, if the base field has characteristic zero, not even the usual Grothendieck ring of varieties $K _ { 0 , k }$ is a domain. Moreover, if $k = \\\\mathbb { C }$ it was proven in [1] that $\\\\mathbb { L }$ is a zero-divisor in $K _ { 0 , k }$ . It was proven in [7] that $\\\\mathcal { M } _ { k } = K _ { 0 , k } [ \\\\mathbb { L } ^ { - 1 } ]$ is not a domain, the background to this result is presented in Appendix A of [3]; since $\\\\mathcal { M } _ { k }$ injects into $\\\\mathcal { M } _ { k } ^ { \\\\hat { \\\\mu } }$ (as varieties with trivial $\\\\widehat { \\\\mu }$ action) it follows that the latter is not a domain either. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[779,54,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"143\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"L. Lunardon \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,152,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"143\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"140 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,876,124]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"144\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"At this point we have all the background needed to explain what the monodromy property for Calabi-Yau varieties is. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,133,879,207]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"144\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3.19 ([9], Definition 2.3.5) Given a Calabi-Yau variety $X$ over $K$ and a volume form $\\\\omega$ on $X$ we say that the couple ' ( $( X , \\\\omega )$ satisfies the monodromy property if there exists a finite set $S \\\\subset \\\\mathbb { Z } \\\\times \\\\mathbb { Z } _ { > 0 }$ such that $Z _ { X , \\\\omega } ( T )$ is an element of the sub-ring \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[307,220,690,265]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0821/580c61c42f1413625ed4f22c559d5f120a1a903d4bad7992b33d27b7693117cf.jpg\",\n \"list_items\": null,\n \"page_idx\": \"144\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\mathcal { M } _ { k } ^ { \\\\widehat { \\\\mu } } \\\\left[ T , \\\\frac { 1 } { 1 - \\\\mathbb { L } ^ { a } T ^ { b } } \\\\right] _ { ( a , b ) \\\\in S } \\\\subset \\\\mathcal { M } _ { k } ^ { \\\\widehat { \\\\mu } } [ [ T ] ] ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,280,881,337]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"144\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and, for any couple $( a , b ) \\\\ \\\\in \\\\ S$ , we have that $\\\\textstyle \\\\exp { \\\\bigl ( } 2 \\\\pi { \\\\sqrt { - 1 } } { \\\\frac { a } { b } } { \\\\bigr ) }$ is a monodromy eigenvalue as in definition 3.1; which means that it is an eigenvalue of the action of $\\\\operatorname { G a l } ( { \\\\overline { { K } } } / K )$ on $H _ { e t } ^ { i } ( X \\\\times _ { K } { \\\\overline { { K } } } , \\\\mathbb { Q } _ { l } )$ for some $i \\\\geq 0$ and every embedding of $\\\\mathbb { Q } _ { l }$ in $\\\\mathbb { C }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,381,505,402]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"144\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 Some Well-Understood Cases \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,426,881,642]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"144\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We are far from a complete understanding of whether or not Calabi-Yau varieties satisfy the monodromy property. It was proven for certain families, under some more restrictive hypothesis on the type of Calabi-Yau variety or on the type of degeneration. The conjecture was first proven for abelian surfaces in [8], this proof uses the theory of Néron models. A generalization of this result was obtained in [9]; in this paper it was proven that degenerations of Calabi-Yau varieties which admit equivariant Kulikov models satisfy the monodromy property and that abelian varieties admit such models. Using the aforementioned result of [9], it was also possible to prove the monodromy property for Kummer surfaces: it was showed in [18] that they admit equivariant Kulikov models. All the examples we mentioned have in common that the motivic zeta function has a unique pole. This is not by\\b chance, in fact, we have the following, more general, result. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,651,879,746]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"144\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 4.1 ([9, Theorem 3.3.3]) Let $X$ be a Calabi-Yau variety of dimension $n$ over $K$ , with a volume form ω. Choose an snc model for $X$ , let $\\\\begin{array} { r } { \\\\mathcal { X } _ { k } = \\\\sum _ { i } N _ { i } E _ { i } } \\\\end{array}$ be the special fiber of this model and denote with $\\\\nu _ { i }$ the vanishing order of $\\\\omega$ on $E _ { i }$ . $\\\\begin{array} { r } { L e t \\\\operatorname* { m i n } ( \\\\omega ) = \\\\operatorname* { m i n } _ { i } \\\\frac { \\\\nu _ { i } } { N _ { i } } } \\\\end{array}$ . Then exp $\\\\left( - 2 \\\\pi i \\\\operatorname* { m i n } ( \\\\omega ) \\\\right)$ ) is an eigenvalue of the action of $\\\\sigma$ on $H _ { e t } ^ { n } ( X \\\\times _ { K } { \\\\overline { { K } } } , \\\\mathbb { C } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,755,881,881]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"144\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For any Calabi-Yau varieties $X$ and any volume form $\\\\omega$ on $X$ , we have that $1 - \\\\operatorname* { m i n } ( \\\\omega )$ is always a pole of $Z _ { X , \\\\omega }$ . Indeed, in all the cases we mentioned at the beginning of this section, it was proven that $1 - \\\\operatorname* { m i n } ( \\\\omega )$ is the unique pole, and since $\\\\exp ( - 2 \\\\pi i \\\\operatorname* { m i n } ( \\\\omega ) )$ is a monodromy eigenvalue, then the monodromy property holds. In Sect. 5 we present a family of degenerations of K3 surfaces that satisfies the monodromy property, but whose motivic zeta functions may have more than one pole, most of these results are the work of [11] and [12]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,719,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"144\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"144\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"141 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"145\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 4.2 ([9, Theorem 4.2.2] ) Fix an abelian variety A with a volume form $\\\\omega$ . Choose an snc model $\\\\mathcal { A }$ , and let' $\\\\begin{array} { r } { \\\\mathcal { A } _ { k } = \\\\sum _ { i } N _ { i } E _ { i } } \\\\end{array}$ be the special fiber. We denote with $\\\\nu _ { i }$ the vanishing order of $\\\\omega$ on $E _ { i }$ . The motivic zeta function $Z _ { A , \\\\omega } ( T )$ has $a$ unique pole at $q = 1 - \\\\operatorname* { m i n } ( \\\\omega )$ . More precisely: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[225,175,774,220]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0099/b7cc313c7eae395075086f47f25823a22aee0254a341e58377c2128f7d0485a2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"145\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nZ _ { A , \\\\omega } ( T ) \\\\in \\\\mathcal { M } _ { k } ^ { \\\\widehat { \\\\mu } } \\\\left[ T , \\\\frac { 1 } { 1 - \\\\mathbb { L } ^ { a } T ^ { b } } \\\\right] _ { ( a , b ) \\\\in S ; a / b = q } \\\\subset \\\\mathcal { M } _ { k } ^ { \\\\widehat { \\\\mu } } [ [ T ] ] .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[123,234,876,270]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"145\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It follows that the monodromy property holds for abelian varieties from Theorems 4.2 and 4.1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,270,876,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"145\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now we are ready to introduce the definition of equivariant Kulikov models for Calabi-Yau varieties. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,315,881,369]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"145\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 4.3 (Equivariant Kulikov Model) Let $X$ be a Calabi-Yau variety over $K$ , and fix a positive integer $d$ . A Kulikov model for $X$ over $R ( d )$ is a regular, proper and flat algebraic space $\\\\mathcal { X }$ over $R ( d )$ such that: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,378,551,396]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"145\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(1) there is an isomorphism of $K ( d )$ -schemes: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[318,414,710,500]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0929/31030d739e4572b61c1e78e13af6a98e2480be0c67aff328a11121589078d6d8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"145\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l r } { \\\\mathcal { X } _ { K ( d ) } \\\\xrightarrow { \\\\sim \\\\sim \\\\sim X \\\\times _ { K } K ( d ) } \\\\xrightarrow { \\\\sim X } X } & { { } \\\\mathrm { ~ , ~ } } & { } \\\\\\\\ { \\\\bigoplus _ { \\\\substack { \\\\operatorname * { s p e c } ( K ( d ) ) \\\\xrightarrow [ ] { } \\\\sim \\\\operatorname { S p e c } K } } \\\\mathcal { Y } } & { { } \\\\quad } & { } \\\\\\\\ { \\\\operatorname { S p e c } ( K ( d ) ) \\\\xrightarrow [ ] { } \\\\operatorname { S p e c } K } & { { } \\\\quad } & { } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,519,826,557]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"145\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(2) the special fiber $\\\\mathcal { X } _ { k }$ is a divisor with normal crossing (3) the logarithmic relative canonical bundle $\\\\omega \\\\chi _ { / R ( d ) } ( \\\\mathcal { X } _ { k , r e d } - \\\\mathcal { X } _ { k } )$ is trivial. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,564,876,600]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"145\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We say that the Kulikov model $\\\\mathcal { X }$ is equivariant if the Galois action of $\\\\mu _ { d }$ on $X \\\\times _ { K } K ( d )$ extends to $\\\\mathcal { X }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,609,881,719]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"145\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 4.4 Not all snc models are Kulikov models; for instance, the model we constructed in Example 2.10 is not a Kulikov model over $K$ . It is easy to check that the K3 surface of Example 2.10 admits a Kulikov model over $K ( 2 )$ , however, we will show in Example 5.8 that this model is not equivariant. Indeed, in Example 5.8 we show something stronger, i.e. that the K3 surface of Example 2.10 does not admit an equivariant Kulikov model for any $d$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,725,881,801]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"145\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 4.5 ([9, Theorem 5.3.2]) Let $X$ be a Calabi-Yau variety, with a volume form ω. Assume that $X$ admits an equivariant Kulikov model over ' ( $R ( d )$ for some positive $d$ . Then the motivic zeta function of $( X , \\\\omega )$ has a unique pole at $q = 1 -$ $\\\\begin{array} { r } { \\\\operatorname* { m i n } _ { i } \\\\frac { \\\\nu _ { i } } { N _ { i } } } \\\\end{array}$ . More precisely: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[227,816,772,863]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0213/6a45d45e52ebde4dbcd369da15989464c4df1d6802e33f12f15e3606ee7675cc.jpg\",\n \"list_items\": null,\n \"page_idx\": \"145\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nZ _ { X , \\\\omega } ( T ) \\\\in \\\\mathcal { M } _ { k } ^ { \\\\widehat { \\\\mu } } \\\\left[ T , \\\\frac { 1 } { 1 - \\\\mathbb { L } ^ { a } T ^ { b } } \\\\right] _ { ( a , b ) \\\\in S , a / b = q } \\\\subset \\\\mathcal { M } _ { k } ^ { \\\\widehat { \\\\mu } } [ [ T ] ] .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[779,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"145\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"L. Lunardon \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,152,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"145\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"142 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,879,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"146\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 4.6 As in the previous case, the monodromy property for Calabi-Yau varieties admitting an equivariant Kulikov model follows from Theorems 4.5 and 4.1. This result generalize Theorem 4.2, in fact it was proven that abelian varieties admit an equivariant Kulikov model in [9, Theorem 5.1.6]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,171,876,205]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"146\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Another consequence of Theorem 4.5 is that the monodromy property holds for-Kummer surfaces, this was proven in [18].- \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,216,879,304]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"146\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 4.7 (Kummer Surface) Let $A$ be an abelian surface over $K$ , consider an involution $\\\\iota$ , and call $A _ { \\\\iota }$ the fixed point scheme of $\\\\iota$ . Let $\\\\widetilde { A }$ be the blow of $A$ at $A _ { \\\\iota }$ ; the involution $\\\\iota$ acts regularly on $\\\\dot { A }$ . Call $X$ the quotient of $\\\\widetilde { A }$ by the action of $\\\\iota$ ; $X$ is a smooth K3 surface over $K$ . Any K3 surface that can be obtained in this way is called a Kummer surface. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,315,879,385]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"146\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 4.8 ([18, Theorem 6.2]) Let $X$ be a Kummer surface, then there exists a minimal $d _ { 0 } > 0$ such that $X ( d _ { 0 } )$ has an equivariant Kulikov model, moreover, if $d \\\\ > \\\\ 0$ is such that $X ( d )$ admits an equivariant Kulikov model, then $d _ { 0 } | d$ . In particular, Kummer surfaces satisfy the monodromy property. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,396,879,466]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"146\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 4.1 was also used in [9] to prove the monodromy property for some Calabi-Yau varieties which were not in any of these families. As far as we know, the cases we listed in this section are the only ones where Theorem 4.1 was used to prove the monodromy property. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,513,708,555]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"146\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 Triple-Points-Free Models of K3 and Why they are Interesting \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,578,881,668]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"146\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section we discuss triple-points-free models of K3 surfaces. If a K3 surface admits a triple-points-free model, it satisfies the monodromy property, however, the motivic zeta function of this K3 surface may have more then a single pole. In particular K3 surfaces admitting a triple-points-free model may not admit an equivariant Kulikov model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,669,881,812]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"146\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In [4] Crauder and Morrison described special the fiber of relatively minimal, triple-points-free snc models of surfaces with trivial pluricanonical bundles. Many additional results to describe the combinatorics of the special fiber were given by Jaspers in [12]. The motivic zeta function and the monodromy property for these surfaces were studied in [11] and [12]. In Corollary 4.2.4 of [12] it was shown that poles of the motivic zeta function may be recovered from the combinatorial data of the model, in particular it was shown that besides $1 - \\\\operatorname* { m i n } ( \\\\omega )$ , there are additional poles as soon as the triple-points-free model has a so-called conic flower. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,813,879,884]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"146\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Regarding the monodromy property, it was proven that under some additional conditions on the special fiber, K3 surfaces admitting a triple-points-free model satisfy the monodromy property. In Appendix B of [12] there is a classification of the possible special fibers of triple-points-free models that do not satisfy the monodromy property; such surfaces are known as combinatorial countercandidates. In [15] we proved that these combinatorial countercandidates do not exist. As a consequence, the monodromy property holds for K3 admitting triple-points-free models. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,719,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"146\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"146\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"147\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,169,879,225]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"147\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 5.1 (Triple-Points-Free Model) Given a K3 surface $X$ , a triple-pointsfree model $\\\\mathcal { X }$ of $X$ is a relatively minimal snc model such that given any three distinct irreducible components $E _ { i } , E _ { j }$ , $E _ { k }$ of the special fiber $\\\\mathcal { X } _ { k }$ , \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[407,241,589,262]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0483/8bad81a2d70f0cf9a04a0b0cef8bba63bbe72af939d972dc760b82e8eb3fc7a0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"147\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nE _ { i } \\\\cap E _ { j } \\\\cap E _ { k } = \\\\emptyset .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,277,881,315]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"147\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since there are not triple intersections, the dual complex of the special fiber of this model is a graph, we call it $\\\\Gamma$ ; to each vertex of $\\\\Gamma$ we associate the weight \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[432,328,562,364]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0891/5e00c990411023c2c8b51938b5130f59197b466dc53c4d91a404705464386534.jpg\",\n \"list_items\": null,\n \"page_idx\": \"147\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\rho _ { i } = \\\\frac { \\\\nu _ { i } } { N _ { i } } + 1 ;\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,378,779,397]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"147\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"we denote by $\\\\Gamma _ { \\\\mathrm { m i n } }$ the subgraph of $\\\\Gamma$ of components of minimal weight. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,405,876,441]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"147\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There is a very explicit description of possible special fibers of triple-points-free models for K3 surfaces. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,450,879,506]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"147\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 5.2 (Crauder-Morrison Classification for K3 Surfaces, [4] and [12]) Let X be a smooth, proper $K 3$ surface over $K$ , let $\\\\mathcal { X }$ be a relatively minimal triplepoints-free model of $X$ , then $\\\\mathcal { X }$ has the following properties: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,513,760,533]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"147\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(1) $\\\\Gamma _ { \\\\mathrm { m i n } }$ is a connected subgraph of \\b. It is either a vertex or a chain. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,533,879,639]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"147\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(2) Each connected component of $\\\\Gamma \\\\backslash \\\\Gamma _ { \\\\mathrm { m i n } }$ is a chain (called flower) $F _ { 0 } - F _ { 1 } - \\\\cdot \\\\cdot \\\\cdot -$ $F _ { l }$ where only $F _ { l }$ meets $\\\\Gamma _ { \\\\mathrm { m i n } }$ . The weights strictly decrease along these flowers, $F _ { 0 }$ being the one with maximal weight. The surface $F _ { 0 }$ is either minimal ruled or isomorphic to $\\\\mathbb { P } ^ { 2 }$ . If it is isomorphic to $\\\\mathbb { P } ^ { 2 }$ then $F _ { 0 } \\\\cap F _ { 1 }$ is either a line or a conic. The other components are minimal ruled surfaces, and $F _ { i } \\\\cap F _ { i + 1 }$ and $F _ { i } \\\\cap F _ { i - 1 }$ are both sections of the ruling. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,641,881,711]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"147\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(3) If $\\\\Gamma _ { \\\\mathrm { m i n } }$ is a single vertex, there are three possible cases for the corresponding surface. It is either isomorphic to $\\\\mathbb { P } ^ { 2 }$ or it is a ruled surface or the canonical divisor is numerically trivial. If $\\\\Gamma _ { \\\\mathrm { m i n } }$ is a point, we call the model a flowerpot degeneration. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,713,881,801]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"147\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(4) If $\\\\Gamma _ { \\\\mathrm { m i n } }$ is a chain $V _ { 0 } - V _ { 1 } - \\\\cdot \\\\cdot \\\\cdot - V _ { k + 1 }$ , then we can describe the components of the chain. I $f i \\\\neq 0 , k + 1 ,$ , $V _ { i }$ is an elliptic ruled surface, and $V _ { i - 1 } \\\\cap V _ { i }$ and $V _ { i } \\\\cap V _ { i + 1 }$ are both sections of the ruling; $i f i = 0 , k + 1$ , then $V _ { i }$ is isomorphic to $\\\\mathbb { P } ^ { 2 }$ , or it is $a$ , rational or elliptic, ruled surface. If $\\\\Gamma _ { \\\\mathrm { m i n } }$ is a chain we call the model a chain degeneration. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,810,881,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"147\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 5.3 In [4] there is an even stronger result. Indeed, they classified the special fiber of triple-points-free models with generic fiber $\\\\mathcal { X } _ { K }$ with trivial pluricanonical bundle. In [4] there is also a complete classification of the possible flowers, divided in 21 combinatorial classes. If $\\\\mathcal { X } _ { K }$ is a K3 surface, this classification was refined in Chapter 3 of [12]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[779,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"147\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"L. Lunardon \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"147\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"144 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"148\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"One of the main results of [12] is the description of the motivic zeta functions of these models. This description is extremely explicit and using it, it was possible to study the poles. It turned out that poles of the motivic zeta function are closely related to the presence of conic flowers (i.e. $F _ { 0 }$ is isomorphic to $\\\\mathbb { P } ^ { 2 }$ and $F _ { 0 } \\\\cap F _ { 1 }$ is a conic). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,187,881,261]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"148\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 5.4 ([12, Theorem 4.3.8]) Let $X$ be a $K 3$ surface over $K$ with a volume form ω. Choose a triple-points-free model $\\\\mathcal { X }$ , whose special fiber is $\\\\begin{array} { r } { \\\\mathcal { X } _ { k } = \\\\sum _ { i } N _ { i } E _ { i } } \\\\end{array}$ , with numerical data $( N _ { i } , \\\\nu _ { i } )$ . Then $q \\\\in \\\\mathbb { Q }$ is a pole of $Z _ { X , \\\\omega } ( T )$ if and only if there exists an $i$ such that the numerical data of $E _ { i }$ satisfy $q = - \\\\nu _ { i } / N _ { i }$ and such that: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,270,462,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"148\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(1) either $\\\\rho _ { i }$ is minimal (2) or $E _ { i }$ is the top of a conic flower. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,315,879,367]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"148\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moreover, while in the second case the pole is always of order 1, in the first one it is of order $I$ if $\\\\mathcal { X }$ is a flowerpot degeneration and of order 2 if it is a chain degeneration. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,378,874,412]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"148\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 5.5 In [12, Appendix A] there is a Python code that describes the contribution of each flower to the motivic zeta function. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,421,881,531]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"148\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 5.6 In [12] it was also discussed whether or not a K3 surface $X$ admitting a triple-points-free model $\\\\mathcal { X }$ satisfies the monodromy property. In [12, Theorem 5.2.1] it was proven that flowerpot degenerations satisfy the monodromy property. In [12, Theorem 5.3.1] it was proven that chain degenerations with some extra assumptions satisfy the monodromy property. However, it was not clear if those assumptions were enough to prove the monodromy property for any chain degeneration. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,531,881,710]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"148\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The strategy adopted in [12] was to try to prove the conjecture by contradiction. The assumption was that there exists a chain degeneration that does not satisfy the monodromy property, this mean that some poles of the motivic zeta function of this degeneration do not correspond to monodromy eigenvalues. The first step was to use A’Campo’s formula and Denef and Loeser’s formula to deduce some information on the geometry of the special fiber of this triple-points-free model. The last step would have been to prove that these surfaces—called combinatorial countercandidates— do not appear as the special fiber of any triple-points-free model. This would have proven the monodromy property for all K3 surfaces admitting a triple-points-free model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,711,879,782]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"148\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In [12, Section 6] there is a description of some topological properties of these combinatorial countercandidates, and this description become even more explicit in [12, Appendix B], where all the possible countercandidates are listed. The kind of information we have about the numerical countercandidates is: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[113,791,815,810]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"148\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(1) the central fiber of the degeneration is a chain degeneration, the chain is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[412,827,626,846]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0472/c843f2cca4a856936b94dc62f811a97a6f576689a25d7648053dbcd47be230ae.jpg\",\n \"list_items\": null,\n \"page_idx\": \"148\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nV _ { 0 } - V _ { 1 } - \\\\cdot \\\\cdot \\\\cdot - V _ { k + 1 } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,52,719,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"148\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"148\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,605,108]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"149\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(2) The surface $V _ { 0 }$ satisfies the following properties: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[157,115,879,225]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"149\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(a) it contains a smooth elliptic curve $D$ such that $D = - K _ { V _ { 0 } }$ ; \\n(b) it is a smooth rational ruled non-minimal surface obtained from an Hirzebruch surface by $l _ { 0 }$ blow-ups whose centers lies in the image of $D$ ; \\n(c) it contains at least $h$ smooth rational curves of self intersection $^ { - 2 }$ , we denote them by $C _ { i }$ . The curves $C _ { i }$ are all disjoint, moreover from the adjunction formula it follows that they do not intersect $D$ ; \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,232,881,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"149\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(3) the other internal components of the chain are elliptic or rational ruled surfaces, and they are obtained from minimal surface by some blow-up along the intersection of two components. Moreover, we know that there is a certain number of $^ { - 2 }$ rational curves, disjoint from the sections $V _ { i } \\\\cap V _ { i + 1 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,315,881,369]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"149\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Whether or not any combinatorial countercandidate exists was left as an open problem in [12]. It was proven in [15] that they do not exist, all the information required to prove this result are the one regarding the surface $V _ { 0 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,378,876,414]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"149\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 5.7 ([12, 15]) Let $X$ be a $K 3$ surface that admits a triple-points-free model $\\\\mathcal { X }$ . Then the monodromy property holds for $X$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,421,838,441]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"149\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To conclude this section we come back to the quartic surface of Example 2.10. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,448,867,468]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"149\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 5.8 ([9, 11, 12]) Let $\\\\mathrm { X }$ be the K3 surfaces in $\\\\mathbb { P } _ { K } ^ { 3 }$ , given by the equation: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[261,483,733,507]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0361/e220eee700cd14e56fa8e0fb09f7c28c4efc15e23bad1b6f12bf7d53d446b1e2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"149\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nx ^ { 2 } w ^ { 2 } + y ^ { 2 } w ^ { 2 } + z ^ { 2 } w ^ { 2 } + x ^ { 4 } + y ^ { 4 } + z ^ { 4 } + t w ^ { 4 } = 0 .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,522,881,650]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"149\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From Example 2.10 we already know that there exists a model $\\\\mathcal { X } ^ { \\\\prime }$ such that the special fiber is $\\\\mathcal { X } _ { \\\\mathbb { C } } ^ { \\\\prime } = D + 2 E$ , where $D$ is a K3 surface and $E \\\\cong \\\\mathbb { P } _ { \\\\mathbb { C } } ^ { 2 }$ . The curve $C = D \\\\cap E$ is a smooth conic in $E$ ; thus $E$ is a conic flower of type 2B. We can now choose a volume form on $\\\\mathrm { X }$ (for instance the natural volume form induced by the embedding in $\\\\mathbb { P } ^ { 3 }$ ). This form extends to a relative volume form on $\\\\mathcal { X } ^ { \\\\prime }$ . We can now compute the numerical data of this relative volume form and we obtain- - - $\\\\nu _ { D } = 0$ and $\\\\nu _ { E } = 1$ . The motivic zeta function of this K3 is: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[125,663,870,705]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0069/380d890f0d550b67df66138a84ad1e10a8e694721526e9bcd2faed6840083136.jpg\",\n \"list_items\": null,\n \"page_idx\": \"149\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nZ _ { X , \\\\omega } ( T ) = [ \\\\widetilde { D } ^ { \\\\circ } ] \\\\frac { T } { 1 - T } + [ \\\\widetilde { E } ^ { \\\\circ } ] \\\\frac { \\\\mathbb { L } ^ { - 1 } T ^ { 2 } } { 1 - \\\\mathbb { L } ^ { - 1 } T ^ { 2 } } + [ \\\\widetilde { C } ] ( \\\\mathbb { L } - 1 ) \\\\frac { \\\\mathbb { L } ^ { - 1 } T ^ { 3 } } { ( 1 - T ) ( 1 - \\\\mathbb { L } ^ { - 1 } T ^ { 2 } ) } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,723,740,743]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"149\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The motivic zeta function of this K3 has two poles, namely 0 and $\\\\frac { 1 } { 2 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,750,881,804]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"149\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 5.9 The quartic surface of Example 5.8 does not admit any equivariant Kulikov model. This follows from Theorem 4.5, since its motivic zeta function has two poles. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[779,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"149\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"L. Lunardon \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,152,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"149\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"146 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,307,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"150\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 What’s Next \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,133,879,205]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"150\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this section we describe a possible approach to future research in this area. Our idea is to construct further examples of K3 surfaces whose motivic zeta function has more than one pole. Once we have done this, the natural question is whether or not these K3 surfaces satisfy the monodromy property. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,205,881,331]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"150\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From the results of Theorem 5.4 it is clear that, in the case of triple-pointsfree models, additional poles come only from conic flowers. However, it is not immediately clear what distinguishes these flowers geometrically. To obtain some understanding of this, we tried to study what happens to the model if we contract some flowers. The idea of contracting flowers already appears in [5], in this paper there is also a list of the singularities caused by the contraction of some family of flowers. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,333,881,456]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"150\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From the computations of Example 5.8, we can see that the contraction of the flower of type 2B gives a regular model with a special fiber which is irreducible but with a singular point. The situation with non-conic flowers is different, L. Halle showed me that, up to a finite base change, it is possible to contract them smoothly— by this we mean that the resulting model is an snc model. This suggested to us that there might be a relation between poles of the motivic zeta function and singularities of the models. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,459,881,636]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"150\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Assume now that $X$ is a K3 surface over $K$ , $\\\\omega$ a volume form on $X$ and let $\\\\mathcal { X }$ be a model for $X$ . To $X$ we can associate a motivic zeta function $Z _ { X , \\\\omega }$ ; however, it may not be possible to use Denef and Loeser’s formula with the model $\\\\mathcal { X }$ . In general it is not regular, and even if it were, the irreducible components of the special fiber $\\\\mathcal { X } _ { k }$ might be singular. Of course, by Hironaka’s resolution of singularities it is possible to construct an snc model $\\\\mathcal { V }$ which dominates $\\\\mathcal { X }$ , but what we would like to understand is whether or not it is possible to deduce the presence of additional poles of $Z _ { X , \\\\omega }$ from the singularities of $\\\\mathcal { X }$ . This problem, however, is very generic, and thus pretty tough to approach. So we had to restrict the class of singularities we would consider. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,647,879,737]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"150\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 6.1 (Rational Double Points) Given $X$ a normal surface over $\\\\mathbb { C }$ , we say that a point $x \\\\in X$ is a rational double point (or ADE or Du Val singularity) if it is a canonical singularity. Given a rational double point singularity, we have that étale locally around the point the surface is isomorphic to the closed subset of $\\\\mathbb { A } ^ { 3 }$ given by one of the following equations: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,744,719,836]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"150\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(1) $x ^ { 2 } + y ^ { 2 } + z ^ { n + 1 } = 0$ ; these are $A _ { n }$ singularities; \\n(2) $x ^ { 2 } + y ^ { 2 } z + z ^ { n - 1 } = 0$ , with $n > 3$ ; these are $D _ { n }$ singularities; \\n(3) $x ^ { 2 } + y ^ { 3 } + z ^ { 4 } = 0$ ; this is the $E _ { 6 }$ singularity; \\n(4) $x ^ { 2 } + y ^ { 3 } + z ^ { 3 } y = 0$ ; this is the $E _ { 7 }$ singularity; \\n(5) $x ^ { 2 } + y ^ { 3 } + z ^ { 5 } = 0$ ; this is the $E _ { 8 }$ singularity. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,845,881,899]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"150\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Fix a K3 surface $X$ which admits a regular model $\\\\mathcal { X }$ whose special fiber $\\\\mathcal { X } _ { k }$ has some rational double points (for instance, the K3 surface of Example 5.8). Assume, furthermore, that none of the rational double points lie the intersection of some irreducible components of the special fiber and call the set of rational double points S. Then we can construct an snc model $\\\\mathcal { V }$ for $X$ , with a morphism $\\\\pi \\\\colon \\\\mathcal { V } \\\\mathcal { X }$ blowing-up $\\\\mathcal { X }$ at some smooth points and at some smooth curves contained in $\\\\mathcal { X } _ { k }$ . Once we have constructed such an snc model, it is possible to compute the motivic zeta function using Denef and Loeser’s formula. We can split the motivic zeta function $Z _ { X , \\\\omega }$ in two parts, the one which depends on the strata contained in $\\\\pi ^ { - 1 } ( { \\\\mathcal { X } } _ { k } \\\\setminus S )$ and the one which depends on the strata contained in $\\\\pi ^ { - 1 } ( S )$ . We denote the second part by $Z _ { X , \\\\omega , s i n g }$ , and we refer to it as the contribution of the singularities to $Z _ { X , \\\\omega }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,719,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"150\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"150\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,252]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"151\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,259,876,279]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"151\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 6.2 For instance, in Example 5.8, the contribution of the $A _ { 1 }$ singularity is: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[191,292,801,336]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0745/a38e94cadbcd7cd175077178521a38010bf49c943b546ac7a0c61edb0d330fd8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"151\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nZ _ { X , \\\\omega , s i n g } = [ \\\\widetilde { E } ^ { \\\\circ } ] \\\\frac { \\\\mathbb { L } ^ { - 1 } T ^ { 2 } } { 1 - \\\\mathbb { L } ^ { - 1 } T ^ { 2 } } + [ \\\\widetilde { C } ] ( \\\\mathbb { L } - 1 ) \\\\frac { \\\\mathbb { L } ^ { - 1 } T ^ { 3 } } { ( 1 - T ) ( 1 - \\\\mathbb { L } ^ { - 1 } T ^ { 2 } ) } ;\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,349,879,402]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"151\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"since the motivic zeta function in this example has more than one pole, then $X$ can’t have good reduction over $K$ ; in particular we have that the action of $\\\\operatorname { G a l } ( { \\\\overline { { K } } } / K )$ is not trivial. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,412,881,521]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"151\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We are studying the contributions of ADE singularities to the motivic zeta function. In the last part of this paper we briefly discuss some of the questions we would like to answer. Questions 6.3, 6.4, and 6.5 may be considered short term goals, while Question 6.7 is much wilder, and should be considered a long term goal. Question 6.9 was for us a further motivation to study this topic, and a natural follow-up question to Question 6.7. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,530,879,584]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"151\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Question 6.3 Assume that $X$ is a K3 surface that admits a regular model whose special fiber has rational double points, is it true that the motivic zeta function of $X$ has at least two poles? If not, for which singularities is this true? \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,584,876,620]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"151\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A positive answer to this question would provide more examples of Calabi-Yau varieties whose motivic zeta function has multiple poles. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,629,876,665]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"151\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Question 6.4 In the above setting, can we describe explicitly the contributions of the various singularities? \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,674,879,728]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"151\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Having an explicit description of the motivic zeta function would be not only interesting by itself, but it is a necessary step in a better understanding of the monodromy property for K3 surfaces. This brings us to the next question. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,737,876,773]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"151\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Question 6.5 In the above setting, does $X$ satisfy the monodromy property? If not, which obstructions are encountered? \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,782,879,834]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"151\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Requiring that the model $\\\\mathcal { X }$ is regular is a strong assumption; we would like to relax this hypothesis. The natural class of threefold singularities that extends Du Val singularities is known as compound Du Val singularities. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,845,879,899]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"151\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 6.6 (Compound Du Val Singularities) Given $X$ a normal threefold over $\\\\mathbb { C }$ , we say that a point $x \\\\in X$ is a compound Du Val singularity if for some general hyperplane section $H$ through $x$ we have that $x \\\\in H$ is a Du Val singularity. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[779,54,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"151\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"L. Lunardon \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,54,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"151\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,879,142]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"152\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Question 6.7 Assume that $X$ is a K3 surface that admits a model $\\\\mathcal { X }$ whose singularities are compound Du Val singularities. What can we say of the analogs of Questions 6.3, 6.4, 6.5 under these less restrictive hypothesis? \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,151,772,169]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"152\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There exists a criterion to show that a K3 surface $X$ has good reduction. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,178,876,216]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"152\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 6.8 ([14], Theorem 6.1) Let $X$ be a $K 3$ surface over $K$ , then $X$ has good reduction if and only if the action of $\\\\operatorname { \\\\overrightarrow { { r a l } } } ( \\\\overline { { K } } / K )$ on $H _ { \\\\mathit { e t } } ^ { 2 } ( X _ { \\\\overline { { K } } } , \\\\mathbb { Q } _ { l } )$ is trivial. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,223,876,259]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"152\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An interesting problem is to find sufficient conditions to prove that a K3 surface does not have good reduction. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,270,881,358]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"152\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Question 6.9 Assume that $X$ is a K3 surface that admits a model $\\\\mathcal { X }$ whose singularities are compound Du Val singularities or a regular model whose special fiber has rational double points. Is that enough to prove that $X$ does not have good reduction? If $X$ satisfies the monodromy property, can this obstruction be detected from the motivic zeta function? \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,379,879,439]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"152\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acknowledgements I would like to thank Prof. J. Nicaise for his help during the preparation of this paper. This work was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1]. The EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,486,248,506]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"152\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[125,530,881,887]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"152\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Borisov, L.A.: The class of the affine line is a zero divisor in the Grothendieck ring. J. Algebraic Geom. 27(2), 203–209 (2018) \\n2. Bultot, E., Nicaise, J.: Computing zeta functions on log smooth models. C. R. Math. Acad. Sci. Paris 353(3), 261–264 (2015) \\n3. Cauwbergs, T.: Splicing motivic zeta functions. Rev. Mat. Complut. 29(2), 455–483 (2016) \\n4. Crauder, B., Morrison, D.R.: Triple-point-free degenerations of surfaces with Kodaira number zero. In: The Birational Geometry of Degenerations (Cambridge, Mass, 1981). Progress in Mathematics, vol. 29, pp. 353–386. Birkhäuser, Boston (1983) \\n5. Crauder, B., Morrison, D.R.: Minimal models and degenerations of surfaces with Kodaira number zero. Trans. Am. Math. Soc. 343(2), 525–558 (1994) \\n6. Denef, J., Loeser, F.: Motivic Igusa zeta functions. J. Algebraic Geom. 7(3), 505–537 (1998) \\n7. Ekedahl, T.: The Grothendieck group of algebraic stacks. arXiv:0903.3143 [math.AG] (2009) \\n8. Halvard Halle, L., Nicaise, J.: Motivic zeta functions of abelian varieties, and the monodromy conjecture. Adv. Math. 227(1), 610–653 (2011) \\n9. Halvard Halle, L., Nicaise, J.: Motivic zeta functions of degenerating Calabi-Yau varieties. Math. Ann. 370(3–4), 1277–1320 (2018) \\n10. Hartmann, A.: The quotient map on the equivariant Grothendieck ring of varieties. Manuscripta Math. 151(3–4), 419–451 (2016) \\n11. Jaspers, A.: The global monodromy property for $K 3$ surfaces allowing a triple-point-free model. C. R. Math. Acad. Sci. Paris 355(2), 200–204 (2017) \\n12. Jaspers, A.: The Monodromy Property for K3 Surfaces Allowing a Triple-Point-Free Model (2017). Ph. D. Thesis, KU Leuven. arXiv:1706.07086 \\n13. Kempf, G. Knudsen, F.F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings. I. Lecture Notes in Mathematics, vol. 339. Springer, Berlin (1973) \\n14. Liedtke, C., Matsumoto, Y.: Good reduction of K3 surfaces. Compos. Math. 154(1), 1–35 (2018) \\n15. Lunardon, L.: The monodromy property for certain families of K3 surfaces and higher dimesional Calabi-Yau varieties, Ph.D. Thesis, Imperial College London (2020) \\n16. Nicaise, J.: Geometric criteria for tame ramification. Math. Z. 273(3–4), 839–868 (2013) \\n17. Nicaise, J., Xu, C.: Poles of maximal order of motivic zeta functions. Duke Math. J. 165(2), 217–243 (2016) \\n18. Overkamp, O.: Degeneration of Kummer surfaces. arXiv:1806.10105 [math.AG] (2018) \\n19. Poonen, B.: The Grothendieck ring of varieties is not a domain. Math. Res. Lett. 9(4), 493–497 (2002) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,719,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"152\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"What is the Monodromy Property for Degenerations of Calabi-Yau Varieties? \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"152\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,91,881,240]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"153\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[779,54,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"153\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"L. Lunardon \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,52,152,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"153\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"150 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,676,139]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"154\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Examples of Irreducible Symplectic Varieties \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,175,250,193]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"154\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Arvid Perego \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,286,881,412]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"154\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract Irreducible symplectic manifolds are one of the three building blocks of compact Kähler manifolds with numerically trivial canonical bundle by the Beauville-Bogomolov decomposition theorem. There are several singular analogues of irreducible symplectic manifolds, in particular in the context of compact Kähler orbifolds, and in the context of normal projective varieties with canonical singularities. In this paper we will collect their definitions, analyze their mutual relations and provide a list of known examples. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,432,824,465]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"154\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords Irreducible symplectic varieties $\\\\cdot$ Moduli spaces of sheaves $\\\\cdot$ K3 surfaces \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,504,309,525]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"154\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,549,881,638]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"154\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A central problem in complex geometry is the classification of Ricci-flat compact Kähler manifolds. By Yau’s theorem [51], these are exactly the compact Kähler manifolds whose first Chern class is zero in $H ^ { 2 } ( X , \\\\mathbb { R } )$ or, equivalently, whose canonical bundle is numerically trivial. This implies that the Kodaira dimension is zero. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,639,879,710]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"154\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In dimension 1, compact Kähler manifolds of Kodaira dimension zero are exactly elliptic curves. The birational classification of compact complex surfaces shows that compact Kähler surfaces of Kodaira dimension zero are K3 surfaces, 2-dimensional complex tori, Enriques surfaces and bielliptic surfaces. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,711,879,764]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"154\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Among compact Kähler manifolds of Kodaira dimension zero, a very special role is played by those manifolds whose canonical bundle is trivial (which are sometimes called Calabi-Yau manifolds). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,834,630,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"154\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. Perego $( \\\\boxtimes )$ Dipartimento di Matematica, Università di Genova, Genova, Italy e-mail: perego@dima.unige.it \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,897,692,941]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"154\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_9 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[804,91,876,139]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"154\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,899,876,909]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"154\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,196]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"155\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The first family of examples is given by complex tori, i.e. quotients of a complex vector space $V$ of dimension $n$ by a rank $2 n$ lattice $\\\\Gamma$ in $V$ . A projective complex torus is called Abelian variety: in dimension 1 all complex tori are projective, and they are exactly the elliptic curves. In higher dimension there are complex tori which are not projective. In any case, if $X$ is a complex torus of dimension $n$ , then $\\\\pi _ { 1 } ( X ) \\\\simeq$ $\\\\mathbb { Z } ^ { 2 n }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,198,881,304]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"155\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A second family of examples is given by special unitary manifolds, i.e. compact Kähler manifolds with trivial canonical bundle, whose every finite étale covering has no non-trivial holomorphic $p$ -form for $0 ~ < ~ p ~ < ~ n$ (where $n$ is the complex dimension). They are projective for $n \\\\geq 3$ , and have finite fundamental group, which is trivial if $n$ is even (see Proposition 2 of [2]). Simply connected special unitary manifolds are called irreducible Calabi-Yau manifolds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,306,881,412]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"155\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A third family of examples is given by irreducible symplectic manifolds, that will be our main interest. We recall that if $X$ is a complex manifold, a holomorphic symplectic form on $X$ is a closed, holomorphic 2-form $\\\\sigma$ on $X$ which is everywhere non-degenerate. A holomorphic symplectic manifold is a complex manifold admitting a holomorphic symplectic form. Among holomorphic symplectic manifolds we find all even-dimensional complex tori. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,414,881,504]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"155\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $X$ is a holomorphic symplectic manifold, then its complex dimension is even. If $2 n$ is the complex dimension of $X$ , and $\\\\sigma$ is a holomorphic symplectic form on $X$ , then $\\\\sigma ^ { n }$ is a nowhere vanishing section of $K _ { X }$ : it follows that a holomorphic symplectic manifold has trivial canonical bundle. Moreover, as $\\\\sigma$ is closed, then it defines a nontrivial cohomology class in $H ^ { 0 } ( X , \\\\Omega _ { X } ^ { 2 } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,513,879,549]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"155\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 1 An irreducible symplectic manifold is a compact, Kähler, holomorphic symplectic manifold $X$ which is simply connected and such that $h ^ { 2 , 0 } ( X ) = 1$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,557,876,594]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"155\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark $I$ As shown in Propositions 3 and 4 of [2], for an irreducible symplectic manifold $X$ of dimension $n$ and for every $0 \\\\leq p \\\\leq n$ we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[369,609,626,651]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0546/8c02461fecb4973f964bf2160918d6d2edc690c5f490c3aa91167064ec0948ad.jpg\",\n \"list_items\": null,\n \"page_idx\": \"155\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nh ^ { p , 0 } ( X ) = { \\\\left\\\\{ \\\\begin{array} { l l } { 0 , { \\\\mathrm { i f ~ } } p { \\\\mathrm { ~ i s ~ o d d } } } \\\\\\\\ { 1 , { \\\\mathrm { o t h e r w i s e } } } \\\\end{array} \\\\right. }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,666,879,720]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"155\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conversely, by Proposition A.1 of [21] every compact, Kähler, holomorphic symplectic manifold $X$ of complex dimension $n$ such that $h ^ { p , 0 } ( X )$ is as in (1) is simply connected, and hence an irreducible symplectic manifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,729,879,801]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"155\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 2 As shown in Proposition 4 of [2], a compact Kähler manifold of complex dimension $2 n$ is a holomorphic symplectic manifold if and only if its holonomy group is contained in the symplectic group $S p ( r )$ . The holonomy group is precisely $S p ( r )$ if and only if $X$ is an irreducible symplectic manifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,810,879,864]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"155\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As already recalled, Ricci-flat compact Kähler surfaces are K3 surfaces, complex tori of dimension 2, Enriques surfaces and bielliptic surfaces. Among them, K3 surfaces are both irreducible Calabi-Yau and irreducible symplectic. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,866,876,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"155\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An Enriques surface is a finite quotient of a K3 surface by a fixed point free involution, while a bielliptic surface is quotient of an Abelian surface (product of two elliptic curves) by the free action of a finite Abelian group. It follows that a Ricci-flat compact Kähler surface has a finite étale covering which is either an irreducible symplectic (Calabi-Yau) surface or a complex torus of dimension 2. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"155\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"152 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[799,54,879,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"155\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,879,142]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"156\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,144,881,252]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"156\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The same phenomenon occurs for higher dimensional Ricci-flat compact Kähler manifolds, giving to complex tori, irreducible Calabi-Yau manifolds and irreducible symplectic manifolds a special role in the classification. This is the content of the following, which goes under the name of Beauville-Bogomolov Decomposition Theorem, and whose proof (based on several results of differential geometry) is contained in [6] and [2]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,261,879,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"156\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 1 Let $X$ be a Ricci-flat compact Kähler manifold. Then $X$ has a finite étale covering $f : Y \\\\longrightarrow X$ , where $Y$ is a product of complex tori, irreducible Calabi-Yau manifolds and irreducible symplectic manifolds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,324,881,378]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"156\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"While it is not difficult to provide examples of complex tori and irreducible Calabi-Yau manifolds, it is a hard problem to give examples of irreducible symplectic manifolds. The list of all the known deformation classes is very short: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,387,881,584]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"156\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Irreducible symplectic surfaces are exactly K3 surfaces. \\n2. For $n \\\\geq 2$ , the Hilbert scheme $H i l b ^ { n } ( S )$ of $n$ points on a K3 surface $s$ is an irreducible symplectic manifold of dimension $2 n$ (see Théorème 3 of [2]). \\n3. For $n \\\\geq 2$ , let $T$ be a complex torus of dimension 2 and $s : H i l b ^ { n + 1 } ( T ) \\\\longrightarrow T$ the sum morphism. Then $K u m ^ { n } ( T ) : = s ^ { - 1 } ( 0 _ { S } )$ , called generalized Kummer variety, is an irreducible symplectic manifold of dimension $2 n$ (see Théorème 4 of [2]). \\n4. Two more deformation classes, $O G _ { 6 }$ of dimension 6 and $O G _ { 1 0 }$ of dimension 10, were constructed by O’Grady (in [44] and [43] respectively) as a symplectic resolution of some singular moduli spaces of semistable sheaves on a projective K3 or on an Abelian surface. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,594,879,647]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"156\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The previous examples form different deformation classes because even if they can have the same dimension, they have different second Betti number (which is, following the previous ordering, 22, 23, 7, 8 and 24, see [2, 43, 44] and [48]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,656,881,782]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"156\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 3 Different constructions of examples of irreducible symplectic manifolds were presented in [4, 9, 22, 45] (deformation equivalent to $H i l b ^ { 2 } ( K 3 ) )$ and [28] (deformation equivalent to $H i l b ^ { 4 } ( K 3 ) )$ . Moduli spaces of stable sheaves or of Bridgeland stable complexes on projective K3 surfaces or on Abelian surfaces give rise to examples of irreducible symplectic manifolds which are deformation equivalent to Hilbert schemes of points on K3 surfaces or to generalized Kummer varieties on Abelian surfaces. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,792,881,899]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"156\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 1 is stated only for compact Kähler manifolds. Anyway, if $X$ is a complex projective manifold with Kodaira dimension zero, the minimal model $Y$ of $X$ (whose conjectural existence is predicted by the Minimal Model Program) is a projective variety which is birational to $X$ , and has terminal singularities and nef canonical divisor. Assuming the abundance conjecture, it follows that a multiple of the canonical divisor $K _ { Y }$ of $Y$ is trivial. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"156\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,471,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"156\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Examples of Irreducible Symplectic Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,881,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"157\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For the classification of projective varieties whose Kodaira dimension is 0 it is then central to extend Theorem 1 to normal projective varieties having terminal singularities and torsion (i.e. numerically trivial by Theorem 8.2 of [26]) canonical divisor. This implies the need for a definition of singular analogues of irreducible Calabi-Yau and symplectic manifolds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,180,881,252]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"157\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Various definitions have been proposed and studied over the years, and the main purpose of this survey is to present a list of definitions of irreducible Calabi-Yau and symplectic varieties which can be found in the literature, together with a list of known example. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,297,537,318]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"157\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 Irreducible Symplectic Varieties \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,340,881,412]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"157\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There are two main settings we will consider: orbifolds and varieties with canonical singularities. The second one is more natural for the purposes of the Minimal Model Program, while the first one has the advantage to be more similar to the smooth case (and it was the first one to be considered). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,459,287,478]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"157\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.1 Orbifolds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,503,879,557]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"157\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A first generalization of the decomposition theorem was obtained by Campana in [8], in the setting of orbifolds. We refer the reader to [10] or to [15] (where orbifolds are called $V .$ -manifolds) for precise definitions and results on these analytic spaces. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,558,879,665]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"157\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An orbifold is a connected, para-compact analytic space $X$ such that for every $x \\\\in X$ there is an open neighborhood $U$ of $x$ in $X$ , an open subset $V$ of $\\\\mathbb { C } ^ { n }$ (with respect to the Euclidean topology) and a finite group $G$ of automorphisms of $V$ such that there is an isomorphism $\\\\phi : V / G \\\\longrightarrow U$ . The composition $\\\\pi : V \\\\longrightarrow U$ of the projection $V \\\\longrightarrow V / G$ with $\\\\phi$ is called uniformization map, and the triple $( V , G , \\\\pi )$ a uniformizing system for $U$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,666,874,701]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"157\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It follows from Proposition 1.3 of [5] and by Proposition 5.15 of [27] that an orbifold is normal, $\\\\mathbb { Q }$ -factorial, Cohen-Macaulay and has rational singularities. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,702,881,810]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"157\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On an orbifold $X$ of dimension $n$ one can then always find a uniformizing open cover $\\\\{ U _ { i } \\\\} _ { i \\\\in I }$ , i.e. for each $i \\\\in I$ there is a uniformizing system $( V _ { i } , G _ { i } , \\\\pi _ { i } )$ for $U _ { i }$ . The sheaf of differential $p$ -forms on $X$ is denoted $\\\\mathcal { A } _ { X } ^ { p }$ , and by definition it is the sheaf which restricted to $U _ { i }$ is $\\\\pi _ { i * } ( \\\\mathcal { A } _ { V _ { i } } ^ { p } ) ^ { G _ { i } }$ , i.e. the push-forward under $\\\\pi _ { i }$ of the $G _ { i }$ -invariant part of the sheaf of differential $p$ -forms on $V _ { i }$ . Similarly one defines the sheaf of holomorphic $p$ -forms on $X$ , denoted $\\\\Omega _ { X } ^ { p }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,810,879,864]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"157\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A differential 2-form $\\\\omega \\\\in \\\\mathcal { A } _ { X } ^ { 2 } ( X )$ is a Kähler form for $X$ if for every $x \\\\in X$ and every uniformizing system $( V , G , \\\\pi )$ of an open neighborhood $U$ of $x$ , we have that $\\\\omega _ { | U } \\\\in ( \\\\mathcal { A } _ { V } ^ { 2 } ) ^ { G } ( V )$ is a Kähler form on $V$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,866,876,900]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"157\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moreover, one can still define the notion of first Chern class for orbifolds: the canonical sheaf $K _ { X }$ (which, restricted to every uniformized open subset $U$ with uniformizing system $( V , G , \\\\pi )$ , is just $\\\\pi _ { * } K _ { V }$ ) is not in general locally free, but $K _ { X } ^ { m }$ is if $m$ is a multiple of $| G |$ . It follows that if $X$ is compact, then there is $m \\\\gg 0$ such that $K _ { X } ^ { m }$ is locally free, and one defines $\\\\begin{array} { r } { c _ { 1 } ( X ) : = - \\\\frac { \\\\bar { 1 } } { m } c _ { 1 } ( K _ { X } ^ { m } ) } \\\\end{array}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,54,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"157\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[799,54,879,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"157\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,879,144]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"158\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,145,879,196]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"158\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As shown by Theorem 1.1 of [8], if $X$ is a compact, connected Kähler orbifold such that $c _ { 1 } ( X ) = 0$ in $H ^ { 2 } ( X , \\\\mathbb { R } )$ , every Kähler class is represented by a unique Ricci-flat Kähler metric on $X$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,232,605,250]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"158\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.1.1 The Decomposition Theorem for Orbifolds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,270,879,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"158\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As in the smooth case, one has two important kinds of Ricci-flat compact Kähler orbifolds which arise. The first one generalizes irreducible Calabi-Yau manifolds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,313,881,367]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"158\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 2 An irreducible Calabi-Yau orbifold is a compact, connected Kähler orbifold $X$ having simply connected smooth locus, $K _ { X } \\\\simeq O _ { X }$ and $H ^ { 0 } ( X , \\\\Omega _ { X } ^ { p } ) = 0$ for every $0 < p < \\\\dim ( X )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,378,881,448]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"158\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By Proposition 6.6 of [8] an irreducible Calabi-Yau orbifold of dimension $n$ is equivalently a compact, connected Kähler orbifold whose smooth locus is simply connected, and which has a Ricci-flat Kähler metric whose holonomy group is $S U ( n )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,450,881,557]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"158\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The second kind of Ricci-flat compact Kähler orbifolds is a generalization of irreducible symplectic manifolds. A closed holomorphic 2-form on $X$ , i.e. a global section of $\\\\Omega _ { X } ^ { 2 }$ , which is everywhere non-degenerate is called symplectic form. An orbifold admitting a holomorphic symplectic form is called holomorphic symplectic orbifold. As for holomorphic symplectic manifolds, a holomorphic symplectic orbifold has even complex dimension and trivial canonical sheaf. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,567,881,621]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"158\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3 An irreducible symplectic orbifold is a compact, connected Kähler orbifold which is holomorphic symplectic, has simply connected smooth locus and $h ^ { 0 } ( X , \\\\Omega _ { X } ^ { p } ) = 1$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,630,879,702]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"158\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By Proposition 6.6 of [8] an irreducible symplectic orbifold of dimension $2 n$ is equivalently a compact, connected Kähler orbifold whose smooth locus is simply connected, and which has a Ricci-flat Kähler metric whose holonomy group is $S p ( n )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,702,874,738]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"158\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The decomposition theorem for Ricci-flat compact Kähler orbifolds is the following (see Théorème 6.4 of [8]): \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,746,879,801]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"158\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2 Let $X$ be a Ricci-flat compact Kähler orbifold. Then $X$ has a finite quasi-étale covering $f : Y \\\\longrightarrow X$ , where $Y$ is a product of complex tori, irreducible Calabi-Yau orbifolds and irreducible symplectic orbifolds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,810,879,864]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"158\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A finite quasi-étale morphism is a finite morphism which is étale in codimension 1. The statement of Théorème 6.4 of [8] is more precise about the finite quasi-étale covering: it is indeed a finite orbifold covering (see Définition 5.1 of [8]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,864,876,900]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"158\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The proof of Theorem 2 relies first on a generalization to orbifolds of the de Rham decomposition theorem (Proposition 5.4 of [8]), which provides a decomposition of the universal orbifold covering $\\\\widetilde { X }$ of $X$ as a product $M _ { 0 } \\\\times M _ { 1 } \\\\times \\\\cdots \\\\times M _ { k }$ , where $M _ { 0 }$ is a Euclidean space, and $M _ { 1 } , \\\\dots , M _ { k }$ have all irreducible holonomy representation. The orbifold version of the Cheeger-Gromoll theorem is provided by Borzellino and Zhu (see [7]), and implies that $M _ { 1 } , \\\\dots , M _ { k }$ are all compact and their Ricci curvature is zero. The remaining part of the proof is similar to the one for compact manifolds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,471,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"158\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Examples of Irreducible Symplectic Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"158\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,881,199]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"159\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,235,487,253]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"159\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.1.2 Related Notions and Examples \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,271,849,289]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"159\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There are other definitions which are related to irreducible symplectic orbifolds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,298,879,369]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"159\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 4 A primitively symplectic V-manifold is a holomorphic symplectic orbifold $X$ such that $h ^ { 2 , 0 } ( X ) \\\\ = \\\\ 1$ . An irreducible symplectic V-manifold is a compact, connected Kähler holomorphic symplectic orbifold $X$ which is simply connected and such that $h ^ { 2 , 0 } ( X ) = 1$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,379,879,468]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"159\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Primitively symplectic V-manifolds are introduced in (section 2.1, [15]), while irreducible symplectic V-manifolds are introduced in (Definition 1.3.(iv), [30]). Clearly an irreducible symplectic V-manifold is a compact, connected Kähler primitively symplectic V-manifold which is simply connected. Moreover, an irreducible symplectic orbifold is an irreducible symplectic V-manifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,471,881,542]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"159\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We will discuss in the following several examples of irreducible symplectic V-manifolds/orbifolds of dimension 4 and 6 which appear in the literature. In particular, we will see that irreducible symplectic V-manifolds are not always irreducible symplectic orbifolds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,578,448,594]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"159\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Symmetric Products of K3 Surfaces \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,614,881,722]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"159\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $s$ is a K3 surface and $m \\\\ \\\\geq \\\\ 2$ , the symmetric product $X \\\\ : = \\\\ S y m ^ { m } ( S )$ is an irreducible symplectic $\\\\mathrm { V } .$ -manifold. Indeed, it is a compact, connected Kähler orbifold having $Y \\\\ : = \\\\ H i l b ^ { m } ( S )$ as a resolution of the singularities. As $Y$ is an irreducible symplectic manifold, it is simply connected and has $h ^ { 2 , 0 } ( Y ) \\\\ =$ 1: it follows that $X$ is simply connected and $h ^ { 2 , 0 } ( X ) \\\\ = 1$ (see as instance Proposition 2.13 of [15]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,723,876,756]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"159\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We notice that the smooth locus $X ^ { s }$ of $X$ is not simply connected, since $\\\\pi _ { 1 } ( X ^ { s } ) \\\\simeq$ $\\\\Sigma _ { m }$ . It follows that $X$ is not an irreducible symplectic orbifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,794,287,812]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"159\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Fujiki’s Examples \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,830,879,884]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"159\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Tables I and $\\\\mathrm { I I }$ in section 13 of [15] present a list of 18 examples of irreducible symplectic V-manifolds of dimension 4 constructed as follows: let $s$ be a K3 surface or a 2-dimensional complex torus, $H$ a finite group acting symplectically on $s$ (i.e. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,54,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"159\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[799,54,879,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"159\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"160\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"each element $h \\\\in H$ preserves the holomorphic symplectic form of $s$ ) and $\\\\tau$ an automorphism of $H$ of order 2. Then $H$ acts on $S \\\\times S$ via the action mapping $( h , \\\\left( s , t \\\\right) )$ to $( h ( s ) , \\\\tau ( h ) ( t ) )$ . Let $G ( H )$ be the subgroup of $A u t ( S \\\\times S )$ generated by $H$ and by the involution $\\\\iota$ mapping $( s , t )$ to $( t , s )$ , and consider $Y : = S \\\\times S / G ( H )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,162,881,250]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"160\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The fixed locus of the action of $G ( H )$ may have 2-dimensional components, and apart from that it has only isolated points: the singular locus of $Y$ is then given by a (possibly empty) 2-dimensional locus $\\\\Sigma$ and by a finite number of points. Blowing up $\\\\Sigma$ , one gets a $\\\\mathrm { V } .$ -manifold $X$ whose singular locus is given by a finite number of points. Theorem 13.1 of [15] shows that $X$ is an irreducible symplectic V-manifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,252,881,363]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"160\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We reproduce here Tables I and $\\\\mathrm { I I }$ for the convenience of the reader: here $b _ { 2 }$ is the second Betti number of the variety, and $a _ { k }$ is the number of singular points of type $\\\\widehat { A } _ { k } = ( \\\\mathbb { C } ^ { 4 } / g _ { k } , 0 )$ , where $g _ { k } = ( \\\\zeta _ { k } , \\\\zeta _ { k } , \\\\zeta _ { k } ^ { - 1 } , \\\\zeta _ { k } ^ { - 1 } )$ and $\\\\zeta _ { k } : = e ^ { \\\\frac { 2 \\\\pi i } { k } }$ (the sum $a _ { 2 } + a _ { 3 } + a _ { 4 } + a _ { 6 }$ is the number of singular points of the variety). The varieties $X _ { p }$ are obtained starting from a K3 surface $s$ , while the varieties $Y _ { n }$ are obtained starting from a 2-dimensional complex torus. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[259,409,738,756]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0612/03d34fa0df96f0bb4d725c2cb0f41ddd22363529baaaebb51379b19812144989.jpg\",\n \"list_items\": null,\n \"page_idx\": \"160\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=1 colspan=1>Symbol</td><td rowspan=1 colspan=1>H</td><td rowspan=1 colspan=1>b</td><td rowspan=1 colspan=1>a2</td><td rowspan=1 colspan=1>a3</td><td rowspan=1 colspan=1>a4</td><td rowspan=1 colspan=1>a6</td></tr><tr><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>ids</td><td rowspan=1 colspan=1>23</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>L</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>Z/2Z</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>D</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>Z/2Z)2</td><td rowspan=1 colspan=1>14</td><td rowspan=1 colspan=1>36</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>D</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X4</td><td rowspan=1 colspan=1>Z/2Z)3</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X5</td><td rowspan=1 colspan=1>Z/3Z</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>15</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X6</td><td rowspan=1 colspan=1>（Z/2Zz)2</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X7</td><td rowspan=1 colspan=1>Z/4Z</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X8</td><td rowspan=1 colspan=1>(Z/4Z)2</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X9</td><td rowspan=1 colspan=1>Z/6Z</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>X10</td><td rowspan=1 colspan=1>(Z/2Z)(Z/4Z)</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>X11</td><td rowspan=1 colspan=1>(Z/2Z)(Z/6Z)</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Y1</td><td rowspan=1 colspan=1>Z/3Z</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>36</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Y</td><td rowspan=1 colspan=1>(Z/3Z)2</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>27</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Y</td><td rowspan=1 colspan=1>Z/3Z)3</td><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Y4</td><td rowspan=1 colspan=1>Z/4Z</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>54</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Y5</td><td rowspan=1 colspan=1>(Z/4Z)(Z/2Z)</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>52</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Y</td><td rowspan=1 colspan=1>(Z/4Z) (Z/2Z)2</td><td rowspan=1 colspan=1>14</td><td rowspan=1 colspan=1>48</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Y7</td><td rowspan=1 colspan=1>Z/6Z</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>35</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 1 Fujiki’s examples \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[120,776,881,848]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"160\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In all the previous examples, the singular locus of $S \\\\times S / G ( H )$ has always at least one irreducible component of dimension 2: we have 1 component for $X _ { 1 } , X _ { 5 }$ , $X _ { 6 } , Y _ { 1 }$ , $Y _ { 2 }$ and $Y _ { 3 }$ ; 2 components for $X _ { 2 }$ , $X _ { 7 }$ , $X _ { 9 }$ , $Y _ { 4 }$ and $Y _ { 7 }$ ; 4 components for $X _ { 3 }$ , $X _ { 8 }$ , $X _ { 1 0 }$ , $X _ { 1 1 }$ and $Y _ { 5 }$ ; 8 components for $X _ { 4 }$ and $Y _ { 6 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,849,876,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"160\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The varieties $X _ { 1 }$ and $Y _ { 3 }$ are the only smooth examples we get, and they are both irreducible symplectic manifolds: $X _ { 1 }$ is the Hilbert scheme of two points on $s$ , while \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,471,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"160\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Examples of Irreducible Symplectic Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"160\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,91,876,127]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"161\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$Y _ { 3 }$ is deformation equivalent to $K u m ^ { 2 } ( T )$ (see Remark 13.2.3 and Proposition 14.3 of [15]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,129,881,235]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"161\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $p \\\\neq 1$ and $n \\\\neq 3$ , then $X _ { p }$ and $Y _ { n }$ are singular symplectic varieties whose singular locus has codimension 4: by Corollary 1 of [39] they all have terminal singularities, and by the Main Theorem of [41] their deformations are locally trivial. It follows that if $p \\\\neq q$ and $( p , q ) \\\\neq ( 2 , 4 )$ then $X _ { p }$ and $X _ { q }$ are not deformation equivalent; if $n \\\\neq m$ then $Y _ { n }$ is not deformation equivalent to $Y _ { m }$ , and for every $p , n$ we have that $X _ { p }$ is not deformation equivalent to $Y _ { n }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,237,881,325]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"161\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is not known if $X _ { 2 }$ and $X _ { 4 }$ give different deformation classes. Anyway, the Fujiki examples provide at least 17 different deformation classes of irreducible symplectic V-manifolds in dimension 4, 15 of which are singular. The values of $b _ { 2 }$ of these examples are 7, 8, 10, 11, 14, 16 and 23. It is still an open problem to determine if $X _ { p }$ and $Y _ { n }$ are irreducible symplectic orbifolds if $p \\\\neq 1$ and $n \\\\neq 3$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,361,348,379]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"161\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Quotients of $H i l b ^ { n } ( K 3 )$ \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,397,881,506]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"161\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We now consider quotients of $H i l b ^ { n } ( S )$ where $s$ is a projective K3 surface. We start with the case $n = 2$ , so that the fixed locus $F i x ( G )$ of $G$ is a union of a finite number of points (depending on the order of $G$ ) and a (possibly empty) holomorphic symplectic surface, and the same holds for the singular locus of $H i l b ^ { 2 } ( S ) / G$ . By blowing up the singular surface one then gets an irreducible symplectic V-manifold $M _ { G }$ (see [15]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,507,881,596]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"161\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A particular case is when $G = \\\\langle \\\\phi \\\\rangle$ , where $\\\\phi$ is a symplectic automorphism of prime order. By Corollary 2.13 of [36], the order $p$ of $\\\\phi$ can only be 2, 3, 5, 7 or 11. The case of $p = 2$ , i.e. $\\\\phi$ is a symplectic involution, is studied in [30] and [32]. The case $p = 3$ is studied in [32]. The case $p = 1 1$ is studied in [33]. The cases $p = 5 , 7$ are treated in [35]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,633,851,723]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"161\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 1 (Quotient by a Symplectic Involution) If $G ~ = ~ \\\\langle \\\\phi \\\\rangle$ where $\\\\phi$ is a symplectic involution, then $M _ { 2 } \\\\ : = \\\\ M _ { G }$ is an irreducible symplectic Vmanifold with 28 isolated singular points, and by Theorem 2.5 of [32] we have $b _ { 2 } ( M _ { 2 } ) = 1 6$ . By Corollary 3.23 of [34] and Proposition 5.1 of [30] we have that the topological Euler number of $M _ { 2 }$ is 268. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,725,851,795]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"161\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It follows that $M _ { 2 }$ is not deformation equivalent to any of the Fujiki examples above: the only examples in Table 1 having 28 singular points are $X _ { 2 }$ and $X _ { 4 }$ (for both of which $b _ { 2 } = 1 6$ ), whose topological Euler number is 226 (see Remark 13.2.4 of [15]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,54,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"161\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[799,54,879,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"161\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[145,109,854,180]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"162\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 2 (Quotient by a Natural Symplectic Automorphism of Order 3) If ${ \\\\cal G } ~ = ~ \\\\langle \\\\phi \\\\rangle$ where $\\\\phi$ is a natural symplectic automorphism of order 3, then $F i x ( G )$ is given by 27 isolated points. We get that $M _ { 3 } : = M _ { G }$ is an irreducible symplectic V-manifold with 27 isolated singular points. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,183,851,252]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"162\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This example is then not deformation equivalent to $M _ { 2 }$ , and neither to any of the Fujiki examples in Table 1: the only one of that list having 27 singular points is $Y _ { 2 }$ , whose $b _ { 2 } = 7 .$ . But by Theorem 1.3 of [33] we have $b _ { 2 } ( M _ { 3 } ) = 1 1$ , so $M _ { 3 }$ is a new deformation class. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,309,854,399]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"162\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example 3 (Quotient by a Symplectic Automorphism of Order $1 l$ ) Examples 4.5.1 and 4.5.2 of [36] provide two K3 surfaces $S _ { 1 }$ and $S _ { 2 }$ such that $H i l b ^ { 2 } ( S _ { i } )$ has an automorphism $\\\\sigma _ { i }$ of order 11. Let $G _ { i } : = \\\\langle \\\\sigma _ { i } \\\\rangle$ : in both cases $F i x ( G _ { i } )$ is given by 5 isolated points. Then $M _ { 1 1 } ^ { i } : = M _ { G _ { i } }$ is an irreducible symplectic V-manifold with 5 singular points. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,399,851,435]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"162\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By [33] we have $b _ { 2 } ( M _ { 1 1 } ^ { i } ) = 3$ and that $M _ { 1 1 } ^ { 1 }$ and $M _ { 1 1 } ^ { 2 }$ are not deformation equivalent, so they provide two more deformation classes. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,472,881,633]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"162\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $G = \\\\phi$ where $\\\\phi$ has order 5, then $M _ { 5 } : = M _ { G }$ is an irreducible symplectic V-manifold with 14 singular points and $b _ { 2 } ( M _ { 5 } ) = 7$ . If $G = \\\\phi$ where $\\\\phi$ has order 7, then $M _ { 7 } : = M _ { G }$ is an irreducible symplectic V-manifold with 9 singular points and $b _ { 2 } ( M _ { 7 } ) = 5$ (see Theorems 1.2 and 1.3 of [35] and Examples 4.3.1 and 4.4.1 of [36]). We then get 6 more deformation classes of singular irreducible symplectic V-manifolds of dimension 4 to add to the 15 classes presented by Fujiki. If we now consider a K3 surface $s$ and $H i l b ^ { n } ( S )$ for $n \\\\geq 3$ , let $\\\\phi$ be a symplectic involution on $H i l b ^ { n } ( S )$ . The quotient $H _ { n }$ of $H i l b ^ { n } ( S )$ by the action of $\\\\phi$ is again an irreducible symplectic $V$ -manifold (see [15], or Lemma 1 below). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,635,876,671]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"162\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We are in the position to describe which of these examples are irreducible symplectic orbifolds, as the following shows: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,680,876,716]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"162\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 1 Let $s$ be a projective $K 3$ surface and $G$ a finite group of automorphisms of $H i l b ^ { n } ( S )$ acting symplectically. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,723,879,759]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"162\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. If the codimension of $F i x ( G )$ is at least 4, then $M _ { G } : = H i l b ^ { 2 } ( S ) / G$ is not an irreducible symplectic orbifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,761,881,833]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"162\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. If $n = 2$ and $G$ is generated by a symplectic involution, then the singular locus of $H i l b ^ { 2 } ( S ) / G$ is given by 28 isolated points and a K3 surface $\\\\Sigma$ . Then the partial resolution $M _ { G }$ of $H i l b ^ { 2 } ( S ) / G$ obtained by blowing-up $\\\\Sigma$ is an irreducible symplectic orbifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,842,879,896]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"162\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In particular $M _ { 2 }$ is an irreducible symplectic orbifold, while $M _ { 3 }$ , $M _ { 5 }$ , $M _ { 7 }$ , $M _ { 1 1 } ^ { 1 }$ , $M _ { 1 1 } ^ { 2 }$ and $H _ { n }$ are irreducible symplectic $V$ -manifolds which are not irreducible symplectic orbifolds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,471,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"162\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Examples of Irreducible Symplectic Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"162\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,879,196]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"163\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Suppose first that $F i x ( G )$ has codimension at least 4 in $H i l b ^ { n } ( S )$ , so the singular locus of $M _ { G }$ has codimension at least 4. We know that $M _ { G }$ is an irreducible symplectic V-manifold: we let $U$ be its smooth locus and $f : H i l b ^ { n } ( S ) \\\\longrightarrow M _ { G }$ the quotient morphism. Hence $f : f ^ { - 1 } ( U ) \\\\longrightarrow U$ is a finite étale covering whose degree is the order of $G$ : hence $U$ is not simply connected, and $M _ { G }$ is not an irreducible symplectic orbifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,196,881,286]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"163\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We notice that for $M _ { 3 } , M _ { 5 } , M _ { 7 } , M _ { 1 1 } ^ { 1 }$ and $M _ { 1 1 } ^ { 2 }$ we have that $F i x ( G )$ is given by isolated points (see Examples 2 and 3 above, and Examples 4.3.1 and 4.4.1 of [36]), so they are not irreducible symplectic orbifolds. The fixed locus $F i x ( G )$ in the case of $H _ { n }$ has codimension at least 4 by Theorem 1.1 of [24], so that $H _ { n }$ is not an irreducible symplectic orbifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,288,879,360]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"163\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The case of $n = 2$ and $G$ generated by a symplectic involution is due to Menet (see Remark 3.22 of [34]). We need to show that the smooth locus $U$ of $M _ { G }$ is simply connected. Let $D$ be the exceptional divisor of $M _ { G }$ coming from the blowup $b : M _ { G } \\\\longrightarrow X _ { G } : = H i l b ^ { 2 } ( S ) / G$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,360,881,484]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"163\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now, let $\\\\Sigma ^ { \\\\prime }$ be the 2-dimensional component of the singular locus of $X _ { G }$ , and $U ^ { \\\\prime }$ the smooth locus of $X _ { G }$ . We notice that the blow-up morphism $^ b$ gives an isomorphism between $U \\\\backslash D$ and $U ^ { \\\\prime }$ . Moreover the quotient morphism $f :$ $H i l b ^ { 2 } ( S ) \\\\longrightarrow X _ { G }$ gives a double étale covering $H i l b ^ { 2 } ( S ) \\\\setminus F i x ( G ) \\\\longrightarrow U ^ { \\\\prime }$ . As $F i x ( G )$ has codimension 2 in $H i l b ^ { 2 } ( S )$ , and $H i l b ^ { 2 } ( S )$ is smooth, it follows that $\\\\pi _ { 1 } ( H i l b ^ { 2 } ( S ) ) \\\\simeq \\\\pi _ { 1 } ( H i l b ^ { 2 } ( S ) \\\\setminus F i x ( G ) )$ , so that $H i l b ^ { 2 } ( S ) \\\\setminus F i x ( G )$ is simply connected. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,486,879,557]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"163\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As a consequence, we see that $\\\\pi _ { 1 } ( U \\\\setminus D ) \\\\simeq \\\\pi _ { 1 } ( U ^ { \\\\prime } ) \\\\simeq \\\\mathbb { Z } / 2 \\\\mathbb { Z } .$ . Hence, there is an étale covering $\\\\pi ^ { \\\\prime } : Y ^ { \\\\prime } \\\\longrightarrow U \\\\setminus D$ of degree 2, which extends to a finite covering-$\\\\pi : Y \\\\longrightarrow U$ branched along $D$ . Notice that $Y ^ { \\\\prime }$ is simply connected, hence it follows that $Y$ is simply connected as well. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,558,881,648]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"163\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let now $x _ { 0 } \\\\in D$ and consider a loop - $\\\\gamma$ in $U$ pointed at $x _ { 0 }$ . Let $\\\\widetilde { \\\\gamma }$ be a lift of $\\\\gamma$ to $Y$ : notice that as $x _ { 0 }$ is a branching point, the fiber of $\\\\pi$ over $x _ { 0 }$ is given by a unique point $y _ { 0 }$ , so $\\\\widetilde { \\\\gamma }$ is a loop in $Y$ pointed at $y _ { 0 }$ . Since $Y$ is simply connected, it follows that the homotopy class of $\\\\widetilde { \\\\gamma }$ is zero in $\\\\pi _ { 1 } ( Y , y _ { 0 } )$ , so the homotopy class of $\\\\gamma$ is zero as well in $\\\\pi _ { 1 } ( U , x _ { 0 } )$ , and we are done. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,671,330,689]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"163\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Quotients of $K u m ^ { 2 } ( S )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,707,879,761]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"163\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Similar considerations are done in [25] for quotients of $K u m ^ { 2 } ( S )$ for an Abelian surface S. If $\\\\sigma$ is a symplectic involution on $K u m ^ { 2 } ( S )$ , by Theorem 7.5 of [25] the fixed locus of $\\\\sigma$ is given by 36 points together with a K3 surface $\\\\Sigma$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,762,881,854]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"163\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The quotient $K u m ^ { 2 } ( S ) / \\\\sigma$ has then a singular locus given by 36 points and a K3 surface. Blowing up the image of $\\\\Sigma$ in $K u m ^ { 2 } ( S ) / \\\\sigma$ one gets a $V$ -manifold $K$ which has 36 singular points. Again $K$ is an irreducible symplectic V-manifold, and $b _ { 2 } ( K ) = 8$ (see [25]), hence $K$ is not deformation equivalent to $M _ { 2 } , M _ { 3 } , M _ { 1 1 } ^ { 1 }$ and $M _ { 1 1 } ^ { 2 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,854,876,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"163\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moreover, the only examples in Table 1 having 36 singular points are $X _ { 3 }$ and $Y _ { 1 }$ , whose $b _ { 2 }$ is 14 and 7 respectively, so $K$ is not deformation equivalent to any of the Fujiki examples: we then have another deformation class of singular irreducible symplectic V-manifolds to add the previously mentioned 21 classes. The same proof of point 2 of Proposition 1 gives the following: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,54,152,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"163\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[799,54,879,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"163\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,879,142]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"164\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,153,731,169]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"164\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 2 The variety $K$ is an irreducible symplectic orbifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,198,487,213]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"164\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Examples of Markushevich-Tikhomirov \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,232,881,358]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"164\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Markushevich and Tikhomirov present in [30] a different construction of irreducible symplectic $\\\\mathrm { V } .$ -manifolds of dimension 4. Let $X$ be a del Pezzo surface which is the double cover of $\\\\mathbb { P } ^ { 2 }$ branched along a smooth quartic $B _ { 0 }$ with 28 bitangent lines, and let $s$ be the double cover of $X$ branched along the curve $\\\\Delta _ { 0 }$ such that $\\\\Delta _ { 0 } + i ( \\\\Delta _ { 0 } )$ is the inverse image of a smooth quartic curve of $\\\\mathbb { P } ^ { 2 }$ which is totally tangent to $B _ { 0 }$ at eight distinct points (here $i$ is the involution on $X$ induced by the double cover $X \\\\longrightarrow \\\\mathbb { P } ^ { 2 }$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,360,879,468]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"164\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The surface $s$ is a K3 surface, and let $M _ { k }$ be the moduli space of torsion sheaves on $s$ with first Chern class $H$ (the pull-back of $- K _ { X } )$ and Euler character $k -$ 2 for $k \\\\in 2 { \\\\mathbb { Z } }$ , i.e. of Mukai vector $\\\\ : v = ( 0 , H , k - 2 ) \\\\ :$ . We notice that $M _ { k }$ is a projective variety of dimension 6: as $k$ is even, $M _ { k }$ has exactly 28 singular points (see Proposition 1.12.(iii) of [30]). Moreover, the moduli space $M _ { k }$ has an involution $\\\\sigma$ mapping $L \\\\in M _ { k }$ to $\\\\mathcal { E } x t _ { O _ { S } } ^ { 1 } ( L , O _ { S } ( - H ) )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,468,879,539]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"164\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $\\\\tau$ Ois the involution on $S$ induced by the double cover $s \\\\longrightarrow X$ , we let $\\\\kappa : =$ $\\\\tau ^ { * } \\\\circ \\\\sigma$ : the fixed locus of $\\\\kappa$ has a 4-dimensional irreducible component, denoted $\\\\mathcal { P } ^ { k }$ . The morphism mapping $L \\\\in { \\\\mathcal { P } } ^ { k }$ to $L ( H ) \\\\in \\\\mathcal { P } ^ { k + 2 }$ is an isomorphism, hence we get at most two non-isomorphic varieties 0 and 2. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,540,879,611]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"164\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By Markushevich and Tikhomirov [30] we know that $\\\\mathcal { P } ^ { 0 }$ and $\\\\mathcal { P } ^ { 2 }$ are both irreducible symplectic V-manifolds having 28 singular points. By Lemma 5.2 and Corollary 5.7 of [30] we have that $\\\\mathcal { P } ^ { 0 }$ is birational to $M _ { 2 }$ via a Mukai flop, and by Corollary 3.23 of [34]: it follows that $\\\\mathcal { P } ^ { 0 }$ is an irreducible symplectic orbifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,612,879,647]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"164\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is not known if $\\\\mathcal { P } ^ { 0 }$ and $\\\\mathcal { P } ^ { 2 }$ are birational, isomorphic nor deformation equivalent, and it is still an open question if $\\\\mathcal { P } ^ { 2 }$ is an irreducible symplectic orbifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,675,309,692]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"164\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Example of Matteini \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,711,879,800]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"164\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A similar construction to that of Markushevich and Tikhomirov is presented in [31]. Take a K3 surface $s$ which is a double cover of a generic cubic surface $Y$ with involution $\\\\tau$ , and take $M$ to be the moduli space of semistable torsion sheaves with first Chern class $H$ (the pull-back of $- K _ { Y }$ ) and Euler character $^ { - 3 }$ : this is a singular projective variety of dimension 8. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,801,879,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"164\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"One still has an involution $\\\\sigma$ on $M$ obtained as before, and we let $\\\\kappa : = \\\\tau ^ { * } \\\\circ \\\\sigma$ . The fixed locus of $\\\\kappa$ has a 6-dimensional irreducible component $\\\\mathcal { P }$ , whose singular locus is the union of 27 singular K3 surfaces. In [31] it is shown that $\\\\mathcal { P }$ is an irreducible symplectic V-manifold of dimension 6. It is not known if this example is an irreducible symplectic orbifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,471,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"164\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Examples of Irreducible Symplectic Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,874,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"164\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[861,156,879,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"164\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"□ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,610,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"165\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2 Varieties with Canonical Singularities \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,132,881,205]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"165\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A further generalization of the Beauville-Bogomolov decomposition theorem was obtained for projective varieties with canonical singularities by the works of Druel, Greb, Guenancia, Höring, Kebekus and Peternell, in particular [11, 12, 16, 18] and [19]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,240,697,259]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"165\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2.1 The Decomposition for Singular Projective Varieties \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,276,881,331]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"165\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We introduce the following notation: if $X$ is a normal variety and \\u0003 \\u0004 $X _ { r e g }$ is the smooth locus of $X$ whose open embedding in $X$ is $j : X _ { r e g } \\\\longrightarrow X$ , for every $p \\\\in \\\\mathbb N$ such that $0 \\\\leq p \\\\leq \\\\dim ( X )$ we let \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[348,345,649,375]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0251/03ecdb855e65b1d597a8e39742521892ca6834ceaf7901fb6aeee798e178c074.jpg\",\n \"list_items\": null,\n \"page_idx\": \"165\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\Omega _ { X } ^ { [ p ] } : = j _ { * } \\\\Omega _ { X _ { r e g } } ^ { p } = \\\\big ( \\\\wedge ^ { p } \\\\Omega _ { X } \\\\big ) ^ { * * } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,388,633,406]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"165\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"whose global sections are called reflexive- $p$ -forms on $X$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,406,881,507]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"165\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Notice that by definition of $\\\\Omega _ { X } ^ { [ p ] }$ we have $H ^ { 0 } ( X , \\\\Omega _ { X } ^ { [ p ] } ) \\\\ = \\\\ H ^ { 0 } ( X _ { r e g } , \\\\Omega _ { X _ { r e g } } ^ { p } )$ Theorem 1.4 of [17] shows that if -- $X$ is a quasi-projective variety with klt singularities and $\\\\pi : { \\\\widetilde { X } } \\\\longrightarrow X$ is a log-resolution, then for every $p \\\\in \\\\mathbb { N }$ such that $0 \\\\leq p \\\\leq \\\\dim ( X )$ the sheaf $\\\\pi _ { * } \\\\Omega _ { \\\\widetilde { X } } ^ { p }$ is reflexive. This implies in particular that $H ^ { 0 } ( X , \\\\Omega _ { X } ^ { [ p ] } ) \\\\simeq H ^ { 0 } ( \\\\widetilde { X } , \\\\Omega _ { \\\\widetilde { X } } ^ { p } )$ (see Observation 1.3 therein). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,509,881,603]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"165\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As shown in [17], if ${ \\\\hat { f } } : Y \\\\longrightarrow X$ is a finite, dominant morphism between two irreducible normal varieties, then there is a morphism of reflexive sheaves $f ^ { * } \\\\Omega _ { X } ^ { [ p ] } \\\\longrightarrow \\\\Omega _ { Y } ^ { [ p ] }$ , induced by the usual pull-back morphism of forms on the smooth loci, giving a morphism $f ^ { [ * ] } : H ^ { 0 } ( X , \\\\Omega _ { X } ^ { [ p ] } ) \\\\longrightarrow H ^ { 0 } ( Y , \\\\Omega _ { Y } ^ { [ p ] } )$ , called reflexive pullback morphism. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[138,603,845,621]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"165\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We recall the definitions of symplectic form and symplectic variety (see [3]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,630,487,648]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"165\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 5 Let $X$ be a normal variety. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,657,881,803]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"165\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. A symplectic form on $X$ is a closed reflexive 2-form $\\\\sigma$ on $X$ which is nondegenerate at each point of $X _ { r e g }$ . 2. If $\\\\sigma$ is a symplectic form on $X$ , the pair $( X , \\\\sigma )$ is a symplectic variety if for everyresolution $f : { \\\\widetilde { X } } \\\\longrightarrow X$ of the singularities of $X$ , the holomorphic symplectic form $\\\\sigma _ { r e g } : = \\\\sigma _ { | X _ { r e g } }$ extends to a holomorphic 2-form on $\\\\widetilde { X }$ . 3. If $( X , \\\\sigma )$ is a symplectic variety and $f : { \\\\widetilde { X } } \\\\ \\\\longrightarrow \\\\ X$ is a resolution of the singularities over which $\\\\sigma _ { r e g }$ extends to a holomorphic symplectic form, we say that $f$ is a symplectic resolution. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,812,879,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"165\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A normal variety having a symplectic form and whose singular locus has codimension at least 4 is a symplectic variety (see [14]), and a symplectic variety has terminal singularities if and only if its singular locus has codimension at least 4 (Corollary 1 of [39]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[799,54,879,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"165\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"165\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"162 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,879,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"166\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We now define irreducible Calabi-Yau and irreducible symplectic varieties following [18]. If $X$ and $Y$ are two irreducible normal projective varieties, a finite quasi-étale morphism $f ~ : ~ Y ~ \\\\longrightarrow ~ X$ is a finite morphism which is étale in codimension one. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,169,876,205]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"166\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 6 Let $X$ be an irreducible normal projective variety with trivial canonical divisor and canonical singularities, of dimension $d \\\\ge 2$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,216,881,309]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"166\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. The variety $X$ is irreducible Calabi-Yau if for every $0 < p < d$ and for every finite quasi-étale morphism $Y \\\\longrightarrow X$ , we have $H ^ { 0 } ( Y , \\\\Omega _ { Y } ^ { [ p ] } ) = 0$ . 2. The variety $X$ is irreducible symplectic if it has a symplectic form $\\\\sigma \\\\in$ $H ^ { 0 } ( X , \\\\Omega _ { X } ^ { [ 2 ] } )$ , and for every finite quasi-étale morphism $f : Y \\\\longrightarrow X$ the exterior algebra of reflexive forms on $Y$ is spanned by $f ^ { [ * ] } \\\\sigma$ . □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,318,881,390]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"166\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A description of irreducible Calabi-Yau and irreducible symplectic varieties in terms of holonomy is available by the work of Greb et al. [16]. More precisely, if $H$ is an ample divisor on $X$ , by Eyssidieux et al. [13] there is a singular Ricci-flat Kähler metric $\\\\omega _ { H }$ in $c _ { 1 } ( H )$ , inducing a Riemannian metric $g _ { H }$ on $X _ { r e g }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,391,881,480]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"166\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We let $H o l ( X _ { r e g } , g _ { H } )$ be the holonomy group of this metric. Proposition F of [18] shows that a normal projective variety of dimension $n$ with klt singularities and trivial canonical bundle is an irreducible Calabi-Yau variety if and only if $H o l ( \\\\boldsymbol { X _ { r e g } } , g _ { H } )$ is isomorphic to $S U ( n )$ , and it is an irreducible symplectic variety if and only if $H o l ( X _ { r e g } , g _ { H } )$ is isomorphic to $S p ( n / 2 )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[141,481,847,498]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"166\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The decomposition theorem for singular projective varieties is the following: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,507,881,596]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"166\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3 Let $X$ be a normal projective varieties with klt singularities and numerically trivial canonical bundle. Then $X$ has a finite quasi-étale covering $f : Y \\\\longrightarrow X$ , where $Y$ is a normal projective variety with canonical singularities which is a product of complex tori, irreducible Calabi-Yau varieties and irreducible symplectic varieties. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,606,881,750]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"166\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This is Theorem 1.15 of [19], and the proof can be found therein. It consists of three major parts: one is the holonomy decomposition obtained by Greb, Guenancia and Kebekus in [16] (namely Theorem B and Proposition D therein); a second one is an algebraic integrability theorem of Druel, which is Theorem 1.4 of [11]; the final ingredient is Theorem 1.1 of [19]. Less general versions of the Bogomolov decomposition theorem in the projective singular setting were previously obtained in [11, 18] and [12]. The complete proof can be found in section 4 of [19] (the proof of Theorem 1.5 therein). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,759,879,831]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"166\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 4 We notice that even if the statement of Theorem 1.15 of [19] gives the existence of a quasi-étale covering $f : Y \\\\implies X$ , this is consistent with the statement of Theorem 3 since they define quasi-étale morphisms as finite morphisms whose ramification divisor is empty. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,471,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"166\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Examples of Irreducible Symplectic Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"166\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,567,108]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"167\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2.2 Relation Between the Previous Notions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,126,879,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"167\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The first result we state is about the relation between irreducible symplectic manifolds, orbifolds and varieties. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,169,539,189]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"167\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 3 The following properties hold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,198,867,252]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"167\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Irreducible symplectic manifolds are irreducible symplectic orbifolds. \\n2. Projective irreducible symplectic orbifolds are irreducible symplectic varieties. \\n3. Smooth irreducible symplectic varieties are irreducible symplectic manifolds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,279,879,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"167\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof The first point is trivial, and the last is a consequence of Proposition A.1 of [21]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,315,881,421]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"167\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Suppose that $X$ is a projective irreducible symplectic orbifold. Then $X$ is an normal projective variety with rational Cohen-Macaulay singularities and trivial canonical bundle. It follows that $X$ has rational Gorenstein singularities. Moreover, it has a holomorphic symplectic form on its singular locus, so by Theorem 6 of [40] it follows that $X$ is a projective symplectic variety. In particular it has canonical singularities (see [3]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,423,881,530]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"167\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By Theorem 3 there is then a finite quasi-étale covering $f : Y \\\\longrightarrow X$ , where $Y$ is a product of Abelian varieties, irreducible Calabi-Yau varieties and irreducible symplectic varieties. As $X _ { r e g }$ is simply connected by definition of irreducible symplectic orbifold, it follows that $f$ is an isomorphism. As a consequence $Y$ is simply connected, so it has no factor which is an Abelian variety, and it is a symplectic variety, hence it has no factor which is an irreducible Calabi-Yau variety. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,533,879,624]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"167\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hence $Y$ is a product of irreducible symplectic varieties. But as $H ^ { 0 } ( X , \\\\Omega _ { X } ^ { [ 2 ] } )$ is one dimensional by definition of irreducible symplectic orbifold, the same holds for $Y$ . If $Y$ is a product of $m$ irreducible symplectic varieties, we have that $H ^ { 0 } ( Y , \\\\Omega _ { Y } ^ { [ 2 ] } )$ has dimension $m$ : it follows that $m = 1$ , so $Y$ is an irreducible symplectic variety. As $X$ is isomorphic to $Y$ , we are done. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,632,881,723]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"167\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Irreducible symplectic $\\\\mathrm { V } .$ -manifolds are not necessarily irreducible symplectic varieties. A first example of this is given by symmetric products of K3 surfaces: if $s$ is a K3 surface and $m \\\\in \\\\mathbb { N } , m \\\\geq 2$ , then $X : = S y m ^ { m } ( S )$ is an irreducible symplectic V-manifold. Anyway it has a finite quasi-étale covering $S ^ { m } \\\\longrightarrow X$ , and $h ^ { 0 } ( S ^ { m } , \\\\Omega _ { S ^ { m } } ^ { 2 } ) = m$ , so $X$ is not an irreducible symplectic variety. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,723,881,831]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"167\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By Theorem I of [18] we know that all irreducible symplectic varieties are simply connected, so if $X$ is a primitively symplectic V-manifold which is an irreducible symplectic variety, then $X$ is an irreducible symplectic V-manifold. Anyway, as symplectic singularities are not, in general, quotient singularities, we cannot expect that an irreducible symplectic variety is an irreducible symplectic V-manifold: an example will be given in the last section. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,833,881,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"167\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The following is a criterion to guarantee that some quotients of an irreducible symplectic manifold are irreducible symplectic varieties (and irreducible symplectic $V$ -manifolds as well). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[799,54,879,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"167\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[861,255,876,267]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"167\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"□ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"167\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"164 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,876,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"168\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 1 Let $X$ be an irreducible symplectic manifold, $G \\\\subseteq A u t ( X )$ a finite subgroup acting symplectically on $X$ and $Y : = X / G$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,135,881,187]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"168\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. The quotient Y is an irreducible symplectic $V$ -manifold. 2. If Y has terminal singularities and $X$ is projective, then $Y$ is an irreducible symplectic variety. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,196,881,340]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"168\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof The proof of the first part is basically contained in [15], but we present it here for the reader’s sake. The fact that $X$ is a compact, connected Kähler orbifold is Lemma 1.4 of [15], and by Lemma 2.4 of [15] we have that $Y$ is symplectic. As $X$ is simply connected, by Lemma 1.2 of [15] it follows that $Y$ is simply connected as well. Finally, we have $H ^ { 2 , 0 } ( Y ) \\\\simeq H ^ { 2 , 0 } ( X ) ^ { G }$ , where $H ^ { 2 , 0 } ( X ) ^ { G }$ is the space of $G$ - invariant sections. But as $G$ acts symplectically and $X$ is an irreducible symplectic manifold, it follows that $H ^ { 2 , 0 } ( Y )$ is 1-dimensional, so that $Y$ is an irreducible symplectic $V$ -manifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,342,879,448]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"168\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For the second part, as we know that $Y$ is a normal projective symplectic variety with terminal singularities, in order to show that it is an irreducible symplectic variety we just need to look at its finite quasi-étale coverings. So, let $f : Y ^ { \\\\prime } \\\\longrightarrow Y$ be a finite quasi-étale covering of Y . As the singularities of $Y$ are terminal and as $X$ is simply connected (being an irreducible symplectic manifold), it follows that $X$ is the universal covering of $Y$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,450,881,539]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"168\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In particular, it follows that we have a finite quasi-étale covering $\\\\pi ^ { \\\\prime } : X \\\\longrightarrow Y ^ { \\\\prime }$ given by the quotient by a subgroup $G ^ { \\\\prime }$ of $G$ , and if we let $\\\\pi : X \\\\longrightarrow Y$ , then $\\\\pi = f \\\\circ \\\\pi ^ { \\\\prime }$ . As $X$ is an irreducible symplectic manifold, its exterior algebra of holomorphic forms is spanned by the symplectic form on $X$ , which is the reflexive pull-back of the one on $Y$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,539,881,639]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"168\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moreover, we notice that $H ^ { 0 } ( Y ^ { \\\\prime } , \\\\Omega _ { Y ^ { \\\\prime } } ^ { [ p ] } ) \\\\simeq H ^ { p , 0 } ( X ) ^ { G ^ { \\\\prime } }$ , and as $X$ is an irreducible symplectic manifold and $G$ (and hence $G ^ { \\\\prime }$ ) acts symplectically we see that if $p$ is even then $H ^ { 0 } ( Y ^ { \\\\prime } , \\\\Omega _ { Y ^ { \\\\prime } } ^ { [ p ] } )$ is spanned by $f ^ { [ * ] } \\\\sigma ^ { p / 2 }$ (where $\\\\sigma$ is the symplectic form on $Y$ ) and if $p$ is odd then $H ^ { 0 } ( Y ^ { \\\\prime } , \\\\Omega _ { Y ^ { \\\\prime } } ^ { [ p ] } ) = 0 .$ . But this shows that $Y$ is an irreducible symplectic variety. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,647,881,701]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"168\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As an application of this, we see that the examples $M _ { 3 }$ , $M _ { 5 }$ , $M _ { 7 }$ , $M _ { 1 1 } ^ { 1 }$ , $M _ { 1 1 } ^ { 2 }$ and $H _ { n }$ presented in Sect. 2.1.2 are all examples of irreducible symplectic $V$ -manifolds which are irreducible symplectic varieties (but not irreducible symplectic orbifolds). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,702,881,791]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"168\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The partial resolution $M _ { 2 }$ of the quotient of $H i l b ^ { 2 } ( S )$ by a symplectic involution (where $s$ is a K3 surface), the partial resolution $K _ { 2 }$ of the quotient of $K u m ^ { 2 } ( T )$ by a symplectic involution (where $T$ is a 2-dimensional complex torus), and the example $\\\\mathcal { P } ^ { 0 }$ of [30] (which is deformation equivalent to $M _ { 2 }$ ) are all irreducible symplectic varieties as they are irreducible symplectic orbifolds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,792,876,827]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"168\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For all other examples in Sect. 2.1.2 (those of Fujiki, the example $\\\\mathcal { P } ^ { 2 }$ of [30] and the example of Matteini), it is not known if they are irreducible symplectic varieties. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,471,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"168\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Examples of Irreducible Symplectic Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"168\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,346,106]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"169\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2.3 Related Notions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,126,879,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"169\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Irreducible symplectic varieties appear in several papers under different definitions. A first one appears in [1], and it is defined as follows (the name given to these varieties in [1] is irreducible symplectic varieties). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,187,879,259]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"169\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 7 A resolvable symplectic variety is a normal, compact Kähler space $X$ whose smooth locus has a holomorphic symplectic form $\\\\sigma$ , and which has a symplectic resolution of the singularities which is an irreducible symplectic manifold. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,268,881,358]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"169\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A projective resolvable symplectic variety is not always an irreducible symplectic variety: if $s$ is a projective K3 surface and $m \\\\ge 2$ , then $S y m ^ { m } ( S )$ is a projective resolvable symplectic variety (since it is a normal projective symplectic variety having $H i l b ^ { m } ( S )$ as a symplectic resolution), but it is not an irreducible symplectic variety. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,360,879,448]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"169\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Similarly, singular Kummer surfaces (i.e. a surface $s$ obtained as quotient of an Abelian surface $A$ by the involution mapping $p \\\\in A$ to $- p \\\\in A$ ) are resolvable symplectic surfaces (they have a symplectic resolution which is a K3 surface), irreducible symplectic V-manifolds but not irreducible symplectic varieties (as the quotient map $A \\\\longrightarrow S$ is a finite quasi-étale covering and $h ^ { 1 , 0 } ( A ) \\\\neq 0 \\\\}$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,450,879,503]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"169\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We will see in Sect. 2.2.4 examples of irreducible symplectic varieties which are not resolvable symplectic varieties, and of resolvable symplectic varieties which are not irreducible symplectic V-manifolds. Anyway we have the following: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,513,879,566]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"169\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 4 If $X$ is an irreducible symplectic variety (resp. an irreducible symplectic $V .$ -manifold) having $a$ symplectic resolution Y , then $Y$ is an irreducible symplectic manifold. In particular, $X$ is a resolvable symplectic variety. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,575,879,666]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"169\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof If $X$ is an irreducible symplectic variety, this is Remark 1.16 of [47]. If $X$ is an irreducible symplectic $\\\\mathrm { v } .$ -manifold, then $X$ has canonical singularities. By Takayama [50] we get $\\\\pi _ { 1 } ( X ) \\\\ \\\\simeq \\\\ \\\\pi _ { 1 } ( Y )$ , so $Y$ is simply connected. Finally, by Theorem 1.4 of [17] we have $h ^ { 0 } ( Y , \\\\Omega _ { Y } ^ { 2 } ) = h ^ { 0 } ( X , \\\\Omega _ { X } ^ { [ 2 ] } )$ , which is 1 by definition, and we are done. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,677,879,750]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"169\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A further definition of irreducible symplectic variety appears in [49], where it is defined as a projective symplectic variety $X$ such that $h ^ { 1 } ( X , O _ { X } ) \\\\ = \\\\ 0$ and $h ^ { 0 } ( X , \\\\Omega _ { X } ^ { [ 2 ] } ) = \\\\bar { 1 }$ O. It is called Namikawa symplectic variety if it is moreover $\\\\mathbb { Q }$ factorial and its singular locus has codimension at least 4 (see Definition 1 therein). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,759,876,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"169\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 8 We will call Namikawa symplectic variety a normal, compact Kähler complex space $X$ such that $h ^ { 1 } ( X , O _ { X } ) = 0$ and $h ^ { 0 } ( X , \\\\Omega _ { X } ^ { [ 2 ] } ) = 1$ . □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,806,876,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"169\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Namikawa symplectic varieties are the most general kind of varieties we will deal with. Namely: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,851,881,887]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"169\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 5 Irreducible symplectic varieties, resolvable symplectic varieties and irreducible symplectic V-manifolds are all Namikawa symplectic varieties. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[799,54,879,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"169\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,152,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"169\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"166 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"170\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof The proof that irreducible symplectic and resolvable symplectic varieties are Namikawa symplectic is given in Propositions 1.9 and 1.10 of [47]. If $X$ is an irreducible symplectic V-manifold, then - $X$ has rational Gorenstein singularities and has a symplectic form on its smooth locus, hence by Theorem 6 of [40] it is a symplectic variety. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,180,881,291]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"170\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moreover, if ${ \\\\check { f } } : { \\\\widetilde { X } } \\\\longrightarrow X$ is a resolution of the singularities, as- $X$ has rational singularities we have an isomorphism between- $H ^ { 1 } ( X , { \\\\bar { O } } _ { X } )$ and $H ^ { 1 } ( \\\\widetilde { X } , O _ { \\\\widetilde { X } } )$ . As $X$ has klt singularities we have- $\\\\pi _ { 1 } ( X ) \\\\simeq \\\\pi _ { 1 } ( \\\\widetilde { X } )$ (by Takayama [50]), and since $X$ is simply connected it follows that $\\\\tilde { X }$ is simply connected. As a consequence of this we see that $H ^ { 1 } ( \\\\widetilde { X } , O _ { \\\\widetilde { X } } ) = 0$ , and hence $H ^ { 1 } ( X , O _ { X } ) = 0$ . As by definition of irreducible symplectic V-manifold we have that $H ^ { 0 } ( X , \\\\Omega _ { X } ^ { [ 2 ] } )$ is one dimensional, we are done. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,298,881,352]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"170\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"No example of Namikawa symplectic variety which is not an irreducible symplectic variety, nor an irreducible symplectic V-manifold, nor a resolvable symplectic variety is known. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,388,287,406]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"170\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2.4 Examples \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,424,879,496]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"170\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We now introduce two families of examples of irreducible symplectic varieties. In what follows $s$ will denote a projective K3 surface or an Abelian surface, and welet $\\\\epsilon ( S ) : = 1$ if $s$ is K3, and 0 if $s$ is Abelian. We let $\\\\rho ( S )$ be the rank of the Néron-Severi group $N S ( S )$ of $s$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,496,879,569]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"170\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An element $v \\\\in \\\\widetilde { H } ( S , \\\\mathbb { Z } ) : = H ^ { 2 * } ( S , \\\\mathbb { Z } )$ will be written $\\\\boldsymbol { v } = ( v _ { 0 } , v _ { 1 } , v _ { 2 } )$ , where $v _ { i } \\\\in H ^ { 2 i } ( S , \\\\mathbb { Z } )$ , and - $v _ { 0 } , v _ { 2 } \\\\ \\\\in \\\\ \\\\mathbb { Z }$ . It will be called Mukai vector if $v _ { 0 } \\\\geq 0$ , $v _ { 1 } ~ \\\\in$ $N S ( S )$ and if $v _ { 0 } = 0$ , then either $v _ { 1 }$ is the first Chern class of an effective divisor, or $v _ { 1 } = 0$ and $v _ { 2 } > 0$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,570,881,659]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"170\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The $\\\\mathbb { Z }$ -module $\\\\widetilde { H } ( S , \\\\mathbb { Z } )$ has a pure weight-two Hodge structure and a lattice structure with respect to the Mukai pairing $\\\\left( . , . \\\\right)$ (see [20], Definitions 6.1.5 and 6.1.11). We let $\\\\boldsymbol { v } ^ { 2 } : = ( v , v )$ for every Mukai vector $v$ , and we refer to ${ \\\\widetilde { H } } ( S , \\\\mathbb { Z } )$ as the Mukai lattice of $s$ . We will always write 1 $v \\\\ = \\\\ m w$ , where $m \\\\in \\\\mathbb { N }$ and $w$ is a primitive Mukai vector on $s$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,659,681,677]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"170\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To any coherent sheaf $\\\\mathcal { F }$ on $s$ we associate a Mukai vector \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[337,692,658,717]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0700/6b5f1e7818b4194258c2ae7cdc920554ee04251afd90d10daa2ab472773cb867.jpg\",\n \"list_items\": null,\n \"page_idx\": \"170\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\boldsymbol { v } ( \\\\mathcal { F } ) : = c h ( \\\\mathcal { F } ) \\\\sqrt { t d ( S ) } \\\\in \\\\widetilde { H } ( S , \\\\mathbb { Z } ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,732,881,824]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"170\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Taking $v$ a Mukai vector on $s$ and supposing that $H$ is a $v$ -generic polarization (see as instance section 2.1 of [47] for the precise definition), we consider the moduli space $M _ { v } ( S , H )$ (resp. $M _ { v } ^ { s } ( S , H ) _ { , } ^ { \\\\setminus }$ ) of Gieseker $H$ -semistable (resp. $H$ - stable) sheaves on $s$ with Mukai vector $v$ . Then $M _ { v }$ is a projective variety and $M _ { v } ^ { s } \\\\subseteq M _ { v }$ is open (see [20]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,471,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"170\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Examples of Irreducible Symplectic Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"170\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[145,88,425,108]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"171\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The following properties hold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,117,881,169]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"171\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. If $s$ is a K3 surface and $v = m w$ for a primitive Mukai vector $w$ , then $M _ { v } \\\\ne \\\\emptyset$ if and only if $w ^ { 2 } \\\\geq - 2$ (see [38] and [52]). If $s$ is an Abelian surface, then $M _ { v } \\\\ne \\\\emptyset$ if and only $w ^ { 2 } \\\\geq 0$ (see [53]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,171,876,205]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"171\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. If $s$ is a K3 surface and $v = m w$ for a primtive Mukai vector $w$ such that $w ^ { 2 } =$ $^ { - 2 }$ , then $M _ { v }$ is a point (see [38]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,207,881,295]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"171\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3. If $s$ is a K3 surface, $v = m w$ and $w ^ { 2 } = 0$ , then $M _ { v } \\\\simeq S y m ^ { m } ( S ^ { \\\\prime } )$ for a projective K3 surface $S ^ { \\\\prime }$ (see [38] and [23]). If $m = 1$ we then get a projective K3 surface, while if $m \\\\ \\\\geq \\\\ 2$ we then get an irreducible symplectic V-manifold, which is a resolvable symplectic variety but which is neither an irreducible symplectic variety nor an irreducible symplectic orbifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,297,881,421]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"171\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4. If $s$ is an Abelian surface, $v = m w$ and $w ^ { 2 } = 0$ , then $M _ { v } \\\\simeq S y m ^ { m } ( T )$ for an Abelian surface $T$ (see [38] and [23]). If $m = 1$ we then get an Abelian surface, and hence a K3 surface via the Kummer construction. If $m \\\\ge 2$ we consider the natural sum morphism $s : M _ { v } \\\\longrightarrow T$ , and let $K : = s ^ { - 1 } ( 0 )$ : then $K$ is an irreducible symplectic V-manifold, which is a resolvable symplectic variety but which is neither an irreducible symplectic variety nor an irreducible symplectic orbifold (see Example 1.13 of [47]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,423,881,493]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"171\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5. If $s$ is a K3 surface or an Abelian surface and $v = m w$ for a primitive Mukai vector $w$ such that $w ^ { 2 } > 0$ , then $M _ { v }$ is a normal, irreducible projective variety of dimension $v ^ { 2 } + 2$ whose smooth locus is $M _ { v } ^ { s }$ (see [52]). By [37] $M _ { v }$ has a symplectic form. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,495,881,585]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"171\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6. If $s$ is an Abelian surface and $v \\\\ = \\\\ m w$ for a primitive Mukai vector $w$ such that $w ^ { 2 } > 0$ , by section 4.1 of [53] we have a dominant isotrivial fibration $a _ { v } :$ $M _ { v } ( S , H ) \\\\longrightarrow S \\\\times { \\\\widehat S }$ , where $\\\\widehat { S }$ is the dual of $s$ . We let $K _ { v } : = a _ { v } ^ { - 1 } ( 0 _ { S } , O _ { S } )$ , and $K _ { v } ^ { s } : = K _ { v } \\\\cap M _ { v } ^ { s }$ . The restriction of the symplectic form of $M _ { v }$ to $K _ { v }$ is a symplectic form (see [53]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,594,881,648]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"171\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The moduli spaces $M _ { v }$ and $K _ { v }$ described above give us examples of irreducible symplectic varieties if $v ^ { 2 } > 0$ . This is the content of the following result, which is Theorem 1.19 of [47]: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,656,879,711]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"171\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 6 Let $s$ be a projective $K 3$ surface of an Abelian surface, v a Mukai vector on $s$ such that $v = m w$ for a primitive Mukai vector $w$ with $w ^ { 2 } > 0$ , and $H$ a v-generic polarization. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,719,715,738]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"171\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. If $s$ is $K 3$ , then $M _ { v } ( S , H )$ is an irreducible symplectic variety. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[143,746,881,819]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"171\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"a. If $m = 1$ , it is an irreducible symplectic manifold. \\n$b _ { \\\\cdot }$ . If $m = 2$ and $w ^ { 2 } = 2$ , it has a symplectic resolution which is an irreducible symplectic manifold. \\nc. In all other cases it has terminal singularities. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,827,881,864]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"171\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. Suppose that $s$ is Abelian. If $m = 1$ and $w ^ { 2 } = 2$ then $K _ { v } ( S , H )$ is a point. In all other cases $K _ { v } ( S , H )$ is an irreducible symplectic variety. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,872,742,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"171\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"a. If $m = 1$ and $w ^ { 2 } > 2$ , it is an irreducible symplectic manifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[799,54,879,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"171\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"171\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"168 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[143,90,879,145]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"172\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"b. If $m = 2$ and $w ^ { 2 } = 2$ , it has a symplectic resolution which is an irreducible symplectic manifold. \\nc. In all other cases it has terminal singularities. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,154,881,244]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"172\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The case $m = 1$ was proved in its final form by Yoshioka in [52] and [53] (but with important steps towards the complete proof given in [2, 37, 42]). The case $m = 2$ , $w ^ { 2 } = 2$ was studied first by O’Grady in [43] and [44] for $v = 2 ( 1 , 0 , - 1 )$ , where it was shown that $M _ { v }$ (resp. $K _ { v }$ ) has a symplectic resolution which is an irreducible symplectic manifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,246,879,316]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"172\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In [29] it is shown that the symplectic resolution exists for all $v = 2 w$ , where $w$ is primitive and $w ^ { 2 } = 2$ . In [46] it is shown that such a symplectic resolution is an irreducible symplectic manifold, deformation equivalent to $O G _ { 1 0 }$ (resp. $O G _ { 6 }$ ). The proof of the statement for all other cases is contained in [47]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,318,881,424]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"172\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The cases $m = 1$ and $m = 2$ , $w ^ { 2 } = 2$ then recover all the known deformation classes of irreducible symplectic manifolds. Moreover, the case $m = 2$ , $w ^ { 2 } = 2$ gives examples of irreducible symplectic varieties which are resolvable symplectic varieties. We notice in particular that if $s$ is Abelian, then the smooth locus of $K _ { v }$ in this case is not simply connected (see Theorem 3.7 of [47]), giving then an example of irreducible symplectic variety which is not an irreducible symplectic orbifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,426,879,478]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"172\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The remaining cases give examples of irreducible symplectic varieties having no symplectic resolution, hence they are not resolvable symplectic varieties. Moreover, we have the following result: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,487,881,576]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"172\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 7 Let $S _ { 1 }$ and $S _ { 2 }$ be two projective $K 3$ surfaces (resp. Abelian surfaces), $v _ { i } = m _ { i } w _ { i }$ a Mukai vector on $S _ { i }$ for $m _ { i } > 0$ and $w _ { i }$ primitive Mukai vector, and $H _ { i } a$ $v _ { i }$ -generic polarization on $S _ { i }$ . Then $M _ { v _ { 1 } } ( S _ { 1 } , H _ { 1 } )$ (resp. $K _ { v _ { 1 } } ( S _ { 1 } , H _ { 1 } ) ,$ ) is deformation equivalent to $M _ { v _ { 2 } } ( S _ { 2 } , H _ { 2 } )$ (resp. $K _ { v _ { 2 } } ( S _ { 2 } , H _ { 2 } ) )$ 1 if and only if $m _ { 1 } = m _ { 2 }$ and $w _ { 1 } ^ { 2 } = w _ { 2 } ^ { 2 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,587,881,696]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"172\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The sufficient condition is essentially proved by Yoshioka in [54] and [55] (see even [47]). The converse is proved in a forthcoming paper of the author and Rapagnetta. As a consequence we see that in dimension $2 n$ we find a deformation class of irreducible symplectic varieties of dimension $2 n$ for each pair $( m , k )$ such that $m ^ { 2 } k + 1 = n$ or ${ \\\\dot { m } } ^ { 2 } { \\\\dot { k } } - 1 = n$ (just take the moduli space of semistable sheaves of Mukai vector $( m , 0 , - m k ) )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,704,881,831]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"172\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Remark 5 The deformation classes in dimension 4 and 6 we obtain in this way are different from those of the examples of singular irreducible symplectic V-manifolds we presented in Sect. 2.1.2. Indeed, if $s$ is K3 we have $\\\\dim ( M _ { v } ) \\\\ : = \\\\ : 4 , 6$ only if $v ^ { 2 } = 2 , 4$ , so $v$ must be primitive and $M _ { v }$ is then smooth. If $s$ is Abelian we have $\\\\mathrm { d i m } ( K _ { v } ) = 4 , 6$ only if $v ^ { 2 } = 6 , 8$ . If $\\\\upsilon ^ { 2 } = 6$ then $v$ is primitive and $K _ { v }$ smooth; if $\\\\upsilon ^ { 2 } = 8$ then either $v$ is primitive and $K _ { v }$ is smooth, or $v = 2 w$ for $w$ primitive with $w ^ { 2 }$ , and $K _ { v }$ has a symplectic resolution. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,840,881,894]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"172\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally, we remark that the singular moduli spaces in the statement of Proposition 6 give examples of irreducible symplectic varieties which are not irreducible symplectic V-manifolds, as their singularities are not quotient singularities. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,471,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"172\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Examples of Irreducible Symplectic Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"172\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[861,130,879,144]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"172\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"□ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[861,564,876,576]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"172\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"□ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,91,876,121]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"173\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acknowledgements The author wishes to thank Chiara Camere and Baohua Fu for useful discussions, and especially Grégoire Menet and Antonio Rapagnetta for all the fruitful exchanges. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,168,248,187]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"173\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,202,881,900]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"173\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Bakker, B., Lehn, C.: A Global Torelli Theorem for Singular Symplectic Varieties (2016). Preprint arxiv:1612.07894 \\n2. Beauville, A.: Variétés kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18, 755–782 (1983) \\n3. Beauville, A.: Symplectic singularities. Invent. Math. 139(3), 541–549 (2000) \\n4. Beauville, A., Donagi, R.: La variété des droites d’une hypersurface cubique de dimension 4. C. R. Acad. Sci. Paris Sr. I Math. 301(14), 703–706 (1985) \\n5. Blache, R.: Chern classes and Hirzeburch-Riemann-Roch theorem for coherent sheaves on complex projective orbifolds with isolated singularities. Math. Z. 222, 7–57 (1996) \\n6. Bogomolov, F.: On the decomposition of Kähler manifolds with trivial canonical class. Math. USSR-Sb. 22, 580–583 (1974) \\n7. Borzellino, J., Zhu, S.-H.: The Splitting Theorem for Orbifolds. Ill. J. Math. 38, 679–691 (1994) \\n8. Campana, F.: Orbifoldes à première classe de Chern nulle. In: The Fano Conference, pp. 339– 351, University Torino, Torino (2004) \\n9. Debarre, O., Voisin, C.: Hyper-Kähler fourfolds and Grassmann Geometry. J. Reine Angew. Math. 649, 63–87 (2010) \\n10. Demailly, J.-P., Kollár, J.: Semi-continuity of Complex Singularity Exponents and Kähler-Einstein Metrics on Fano Orbifolds. Ann. Sci. Ec. Norm. Sup. Série 4 34(4), 525–556 (2001) \\n11. Druel, S.: A decomposition theorem for singular spaces with trivial canonical class of dimension at most five. Invent. Math. 211(1), 245–296 (2018) \\n12. Druel, S., Guenancia, H.: A decomposition theorem for smoothable varieties with trivial canonical class. J. de l’École Polytechnique—Mathématiques 5, 117–147 (2018) \\n13. Eyssidieux, P., Guedj, V., Zeriahi, A.: Singular Kähler-Einstein metrics. J. Am. Math. Soc. 22(3), 607–639 (2009) \\n14. Flenner, H.: Extendability of differential forms on non-isolated singularities. Invent. Math. 94, 317–326 (1988) \\n15. Fujiki, A.: On primitively symplectic compact Kähler V-Manifolds of dimension four. In: Classification of Algebraic and Analytic Manifolds. Progress in Mathematics 39, 71–250 (1983) \\n16. Greb, D., Guenancia, H., Kebekus, S.: Klt with Trivial Canonical Class—Holonomy, Differential Forms, and Fundamental Group. Geom. Topol. 23, 2051–2124 (2019) \\n17. Greb, D., Kebekus, S., Kovács, S., Peternell, T.: Differential forms on log-canonical spaces. Publ. Math. IHES 114(1), 87–169 (2011) \\n18. Greb, D., Kebekus, S., Peternell, T.: Singular spaces with trivial canonical class. In: Proceedings of the conference in honor of Shigefumi Mori’s 60th birthday, minimal models and extreme rays. Advanced Studies in Pure Mathematics. Kinokuniya Publishing House, Tokyo (2011) \\n19. Höring, A., Peternell, T.: Algebraic Integrability of Foliations with Numerically Trivial Canonical Bundle. Invent. Math. 216, 395–419 (2019) \\n20. Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. Aspects of Mathematics E, vol. 31. Vieweg, Braunschweig (1997) \\n21. Huybrechts, D., Nieper-Wisskirchen, M.: Remarks on derived equivalence of Ricci-flat manifolds. Math. Z. 267(3–4), 939–963 (2011) \\n22. Iliev, A., Ranestad, K.: K3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds. Trans. Am. Math. Soc. 353, 1455–1468 (2001) \\n23. Kaledin, D., Lehn, M., Sorger, C.: Singular symplectic moduli spaces. Invent. Math. 164, 591– 614 (2006) \\n24. Kamenova, L., Mongardi, G., Oblomkov, A.: Symplectic involutions of $K 3 ^ { [ n ] }$ type and Kummer n type manifolds (2018). Preprint arxiv:1809.02810 \\n25. Kapfer, S., Menet, G.: Integral cohomology of the generalized Kummer fourfold. Algebr. Geom. 5(5), 523–567 (2018) \\n26. Kawamata, Y.: Minimal models and the Kodaira dimension of algebraic fiber spaces. J. Reine Angew. Math. 363, 1–46 (1985) \\n27. Kollár, J., Mori, S.: Birational geometry of algebraic varieties. In: Cambridge Tracts in Mathematics, vol. 134. Cambridge University, Cambridge (1998) \\n28. Lehn, C., Lehn, M., Sorger, C., van Straten, D.: Twisted cubics on cubic fourfolds. J. Reine Angew. Math. 731, 87–128 (2017) \\n29. Lehn, M., Sorger, C.: La singularité de O’Grady. J. Algebraic Geom. 15, 756–770 (2006) \\n30. Markushevich, D., Tikhomirov, A.S.: New symplectic V-manifolds of dimension four via the relative compactified Prymian. Int. J. Math. 18(10), 1187–1224 (2007) \\n31. Matteini, T.: A new singular symplectic variety of dimension 6 with a Lagrangian Prym fibration. Manuscripta Math. 149(1), 131–151 (2016) \\n32. Menet, G.: Beauville-Bogomolov lattice for a singular symplectic variety of dimension 4. J. Pure App. Math. 219(5), 1455–1495 (2014) \\n33. Menet, G.: On the integral cohomology of quotients of manifolds by cycling groups. J. Math. Pures Appl. 119(9), 280–325 (2018) \\n34. Menet, G.: A Global Torelli Theorem for irreducible symplectic orbifolds. Preprint arxiv:1805.12426 \\n35. Menet, G.: Integral cohomology of quotients via toric geometry (2019). Preprint \\n36. Mongardi, G.: Automorphisms of Hyperkähler Manifolds. PhD Thesis arxiv:1303.4670 \\n37. Mukai, S.: Symplectic structure of the moduli space of sheaves on an Abelian or K3 surface. Invent. Math. 77, 101–116 (1984) \\n38. Mukai, S.: On the moduli space of bundles on K3 surfaces I. In: Vector Bundles on Algebraic Varieties, Bombay (1984). Tata Institute of Fundamental Research Studies on Mathematics, vol. 11, , pp. 341–413. Tata Institute of Fundamental Research, Bombay (1987) \\n39. Namikawa, Y.: A Note on Symplectic Singularities (2001). Preprint arxiv:0101028 \\n40. Namikawa, Y.: Extension of 2-forms and symplectic varieties. J. Reine Angew. Math. 539, 123–147 (2001) \\n41. Namikawa, Y.: On deformations of $\\\\mathbb { Q }$ -factorial symplectic varieties. J. Reine Angew. Math. 599, 97–110 (2006) \\n42. O’Grady, K.: The Weight-two Hodge Structure of Moduli Spaces of Sheaves on a K3 Surface. J. Algebraic Geom. 6(4), 599–644 (1997) \\n43. O’Grady, K.: Desingularized Moduli spaces of sheaves on a K3 surface. J. Reine. Angew. Math. 512, 49–117 (1999) \\n44. O’Grady, K.: A new six dimensional irreducible symplectic variety. J. Algebraic Geom. 12, 435–505 (2003) \\n45. O’Grady, K.: Irreducible symplectic 4-folds and Eisenbud-Popescu-Walters sextics. Duke Math. J. 134, 99–137 (2006) \\n46. Perego, A., Rapagnetta, A.: Deformation of the O’Grady moduli spaces. J. Reine Angew. Math. 678, 1–34 (2013) \\n47. Perego, A., Rapagnetta, A.: The moduli spaces of sheaves on K3 surfaces are irreducible symplectic varieties. Preprint arxiv:1802.01182 \\n48. Rapagnetta, A.: On the Beauville form of the known irreducible symplectic varieties. Math. Ann. 340(1), 77–95 (2008) \\n49. Schwald, M.: Fujiki Relations and Fibrations of Irreducible Symplectic Varities (2017). Preprint arxiv:1701.09069 \\n50. Takayama, S.: Local simple connectedness of resolutions of log-terminal singularities. Int. J. Math. 14(8), 825–836 (2003) \\n51. Yau, S.-T.: On the ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. Commun. Pure App. Math 31, 339–411 (1978) \\n52. Yoshioka, K.: Irreducibility of Moduli Spaces of Vector Bundles on K3 Surfaces (1999). Preprint arxiv:math/9907001 \\n53. Yoshioka, K.: Moduli spaces of stable sheaves on Abelian surfaces. Math. Ann. 321(4), 817– 884 (2001) \\n54. Yoshioka, K.: Twisted stability and Fourier-Mukai functor I. Compos. Math. 138(3), 261–288 (2003) \\n55. Yoshioka, K.: Twisted stability and Fourier-Mukai functor II. Compos. Math. 145, 112–142 (2009) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,54,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"173\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[799,54,879,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"173\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[116,99,883,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"174\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,471,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"174\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Examples of Irreducible Symplectic Varieties \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"174\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"171 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[116,91,881,240]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"175\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[799,54,879,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"175\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"175\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,644,141]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"176\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An Example of Mirror Symmetry for Fano Threefolds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,175,280,193]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"176\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Andrea Petracci \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,288,881,340]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"176\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract In this note we illustrate the Fanosearch programme of Coates, Corti, Galkin, Golyshev, and Kasprzyk in the example of the anticanonical cone over the smooth del Pezzo surface of degree 6. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,360,835,394]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"176\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords Mirror symmetry $\\\\cdot$ Fano varieties $\\\\cdot$ Toric varieties $\\\\cdot$ Deformation theory \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,412,797,432]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"176\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2010 Mathematics Subject Classification 14J45, 14M25, 14J33, 14D15 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,468,309,489]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"176\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,507,230,527]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"176\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1.1 Aim \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,552,881,731]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"176\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Fanosearchprogramme of Coates et al. [8] studies Fano varieties via Mirror Symmetry. In this context it is crucial to study toric degenerations of smooth Fano varieties, or conversely smoothings of toric Fano varieties. Toric Fano varieties are associated to certain lattice polytopes, called Fano polytopes; some combinatorial input on a Fano polytope conjecturally allows to construct a deformation of the corresponding toric Fano. This is also reflected by Mirror Symmetry, where the combinatorial input is encoded by certain special Laurent polynomials. The goal of this note is to illustrate this programme in a specific example where two different combinatorial inputs on the same polytope produce two different smoothings of the same toric Fano threefold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,834,635,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"176\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. Petracci $( \\\\boxtimes )$ Institut für Mathematik, Freie Universität Berlin, Berlin, Germany e-mail: andrea.petracci $@$ fu-berlin.de \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,897,706,941]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"176\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_10 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[804,91,876,139]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"176\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,899,876,909]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"176\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,332,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"177\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1.2 The Example \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,132,881,277]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"177\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The example we consider is the projective cone over the anticanonical embedding of the smooth del Pezzo surface of degree 6. This threefold, denoted by $X$ , is a toric Fano and has an isolated Gorenstein canonical non-terminal singularity at the vertex of the cone. The deformations of this singularity have been studied by Altmann [4]; we will recall Altmann’s results in Sect. 2.3. In Sect. 2.4 we will see that the base of the miniversal deformation (or equivalently the Kuranishi family) of the projective threefold $X$ has two irreducible components, which deform $X$ to two different smooth Fano threefolds, namely: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,286,669,321]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"177\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"• a general element $X _ { 2 }$ of the linear system $| O _ { \\\\mathbb { P } ^ { 2 } \\\\times \\\\mathbb { P } ^ { 2 } } ( 1 , 1 ) |$ , • $\\\\bar { X _ { 3 } } = \\\\bar { \\\\mathbb { P } } ^ { 1 } \\\\times \\\\bar { \\\\mathbb { P } } ^ { 1 } \\\\times \\\\bar { \\\\mathbb { P } } ^ { 1 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,331,881,384]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"177\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(The reason for the subscripts 2 and 3 will be evident later.) These two smooth Fanos are connected via deformation through $X$ , but cannot be connected via a deformation with smooth fibres, as their Betti numbers are different. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,385,881,530]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"177\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As $X$ is toric, by means of toric geometry, we can associate to $X$ a 3-dimensional lattice polytope $P \\\\subseteq \\\\mathbb { R } ^ { 3 }$ which is a hexagonal pyramid (see the precise definition in (2) and Fig. 1). The hexagonal facet of $P$ is denoted by $F$ (see (1) and the left part of Fig. 2). In Proposition 2.2 we will see that the two smoothings of $X$ are associated to some combinatorial additional data on the polytope $P$ . More precisely, they correspond to the two maximal Minkowski decompositions of the hexagon $F$ (see (3) and (4), and Fig. 2). We will introduce the notion of Minkowski sum and Minkowski decomposition in Sect. 2.3. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[332,561,662,744]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 1 The 3-dimensional lattice polytope $P$ associated to $X$ \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0842/aea482f8093324f43b4758359351cedb8a78d6dddeb3b74abb5eb9f1254739c7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"177\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[177,785,820,869]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 2 The two maximal Minkowski decompositions of the hexagon $F$ \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0925/dedcce463875c0ebf0feb24831a72d2b18f5490860875bebdd9a565458742ce4.jpg\",\n \"list_items\": null,\n \"page_idx\": \"177\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[790,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"177\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. Petracci \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"177\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"174 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,198]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"178\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now we consider the Laurent polynomials in 3 variables which are supported on $P$ , i.e. Laurent polynomials $f ~ \\\\in ~ \\\\mathbb { C } [ x ^ { \\\\pm } , y ^ { \\\\pm } , z ^ { \\\\pm } ]$ such that if the monomial $x ^ { i } y ^ { j } z ^ { k }$ appears in $f$ then the point $( i , j , k ) \\\\in \\\\mathbb { Z } ^ { 3 }$ lies in $P$ . Among these Laurent polynomials, we consider those which have coefficient 1 on the vertices of2 3 $P$ and have coefficient 0 on the origin of ${ \\\\mathbb { R } } ^ { 3 }$ ; this gives rise to the following 1-dimensional family: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[277,211,722,253]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0858/8288fcbf3aee22daf5ba42e9b14f861ff07b45f7537e611d081ae3c98ca1a5a4.jpg\",\n \"list_items\": null,\n \"page_idx\": \"178\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nf _ { a } = z \\\\left( a + x + x y + y + { \\\\frac { 1 } { x } } + { \\\\frac { 1 } { x y } } + { \\\\frac { 1 } { y } } \\\\right) + { \\\\frac { 1 } { z } }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,268,325,285]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"178\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"with parameter $a \\\\in \\\\mathbb { C }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,286,881,411]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"178\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"One can show that $f _ { 2 }$ is mirror to $X _ { 2 }$ and $f _ { 3 }$ is mirror to $X _ { 3 }$ . In other words, a certain generating function for some Gromov–Witten invariants of $X _ { 2 }$ , called quantum period (see Sect. 3.1), is equal to a certain power series, called classical period (see Sect. 3.3), associated to $f _ { 2 }$ , and the same holds for $X _ { 3 }$ and $f _ { 3 }$ . Here we are using the formulation of the Mirror Symmetry correspondence between Fanos and Landau–Ginzburg models that is given in [8, 31, 32] and summarised in Sect. 3.4. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,412,881,483]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"178\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We will see that the Laurent polynomial $f _ { 2 }$ is closely related to the combinatorial input given by the Minkowski decomposition of the hexagon $F$ which is associated to the smoothing of $X$ to $X _ { 2 }$ . Analogously, $f _ { 3 }$ is closely related to the Minkowski decomposition of the hexagon $F$ which is associated to the smoothing of $X$ to $X _ { 3 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,530,412,549]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"178\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1.3 The General Picture \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,575,879,609]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"178\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"What we have described in the case of the projective cone over the del Pezzo surface of degree 6 is an instance of the following conjecture, which is still slightly vague. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,618,881,672]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"178\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conjecture 1.1 ([8]) Let $Q$ be a Fano polytope of dimension 3 and let $X _ { Q }$ be the corresponding toric Fano threefold. Assume that $X _ { Q }$ has Gorenstein singularities. From some “combinatorial input” on $Q$ one constructs \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,683,528,717]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"178\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(i) a smoothing $V$ of $X _ { Q }$ and (ii) a Laurent polynomial $f$ supported on $Q$ such that $f$ is mirror to $V$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,726,362,744]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"178\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,753,879,788]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"178\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The definition of “mirror” that we are using comes from [8, 31, 32] and is given in Definition 3.4. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,791,881,881]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"178\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If the toric variety $X _ { Q }$ is smooth (there are 18 cases), then the polytope $Q$ has only triangular facets which are standard simplices and $X _ { Q }$ is rigid. Thus $V = X _ { Q }$ and $f$ is uniquely determined by insisting that it has coefficient 1 on vertices of $Q$ and coefficient 0 on the origin. This case was already known by Givental [17, 18] who proved that $f$ is mirror to $X _ { Q }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,539,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"178\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An Example of Mirror Symmetry for Fano Threefolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"178\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"175 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,879,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"179\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the example considered in this note, the combinatorial input on $P$ is the choice of a maximal Minkowski decomposition of the facet $F$ of $P$ . There are two such choices which lead to two different smoothings of $X$ and to two different Laurent polynomials. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,162,881,394]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"179\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An interesting case, which is not too restrictive, is the following: the combinatorial input is the choice of a Minkowski decomposition of each facet of $Q$ into $A$ -triangles. Here an $A$ -triangle is either a unitary segment or a lattice triangle which is $\\\\mathbb { Z } ^ { 2 } \\\\rtimes \\\\mathbf { G L } _ { 2 } ( \\\\mathbb { Z } )$ -equivalent to the convex hull of the points $( 0 , 0 ) , ( 0 , 1 ) , ( \\\\ell , 0 )$ , for some integer $\\\\ell \\\\geq 1$ . For example, both maximal Minkowski decompositions of the hexagon $F$ are decompositions into $A$ -triangles. In these circumstances one can easily construct a Laurent polynomial $f$ which is supported on $Q$ and depends on the choice of the Minkowski decompositions of the facets of $Q$ (see [1], where such Laurent polynomial $f$ is called a Minkowski polynomial). In joint work with Corti and Hacking [11], we construct a smoothing $V$ of $X _ { Q }$ , under a slight additional assumption which is necessary by Petracci [29]. It is conjectured that $f$ is mirror to $V$ . However, even in this situation we completely lack a conceptual way to prove that $f$ is mirror to $V$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,396,881,448]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"179\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Unfortunately, there exist polytopes $Q$ which have facets without Minkowski decompositions into $A$ -triangles. So, at the moment, it is not clear what sort of combinatorial input we should consider on $Q$ in the general case. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,450,879,521]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"179\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Another approach to construct smoothings of the toric Fano variety $X _ { Q }$ is pursued by Coates et al. [10, 30]; they embed $X _ { Q }$ into a bigger toric variety $Z$ and try to deform it inside Z. This works very well in many explicit examples, but a general framework has yet to be discovered. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,522,879,575]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"179\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally, it is worth mentioning that Conjecture 1.1 can be stated in all dimensions. Therefore, this might be a way to classify smooth Fano varieties that admit a toric degeneration. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,621,476,641]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"179\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1.4 Notation and Conventions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,665,881,737]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"179\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In a real vector space of finite dimension, a polyhedron is the intersection of finitely many closed half-spaces and a polytope is a compact polyhedron; equivalently, a polytope is the convex hull of a finite set. We denote by conv $\\\\{ \\\\cdot \\\\}$ the convex hull of a set. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[113,738,876,773]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"179\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"All varieties and schemes are defined over $\\\\mathbb { C }$ . We always use the following notation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,783,829,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"179\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\mathrm { d } \\\\mathrm { P } _ { 6 }$ the smooth del Pezzo surface of degree 6 \\n$X$ the projective cone over the anticanonical embedding of $\\\\mathrm { d } \\\\mathrm { P } _ { 6 }$ \\n$U$ the affine cone over the anticanonical embedding of $\\\\mathrm { d } \\\\mathrm { P } _ { 6 }$ \\nX2 a general effective divisor of type $( 1 , 1 )$ in $\\\\mathbb { P } ^ { 2 } \\\\times \\\\mathbb { P } ^ { 2 }$ \\n$X _ { 3 }$ $\\\\mathbb { P } ^ { \\\\bar { 1 } } \\\\times \\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 }$ \\n$F$ the lattice polygon associated to $\\\\mathrm { d } \\\\mathrm { P } _ { 6 }$ (see (1) and the left part of Fig. 2) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,54,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"179\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[790,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"179\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. Petracci \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,799,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"180\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$P$ the lattice polytope associated to $X$ (see (2) and Fig. 1) $f _ { a }$ $z \\\\left( a + x + x y + y + x ^ { - 1 } + x ^ { - 1 } y ^ { - 1 } + y ^ { - 1 } \\\\right) + z ^ { - 1 }$ , for each $a \\\\in \\\\mathbb { C }$ $Q$ an arbitrary Fano polytope (see Definition 2.1) XQ the Fano toric variety associated to the Fano polytope $Q$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,205,389,226]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"180\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 The Geometry of $X$ \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,246,357,265]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"180\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.1 Toric Geometry \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,289,876,327]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"180\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We now recall the basics of Fano toric varieties. We refer the reader to [12, $\\\\ S 8 . 3 ]$ [15, p. 25], and [22]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,334,881,388]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"180\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 2.1 Let $N$ be a lattice of rank $n$ . A Fano polytope in $N$ is an $n$ -dimensional polytope $Q \\\\subseteq N _ { \\\\mathbb { R } }$ such that the origin $0 \\\\in N$ lies in the interior of $Q$ and every vertex of $Q$ is a primitive lattice point of $N$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,390,881,442]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"180\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The spanning fan of a Fano polytope $Q$ in $N$ is the complete fan whose cones are the cones over the proper faces of $Q$ . We denote by $X _ { Q }$ the toric variety associated to the spanning fan of a Fano polytope $Q$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,451,881,524]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"180\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For brevity, we say that $X _ { Q }$ is associated to $Q$ , and conversely. If $Q$ is a Fano polytope of dimension $n$ , then $X _ { Q }$ is an $n$ -dimensional complete toric variety which is Fano, i.e. its anticanonical divisor is \\u001f2 3 2 3 2 3 $\\\\mathbb { Q }$ -Cartier and ample. Every Fano toric variety3 2 3 2 3! arises in this way from a Fano polytope. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,525,466,542]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"180\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For example, consider the hexagon \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[186,558,760,602]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0456/0a57f824ea6fe11db3fd6086ecaeae26c6dc96c10fd1b802234f30777549465f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"180\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nF = \\\\operatorname { c o n v } \\\\{ ( { \\\\binom { 1 } { 0 } } , ( { \\\\binom { 1 } { 1 } } ) , ( { \\\\binom { 0 } { 1 } } ) , ( { \\\\binom { - 1 } { 0 } } ) , ( { \\\\binom { - 1 } { - 1 } } ) , ( { \\\\binom { 0 } { - 1 } } ) \\\\} \\\\subseteq \\\\mathbb { R } ^ { 2 } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,618,881,689]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"180\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"which is depicted on the left of Fig. 2. It is clear that $F$ is a Fano polytope in $\\\\mathbb { Z } ^ { 2 }$ . The toric variety associated to its spanning fan is the smooth del Pezzo surface of degree 6, denoted by $\\\\mathrm { d } \\\\mathrm { P } _ { 6 }$ . The anticanonical map of $\\\\mathrm { d } \\\\mathrm { P } _ { 6 }$ is a closed embedding into $\\\\mathbb { P } ^ { 6 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,690,876,728]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"180\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now imagine to put the hexagon⎨ $F$ into the plane $\\\\mathbb { R } ^ { 2 } \\\\times \\\\{ 1 \\\\}$ in $\\\\mathbb { R } ^ { 3 }$ and create the⎬ pyramid over it with apex at the point ⎩⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ $( 0 , 0 , - 1 )$ : this is the polytope⎠ ⎝ ⎠ ⎝ ⎠ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[129,743,820,804]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0267/9bd01abfab559a2450127e4bf33db11a61fed6654fcfaf4491db5030b2a4917c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"180\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nP = \\\\mathrm { c o n v } \\\\{ ( { \\\\binom { 1 } { 1 } } , ( { \\\\binom { 1 } { 1 } } ) , ( { \\\\binom { 0 } { 1 } } ) , ( { \\\\binom { 1 } { 1 } } ) , ( { \\\\binom { - 1 } { 0 } } ) , ( { \\\\binom { - 1 } { 1 } } ) , ( { \\\\binom { 0 } { - 1 } } ) , ( { \\\\binom { 0 } { - 1 } } ) \\\\} \\\\subseteq \\\\mathbb { R } ^ { 3 }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,821,881,894]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"180\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and is depicted in Fig. 1. It is clear that $P$ is a Fano polytope in $\\\\mathbb { Z } ^ { 3 }$ . Let $X$ be the toric variety associated to the spanning fan of $P$ . Let $U$ be the affine toric open subscheme of $X$ associated to the hexagonal face of $P$ , i.e. $U$ is the affine toric variety associated to the cone $\\\\mathbb { R } _ { \\\\geq 0 } ( F \\\\times \\\\{ 1 \\\\} )$ . Hence $X$ (resp. $U$ ) is the projective (resp. affine) cone over the anticanonical embedding of $\\\\mathrm { d P } _ { 6 }$ . We have that $X$ is a Fano threefold with an isolated non-terminal canonical Gorenstein singularity at the vertex of the cone. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,539,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"180\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An Example of Mirror Symmetry for Fano Threefolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"180\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"177 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,879,142]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"181\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,189,298,210]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"181\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2 Equations \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,231,881,321]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"181\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The equations of the three closed embeddings $\\\\mathrm { d } \\\\mathsf { P } _ { 6 } \\\\subseteq \\\\mathbb { P } ^ { 6 }$ , $U \\\\subseteq \\\\mathbb { A } ^ { 7 }$ and $X \\\\subseteq \\\\mathbb { P } ^ { 7 }$ are the same and can be conveniently described in two ways. Here, $x _ { 1 } , \\\\ldots , x _ { 7 }$ denote the homogeneous coordinates of $\\\\mathbb { P } ^ { 6 }$ , the affine coordinates of $\\\\mathbb { A } ^ { 7 }$ and the last homogeneous coordinates of $\\\\mathbb { P } ^ { 7 }$ , as⎛ $x _ { 0 }$ is the remaining homogeneous coordinate⎞ of $\\\\mathbb { P } ^ { 7 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,324,296,340]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"181\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The first way is: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[389,358,605,418]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0908/6e3de3a9aabe7b0d3515cea1ea6baa828a2184803e0e5d7d3277768c45523f39.jpg\",\n \"list_items\": null,\n \"page_idx\": \"181\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname { r a n k } { \\\\left( { \\\\begin{array} { l } { x _ { 7 } \\\\ x _ { 1 } \\\\ x _ { 2 } } \\\\\\\\ { x _ { 4 } \\\\ x _ { 7 } \\\\ x _ { 3 } } \\\\\\\\ { x _ { 5 } \\\\ x _ { 6 } \\\\ x _ { 7 } } \\\\end{array} } \\\\right) } \\\\leq 1 .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,433,881,506]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"181\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Note the repetition of $x _ { 7 }$ on the diagonal. If two of the $x _ { 7 }$ ’s had been two extra variables, these would have been the equations of the Segre embedding of $\\\\mathbb { P } ^ { 2 } \\\\times \\\\mathbb { P } ^ { 2 }$ in $\\\\mathbb { P } ^ { 8 }$ . This shows that $X$ is the intersection of the projective cone over the Segre embedding of $\\\\mathbb { P } ^ { 2 } \\\\times \\\\mathbb { P } ^ { 2 }$ with two hyperplanes of $\\\\mathbb { P } ^ { 9 }$ passing through the vertex. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,507,359,524]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"181\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now consider the cube \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[384,542,612,686]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0799/262d1ba485187f1012830774d38852135aadee08ca42a8c3a0b8019d718e2dab.jpg\",\n \"list_items\": null,\n \"page_idx\": \"181\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[118,708,881,836]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"181\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where at the vertices there are the variables $x _ { 1 } , \\\\ldots , x _ { 7 }$ . Note the repetition of $x _ { 7 }$ . The second way to describe the equations is to consider the determinants of all rectangles which can be formed with edges of the cube or with diagonals of faces of the cube. If one of the $x _ { 7 }$ ’s had been an extra variable, these would have been the equations of the Segre embedding of $\\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 }$ into $\\\\mathbb { P } ^ { 7 }$ . This shows that $X$ is the intersection of the projective cone over the Segre embedding of $\\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 }$ with a hyperplane of $\\\\mathbb { P } ^ { 8 }$ passing through the vertex. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[790,54,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"181\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. Petracci \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"181\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"178 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,881,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"182\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The equations above also appear in [20, Example 3.3]. Moreover, these two ways of describing the equations of $X$ in $\\\\mathbb { P } ^ { 7 }$ are called Tom and Jerry, respectively, in [5]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,171,644,192]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"182\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.3 Minkowski Sums and Deformations of U \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,214,881,268]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"182\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We first define the notion of Minkowski sum of polyhedra (for instance see [34, $\\\\ S 1 . 1 ]$ . If $\\\\Pi _ { 1 } , \\\\ldots , \\\\Pi _ { r }$ are polyhedra in a real vector space, we define their Minkowski sum to be the polyhedron \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[218,286,781,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0104/4208566366ec8d72d405618e1b3bfc6769781789870a5a8dc15aaa0a571e3d61.jpg\",\n \"list_items\": null,\n \"page_idx\": \"182\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\Pi _ { 1 } + \\\\cdot \\\\cdot \\\\cdot + \\\\Pi _ { r } : = \\\\{ p _ { 1 } + \\\\cdot \\\\cdot \\\\cdot + p _ { r } \\\\mid p _ { 1 } \\\\in \\\\Pi _ { 1 } , \\\\ldots , p _ { r } \\\\in \\\\Pi _ { r } \\\\} .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,322,881,414]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"182\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"When we have $\\\\Pi = \\\\Pi _ { 1 } + \\\\cdots + \\\\Pi _ { r }$ , we say that we have a Minkowski decomposition of the polyhedron $\\\\Pi$ . We consider Minkowski decompositions up to translation: for instance, we consider the Minkowski decomposition $( p + \\\\Pi _ { 1 } ) + ( - p + \\\\Pi _ { 2 } )$ to be equivalent to $\\\\Pi _ { 1 } + \\\\Pi _ { 2 }$ for every vector $p$ . Moreover, in what follows we require that the summands $\\\\Pi _ { j }$ are lattice polyhedra, i.e. their vertices belong to a fixed lattice.3 2 3! \\u001f2 3 2 3! \\u001f2 3 2 3! \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,414,879,448]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"182\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The hexagon $F$ has two maximal Minkowski decompositions (see Fig. 2): one into 3 unitary segments \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[150,463,799,509]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0159/ef1396ac6f942cb2aba9a2384f26a679c531e6db4d01c309b791a066a3fc05a2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"182\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nF = \\\\operatorname { c o n v } \\\\left\\\\{ { \\\\binom { 0 } { 0 } } , { \\\\binom { 1 } { 0 } } \\\\right\\\\} + \\\\operatorname { c o n v } \\\\left\\\\{ { \\\\binom { 0 } { 0 } } , { \\\\binom { 0 } { 1 } } \\\\right\\\\} + \\\\operatorname { c o n v } \\\\left\\\\{ { \\\\binom { 0 } { 0 } } , { \\\\binom { - 1 } { - 1 } } \\\\right\\\\}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,522,337,540]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"182\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and one into 2 triangles \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[179,555,769,599]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0815/ec2007de04de387266cc3ec966c01ca321b8c15bb6f8a64b71325e21e7369493.jpg\",\n \"list_items\": null,\n \"page_idx\": \"182\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nF = \\\\operatorname { c o n v } \\\\left\\\\{ \\\\left( { 0 } _ { 0 } \\\\right) , \\\\left( { - 1 } _ { 0 } \\\\right) , \\\\left( { - 1 } _ { - 1 } \\\\right) \\\\right\\\\} + \\\\operatorname { c o n v } \\\\left\\\\{ \\\\left( { 0 } _ { 0 } \\\\right) , \\\\left( { 1 } _ { 0 } ^ { 1 } \\\\right) , \\\\left( { 1 } _ { 1 } ^ { 1 } \\\\right) \\\\right\\\\} .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,614,881,704]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"182\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Altmann [3] has noticed that Minkowski decompositions of polytopes induce deformations of affine toric varieties (see also [25] and [28]). More precisely, from a Minkowski decomposition of a polytope $\\\\Pi$ it is possible to construct an unobstructed deformation of the affine toric variety associated to the cone $\\\\mathbb { R } _ { \\\\geq 0 } ( \\\\Pi \\\\times \\\\{ 1 \\\\} )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,704,876,740]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"182\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the case at hand, the Minkowski decomposition (3) induces the deformation of $U$ over Spec $\\\\mathbb { C } [ u , v ]$ given by the equations⎝ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[350,756,646,816]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0455/fddfce9568643a6afc5185e8cdbac7c8bddfee39be16a4d5a1016515209be77c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"182\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname { r a n k } { \\\\left( \\\\begin{array} { l l l } { x _ { 7 } } & { x _ { 1 } } & { x _ { 2 } } \\\\\\\\ { x _ { 4 } } & { x _ { 7 } + u } & { x _ { 3 } } \\\\\\\\ { x _ { 5 } } & { x _ { 6 } } & { x _ { 7 } + v } \\\\end{array} \\\\right) } \\\\leq 1 .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,831,881,887]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"182\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Minkowski decomposition (4) induces the deformation of $U$ over Spec $\\\\mathbb { C } [ s ]$ given by the equations obtained by taking minors of rectangles on edges and diagonals of faces of the following cube. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,51,539,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"182\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An Example of Mirror Symmetry for Fano Threefolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"182\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"179 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[375,94,617,240]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0023/62aad225471d6ec5788405f62cd7ccc4e873d6bbae010dda87760b8a3e6da5be.jpg\",\n \"list_items\": null,\n \"page_idx\": \"183\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[120,261,879,298]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"183\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moreover, Altmann [4] shows that the miniversal deformation of $U$ is (the completion of) the union of these two deformations and its base is $\\\\mathbb { C } [ [ s , u , v ] ] / ( s u , s v )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,342,478,364]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"183\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.4 The Two Smoothings of $X$ \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,387,505,405]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"183\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now we want to study deformations of $X$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,414,881,504]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"183\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 2.2 The base of the miniversal deformation of $X$ is $\\\\mathbb { C } [ [ s , u , v ] ] / ( s u , s v )$ and has two irreducible components. The 2-dimensional component $( s \\\\ = \\\\ 0$ ) is associated to the Minkowski decomposition (3) and deforms $X$ to a general divisor $X _ { 2 } \\\\in | O _ { \\\\mathbb { P } ^ { 2 } \\\\times \\\\mathbb { P } ^ { 2 } } ( 1 , 1 ) |$ . The $^ { l }$ -dimensional component $( u = v = 0$ ) is associated to the Minkowski decomposition (4) and deforms $X$ to $X _ { 3 } = \\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,513,881,585]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"183\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Consider the local-to-global spectral sequence for $\\\\mathrm { E x t } _ { X } ^ { \\\\bullet } ( \\\\Omega _ { X } , O _ { X } )$ : the second page is $E _ { 2 } ^ { p , q } = \\\\mathsf { H } ^ { q } ( X , \\\\mathcal { E } x t _ { X } ^ { p } ( \\\\Omega _ { X } , O _ { X } ) )$ . As $X$ has an isolated singularity, for all $p \\\\geq 1$ , the sheaf $\\\\mathcal { E } x t _ { X } ^ { p } ( \\\\Omega _ { X } , O _ { X } )$ is supported on the singular point of $X$ ; therefore, for all $p \\\\geq 1$ and $q \\\\geq 1$ , $E _ { 2 } ^ { p , q } = 0$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,587,881,678]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"183\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $j \\\\colon W \\\\ \\\\hookrightarrow \\\\ X$ be the smooth locus. The sheaves $\\\\mathcal { H } o m _ { X } ( \\\\Omega _ { X } , O _ { X } )$ and $j _ { * } \\\\Omega _ { W } ^ { 2 } \\\\otimes { \\\\cal O } _ { X } ( - K _ { X } )$ are the same, because they are both reflexive and coincide on $W$ . As $X$ is toric and $- K _ { X }$ is ample, by Bott–Steenbrink–Danilov vanishing [12, Theorem 9.3.1] (see also [6, 14, 26]) one has $E _ { 2 } ^ { 0 , q } = \\\\mathrm { H } ^ { q } ( X , \\\\mathcal { H } o m _ { X } ( \\\\Omega _ { X } , O _ { X } ) ) = 0$ for all $q \\\\geq 1$ . This argument comes from the proof of [33, Theorem 5.1]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,678,876,714]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"183\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Therefore $E _ { 2 }$ is zero outside the line $q = 0$ . This implies that, for all $p \\\\geq 0$ , the natural map \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[302,729,694,753]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0662/2abfa8124514a5d300dc43c54006eea783fe733c2a038d4e88065c9c6556fa94.jpg\",\n \"list_items\": null,\n \"page_idx\": \"183\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname { E x t } _ { X } ^ { p } ( \\\\Omega _ { X } , O _ { X } ) \\\\to \\\\operatorname { H } ^ { 0 } ( X , { \\\\mathcal { E } } x t _ { X } ^ { p } ( \\\\Omega _ { X } , O _ { X } ) )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,768,881,822]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"183\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is an isomorphism. Since the unique singular point of $X$ is contained in $U$ and $U$ is affine, we have $\\\\mathrm { H } ^ { 0 } ( X , \\\\mathcal { E } x t _ { X } ^ { p } ( \\\\Omega _ { X } ^ { \\\\phantom { \\\\dagger } } , O _ { X } ) ) = \\\\mathrm { \\\\bar { E } x t } _ { U } ^ { p } ( \\\\Omega _ { U } ^ { \\\\phantom { \\\\dagger } } , O _ { U } )$ for all $p \\\\geq 1$ . This implies that, for all $p \\\\geq 1$ , the natural map \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[312,837,681,861]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0757/9b730caa7868713ab4d95bfc6530c6379e3879022883ac11fe8e22862ce4ffbd.jpg\",\n \"list_items\": null,\n \"page_idx\": \"183\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\phi _ { p } \\\\colon \\\\mathrm { E x t } _ { X } ^ { p } ( \\\\Omega _ { X } , O _ { X } ) \\\\to \\\\mathrm { E x t } _ { U } ^ { p } ( \\\\Omega _ { U } , O _ { U } ) ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,876,296,894]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"183\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is an isomorphism. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,152,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"183\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"180 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[790,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"183\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. Petracci \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,270]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"184\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We now consider the functors of infinitesimal deformations of $X$ and $U \\\\colon D e f _ { X }$ and $D e f _ { U }$ , which are covariant functors from the category of local finite $\\\\mathbb { C }$ -algebras to the category of sets (see [24, §3]). There is an obvious map $\\\\phi \\\\colon D e f _ { X } D e f _ { U }$ , which restricts a deformation of $X$ to $U$ . Since $X$ is normal, $\\\\mathrm { E x t } _ { X } ^ { 1 } ( \\\\Omega _ { X } , O _ { X } )$ is the tangent space of $D e f _ { X }$ and $\\\\mathrm { E x t } _ { X } ^ { 2 } ( \\\\Omega _ { X } , { \\\\cal O } _ { X } )$ is an obstruction space for $D e f _ { X }$ , and a similar statement holds for $U$ . Since $\\\\phi _ { 1 }$ is bijective and $\\\\phi _ { 2 }$ is injective, we have that $\\\\phi$ induces an isomorphism on tangent spaces and an injection on obstruction spaces. Therefore, by [24, Remark 4.12], $\\\\phi$ is smooth and induces an isomorphism on tangent spaces. In particular, the two functors $D e f _ { X }$ and $D e f _ { U }$ have the same hull, i.e. the bases of the miniversal deformations of $X$ and $U$ are the same. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,271,879,360]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"184\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The equations of the two deformations of $U$ , given in Sect. 2.3, can be projectivised to construct deformations of $X$ : it is enough to replace $s , u$ and $v$ by $s x _ { 0 }$ , $u x _ { 0 }$ and $v x _ { 0 }$ . These are the two components of the miniversal deformation of $X$ . The fact that they are associated to the two Minkowski decompositions (3) and (4) of the hexagon $F$ follows from the discussion in Sect. 2.3. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,361,881,450]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"184\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From Sect. 2.2 we know that $X$ is the intersection of the projective cone over the Segre embedding of $\\\\mathbb { P } ^ { 2 } \\\\times \\\\mathbb { P } ^ { 2 }$ with two hyperplanes of $\\\\mathbb { P } ^ { 9 }$ passing through the vertex of the cone. On the component $( s = 0$ ), in the deformation we are moving these two hyperplanes away from the vertex. Therefore, the general fibre over this component is $X _ { 2 }$ , a general $( 1 , 1 )$ -divisor in $\\\\mathbb { P } ^ { 2 } \\\\times \\\\mathbb { P } ^ { 2 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,451,881,540]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"184\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recall that $X$ is the intersection of the projective cone over the Segre embedding of $\\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 }$ with a hyperplane of $\\\\mathbb { P } ^ { 8 }$ passing through the vertex. On the component $( u ~ = ~ v ~ = ~ 0 )$ ), in the deformation we are moving this hyperplane of $\\\\mathbb { P } ^ { 8 }$ away from the vertex. Therefore, the general fibre on this component is $X _ { 3 } = \\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 } \\\\times \\\\mathbb { P } ^ { 1 }$ . □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,587,373,608]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"184\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 Mirror Symmetry \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,626,735,647]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"184\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.1 Gromov–Witten Invariants and Quantum Periods \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,669,881,723]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"184\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The quantum period of a smooth Fano variety is a generating function for some genus zero Gromov–Witten invariants. The regularised quantum period is a slightly different version, which is convenient for our description of Mirror Symmetry. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,732,876,770]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"184\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3.1 ([8, 9, 31]) The quantum period and the regularised quantum period of a smooth Fano variety $V$ are the following power series \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[225,783,769,827]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0105/926a98c6f4278a20c488c4a094f73ffc6bf3076c79258a85dad835183b4dfdc5.jpg\",\n \"list_items\": null,\n \"page_idx\": \"184\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nG _ { V } ( t ) = 1 + \\\\sum _ { \\\\beta \\\\in \\\\mathrm { H } _ { 2 } ( V , \\\\mathbb { Z } ) } \\\\langle [ \\\\mathsf { p t } ] \\\\psi ^ { - K _ { V } \\\\cdot \\\\beta - 2 } \\\\rangle _ { 0 , 1 , \\\\beta } ^ { V } t ^ { - K _ { V } \\\\cdot \\\\beta } \\\\in \\\\mathbb { Q } \\\\mathbb { I } t ]\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[173,833,824,876]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0772/7de5b065967ed9b99e153be5ae4e152ee58f317705b88ba0c259cad3c9df81f9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"184\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\widehat { G } _ { V } ( t ) = 1 + \\\\sum _ { \\\\beta \\\\in \\\\mathrm { H } _ { 2 } ( V , \\\\mathbb { Z } ) } ( - K _ { V } \\\\cdot \\\\beta ) ! \\\\langle [ \\\\mathrm { p t } ] \\\\psi ^ { - K _ { V } \\\\cdot \\\\beta - 2 } \\\\rangle _ { 0 , 1 , \\\\beta } ^ { V } t ^ { - K _ { V } \\\\cdot \\\\beta } \\\\in \\\\mathbb { Q } [ [ t ] ]\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,52,539,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"184\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An Example of Mirror Symmetry for Fano Threefolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"184\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,881,145]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"185\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where \\u000f[pt]ψ−KV ·β−2\\u0010 V denotes the 1-marked genus zero Gromov–Witten invariant of curve class $\\\\beta$ associated to the cohomology class of a point in $V$ and gravitational descendant of order $- K _ { V } \\\\cdot \\\\beta - 2$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,153,881,262]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"185\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Roughly speaking, $\\\\langle \\\\mathrm { [ p t ] } \\\\psi ^ { - K _ { V } \\\\cdot \\\\beta - 2 } \\\\rangle _ { 0 , 1 , \\\\beta } ^ { V }$ is the number of rational curves in $V$ of class $\\\\beta$ passing through a fixed general point of $V$ and satisfying a certain condition on their complex structure. Therefore, the quantum period $G _ { V }$ gives information about rational curves in $V$ . The series $G _ { V }$ is a symplectic invariant of $V$ , so it does not change if $V$ is deformed to another smooth Fano variety through a deformation with smooth fibres. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,264,881,316]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"185\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If the anticanonical line bundle $O _ { V } ( - K _ { V } )$ is divisible by a positive integer $m$ inside the Picard group of the smooth Fano variety $V$ , then only powers of $t ^ { m }$ appear in the (regularised) quantum period of $V$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,318,876,352]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"185\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is also possible to define quantum periods for Fano varieties with quotient singularities [27, $\\\\ S 3 . 3 ]$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,354,881,426]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"185\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is known how to compute the quantum period of smooth Fano varieties which are either toric or complete intersections in smooth Fano toric varieties [7, 17]. The quantum periods of all smooth Fano varieties of dimension $\\\\leq 3$ have been computed by Coates et al. [9]. In particular, we have the following formulae for $X _ { 2 }$ and $X _ { 3 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,433,876,469]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"185\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 3.2 ([9]) The quantum periods and the regularised quantum periods of $X _ { 2 }$ and $X _ { 3 }$ are the following. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[332,484,665,536]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0478/5a8b09583d2c2ae6de17b302875d7a9cf3bbf0bd0397044ba313b4dfb81a0e33.jpg\",\n \"list_items\": null,\n \"page_idx\": \"185\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nG _ { X _ { 2 } } ( t ) = \\\\sum _ { l = 0 } ^ { \\\\infty } \\\\sum _ { m = 0 } ^ { \\\\infty } \\\\frac { ( l + m ) ! } { ( l ! ) ^ { 3 } ( m ! ) ^ { 3 } } t ^ { 2 l + 2 m }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[277,542,722,594]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0811/9faf04336bfe5a7d8ba7c258ef06919bd987a3ecbd2b7d4af312535c0a72a3ab.jpg\",\n \"list_items\": null,\n \"page_idx\": \"185\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nG _ { X _ { 3 } } ( t ) = \\\\sum _ { l = 0 } ^ { \\\\infty } \\\\sum _ { m = 0 } ^ { \\\\infty } \\\\sum _ { n = 0 } ^ { \\\\infty } \\\\frac { 1 } { ( l ! ) ^ { 2 } ( m ! ) ^ { 2 } ( n ! ) ^ { 2 } } t ^ { 2 l + 2 m + 2 n }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[186,603,810,663]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0951/2be59fb35ccf8833d81e668735bd05dcc03dfd73d220250d1c9b22030e3f9bc7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"185\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\widehat { G } _ { X _ { 2 } } ( t ) = 1 + 4 t ^ { 2 } + 6 0 t ^ { 4 } + 1 1 2 0 t ^ { 6 } + 2 4 2 2 0 t ^ { 8 } + 5 6 7 5 0 4 t ^ { 1 0 } + \\\\cdot \\\\cdot \\\\cdot } \\\\\\\\ & { \\\\widehat { G } _ { X _ { 3 } } ( t ) = 1 + 6 t ^ { 2 } + 9 0 t ^ { 4 } + 1 8 6 0 t ^ { 6 } + 4 4 7 3 0 t ^ { 8 } + 1 1 7 2 5 5 6 t ^ { 1 0 } + \\\\cdot \\\\cdot \\\\cdot } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,704,416,725]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"185\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.2 Laurent Polynomials \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,747,881,824]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"185\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $\\\\mathbb { C } [ x _ { 1 } ^ { \\\\pm } , \\\\ldots , x _ { n } ^ { \\\\pm } ]$ be the ring of Laurent polynomials in $n$ variables with coefficients in $\\\\mathbb { C }$ . To every monomial $x ^ { i } ~ = ~ x _ { 1 } ^ { i _ { 1 } } \\\\cdot \\\\cdot \\\\cdot x _ { n } ^ { i _ { n } }$ we associate the point $\\\\begin{array} { r l } { \\\\mathbf { \\\\boldsymbol { i } } } & { { } = } \\\\end{array}$ $( i _ { 1 } , \\\\ldots , i _ { n } ) \\\\in \\\\mathbb { Z } ^ { n }$ . The Newton polytope of a Laurent polynomial $f$ is the convex hull of the lattice points that correspond to the monomials that appear in $f$ , i.e. if \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[790,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"185\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. Petracci \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"185\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"182 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,321,111]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"186\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\textstyle f = \\\\sum _ { i \\\\in \\\\mathbb { Z } ^ { n } } a _ { i } x ^ { i }$ then \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[300,124,694,148]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0618/33726d82fe924ecd515c77e9d4e13e482b018945d21ff872f15be1f68abc2410.jpg\",\n \"list_items\": null,\n \"page_idx\": \"186\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\operatorname { N e w t } ( f ) = \\\\operatorname { c o n v } \\\\left\\\\{ i \\\\in \\\\mathbb { Z } ^ { n } \\\\mid a _ { i } \\\\neq 0 \\\\right\\\\} \\\\subseteq \\\\mathbb { R } ^ { n } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,163,881,217]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"186\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $Q$ is a lattice polytope in $\\\\mathbb { Z } ^ { n }$ , we say that a Laurent polynomial $\\\\begin{array} { r l } { f } & { { } \\\\in } \\\\end{array}$ $\\\\mathbb { C } [ x _ { 1 } ^ { \\\\pm } , \\\\ldots , x _ { n } ^ { \\\\pm } ]$ is supported on $Q$ if every monomial appearing in $f$ corresponds to a lattice point of $Q$ , or equivalently if Newt $( f ) \\\\subseteq Q$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,219,881,325]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"186\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Given a Fano polytope $Q$ in $\\\\mathbb { Z } ^ { n }$ , Kasprzyk and Tveiten [21] have introduced and studied a particular class of Laurent polynomials supported on $Q$ ; they call them maximally mutable, because these behave well with respect to mutations of Fano polytopes [1]. The definition of maximally mutable Laurent polynomials in dimension 2 can be also found in [2]. We are not going to define maximally mutable Laurent polynomials here, we just mention some properties in a particular case. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,327,881,379]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"186\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In dimension 3, when the Fano toric threefold $X _ { Q }$ has Gorenstein singularities (equivalently $Q$ is a reflexive polytope of dimension 3), every maximally mutable Laurent polynomial on $Q$ is such that: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,388,881,462]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"186\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"• the coefficient of the monomial 1, corresponding to the origin of $\\\\mathbb { Z } ^ { 3 }$ , is 0; • the monomials corresponding to the vertices of $Q$ have coefficients equal to 1; • on the edges of $Q$ there are binomial coefficients. (For example, the 4 lattice points of an edge with lattice length 3 have coefficients 1, 3, 3, 1.) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,469,881,542]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"186\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the case of the polytope $P$ , a Laurent polynomial is supported on $P$ if and only if its monomials are among 1, 2 $x z$ , xyz, yz, $x ^ { - 1 } z , x ^ { - 1 } y ^ { - 1 } z , y ^ { - 1 } z , z ^ { - 1 }$ , which correspond to the lattice points of $P$ . The Laurent polynomials on $P$ which satisfy the three properties above form a 1-dimensional family \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[277,557,722,599]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0368/1144013c06073722afb57a378b845000f6f969f878125c98ee4cf9ef605efd53.jpg\",\n \"list_items\": null,\n \"page_idx\": \"186\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nf _ { a } = z \\\\left( x + x y + y + { \\\\frac { 1 } { x } } + { \\\\frac { 1 } { x y } } + { \\\\frac { 1 } { y } } + a \\\\right) + { \\\\frac { 1 } { z } }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,612,881,684]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"186\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"with parameter $a \\\\in \\\\mathbb { C }$ . Here $a$ is the coefficient of the centre of the hexagonal facet of $P$ . Kasprzyk and Tveiten [21] show that there are exactly two maximally mutable Laurent polynomials on $P$ , namely $f _ { a }$ with $a = 2$ and $a = 3$ . One notices that, in these two cases, the restriction of $f _ { a }$ to the hexagonal facet of $P$ is reducible: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[275,698,724,753]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0608/d0526cd8fe4fb9405f399c15288f7be5217b96b63ee6ef2d1fd7d11a07560680.jpg\",\n \"list_items\": null,\n \"page_idx\": \"186\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { c } { { f _ { 2 } = z ( 1 + x ) ( 1 + y ) ( 1 + x ^ { - 1 } y ^ { - 1 } ) + z ^ { - 1 } , } } \\\\\\\\ { { { } } } \\\\\\\\ { { f _ { 3 } = z ( 1 + x ^ { - 1 } y ^ { - 1 } + y ^ { - 1 } ) ( 1 + x y + y ) + z ^ { - 1 } . } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,770,881,861]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"186\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Newton polytopes of the three factors of $( 1 + x ) ( 1 + y ) ( 1 + x ^ { - 1 } y ^ { - 1 } )$ are the three unitary segments appearing in the Minkowski decomposition (3) of the hexagon $F$ . The Newton polytopes of the factors of $( 1 + x ^ { - 1 } y ^ { - \\\\bar { 1 } } + y ^ { - 1 } ) ( 1 + x y + y )$ are the two triangles appearing in the Minkowski decomposition (4) of the hexagon $F$ . The Laurent polynomials $f _ { 2 }$ and $f _ { 3 }$ are Minkowski polynomials in the sense of [1]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,51,542,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"186\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An Example of Mirror Symmetry for Fano Threefolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"186\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"183 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,375,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"187\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.3 Classical Periods \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,132,797,151]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"187\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We now define the classical period of a Laurent polynomial in $n$ variables. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,157,879,196]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"187\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3.3 ([1, 16]) The classical period of $f \\\\in \\\\mathbb { C } [ x _ { 1 } ^ { \\\\pm } , \\\\ldots , x _ { n } ^ { \\\\pm } ]$ is the power series \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[227,207,769,309]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0860/a3fe8c381cfb92a3260d2d60b91039e5a7631e9b3f1b5e4cfac56f26650f9277.jpg\",\n \"list_items\": null,\n \"page_idx\": \"187\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { l } { \\\\displaystyle \\\\pi _ { f } ( t ) = \\\\left( \\\\frac { 1 } { 2 \\\\pi \\\\mathrm { i } } \\\\right) ^ { n } \\\\int _ { \\\\Gamma _ { \\\\varepsilon } } \\\\frac { 1 } { 1 - t f ( x _ { 1 } , \\\\dots , x _ { n } ) } \\\\frac { \\\\mathrm { d } x _ { 1 } \\\\wedge \\\\dots \\\\wedge \\\\mathrm { d } x _ { n } } { x _ { 1 } \\\\cdot \\\\cdot \\\\cdot x _ { n } } } \\\\\\\\ { = \\\\displaystyle \\\\sum _ { k = 0 } ^ { \\\\infty } \\\\mathrm { c o e f f } _ { 1 } ( f ^ { k } ) t ^ { k } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,321,881,394]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"187\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where in the first formula we are integrating a holomorphic $n$ -form of the torus $( \\\\mathbb { C } ^ { \\\\times } ) ^ { n } = \\\\operatorname { S p e c } \\\\mathbb { C } [ x _ { 1 } ^ { \\\\pm } , \\\\dots , x _ { n } ^ { \\\\pm } ]$ over the real torus $\\\\Gamma _ { \\\\varepsilon } = \\\\{ | x _ { 1 } | = \\\\cdot \\\\cdot \\\\cdot = | x _ { n } | = \\\\varepsilon \\\\} \\\\subseteq$ $( \\\\mathbb { C } ^ { \\\\times } ) ^ { n }$ , for some $0 < \\\\varepsilon \\\\ll 1$ , and coeff $( f ^ { k } ) \\\\in \\\\mathbb { C }$ is the coefficient of the monomial $1 = x _ { 1 } ^ { 0 } \\\\cdots x _ { n } ^ { 0 }$ - 1in the Laurent polynomial $f ^ { k }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,402,881,456]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"187\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The equality between the two formulae in the definition above comes from applying Cauchy’s integral formula $n$ times. The classical period $\\\\pi _ { f }$ is related to the Hodge theory of the fibres of $f \\\\colon ( \\\\mathbb { C } ^ { \\\\times } ) ^ { n } \\\\to \\\\mathbb { A } ^ { 1 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,457,774,475]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"187\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"One can see that the classical period of the Laurent polynomial $f _ { a }$ is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[170,486,826,573]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0752/6f56501d1ee209d7d7f3219318b6d0e3c0fedabf2bdd2eb183f7c0d0588374cd.jpg\",\n \"list_items\": null,\n \"page_idx\": \"187\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { \\\\pi _ { f _ { a } } ( t ) = 1 + 2 a t ^ { 2 } + ( 6 a ^ { 2 } + 3 6 ) t ^ { 4 } + ( 2 0 a ^ { 3 } + 3 6 0 a + 2 4 0 ) t ^ { 6 } } \\\\\\\\ & { \\\\qquad + ( 7 0 a ^ { 4 } + 2 5 2 0 a ^ { 2 } + 3 3 6 0 a + 6 3 0 0 ) t ^ { 8 } } \\\\\\\\ & { \\\\qquad + ( 2 5 2 a ^ { 5 } + 1 5 1 2 0 a ^ { 3 } + 3 0 2 4 0 a ^ { 2 } + 1 1 3 4 0 0 a + 9 0 7 2 0 ) t ^ { 1 0 } + \\\\cdot \\\\cdot \\\\cdot } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,588,394,608]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"187\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for every $a \\\\in \\\\mathbb { C }$ . In particular, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[184,618,815,677]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0419/e49b42d699af52abaf8e2f1e63c273d348590e65b238c095ba5e64648855704c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"187\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { l } { { \\\\pi _ { f _ { 2 } } ( t ) = 1 + 4 t ^ { 2 } + 6 0 t ^ { 4 } + 1 1 2 0 t ^ { 6 } + 2 4 2 2 0 t ^ { 8 } + 5 6 7 5 0 4 t ^ { 1 0 } + \\\\cdots , } } \\\\\\\\ { { \\\\nonumber } } \\\\\\\\ { { \\\\pi _ { f _ { 3 } } ( t ) = 1 + 6 t ^ { 2 } + 9 0 t ^ { 4 } + 1 8 6 0 t ^ { 6 } + 4 4 7 3 0 t ^ { 8 } + 1 1 7 2 5 5 6 t ^ { 1 0 } + \\\\cdots . } } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,717,646,738]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"187\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.4 Fano/Landau–Ginzburg Correspondence \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,761,881,834]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"187\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Mirror Symmetry [8, 23] predicts that the mirror of a smooth Fano $n$ -fold $V$ is a pair $( Y , w )$ , called Landau–Ginzburg model, where $Y$ is an $n$ -fold and $w \\\\in \\\\Gamma ( Y , O _ { Y } )$ is a regular function. The Gromov–Witten theory of $V$ should be related to the Hodge theory of the fibres of $w \\\\colon Y \\\\ \\\\ \\\\mathbb { A } ^ { 1 }$ as follows: the regularised quantum period ${ \\\\widehat { G } } _ { V }$ (see Definition 3.1) of $V$ coincides with the period $\\\\pi _ { w }$ which is defined as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[790,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"187\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. Petracci \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,152,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"187\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"184 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,91,835,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"188\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[403,123,594,163]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0596/15ff247c7df311389ba442076adc66b9b4573b7168c6360296a8e5b141461945.jpg\",\n \"list_items\": null,\n \"page_idx\": \"188\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\pi _ { w } ( t ) = \\\\int _ { \\\\Gamma } \\\\frac { \\\\Omega } { 1 - t w }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,178,879,213]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"188\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $\\\\Omega$ is an appropriate holomorphic $n$ -form on $Y$ and $\\\\Gamma \\\\in \\\\mathrm { H } _ { n } ( Y ; \\\\mathbb { Z } )$ is such that $\\\\begin{array} { r } { \\\\int _ { \\\\Gamma } \\\\Omega = 1 } \\\\end{array}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,214,881,340]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"188\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Under some circumstances (which conjecturally and experimentally should coincide with when there is a toric degeneration of $V$ ) there is an open subset of $Y$ that is isomorphic to the torus $( \\\\mathbb { C } ^ { \\\\times } ) ^ { n } = \\\\operatorname { S p e c } \\\\mathbb { C } [ x _ { 1 } ^ { \\\\pm } , \\\\dots , x _ { n } ^ { \\\\pm } ]$ . In this case the restriction of $w$ 1 to this open subset gives a Laurent polynomial $\\\\ddot { f } \\\\in \\\\mathbb { C } [ x _ { 1 } ^ { \\\\pm } , \\\\ldots , x _ { n } ^ { \\\\pm } ]$ . In this situation the period $\\\\pi _ { w }$ in (5), when $Y = ( \\\\mathbb { C } ^ { \\\\times } ) ^ { n }$ , $\\\\Gamma = \\\\{ | x _ { 1 } | = \\\\cdot \\\\cdot \\\\cdot =$ $\\\\left| x _ { n } \\\\right| = \\\\varepsilon \\\\}$ and $\\\\Omega = ( 2 \\\\pi \\\\mathbf { i } ) ^ { - n } ( x _ { 1 } \\\\cdot \\\\cdot \\\\cdot x _ { n } ) ^ { - 1 } \\\\mathrm { d } x _ { 1 } \\\\cdot \\\\cdot \\\\cdot \\\\cdot \\\\mathrm { d } x _ { n }$ , becomes the classical period of a Laurent polynomial (see Definition 3.3). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,340,874,376]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"188\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Thus, a down-to-earth formulation of Mirror Symmetry between smooth Fano varieties and Laurent polynomials is the following. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,384,881,439]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"188\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 3.4 ([8, 31, 32]) A Laurent polynomial $f \\\\in \\\\mathbb { C } [ x _ { 1 } ^ { \\\\pm } , \\\\ldots , x _ { n } ^ { \\\\pm } ]$ is mirror to a smooth Fano variety $V$ of dimension $n$ if the classical period of the former coincides with the regularised quantum period of the latter: 2 3 - $\\\\widehat { G } _ { V } ^ { ^ { \\\\bullet } } = \\\\pi _ { f }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[143,447,876,466]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"188\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The equality $\\\\widehat { G } _ { V } = \\\\pi _ { f }$ is equivalent to $G _ { V }$ being equal to the oscillatory integral \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[248,483,749,534]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0682/9ee20c42bb2767aa08f510215ce7027cadc87720dba56877467498b6a57a15bb.jpg\",\n \"list_items\": null,\n \"page_idx\": \"188\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\left( { \\\\frac { 1 } { 2 \\\\pi \\\\mathrm { i } } } \\\\right) ^ { n } \\\\int _ { \\\\Gamma _ { \\\\varepsilon } } \\\\mathrm { e } ^ { t f } { \\\\frac { \\\\mathrm { d } x _ { 1 } \\\\wedge \\\\dots \\\\wedge \\\\mathrm { d } x _ { n } } { x _ { 1 } \\\\cdot \\\\cdot \\\\cdot x _ { n } } } = \\\\sum _ { k = 0 } ^ { \\\\infty } { \\\\frac { \\\\mathrm { c o e f f } _ { 1 } ( f ^ { k } ) } { k ! } } t ^ { k } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,549,881,605]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"188\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Moreover, the equality ${ \\\\widehat { G } } _ { V } ~ = ~ \\\\pi _ { f }$ can be upgraded to an equality between the Gauss–Manin connection on the middle cohomology of the fibres of $f$ and the Dubrovin connection of the quantum D-module of $V$ (see [19]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,614,876,651]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"188\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 3.5 ([1, 9]) The Laurent polynomial $f _ { 2 }$ (resp. $f _ { 3 }$ ) is mirror to the smooth Fano threefold $X _ { 2 }$ (resp. $X _ { 3 }$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,659,881,750]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"188\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Set $a = 2$ or $a = 3$ . We need to show that the two power series $\\\\pi _ { f _ { a } }$ and ${ \\\\widehat { G } } _ { X _ { a } }$ coincide. By comparing the formulae given in Proposition 3.2 and at the end of Sect. 3.3, one can check the equality of finitely many coefficients. In order to prove the equality of all coefficients, one has to show that $\\\\pi _ { f _ { a } }$ and ${ \\\\widehat { G } } _ { X _ { a } }$ satisfy the same linear differential equation; this is done in [1] and [9]. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,758,881,810]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"188\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Now we are ready to illustrate Conjecture 1.1 in the example of the projective cone over the smooth del Pezzo surface of degree 6, which is the running example of this note. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,812,879,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"188\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In Proposition 2.2 we saw that the Minkowski decomposition (3) of the hexagonal facet of $P$ into three unitary segments is associated to the smoothing of $X$ to $X _ { 2 }$ . In Sect. 3.2 we saw that the restriction of $f _ { 2 }$ to the hexagonal facet of $P$ is reducible and that the Newton polytopes of its three factors are the three unitary segments appearing in the Minkowski decomposition (3). By Proposition 3.5 we know that $f _ { 2 }$ is mirror to $X _ { 2 }$ . This is an instance of Conjecture 1.1: from the combinatorial input of the Minkowski decomposition of the hexagonal facet of $P$ into three unitary segments we have constructed the smoothing $X _ { 2 }$ of $X$ and the Laurent polynomial $f _ { 2 }$ which is mirror to $X _ { 2 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,51,542,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"188\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An Example of Mirror Symmetry for Fano Threefolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"188\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"185 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,879,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"189\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,180,879,250]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"189\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In a completely analogous manner, we can observe that the Minkowski decomposition (4) of the hexagonal facet of $P$ into two triangles induces the smoothing $X _ { 3 }$ of $X$ and the Laurent polynomial $f _ { 3 }$ . This provides another example for Conjecture 1.1 because $f _ { 3 }$ is mirror to $X _ { 3 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,252,881,394]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"189\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As mentioned in Sect. 1.3, given a reflexive polytope $Q$ of dimension 3, from the combinatorial datum given by the choice of a Minkowski decomposition of each facet of $Q$ into $A$ -triangles, one constructs an associated Laurent polynomial $f$ supported on $Q$ . From the same combinatorial datum on $Q$ (with a slight additional condition which we do not mention here), by Corti et al. [11] it is possible to construct a smoothing $V$ of the toric Fano threefold $X _ { Q }$ associated to $Q$ . It is conjectured that the smooth Fano threefold $V$ is mirror to the Laurent polynomial $f$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,396,879,486]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"189\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This circle of ideas should be considered as an approach to the problem of classifying smooth Fano varieties of dimension $\\\\geq 4$ . Indeed, computers can classify Fano polytopes; therefore, once one has developed a combinatorial technology for smoothing toric Fano varieties, one should be able to construct all smooth Fano varieties which admit a toric degeneration. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,487,881,665]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"189\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There is another difficulty: a smooth Fano variety may have many toric degenerations, hence may arise from several polytopes. For instance, $X _ { 3 } = \\\\bar { \\\\mathbb { P } } ^ { 1 } \\\\times \\\\bar { \\\\mathbb { P } } ^ { 1 } \\\\times \\\\bar { \\\\mathbb { P } } ^ { 1 }$ is itself toric and degenerates to the toric Fano $X$ . Conjecturally, these many toric degenerations of a smooth Fano correspond to many mirror Laurent polynomials; these Laurent polynomials are related via certain birational transformations of the torus $( \\\\mathbb { C } ^ { \\\\times } ) ^ { n }$ , which are called mutations [1, 13, 16] and preserve the classical periods. Therefore, it is conjectured that deformation families of smooth Fano varieties of dimension $n$ are in one-to-one correspondence with mutation-equivalence classes of some “special” Laurent polynomials in $n$ variables. We are not going to expand on this here because otherwise it would lead us far beyond the scope of this note. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,686,879,780]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"189\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acknowledgements I am indebted to Tom Coates, Alessio Corti, Paul Hacking, Alexander Kasprzyk, Thomas Prince, and the other members of the Fanosearch group for countless fruitful conversations about the topics of this note. I wish to thank the organisers of the conference “Birational geometry and moduli spaces”, held in Rome in June 2018, for giving me the opportunity to present a poster about this subject. Finally, I would like to thank Enrica Floris, Luigi Lunardon, and Diletta Martinelli for useful comments on a preliminary version of this note. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,776,724,791]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"189\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The author was funded by Kasprzyk’s EPSRC Fellowship EP/N022513/1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[790,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"189\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. Petracci \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,54,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"189\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,90,248,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"190\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,138,881,899]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"190\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Akhtar, M., Coates, T., Galkin, S., Kasprzyk, A.M.: Minkowski polynomials and mutations. SIGMA Symmetry Integrability Geom. Methods Appl. 8, Paper 094, 17 (2012) \\n2. Akhtar, M., Coates, T., Corti, A., Heuberger, L., Kasprzyk, A., Oneto, A., Petracci, A., Prince, T., Tveiten, K.: Mirror symmetry and the classification of orbifold del Pezzo surfaces. Proc. Am. Math. Soc. 144(2), 513–527 (2016) \\n3. Altmann, K.: Minkowski sums and homogeneous deformations of toric varieties. Tohoku Math. J. 47(2), 151–184 (1995) \\n4. Altmann, K.: The versal deformation of an isolated toric Gorenstein singularity. Invent. Math. 128(3), 443–479 (1997) \\n5. Brown, G., Kerber, M., Reid, M.: Fano 3-folds in codimension 4, Tom and Jerry. Part I. Compos. Math. 148(4), 1171–1194 (2012) \\n6. Buch, A., Thomsen, J.F., Lauritzen, N., Mehta, V.: The Frobenius morphism on a toric variety. Tohoku Math. J. (2) 49(3), 355–366 (1997) \\n7. Coates, T., Givental, A.: Quantum Riemann-Roch, Lefschetz and Serre. Ann. Math. (2) 165(1), 15–53 (2007) \\n8. Coates, T., Corti, A., Galkin, S., Golyshev, V., Kasprzyk, A.: Mirror symmetry and Fano manifolds. In: European Congress of Mathematics, pp. 285–300. European Mathematical Society, Zürich (2013) \\n9. Coates, T., Corti, A., Galkin, S., Kasprzyk, A.: Quantum periods for 3–dimensional Fano manifolds. Geom. Topol. 20(1), 103–256 (2016) \\n10. Coates, T., Kasprzyk, A., Prince, T.: Laurent inversion. Preprint arXiv:1707.05842. https:// arxiv.org/1707.05842 \\n11. Corti, A., Hacking, P., Petracci, A.: Smoothing toric Fano threefolds (in preparation) \\n12. Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011) \\n13. Cruz Morales, J.A., Galkin, S.: Upper bounds for mutations of potentials. SIGMA Symmetry Integrability Geom. Methods Appl. 9, Paper 005, 13 (2013) \\n14. Fujino, O.: Multiplication maps and vanishing theorems for toric varieties. Math. Z. 257(3), 631–641 (2007) \\n15. Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993). The William H. Roever Lectures in Geometry \\n16. Galkin, S., Usnich, A.: Mutations of Potentials. IPMU, Kashiwa, pp. 10–0100 (2010). Preprint \\n17. Givental, A.B.: Equivariant Gromov-Witten invariants. Int. Math. Res. Notices 13, 613–663 (1996) \\n18. Givental, A.: A mirror theorem for toric complete intersections. In: Topological field theory, primitive forms and related topics (Kyoto, 1996). Progress in Mathematics, vol. 160, pp. 141– 175. Birkhäuser, Boston (1998) \\n19. Golyshev, V.V.: Classification problems and mirror duality. In: Surveys in Geometry and Number Theory: Reports on Contemporary Russian Mathematics. London Mathematical Society Lecture Note Series, vol. 338, pp. 88–121. Cambridge University, Cambridge (2007) \\n20. Jahnke, P., Radloff, I.: Terminal Fano threefolds and their smoothings. Math. Z. 269(3–4), 1129–1136 (2011) \\n21. Kasprzyk, A., Tveiten, K.: Maximally mutable Laurent polynomials (in preparation) \\n22. Kasprzyk, A.M., Nill, B.: Fano polytopes. In: Strings, Gauge Fields, and the Geometry Behind, pp. 349–364. World Science Publication, Hackensack (2013) \\n23. Katzarkov, L., Przyjalkowski, V.: Landau-Ginzburg models—old and new. In: Proceedings of the Gökova Geometry-Topology Conference 2011, pp. 97–124. International Press, Somerville (2012) \\n24. Manetti, M.: Differential graded Lie algebras and formal deformation theory. In: Algebraic geometry—Seattle 2005. Part 2. Proceedings of Symposia in Pure Mathematics, vol. 80, pp. 785–810. American Mathematical Society, Providence (2009) \\n25. Mavlyutov, A.R.: Deformations of toric varieties via Minkowski sum decompositions of polyhedral complexes. Preprint arXiv:0902.0967. https://arxiv.org/abs/0902.0967 \\n26. Musta¸ta, M.: Vanishing theorems on toric varieties. Tohoku Math. J. (2) ˘ 54(3), 451–470 (2002) \\n27. Oneto, A., Petracci, A.: On the quantum periods of del Pezzo surfaces with ${ \\\\scriptstyle { \\\\frac { 1 } { 3 } } } ( 1 , 1 )$ singularities. Adv. Geom. 18(3), 303–336 (2018) \\n28. Petracci, A.: Homogeneous deformations of toric pairs. Preprint arXiv:1801.05732. https:// arxiv.org/1801.05732 \\n29. Petracci, A.: Some examples of non-smoothable Gorenstein Fano toric threefolds. Math. Z. https://doi.org/10.1007/s00209-019-02369-8 \\n30. Prince, T.: Cracked polytopes and Fano toric complete intersections. Manuscripta Math. https:// doi.org/10.1007/s00229-019-01149-2 \\n31. Przyjalkowski, V.: On Landau-Ginzburg models for Fano varieties. Commun. Number Theory Phys. 1(4), 713–728 (2007) \\n32. Przyjalkowski, V.V.: Weak Landau-Ginzburg models of smooth Fano threefolds. Izv. Math. 77(4), 772–794 (2013) \\n33. Totaro, B.: Jumping of the nef cone for Fano varieties. J. Algebraic Geom. 21(2), 375–396 (2012) \\n34. Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,52,539,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"190\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"An Example of Mirror Symmetry for Fano Threefolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"190\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[116,90,883,376]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"191\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,54,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"191\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[790,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"191\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A. Petracci \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,88,738,115]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"192\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Chern Numbers of Uniruled Threefolds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,148,451,166]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"192\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Stefan Schreieder and Luca Tasin \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,288,881,376]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"192\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract In this paper we show that the Chern numbers of a smooth Mori fibre space in dimension three are bounded in terms of the underlying topological manifold. We also generalise a theorem of Cascini and the second named author on the boundedness of Chern numbers of certain threefolds to the case of negative Kodaira dimension. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,396,865,430]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"192\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords Minimal model program $\\\\cdot$ Mori fibre spaces $\\\\cdot$ Three-folds $\\\\cdot$ Topological properties of complex manifolds $\\\\cdot$ Characteristic classes and numbers \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,468,309,489]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"192\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,513,881,638]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"192\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"One of the most basic numerical invariants of a compact complex manifold are its Chern numbers. While these numbers depend only on the topological type of the complex structure of the tangent bundle, they are in general not invariants of the underlying topological manifold, but really depend on the complex structure. In fact, answering a question of Hirzebruch from 1954, all linear combinations of Chern and Hodge numbers which are topological invariants of smooth complex projective varieties have recently been determined in [9–11]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,639,881,746]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"192\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Generalising Hirzebruch’s question, Kotschick asked [8] (see also [12]) whether the topology of the underlying smooth manifold determines the Chern numbers of smooth complex projective varieties at least up to finite ambiguity. In [19], we have shown that in dimension at least four, this question has in general a negative answer. That is, there are smooth real manifolds that carry infinitely many complex algebraic structures such that the corresponding Chern numbers are unbounded, except for $c _ { n }$ , $c _ { 1 } c _ { n - 1 }$ and $c _ { 2 } ^ { 2 }$ which are known to be bounded (see [14] for the non-trivial one $c _ { 1 } c _ { n - 1 }$ ). This result left however open the case of threefolds, where it remains unknown whether $c _ { 1 } ^ { 3 }$ is determined up to finite ambiguity by the underlying smooth manifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,897,703,941]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"192\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ Springer Nature Switzerland AG 2020 E. Colombo et al. (eds.), Birational Geometry and Moduli Spaces, Springer INdAM Series 39, https://doi.org/10.1007/978-3-030-37114-2_11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[804,91,876,139]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"192\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,834,804,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"192\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"L. Tasin $( \\\\boxtimes )$ Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, Milano, Italy e-mail: luca.tasin@unimi.it \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,783,822,825]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"192\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"S. Schreieder Mathematisches Institut, Ludwig–Maximilians–Universität München, München, Germany e-mail: schreieder@math.lmu.de \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,899,876,909]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"192\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,91,879,163]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"193\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,165,881,363]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"193\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In [2], Cascini and the second named author started to investigate the boundedness question for Chern numbers via methods from the minimal model program, see also [16, 20] for further developments. In dimension three, the approach in [2] is motivated by the Miyaoka–Yau inequality, which implies that for a minimal smooth complex projective threefold of non-negative Kodaira dimension, $c _ { 1 } ^ { 3 }$ can be bounded in terms of the Betti numbers of $X$ , see e.g. [20, Proposition 9]. This observation makes it natural to approach the boundedness of $c _ { 1 } ^ { 3 } ( X )$ by trying to bound the effect on $c _ { 1 } ^ { 3 } ( X )$ of the steps in the minimal model program for $X$ . This leads to a positive answer for the boundedness question for many smooth projective threefolds of nonnegative Kodaira dimension whose minimal model program is a composition of blow-downs to points and smooth curves in smooth loci, see [2, Corollary 1.5]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,364,881,543]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"193\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this paper we focus on the case of threefolds of negative Kodaira dimension. The main difficulty that we face in this case is that the aforementioned Miyaoka– Yau inequality, which was essential for the case of non-negative Kodaira dimension, does not hold any longer. It is also known by examples of LeBrun [13], that the boundedness does not hold in the non-Kähler case. Nonetheless, for any smooth Kähler threefold $X$ we can run a minimal model program thanks to [6, 7]. If $X$ is uniruled then we arrive at a Mori fibre space $Y B$ , i.e. a Kähler threefold $Y$ with at most terminal singularities together with a morphism of relative Picard rank one with connected fibres to a complex Kähler variety $B$ of smaller dimension whose general fibre is Fano. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,545,555,561]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"193\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The first result of this paper is the following. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,572,879,626]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"193\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 1 Let $( X _ { i } ) _ { i \\\\geq 0 }$ be a sequence of Mori fibre spaces, where $X _ { i }$ are smooth Kähler threefolds. If each $X _ { i }$ is homeomorphic to $X _ { 0 }$ , then the sequence of Chern numbers $c _ { 1 } ^ { 3 } ( X _ { i } )$ is bounded. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,635,881,743]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"193\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The above result should be compared to the fact that all known examples of sequences of homeomorphic varieties with unbounded Chern numbers are Mori fibre spaces, and in fact projective bundles (see [19]). We therefore believe that together with the aforementioned results from [2], the above theorem puts forward strong evidence for the conjecture that the Chern numbers of smooth projective threefolds are determined up to finite ambiguity by the underlying smooth manifold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,743,881,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"193\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $X B$ is a Mori fibre space and $X$ is a smooth Kähler threefold, then there are three main cases to consider, depending on the dimension of $B$ . If $B$ is a point, then $X$ is a Fano variety and we conclude because Fano varieties of fixed dimension form a bounded family. If $B$ is a curve, then it is smooth projective and $X$ is also projective. Since the Pontryagin classes are up to torsion homeomorphism invariants by Novikov’s theorem [17], [20, Proposition 26] proves the above theorem in case all but finitely many of the $X _ { i }$ are Mori fibre spaces over points or curves. Using Novikov’s theorem [17] once again, Theorem 1 thus follows from the following more precise result about Mori fibre spaces over surfaces, where we denote by $H _ { t f } ^ { * } ( X , \\\\mathbb { Z } )$ the quotient of $H ^ { * } ( X , \\\\mathbb { Z } )$ by the subgroup of all torsion classes. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,54,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"193\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[671,54,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"193\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"S. Schreieder and L. Tasin \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[123,88,879,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"194\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,135,881,229]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"194\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 2 Let $( X _ { i } ) _ { i \\\\geq 0 }$ be a sequence of smooth Kähler threefolds admitting $a$ conic bundle structure $f _ { i } : X _ { i } \\\\longrightarrow S _ { i }$ of relative Picard number $^ { l }$ over a smooth Kähler surface $S _ { i }$ . If there is an isomorphism of graded rings $H _ { t f } ^ { * } ( X _ { i } , \\\\mathbb { Z } ) ~ \\\\simeq$ $H _ { t f } ^ { * } ( X _ { 0 } , \\\\mathbb { Z } )$ which respects the first Pontryagin classes, then the sequence of Chern numbers $c _ { 1 } ^ { 3 } ( X _ { i } )$ is bounded. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,237,881,364]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"194\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In view of Theorem 1, it is therefore natural to wonder if we can also bound the Chern numbers of certain threefolds of negative Kodaira dimension which are not necessarily Mori fibre spaces themselves. Our next result achieves this by generalising [2, Corollary 1.5] to the case of negative Kodaira dimension. To state it, recall that for any smooth complex projective threefold $X$ , there is a cubic form $F _ { X }$ on $H ^ { 2 } ( X , \\\\mathbb { Q } )$ , given by cup product. For technical reasons, we will assume that the discriminant $\\\\Delta _ { F _ { X } }$ of the cubic form is non-zero. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,372,881,462]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"194\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Theorem 3 Let $X$ be a smooth complex projective threefold which is uniruled and let $F _ { X }$ be its associated cubic. Assume that $\\\\Delta _ { F _ { X } } \\\\neq 0$ and that there exists $a$ birational morphism $f \\\\colon X \\\\to Y$ onto a Mori fibre space $Y$ , which is obtained as a composition of divisorial contractions to points and blow-downs to smooth curves in smooth loci. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,462,881,496]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"194\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Then there exists a constant $D$ depending only on the topology of the 6-manifold underlying $X$ such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[446,513,553,536]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0565/2fb2e96d403d75f6acaed640dfe0696554e2ad0a5fd50812a9946bae2cb199ae.jpg\",\n \"list_items\": null,\n \"page_idx\": \"194\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n| K _ { X } ^ { 3 } | \\\\le D .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,552,881,642]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"194\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A major step in proving Theorem 3 is Proposition 8 (cf. [2, Theorem 1.3(2)]), where we show that in the assumptions of Theorem 3, most of the topological invariants of $Y$ are determined (up to finite ambiguity) a priori by the smooth manifold underlying $X$ . It would be interesting to understand to what extend this is true in general (see [4] for the case of Betti numbers): \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,651,879,723]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"194\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Question 4 Let $X$ be a smooth complex projective threefold with cubic form $F _ { X }$ and first Pontryagin class $p _ { 1 } ( X )$ . Let $\\\\mathcal { P }$ be the set of pairs $( F _ { Y } , p _ { 1 } ( Y ) )$ , taken up to isomorphism, such that there exists an MMP $X \\\\ \\\\ Y$ . Is the set $\\\\mathcal { P }$ determined by the pair of invariants $( F _ { X } , p _ { 1 } ( X ) )$ of $X$ up to finite ambiguity? \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,770,321,789]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"194\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1.1 Conventions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,813,881,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"194\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"All manifolds are closed and connected. A Kähler manifold is a complex manifold which admits a Kähler metric. For any (Kähler) manifold $X$ , we denote by $H _ { t f } ^ { * } ( X , \\\\mathbb { Z } )$ the quotient $H ^ { * } ( X , \\\\mathbb { Z } ) / H ^ { * } ( X , \\\\mathbb { Z } ) _ { t o r s }$ , where $H ^ { * } ( X , \\\\mathbb { Z } ) _ { t o r s }$ denotes the torsion subgroup of $H ^ { * } ( X , \\\\mathbb { Z } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,52,425,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"194\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Chern Numbers of Uniruled Threefolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,54,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"194\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,551,111]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"195\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 Mori Fibre Spaces Over Surfaces \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,132,687,151]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"195\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The starting point of our investigation is the following lemma. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,159,879,196]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"195\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 5 ([21, Sec. 7.1]) Let $f : X \\\\to S$ be a Mori fibre space such that $X$ is $a$ smooth projective threefold and $s$ is a surface. Then \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,205,881,334]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"195\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(i) $f : X \\\\to S$ is a standard (i.e. relative Picard number $I$ ) conic bundle and $s$ is smooth; \\n(ii) the discriminant $D \\\\subset S$ of $f$ is either empty or a reduced curve with at worst ordinary double points; \\n(iii) $e ( X ) = 2 ( e ( S ) - p _ { a } ( D ) + 1 )$ , $b _ { 1 } ( X ) = b _ { 1 } ( S )$ and $b _ { 3 } ( X ) = 2 ( b _ { 1 } ( X ) + p _ { a } ( D ) -$ 1); \\n(iv) $D \\\\equiv - f _ { \\\\ast } K _ { X / S } ^ { 2 }$ and $- 4 K _ { S } \\\\equiv f _ { * } K _ { X } ^ { 2 } + D .$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[148,340,498,358]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"195\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We will also use the following lemma. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,367,876,403]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"195\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lemma 6 Let $f : X \\\\to S$ be a Mori fibre space such that $X$ is a smooth projective threefold and $s$ is a surface. Then, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[412,421,583,441]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0585/7e3142429574d533dcb200e3dc73560e8dc690a0e2c166a44baf5aa91d0bcf09.jpg\",\n \"list_items\": null,\n \"page_idx\": \"195\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nf _ { * } p _ { 1 } ( X ) \\\\equiv - 3 D ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,456,876,493]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"195\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $p _ { 1 } ( X )$ is the first Pontryagin class of $X$ and $D \\\\subset S$ denotes the discriminant curve of $f$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,501,881,611]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"195\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Since $s$ is smooth projective by Lemma 5, the Néron–Severi group of $s$ is generated by very ample curves. Hence, it suffices to compute the intersection product with a general smooth projective curve $C \\\\subset S .$ The preimage $R : = f ^ { - 1 } ( C )$ is then the blow-up of a minimal ruled surface over $C$ in $C . D$ many points. The normal bundle of $R$ in $X$ is given by $\\\\mathcal { N } _ { R / X } = f ^ { * } \\\\mathcal { O } _ { S } ( C ) | _ { R }$ . Since $T _ { X } | _ { R } = T _ { R } \\\\oplus$ $\\\\textstyle \\\\mathcal { N } _ { R / X }$ , we get \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[138,626,856,650]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0238/e56417eb5cf6b0054884ade9b07af18ed6f782a27a1413d6b02f6828dcfd6cff.jpg\",\n \"list_items\": null,\n \"page_idx\": \"195\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n( 1 + c _ { 1 } ( X ) \\\\vert _ { R } + c _ { 2 } ( X ) \\\\vert _ { R } ) ( 1 - c _ { 1 } ( \\\\mathcal { N } _ { R / X } ) + c _ { 1 } ^ { 2 } ( \\\\mathcal { N } _ { R / X } ) ) = 1 + c _ { 1 } ( R ) + c _ { 2 } ( R ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,666,507,684]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"195\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hence, using $\\\\mathcal { O } _ { X } ( R ) = f ^ { * } \\\\mathcal { O } _ { S } ( C )$ , we get \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[248,699,749,896]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0768/0d739b190cfc81ce7603c18ebe81666b61492906c72503e8ba6d18953db1fb73.jpg\",\n \"list_items\": null,\n \"page_idx\": \"195\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { f _ { * } c _ { 2 } ( X ) . C = c _ { 2 } ( X ) . R } \\\\\\\\ & { \\\\qquad = c _ { 2 } ( X ) | _ { R } } \\\\\\\\ & { \\\\qquad = c _ { 2 } ( R ) - c _ { 1 } ^ { 2 } ( { \\\\mathcal { N } } _ { R / X } ) + c _ { 1 } ( { \\\\mathcal { N } } _ { R / X } ) c _ { 1 } ( X ) | _ { R } } \\\\\\\\ & { \\\\qquad = c _ { 2 } ( R ) - f ^ { * } { \\\\mathcal { O } } _ { S } ( C ) ^ { 3 } + f ^ { * } { \\\\mathcal { O } } _ { S } ( C ) ^ { 2 } c _ { 1 } ( X ) } \\\\\\\\ & { \\\\qquad = 2 - 4 g ( C ) + 2 + C . D + C ^ { 2 } \\\\cdot c _ { 1 } ( { \\\\mathbb { P } } ^ { 1 } ) } \\\\\\\\ & { \\\\qquad = - 2 K _ { S } . C - 2 C ^ { 2 } + C . D + 2 C ^ { 2 } } \\\\\\\\ & { \\\\qquad = - 2 K . C + C . D . } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[671,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"195\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"S. Schreieder and L. Tasin \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,152,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"195\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"192 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,742,111]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By Lemma 5, $f _ { * } c _ { 1 } ^ { 2 } ( X ) \\\\equiv - 4 K _ { S } - D$ . Using $p _ { 1 } = c _ { 1 } ^ { 2 } - 2 c _ { 2 }$ , we get \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[264,126,733,205]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0380/cf071459f8ed6df4f18980c8616644c75876aea5e170c5a280b4f7fcdfbc4977.jpg\",\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array} { r l } & { f _ { * } p _ { 1 } ( X ) . C = f _ { * } c _ { 1 } ^ { 2 } ( X ) . C - 2 f _ { * } c _ { 2 } ( X ) . C } \\\\\\\\ & { \\\\phantom { f _ { * } p _ { 1 } ( X ) . C } = - 4 K _ { S } . C - D . C + 4 K _ { S } . C - 2 D . C } \\\\\\\\ & { \\\\phantom { f _ { * } p _ { 1 } ( X ) . C } = - 3 D . C , } \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,220,348,237]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"which proves the lemma. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,246,879,300]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof of Theorem 2 By [15, Theorem 1.1], any standard conic bundle $f : X \\\\to S$ , where $X$ is a smooth Kähler threefold, has an algebraic deformation. To bound $K _ { X _ { i } } ^ { 3 }$ it thus suffices to assume that $X _ { i }$ and $S _ { i }$ are projective for any $i$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,301,881,396]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By assumptions, there is an isomorphism $H _ { t f } ^ { 2 } ( X _ { i } , \\\\mathbb { Z } ) \\\\ \\\\simeq \\\\ H _ { t f } ^ { 2 } ( X _ { 0 } , \\\\mathbb { Z } ) $ which respects the trilinear forms given by cup products. We use this isomorphism to identify degree two cohomology classes of $X _ { i }$ with those of $X _ { 0 }$ (up to torsion). Using Poincaré duality, we further identify classes of $H _ { t f } ^ { 4 } ( X _ { i } , \\\\mathbb Z )$ with linear forms on $H _ { t f } ^ { 2 } ( X _ { i } , \\\\mathbb { Z } ) \\\\simeq H _ { t f } ^ { 2 } ( X _ { 0 } , \\\\mathbb { Z } )$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,399,881,453]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The codimension one linear subspace $f _ { i } { } ^ { * } \\\\mathbb { P } ( H ^ { 2 } ( S _ { i } , \\\\mathbb { Q } ) )$ of $\\\\mathbb { P } ( H ^ { 2 } ( X _ { 0 } , \\\\mathbb { Q } ) )$ is contained in the cubic hypersurface $\\\\{ \\\\alpha \\\\mid \\\\alpha ^ { 3 } = 0 \\\\}$ . Passing to a suitable subsequence we can therefore assume that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[369,466,630,490]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0193/1448cb259f7151ac30c9f493bf8c11b64f2738f5f6d48ae59d553c5b9485f2ea.jpg\",\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nf _ { i } ^ { * } H ^ { 2 } ( S _ { i } , \\\\mathbb { Q } ) \\\\subset H ^ { 2 } ( X _ { 0 } , \\\\mathbb { Q } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,504,881,582]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"does not depend on $i$ . Let $\\\\ell _ { i } \\\\in H _ { t f } ^ { 4 } ( X _ { 0 } , \\\\mathbb { Z } )$ be the class of a fibre of $f _ { i }$ . The linear form determined by this class on $H ^ { 2 } ( X _ { 0 } , \\\\mathbb { Q } )$ has kernel $f _ { i } ^ { * } H ^ { 2 } ( S _ { i } , \\\\mathbb { Q } )$ , and so $\\\\ell _ { i } \\\\cdot \\\\mathbb { Q }$ is independent of $i$ . Since $\\\\ell _ { i }$ is an integral class with $K _ { X _ { i } } . \\\\ell _ { i } = - 2$ , we may after possibly passing to another subsequence assume that $\\\\ell _ { i } = \\\\ell$ does not depend on $i$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,582,876,618]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since the natural cup product pairing on $H ^ { 2 } ( S _ { i } , \\\\mathbb { Q } )$ can be recovered from the pairing \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,633,676,657]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0718/affcfae030c9eb2442a155661215b127046ddc5b5788e95b061e512cc13d7fca.jpg\",\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nf _ { i } ^ { * } H ^ { 2 } ( S _ { i } , \\\\mathbb { Q } ) \\\\times f _ { i } ^ { * } H ^ { 2 } ( S _ { i } , \\\\mathbb { Q } ) { \\\\longrightarrow } \\\\ell \\\\mathbb { Q } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,671,874,707]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"we get that the pairing on $H ^ { 2 } ( S _ { i } , \\\\mathbb { Q } )$ is determined by the cubic form on $H ^ { 2 } ( X _ { 0 } , \\\\mathbb { Q } )$ and so it does not depend on $i$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,708,879,744]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since $f _ { i } ^ { * } H ^ { 2 } ( S _ { i } , \\\\bar { \\\\mathbb { Q } } ) \\\\subset H ^ { 2 } ( X _ { 0 } , \\\\mathbb { Q } )$ does not depend on $i$ , the same holds for the homomorphism \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[300,759,697,782]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0223/73e9fe583d810546c0f8787a2eee92ea7dc2001f42ede5d5a4b5bf4e6d7b0c5a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\psi _ { i } : H ^ { 2 } ( S _ { i } , \\\\mathbb { Q } ) { \\\\longrightarrow } \\\\mathbb { Q } , \\\\alpha \\\\longmapsto p _ { 1 } ( X _ { i } ) . f _ { i } ^ { * } \\\\alpha\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,798,881,834]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By the projection formula, we have $p _ { 1 } ( X _ { i } ) . f _ { i } ^ { * } \\\\alpha = ( f _ { i } ) _ { * } p _ { 1 } ( X _ { i } ) . \\\\alpha$ . Lemma 6 thus yields \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[412,852,580,872]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0235/94f8a6e09a9cfca2e6cfaf94ae604ff0957bf194c4c5435786521ade368e7ca2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\psi _ { i } ( \\\\alpha ) = - 3 D _ { i } . \\\\alpha ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,51,428,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Chern Numbers of Uniruled Threefolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,879,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"193 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[858,222,879,237]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"196\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"□ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,881,196]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $D _ { i }$ is the discriminant curve of $f _ { i }$ . This shows that the linear form determined by $[ D _ { i } ] \\\\in H ^ { 2 } ( S _ { i } , \\\\mathbb { Q } )$ on $H ^ { 2 } ( S _ { i } , \\\\mathbb { Q } )$ does not depend on i. Since the natural pairing $H ^ { 2 } ( S _ { i } , \\\\mathbb { Q } ) \\\\times H ^ { 2 } ( S _ { i } , \\\\mathbb { Q } ) \\\\mathbb { Q }$ is perfect by Poincaré duality, we get that the class $[ D _ { i } ] \\\\in H ^ { 2 } ( S _ { i } , \\\\mathbb { Q } )$ does not depend on $i$ . Using again the fact that we know the pairing on $H ^ { 2 } ( S _ { i } , \\\\mathbb { Q } )$ , we finally get that the self-intersection $D _ { i } ^ { 2 }$ does not depend on $i$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[132,196,835,216]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For any class $y \\\\in H ^ { 2 } ( X _ { 0 } , \\\\mathbb { Q } )$ , which does not lie in $f _ { i } ^ { * } H ^ { 2 } ( S _ { i } , \\\\mathbb { Q } )$ , we have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[116,229,879,252]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0524/4f5db11f1f0fc440a464f540be24889d7bd4d4a81a02a0c5a3cf6e8b3253b37b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n{ \\\\cal H } ^ { 2 } ( X _ { 0 } , \\\\mathbb { Q } ) = f _ { i } { } ^ { * } { \\\\cal H } ^ { 2 } ( S _ { i } , \\\\mathbb { Q } ) \\\\oplus y \\\\cdot \\\\mathbb { Q } \\\\mathrm { a n d } { \\\\cal H } ^ { 4 } ( X _ { 0 } , \\\\mathbb { Q } ) = f _ { i } { } ^ { * } { \\\\cal H } ^ { 2 } ( S _ { i } , \\\\mathbb { Q } ) \\\\cdot y \\\\oplus \\\\ell \\\\cdot \\\\mathbb { Q } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,267,881,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In particular, $y ^ { 2 } = u y + \\\\lambda \\\\ell$ for some $\\\\lambda \\\\in \\\\mathbb { Q }$ and $u \\\\in f _ { i } { ^ { * } H ^ { 2 } } ( S _ { i } , \\\\mathbb { Q } )$ . Replacing $y$ by a suitable multiple of $\\\\begin{array} { r } { y - \\\\frac { 1 } { 2 } u } \\\\end{array}$ , we may thus assume that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[293,322,699,345]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0201/1f06e5f6efe4418fc7a894d20ff77bbfec828984651172acaf71cf0a01e63282.jpg\",\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\ny . \\\\ell = - 2 \\\\mathrm { ~ a n d ~ } y ^ { 2 } \\\\in { f _ { i } } ^ { * } H ^ { 4 } ( S _ { i } , \\\\mathbb { Q } ) = \\\\ell \\\\cdot \\\\mathbb { Q } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[145,361,369,379]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For any $X _ { i }$ , we then get \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[419,396,578,418]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0330/d1e077581be83014cda209bbf9de95039f2141edc0dbda9a997b4f0ae822d2e3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { X _ { i } } = y + f _ { i } ^ { * } z _ { i }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,433,410,451]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for some $z _ { i } \\\\in H ^ { 2 } ( S _ { i } , \\\\mathbb { Z } )$ . Since \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[416,465,578,490]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0331/05794fdd560ac1676de0bb611a334d55e9bb995285eb068384a2d7e09254ddd2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nK _ { X _ { i } } ^ { 3 } = y ^ { 3 } - 6 z _ { i } ^ { 2 } ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,507,503,527]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"it suffices to prove the boundedness of $z _ { i } ^ { 2 }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,528,876,563]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since $y \\\\cdot \\\\ell = - 2$ , the pushforward of $2 y f _ { i } ^ { * } z _ { i }$ via $f _ { i }$ yields $- 4 z _ { i }$ . Lemma 5 therefore implies that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[337,576,660,603]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0652/6e38a47a10af9d47ca8cdde3303b797578f366e7e25e5b34b0dd53caff8600cf.jpg\",\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n( f _ { i } ) _ { * } K _ { X _ { i } } ^ { 2 } \\\\equiv - 4 z _ { i } \\\\equiv - 4 K _ { S _ { i } } - D _ { i } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,617,186,635]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hence, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[348,648,649,674]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0396/8935d8286be605f5b362d15dc2f23fa053a166b6402c86c0c0219aef92a96e9f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n1 6 z _ { i } ^ { 2 } = 1 6 K _ { S _ { i } } ^ { 2 } + 8 K _ { S _ { i } } D _ { i } + D _ { i } ^ { 2 } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,690,881,728]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since $D _ { i } ^ { 2 }$ does not depend on $i$ and $K _ { S _ { i } } ^ { 2 }$ is bounded in terms of the Betti numbers of $S _ { i }$ , the statement follows from the fact that $p _ { a } ( D _ { i } )$ is bounded by Lemma 5. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,773,398,794]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 Uniruled Threefolds \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,818,876,854]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Before we turn to the proof of Theorem 3, we state few preliminary facts about terminal $\\\\mathbb { Q }$ -factorial threefolds. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[671,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"S. Schreieder and L. Tasin \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"197\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"194 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,88,373,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"198\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.1 Invariant Triples \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,132,501,150]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"198\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $X$ be a terminal $\\\\mathbb { Q }$ -factorial threefold. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,151,876,187]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"198\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There exists a well-defined class $c _ { 2 } ( X ) \\\\ \\\\in \\\\ H ^ { 2 } ( X , \\\\mathbb { Z } ) ^ { \\\\vee } \\\\ = \\\\ \\\\mathrm { H o m } ( H ^ { 2 } ( Z , \\\\mathbb { Z } ) , \\\\mathbb { Z } )$ obtained in the following way (see page 411 in [22]). For any $\\\\alpha \\\\in H ^ { 2 } ( X , \\\\mathbb { Z } )$ set \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[389,201,605,225]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0262/04bb337b3f02b866b87ce809f69d77b86eda6ba62aa37100490a7fbe0e8235f7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"198\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nc _ { 2 } ( X ) . \\\\alpha = c _ { 2 } ( \\\\tilde { X } ) . f ^ { * } \\\\alpha ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[123,240,478,258]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"198\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $f : { \\\\tilde { X } } \\\\to X$ is a resolution of $X$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,259,881,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"198\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We then define the Pontryagin class $p _ { 1 } ( X ) \\\\in H ^ { 2 } ( X , \\\\mathbb { Q } ) ^ { \\\\vee }$ in terms of $c _ { 1 } ( X )$ and $c _ { 2 } ( X )$ in the same way as in the smooth case, where $c _ { 1 } ( X )$ is the class of $- K _ { X }$ in $H ^ { 2 } ( X , \\\\mathbb { Q } )$ : \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[366,327,630,351]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0441/3b5732e3624358f05c3057341db27ec957c68dcf4130fdd9c317a6a5f418d05e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"198\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\np _ { 1 } ( X ) : = c _ { 1 } ( X ) ^ { 2 } - 2 c _ { 2 } ( X ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,366,881,439]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"198\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We also associate to $X$ its cubic form $F _ { X } \\\\ \\\\in \\\\ S ^ { 3 } H ^ { 2 } ( X , \\\\mathbb { Z } ) ^ { \\\\vee }$ , which is induced by the cup product on $H ^ { 2 } ( X , \\\\mathbb { Z } )$ . In this way we can associate to $X$ the triple $( H _ { t . f . } ^ { 2 } ( X , \\\\mathbb { Z } ) , F _ { X } , p _ { 1 } ( X ) )$ . When $X$ is smooth, this triple encodes many geometrical properties of the 6-manifold underlying $X$ (see for instance [18] and [1]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,445,881,540]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"198\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Definition 7 We call $( H _ { t . f . } ^ { 2 } ( X , \\\\mathbb { Z } ) , F _ { X } , p _ { 1 } ( X ) )$ the invariant triple of $X$ . Two triples $( H , F , p )$ and $( H ^ { \\\\prime } , F ^ { \\\\prime } , p ^ { \\\\prime } )$ , where $H$ (resp. $H ^ { \\\\prime }$ ) is a free abelian group, $F \\\\in S ^ { 3 } H ^ { \\\\vee }$ (resp. $F \\\\in S ^ { 3 } H ^ { \\\\prime \\\\vee } )$ ) is a cubic form and $p$ is a linear form on $H \\\\otimes \\\\mathbb { Q }$ (resp. $p ^ { \\\\prime }$ is a linear form on $H ^ { \\\\prime } \\\\otimes \\\\mathbb { Q } )$ are isomorphic if there exists a linear isomorphism $T : H \\\\to H ^ { \\\\prime }$ which identifies $F$ with $F ^ { \\\\prime }$ and its $\\\\mathbb { Q }$ -extension identifies $p$ with $p ^ { \\\\prime }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,585,435,606]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"198\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.2 Terminal Singularities \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[125,630,842,648]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"198\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We now recall few known facts about terminal singularities in dimension three. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,650,881,774]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"198\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Let $( X , p )$ be the germ of a three-dimensional terminal singularity. The index of $p$ is the smallest positive integer $r$ such that $r K _ { X }$ is Cartier. It follows from the classification of terminal singularities, that there exists a deformation of $( X , p )$ into a space with $h \\\\geq 1$ terminal singularities $p _ { 1 } , \\\\ldots , p _ { h }$ which are isolated cyclic quotient singularities of index $r ( p _ { i } )$ (for details see [22, Remark 6.4]). The set $\\\\{ p 1 , \\\\ldots , p _ { h } \\\\}$ is called the basket $B ( X , p )$ of singularities of $X$ at $p$ . As in [5, Section 2.1], we define \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[396,788,601,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0574/5b864ccbd16bff3d0250136291b06708d86f187f8494ba22cd2cb732fbc324c6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"198\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\Xi ( X , p ) = \\\\sum _ { i = 1 } ^ { h } r ( p _ { i } ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,51,428,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"198\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Chern Numbers of Uniruled Threefolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"198\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"195 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,88,879,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"199\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Thus, if $X$ is a projective variety of dimension 3 with terminal singularities and Sing $X$ denotes the finite set of singular points of $X$ , we may define \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[375,139,621,183]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0186/54c91953419cff8c07bf0144af49d1d0563648feb2b7b53179aebfa70624b18a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"199\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\Xi ( X ) = \\\\sum _ { p \\\\in \\\\operatorname { S i n g } X } \\\\Xi ( X , p ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,223,405,244]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"199\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.3 Proof of Theorem 3 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,268,881,324]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"199\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The following result is interesting by itself and leads naturally to the problem of understanding what kind of topological invariants are determined up to finite ambiguity during a running of an MMP, see Question 4. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,331,881,405]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"199\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 8 (Cf. [2, Theorem 1.3(2)]) Let $H$ be a finitely generated free abelian group of rank $n + 1$ , $F \\\\in S ^ { 3 } H ^ { \\\\vee }$ be a cubic form such that $\\\\Delta _ { F } \\\\neq 0$ and $p$ a linear form on $H$ . Consider the set $\\\\mathcal { P }$ of invariant triples $( H ^ { \\\\prime } , F ^ { \\\\prime } , p ^ { \\\\prime } )$ up to isomorphism, such that there exist \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,412,881,486]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"199\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(1) a terminal $\\\\mathbb { Q }$ -factorial threefold $X$ with associated triple $( H , F , p )$ ; (2) a terminal $\\\\mathbb { Q }$ -factorial threefold $Y$ with associated triple $( H ^ { \\\\prime } , F ^ { \\\\prime } , p ^ { \\\\prime } )$ ; (3) a birational morphism $f \\\\colon X \\\\to Y$ which is a divisorial contraction to a point or to a smooth curve contained in the smooth locus of $Y$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,495,330,512]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"199\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Then the set $\\\\mathcal { P }$ is finite. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,521,881,707]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"199\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Note that the proof of this case works also for $\\\\Delta _ { F } = 0$ . Consider the set $\\\\mathcal { A }$ of primitive elements $\\\\alpha \\\\in H$ such that $\\\\alpha$ is proportional to the exceptional divisor $E$ of some divisorial contraction to a point $f : X \\\\to Y$ as in the statement. The elements of $\\\\mathcal { A }$ are points of rank 1 for the Hessian of the cubic form $F$ and so they are finite by Cascini and Tasin [2, Proposition 3.3]. It follows from [2, Proposition 4.7] that for any sub-module $H ^ { \\\\prime } = f ^ { * } H _ { t . f . } ^ { 2 } ( Y , \\\\mathbb { Z } ) \\\\hookrightarrow H$ there is $\\\\alpha \\\\in { \\\\mathcal { A } }$ such that $\\\\alpha ^ { 2 } . H ^ { \\\\prime } = 0$ and such that the index of $H ^ { \\\\prime } \\\\oplus \\\\mathbb { Z } \\\\alpha$ in $H$ is at most $r ^ { n }$ , where $r = | \\\\alpha ^ { 3 } |$ . This implies that for all possible contractions to points $f \\\\colon X \\\\to Y$ as in the statement, the inclusion $f ^ { * } H _ { t . f . } ^ { 2 } ( Y , \\\\mathbb { Z } ) \\\\hookrightarrow H _ { t . f . } ^ { 2 } ( X , \\\\mathbb { Z } )$ : → is determined up to finite ambiguity. This determines also $F ^ { \\\\prime }$ up to finite ambiguity just restricting $F$ to $H ^ { \\\\prime }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,708,874,743]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"199\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To prove the finiteness of $p ^ { \\\\prime }$ consider a divisorial contraction to a point $f : X \\\\to$ $Y$ and write \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[382,759,612,780]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0167/678fb39e53889f14239d2f963b7c2c4524fbe2907dafbea0b24c652555de5bda.jpg\",\n \"list_items\": null,\n \"page_idx\": \"199\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nc _ { 1 } ( X ) = f ^ { * } c _ { 1 } ( Y ) - c E ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,797,881,833]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"199\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $c$ is the discrepancy of the exceptional divisor $E$ . Since $c _ { 2 } ( X ) = f ^ { * } c _ { 2 } ( Y )$ we have that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[298,848,701,872]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0665/fcad983944705a1419887ea7cf0fb9e6dc4e8da91be742e1b2363f2a2d191a94.jpg\",\n \"list_items\": null,\n \"page_idx\": \"199\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\np _ { 1 } ( X ) = f ^ { * } p _ { 1 } ( Y ) - 2 c f ^ { * } c _ { 1 } ( Y ) . E + c ^ { 2 } E ^ { 2 }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,52,152,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"199\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"196 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[671,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"199\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"S. Schreieder and L. Tasin \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,876,130]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and so $p _ { 1 } ( Y )$ is given by the restriction of $p _ { 1 } ( X )$ to $f ^ { * } H _ { t . f . } ^ { 2 } ( Y , \\\\mathbb { Q } )$ . This means that also $p ^ { \\\\prime }$ is determined up to finite ambiguity and we are done. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,130,881,166]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We now look at divisorial contractions to curves. Consider $\\\\mathcal { E }$ the set of pairs $( E , H ^ { \\\\prime } )$ where $E$ is a primitive element in $H$ and $H ^ { \\\\prime } \\\\subset H$ is a submodule such that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[419,181,578,202]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0202/6cb3760bda05462d6adc7bda016fdd694a1df98c9c8b479fc2badd8a4caa2459.jpg\",\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nH = \\\\mathbb { Z } [ E ] \\\\oplus H ^ { \\\\prime }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,220,432,237]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and the cubic $F$ assumes the form \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[309,250,687,301]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0239/3391a63552f7f6a07bb7693845f026153400de7a2b21ec683702740f9debac93.jpg\",\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nF = a x _ { 0 } ^ { 3 } + \\\\sum _ { i = 1 } ^ { n } b _ { i } x _ { 0 } ^ { 2 } x _ { i } + F ^ { \\\\prime } ( x _ { 1 } , \\\\dots , x _ { n } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,316,690,334]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"with respect to any basis $E , \\\\alpha _ { 1 } , \\\\ldots , \\\\alpha _ { n }$ with $\\\\alpha _ { 1 } , \\\\ldots , \\\\alpha _ { n } \\\\in H ^ { \\\\prime }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,336,881,423]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By Cascini and Tasin [2, Thm. 3.1] there are only finitely many possible nonequivalent reduced forms for $F$ . In particular, up to finite ambiguity, we can assume that the coefficients of $F$ in the expression (1) are fixed. Since the isotropy group of a cubic with non-zero discriminant is finite ([18, Thm. 4]), we deduce that $\\\\mathcal { E }$ is finite. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,424,881,478]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"If $f \\\\colon X \\\\ \\\\to \\\\ Y$ is a divisorial contraction which contracts a divisor $E$ to a smooth curve $C$ in the smooth locus of $Y$ , then (see [18, Proposition 14] and [2, Proposition 4.8]) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[341,493,658,516]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0710/b04e8005c373a3ec17392b7df56337f74e7eb1ce8f3174153f24190cb5c64f51.jpg\",\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nH ^ { 2 } ( X , \\\\mathbb { Z } ) = \\\\mathbb { Z } [ E ] \\\\oplus f ^ { * } H ^ { 2 } ( Y , \\\\mathbb { Z } )\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,534,157,549]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,564,676,588]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0470/73fd21445d408290f8772d76bc387788bb0cd8b7284340a7e9495330bddf6929.jpg\",\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\np _ { 1 } ( X ) = f ^ { * } ( p _ { 1 } ( Y ) ) + E ^ { 2 } - 2 f ^ { * } ( C ) .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,603,879,659]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recalling that $E ^ { 2 } . f ^ { \\\\ast } ( \\\\alpha ) = - C . \\\\alpha$ for any $\\\\alpha \\\\in H ^ { 2 } ( Y , \\\\mathbb { Z } )$ we deduce that $p _ { 1 } ( Y )$ is determined by $p _ { 1 } ( X )$ , $E ^ { 2 }$ and by the inclusion $f ^ { * } H ( Y , \\\\mathbb { Z } ) \\\\hookrightarrow H ^ { 2 } ( X , \\\\mathbb { Z } )$ and we conclude using the finiteness of $\\\\mathcal { E }$ . □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,666,874,722]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 9 Let $( X _ { i } ) _ { i \\\\geq 0 }$ be a sequence of terminal $\\\\mathbb { Q }$ -factorial threefolds admitting a conic bundle structure $f _ { i } : X _ { i } \\\\longrightarrow S _ { i }$ of relative Picard number $I$ over $a$ surface $S _ { i }$ . Assume that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,731,774,785]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(1) the Euler characteristics $\\\\chi ( X _ { i } , { \\\\mathcal { O } } _ { X _ { i } } )$ are bounded and $b _ { 2 } ( X _ { i } ) = 2$ ; \\n(2) the invariant triples of $X _ { 0 }$ and $X _ { i }$ are isomorphic for any $i$ ; \\n(3) the sequence $\\\\Xi ( X _ { i } )$ is bounded. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,792,637,813]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Then the sequence of Chern numbers $c _ { 1 } ^ { 3 } ( X _ { i } )$ is bounded. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,821,881,876]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof Let $h$ be an ample generator of $\\\\operatorname { P i c } ( S _ { 0 } )$ and let $x \\\\in H ^ { 2 } ( X _ { 0 } , \\\\mathbb { Z } )$ be a primitive class proportional to $f _ { 0 } ^ { * } h$ . Then $x ^ { 3 } = 0$ , $x ^ { 2 } \\\\neq 0$ and letting $y = c _ { 1 } ( X _ { 0 } )$ we can write \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[364,891,630,914]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0798/16ddc0b06227043d5a9bb0f50beea19daea450798e143e3d9919f83b013c219f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nH ^ { 2 } ( X _ { 0 } , \\\\mathbb { Q } ) = x \\\\cdot \\\\mathbb { Q } \\\\oplus y \\\\cdot \\\\mathbb { Q } .\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,51,428,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Chern Numbers of Uniruled Threefolds \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[847,52,876,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"200\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"197 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[118,90,881,201]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From now on we will use the isomorphism $H ^ { 2 } ( X _ { i } , \\\\mathbb { Q } ) \\\\simeq H ^ { 2 } ( X _ { 0 } , \\\\mathbb { Q } )$ to think about $x$ and $y$ as basis elements of $H ^ { 2 } ( X _ { i } , \\\\mathbb { Q } )$ . Note that $x ^ { 2 } y \\\\neq 0$ since already $x ^ { 3 } = 0$ and $x ^ { 2 } \\\\neq 0$ . Moreover, the space of elements in $H ^ { 2 } ( X _ { 0 } , \\\\mathbb { C } )$ with zero cube is a union of three lines (through 0) and so we may assume without loss of generality that for each $i$ , the pullback of the generator of $H ^ { 2 } ( S _ { i } , \\\\mathbb { Z } )$ to $X _ { i }$ is a multiple of $x$ . In particular, $x ^ { 2 }$ is a multiple of the class of the general fibre of $X _ { i } \\\\longrightarrow S _ { i }$ for all $i$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,201,230,217]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[384,235,610,255]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0658/99b60351af9776feed6a69a37cb46eed7d6d9a4a154da150b8f466e463ef3914.jpg\",\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nc _ { 1 } ( X _ { i } ) = a _ { i } \\\\cdot x + b _ { i } \\\\cdot y ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[118,271,881,325]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"for some $a _ { i } , b _ { i } \\\\in \\\\mathbb { Q }$ . Since $\\\\Xi ( X _ { i } )$ is bounded, there is a positive integer $r$ such that $r K _ { X _ { i } }$ is Cartier for any $i$ . In particular, $r a _ { i } , r b _ { i } \\\\in \\\\mathbb { Z }$ . Since $K _ { X _ { i } } . C = - 2$ where $C$ is a general fibre, we deduce that the sequence of $b _ { i }$ is bounded. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,327,876,361]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We are going to bound the sequence of $a _ { i }$ . By the singular version of Riemann– Roch [22, Corollary 10.3] we get \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[275,376,722,400]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0742/fbb255b268257f25a58c2bf4be066e1196ae54108ae47de8fedeb0472a30b72d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n4 8 \\\\chi ( X _ { i } , \\\\mathcal { O } _ { X _ { i } } ) = c _ { 1 } ( X _ { i } ) . p _ { 1 } ( X _ { i } ) - c _ { 1 } ( X _ { i } ) ^ { 3 } + T _ { i }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[120,417,179,432]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[362,445,633,493]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0668/05ce1a22c31711934356c14526b433d73b5b5e09b98f63a8d92e16a876d2d0e0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nT _ { i } = \\\\sum _ { p _ { \\\\alpha } } \\\\left( r ( p _ { \\\\alpha } ) - \\\\frac { 1 } { r ( p _ { \\\\alpha } ) } \\\\right) ,\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[123,507,879,545]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and the sum runs over all the points of all the baskets of $X _ { i }$ . Note that $T _ { i }$ is a bounded sequence since $\\\\Xi ( X _ { i } )$ is bounded. This implies that \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[123,558,874,582]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"0273/8de54bd10fc79689bd8447dc3898083f6ee2ace40cc4e72023340181c17c08da.jpg\",\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n4 8 \\\\chi ( X _ { i } , \\\\mathcal { O } _ { X _ { i } } ) = - 3 a _ { i } ^ { 2 } b _ { i } x ^ { 2 } y + a _ { i } x ( 2 p _ { 1 } ( X _ { i } ) - 3 b _ { i } ^ { 2 } y ^ { 2 } ) + b _ { i } ^ { 3 } y ^ { 3 } + b _ { i } y p _ { 1 } ( X _ { i } ) + T _ { i }\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[116,597,833,617]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and so the $a _ { i }$ are also bounded, since $b _ { i } x ^ { 2 } y \\\\neq 0$ and $\\\\chi ( X _ { i } , { \\\\mathcal { O } } _ { X _ { i } } )$ are bounded. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,624,881,696]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proof of Theorem 3 Let $f : X \\\\to Y$ be the birational contraction as in Theorem 3. By the proof of [2, Corollary 1.5], we know that $| K _ { X } ^ { 3 } - K _ { Y } ^ { 3 } |$ is bounded by a constant depending only on the Betti numbers of $X$ and on the cubic form $F _ { X }$ . To conclude we need to bound $K _ { Y } ^ { 3 }$ in terms of the topology of $X$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,698,881,786]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since $Y$ is a Mori fibre space and $\\\\Delta _ { F _ { Y } } \\\\neq 0$ , we deduce that either $Y$ is a Fano variety or $Y$ has a conic bundle structure over a surface with second Betti number 1 (otherwise there would be an element in $H ^ { 2 } ( X , \\\\mathbb { C } )$ with square zero, which would imply that $\\\\{ F = 0 \\\\}$ has a singular point and so $\\\\Delta _ { F _ { Y } } = 0 $ ). Since terminal Fano threefolds are bounded, we are left with the conic bundle case. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[118,788,881,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Proposition 8 assures us that the invariant triple of $Y$ is determined up to finite ambiguity by the invariant triple of $X$ . Moreover, the Euler characteristic $\\\\chi ( Y , { \\\\mathcal { O } } _ { Y } ) = \\\\chi ( X , { \\\\mathcal { O } } _ { X } )$ is bounded in terms of the Betti numbers of $X$ and by Cascini and Zhang [3, Prop. 3.3] we also have a bound for $\\\\Xi ( Y )$ depending only on $b _ { 2 } ( X )$ . The result follows then from Proposition 9. □ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[671,52,876,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"S. Schreieder and L. Tasin \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,52,150,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"198 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[861,602,879,614]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"201\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"□ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,91,879,166]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"202\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acknowledgements We thank P. Cascini for many useful conversations on the topic of this manuscript. The first author is supported by DFG grant “Topologische Eigenschaften von Algebraischen Varietäten” (project nr. 416054549). The second author was supported by the DFG grant “Birational Methods in Topology and Hyperkähler Geometry” and is a member of the GNSAGA group of INdAM. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,213,248,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"202\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[120,246,881,858]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"202\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Bisi, C., Cascini, P., Tasin, L.: A remark on the Ueno-Campana’s threefold. Michigan Math. J. 65(3), 567–572 (2016) \\n2. Cascini, P., Tasin, L.:On the Chern numbers of a smooth threefold. Trans. Am. Math. Soc. 370(11), 7923–7958 (2018) \\n3. Cascini, P., Zhang, D.-Q. Effective finite generation for adjoint rings. Ann. Inst. Fourier (Grenoble) 64(1), 127–144 (2014) \\n4. Chen, H.-K.: Betti Numbers in Three Dimensional Minimal Model Program. Preprint arXiv:1605.04372 \\n5. Chen, J.A., Hacon, C.D.: Factoring 3-fold flips and divisorial contractions to curves. J. Reine Angew. Math. 657, 173–197 (2011) \\n6. Höring, A., Peternell, T.: Mori fibre spaces for Kähler threefolds. J. Math. Sci. Univ. Tokyo 22(1), 219–246 (2015) \\n7. Höring, A., Peternell, T.: Minimal models for Kähler threefolds. Invent. Math. 203(1), 217–264 (2016) \\n8. Kotschick, D.: Chern numbers and diffeomorphism types of projective varieties. J. Topol. 1, 518–526 (2008) \\n9. Kotschick, D.: Characteristic numbers of algebraic varieties. Proc. Natl. Acad. Sci. U. S. A. 106, 10114–10115 (2009) \\n10. Kotschick, D.: Topologically invariant Chern numbers of projective varieties. Adv. Math. 229, 1300–1312 (2012) \\n11. Kotschick, D., Schreieder, S.: The Hodge ring of Kähler manifolds. Compositio Math. 149, 637–657 (2013) \\n12. Lafont, J.-F., Sorcar, G., Zheng, F.: Some Kähler structures on products of 2-spheres. L’Enseign. Math. 64, 127–142 (2018) \\n13. LeBrun, C.: Topology versus Chern numbers of complex 3-folds. Pac. J. Math. 191, 123–131 (1999) \\n14. Libgober, A.S., Wood, J.W.: Uniqueness of the complex structure on Kähler manifolds of certain homotopy types. J. Differ. Geom. 32, 139–154 (1990) \\n15. Lin, H.-Y.: Algebraic Approximations of Compact Kähler Threefolds. Preprint arXiv:1710.01083 \\n16. Martinelli, D., Schreieder, S., Tasin, L.: On the number and boundedness of minimal models of general type, to appear in Annales scientifiques de l’ENS \\n17. Novikov, S.P.: On manifolds with free abelian fundamental group and their application. Izv. Akad. Nauk SSSR Ser. Mat. 30, 207–246 (1966) \\n18. Okonek, C., Van de Ven, A.: Cubic forms and complex 3-folds. Enseign. Math. (2) 41(3–4), 297–333 (1995) \\n19. Schreieder, S., Tasin, L.: Algebraic structures with unbounded Chern numbers. J. Topology 9, 849–860 (2016) \\n20. Schreieder, S., Tasin, L.: Kähler structures on spin 6-manifolds. Math. Ann. 373(1–2), 397–419 (2019) \\n21. Shafarevich, I.R.: (ed.) Algebraic Geometry. V. Fano Varieties. Encyclopaedia of Mathematical Sciences, vol. 47, pp. $\\\\mathrm { i } \\\\mathbf { v } { + } 2 4 7$ . Springer, Berlin (1999) \\n22. Reid, M.: Young person’s guide to canonical singularities. In: Algebraic Geometry, Bowdoin, 1985, pp. 345–414. Proceedings of Symposia in Pure Mathematics, vol. 46, Part 1. 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Tasin \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,54,152,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"203\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"200 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.1007/978-3-030-37114-2"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Physics/Mathematics and Computational Science"},"sha256":{"kind":"string","value":"7561d046ed67875a9a772864c664bcf81e8f52b54fec318e50f234af35ed384e"},"title":{"kind":"string","value":"Birational Geometry and Moduli Spaces"}}},{"rowIdx":56,"cells":{"abstract":{"kind":"string","value":"In the 21st century, the world is faced with COVID-19 global pandemic that has had wide-ranging health, social, food, and economic negative impacts throughout the world and also in Indonesia due to the cessation of the mobility of most people and economic engines. The creative empowers the community based on innovation and creativity proven and is one of the pillars of disaster risk reduction recovering the social and economic impact of COVID-19 in the community. The method used is an ethnographic approach starting from the collection of literature sources and supported by field data. The results show that the concept of the creative economy is creative ideas developed by humans supported by the use of technology to produce innovative products and cultures, having regional imagery but are globally competitive and can ultimately provide value to the national economy. Intellectual actors included are not only academics but also include industry players, pioneers, figures in the arts, culture, and sciences who have a role in providing creative ideas, innovations in the creative industry. The result is a triplex helix model in mitigating the effect of the covid19 disaster by connecting intellectuals, businesses, and the government in a creative economic structure."},"author":{"kind":"string","value":"Ratna W.D.P|Rizal Noviansyah|Riza B.S|Fauziyah|Dimyati Muhammad"},"content_list":{"kind":"list like","value":[{"bbox":"[129,156,919,200]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Empowering community through creative economy as a disaster risk reduction strategy in Indonesia ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[127,219,705,235]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ratna W.D.P $^{1*}$ , Noviansyah Rizal $^{1}$ , Riza B.S $^{1}$ , Fauziyah $^{2}$ , and Muhammad Dimyati $^{3}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[127,244,882,282]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"$^{1}$ Institut Teknologi dan Bisnis Widya Gama, Accounting Department, Lumajang East Java Province, Indonesia \",\"2 University of Science and Technology Beijing, Department of Computer and Communication Engineering, Beijing, China \",\"<sup>3</sup> Institute of Economic Science Mandala, Accounting Department, Jember East Java Province, Indonesia \"]","page_idx":"0","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[206,307,826,504]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract. In the 21st century, the world is faced with COVID-19 global pandemic that has had wideranging health, social, food, and economic negative impacts throughout the world and also in Indonesia due to the cessation of the mobility of most people and economic engines. The creative empowers the community based on innovation and creativity proven and is one of the pillars of disaster risk reduction recovering the social and economic impact of COVID-19 in the community. The method used is an ethnographic approach starting from the collection of literature sources and supported by field data. The results show that the concept of the creative economy is creative ideas developed by humans supported by the use of technology to produce innovative products and cultures, having regional imagery but are globally competitive and can ultimately provide value to the national economy. Intellectual actors included are not only academics but also include industry players, pioneers, figures in the arts, culture, and sciences who have a role in providing creative ideas, innovations in the creative industry. The result is a triplex helix model in mitigating the effect of the COVID19 disaster by connecting intellectuals, businesses, and the government in a creative economic structure. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[90,539,230,557]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[87,565,482,715]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Indonesian government has set targets to support an inclusive and sustainable national economy, a full and productive workforce, and decent work for all. The government has designed and implemented policies that support sustainable tourism that enable the creation of jobs while supporting local culture and products by 2030 [1]. However, the spread of COVID-19 has a disastrous impact on local and national economies around the world, many industries and policymakers are researching the impact of COVID-19 on the creative and cultural industries, as well as workers in the sector [2]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[87,717,482,840]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The global recession caused by COVID19 reduce consumption and investment in various countries. In Indonesia in the second quarter of 2020, Indonesia's economic growth was depressed to a negative $5.32\\%$ (yoy) compared to the same period last year. This contraction is more than in the previous quarter of $2.97\\%$ . Only China started showing positive growth to $3.2\\%$ in the second quarter of 2020. Whereas in the first quarter of 2020, economic growth was negative up to $6.8\\%$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[87,840,482,896]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The development of a local culture-based creative economy ecosystem, supported by competent business actors, requires support and cooperation from several parties including the government, the private sector, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,539,907,583]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"community, academia, as well as the media. In addition, the development of the sector can also be carried out with training to improve the welfare of business actors ([3]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[515,590,895,745]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 1. 2020's Global Economic Growth (YoY) \"]","image_footnote":"[]","img_path":"images/0118/619d9fb46e34895993c2b1d162b678b660477a4f2c851a5c41e0896b6e14686e.jpg","list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[510,781,907,892]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The creative economy has been widely developed in major countries such as the US, UK [4], all countries in Australia [5], including in Indonesia [6]. Optimizing the creative economy in Indonesia has become the focus of researchers in recent years [7]. There are challenges concerning creativity and culture that are widely recognized in the academic world. However, it is often invisible to policymakers ([2]. The most populous ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[38,14,312,27]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E3S Web of Conferences 331, 04015 (2021) ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[38,28,127,40]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ICDMM 2021 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[665,14,959,27]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.1051/e3sconf/202133104015 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[127,917,462,933]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"* Corresponding Author: pradnyataj@gmail.com ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[58,969,944,994]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[92,82,478,151]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"population in Indonesia is on the island of Java, along with its potential for cultural diversity and creative economic activities. Local wisdom and cultural events reflect the creativity of the community and can provide economic value for the community [8]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,152,480,329]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The creative economy has started to be discussed in Indonesia and has received proper facilities since 2006. The creative economy has contributed significantly to the economic development in Indonesia. Following the growing business world, the creative economy will also continue to develop along with the increasing creativity of business actors. In the future, it is expected that the creative industry can grow bigger and wider. The government provides various support and assistance for the development of the creative economy and development of this business. The government also holds various important events as a means of exhibiting products created by the community's creativity. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,330,480,451]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Based on the results of research between the Creative Economy Agency (Bekraf) and the Central Statistics Agency (BPS) in 2016, the creative economy contributed to the national economic growth of 922.59 billion rupiahs or 7.44 percent of the national Gross Domestic Product (GDP). Numerous empirical evidence supports that the creative industry has an impact on GDP through the creative industry sub-sector and creates a new form of governance of the cultural industry. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,453,480,737]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There are six main characteristics of the creative economy as quoted from Creative Economy Based on Local Wisdom in 2020. First is the existence of creative economy intellectual creations, emphasizing the importance of creativity. Therefore, creativity and expertise in a sector are a necessity and become one of the main characteristics of the creative economy. Second is the nature of easy to replace. Creations and innovations must continue to be developed under the development of the economic activity. The goal is to be accepted by the market and useful for the consumers. The third is the direct and indirect distribution based on company policies and consumer needs. Fourth, is the requirement of cooperation between all parties, such as between the entrepreneur and the government that regulates the policy. Fifth is based on ideas, that ideas are vital in developing creative industries and will always be related to innovation and creativity. The sixth is the nature of no limitation in product creation which means that innovation and creativity in product creation will always occur and has no definite limits. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,739,478,834]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This research focuses more on innovation and creativity to be developed, which is necessary for the growth of the creative industry in Indonesia. The result shows that the Indonesian economy can compete with other countries and will automatically become an indication that the Indonesian creative economy is one step ahead of its competitors ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,854,183,871]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 Method ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[92,879,478,922]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This study uses an ethnographic approach. The ethnographic method [9] started with literature sources and was complemented by field data obtained through ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,82,904,259]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"observation, participation, and in-depth interviews with selected respondents. Literature sources are obtained from researches that have been carried out starting in 2017 on culture in Banyuwangi. Informants are determined based on their involvement and role in the development of the creative industry, MSME actors, the community, and local government. The validation of the data is performed using triangulation of sources. Data analysis is carried out thoroughly and continuously starting from the process of data inventory, identification, classification, and data interpretation. Interpretation of data is carried out comprehensively by looking at the overall relationship of the data. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,261,904,384]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There are two approaches used in community development, the direct and indirect approaches [10]. The direct approach is carried out by implementing community empowerment, while the indirect approach is carried out with a process of public awareness of the importance of developing creative industries. The model developed is a participatory cycle model, which includes processes that are carried out simultaneously starting from input, process, output, impact, and result. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,403,749,419]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 Result and Discussion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,430,904,757]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The creative industries in Banyuwangi include several creative industries spread across various regions and contribute significantly to the regional economy. One of the creative industries that are developing is the bamboo handicrafts - crafts that use natural products to be used as household products, household accessories, fashion, musical instruments, and others. The creative industry of bamboo handicrafts has shown an increasing trend, attracting the market and opening up wide business opportunities. In certain areas, such as Papring has its prosperous resources of bamboo raw materials. The bamboo craft equipped with ideas and creativity will be able to produce products that are competitive in the global market. Communities in the Papring have started their activities to develop their potential by utilizing the natural potential in their environment by producing handicrafts from bamboo. Papring area has abundant various types of bamboo such as their unique origin Panggonane Pring, which is the source of bamboo plants. The types of bamboo in this area include stone bamboo, apus, petung, nineg, and various other types. The bamboo crafts industry is self-sufficient in terms of raw materials since it did not require supplies from outside of the region for handicraft production. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,758,904,879]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Triple Helix concept integrates the roles of academia, business, and government into creativity, innovation, and technology activities for the creative industry. The creativity and innovation have a close relationship that supports innovation performance. Carayannis and Campbell (2009) stated the importance of government, university, and industry policies and practices to interact intelligently, effectively, and efficiently. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,881,902,922]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The role of intellectuals is to spread and implement science, art, and technology and to form constructive values for the development of creative industries in ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[38,14,312,27]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E3S Web of Conferences 331, 04015 (2021) ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[40,29,127,40]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ICDMM 2021 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[665,14,959,27]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.1051/e3sconf/202133104015 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[492,975,504,986]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[92,84,480,355]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"society. Intellectuals are related to new creating activities that have the bargaining power to the market and the formation of creative people. Academics as part of intellectuals have 3 roles, namely, the role of education aimed at encouraging the birth of future creative generations that support the creation, innovation, and work in the creative industry. The role of research is carried out to provide input on creative industry development policy models and their required instruments, manage the renewal of ideas and creative processes that will produce new products, services, and technology that supports efficient working and use of resources and makes a competitive national creative industry. The role of community service is carried out in forming a society with social institutions and social orders that support the growth of the national creative industry. In realizing the importance of innovation performance in the creative industry, it is necessary to measure the performance of innovation that is appropriate for business actors in the creative industry. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,357,480,425]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Intellectual actors are not only academics but also include culturalists, artists, writers, pioneers, figures in the arts, culture, and sciences whom all have their roles in providing creative ideas, innovations in the creative industry. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,426,480,615]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The business becomes a liaison in the context of economic exchange and the transformation of creativity into economic value. The business function as actors, investors, and creators of new technology, as well as consumers. The creative industry serves as the center of excellence of creators of products, services, and creative culture, as a new market that absorbs products produced and created new jobs for creative individuals or other supporting individuals. The business also plays a role as a community and creative entrepreneur builder which function as a motor that forms a space for thinking, honing creative ideas in doing creative industry business, as well as business coaching or business management training in the creative industry. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,615,480,929]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Government has a major role in the mechanism for providing incentive programs, controlling a conducive business climate, educative directions as well as supporting the development of creative industries for the community and the private sector. The government also acts as a catalyst and facilitator and advocate that provides stimulation, challenges, and encouragement in the development of creative industries so that innovative ideas move to a higher level of competence. This support can be in the form of the government's commitment to using its political power proportionally and by providing good public administration services in addition to financial assistance, incentives, or protection. The government also acts as a regulator that produces policies related to society, industry, institutions, intermediation, and resources and technology. The government can accelerate the development of the creative industry if they were able to make policies that create a conducive business climate for the creative industry. The government as an investor must be able to empower the state's assets to be productive within the scope of the creative industry and to be responsible for industrial infrastructure investment. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,84,904,218]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the creative industry, the government makes a model based on creative individuals with five main pillars, namely: (1) industries involved in the production of creative industries; (2) technology as a supporter of the realization of individual creativity; (3) resources required such as natural resources and land; (4) institutions ranging from norms and values in society, industry associations and supporting communities to the protection of intellectual property; and (5) financial intermediary institutions. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,241,643,255]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 Conclusion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,267,904,388]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Indonesian sustainable tourism policy goals of 2030 which support that enable the creation of jobs while supporting local culture and products have their challenges with the spread of COVID-19. Based on the Triple Helix concept for the creative industry, the roles of academia, business, and government in creativity, innovation, and technology activities to interact intelligently, effectively, and efficiently as required for such goal. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,390,904,661]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"With its nature as being the largest archipelagic country in the world, Indonesia has the potential for a creative economy in terms of unique resources, language, and culture with the support of intellectuals such as academics, culturalists, artists, writers, pioneers, figures in the arts, culture, and sciences whom all have their roles in providing creative ideas, innovations in the creative industry. Academics intellectuals serve their roles in education, research, and community service. The business becomes a liaison in the context of economic exchange and the transformation of creativity into economic value, while the Government has its major role as major support of financial assistance, regulator, business climate, educative directions, catalyst, facilitator, and advocate of creative industries. The development of a local culture-based creative economy ecosystem, supported by competent business actors, requires support and cooperation from several parties including the government, the private sector, community, academia, as well as the media. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,684,626,697]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[515,709,907,926]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. R. B. Sulistyan, & R. W. Paramita., D. Business Location Planning Assistance: Preservation of Traditional Culture of Kampoeng Batara Banyuwangi. Empowerment Society, 4 (1), 17-21. (2021) \",\"2. R. Comunian, & L. England., Creative and cultural work without filters: Covid-19 and exposed precarity in the creative economy. Cultural Trends, 29 (2), 112-128. (2020) \",\"3. I. Hilmiana, Soemaryani, & A. R. Prasetyo., Strategic Partnership Model In Developing The Local Art And Culture-Based Creative Economy. Academy of Strategic Management Journal, 20 (1), 1-9(2021) \",\"4. T. Kemeny, M. Nathan, & D. O'Brien, Creative differences? Measuring creative economy \"]","page_idx":"2","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[40,15,312,27]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E3S Web of Conferences 331, 04015 (2021) ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[40,29,127,40]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ICDMM 2021 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[667,16,957,27]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.1051/e3sconf/202133104015 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[494,975,502,985]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[92,83,482,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"employment in the United States and the UK. Regional Studies, 54 (3), 377-387. (2020) \",\"5. S. Cunningham, M. McCutcheon, G. Hearn, & M. D. Ryan., 'Demand' for culture and 'allied' industries: policy insights from multi-site creative economy research. International Journal of Cultural Policy, 1-14. (2020) \",\"6. Y. Agustina, A. Winarno, H. Pratikto, B. S. Narmaditya, & F. Filianti. A Creative Economy Development Strategy: The Case of Trenggalek Creative Network for Trenggalek Regency, Indonesia. The Journal of Asian Finance, Economics and Business, 7 (12), 1111-1122.(2020) \",\"7. S. Madelan, Optimizing Exports of Indonesian Creative Economy Products Towards Increasing Competitiveness. Journal of Business Economics, \"]","page_idx":"3","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[515,83,904,299]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"Communication, and Social Sciences (BECOSS), 2(3), 273-284. (2020) \",\"8. N. Nurdiani, R. Hendarti, & M. Tedja. Physical Quality of Creative Economic Space on Cultural Tourism Areas in Java Island. IOP Conference Series: Earth and Environmental Science, 452, 012057. (2020) \",\"9. Spradley, Metode Etnografi. Yogyakarta: PT Tiara J. P.Wacana. (1997) \",\"10. N. Anoegrajekti, Asrumi, Macaryus, S., Iskandar, I., Attas, S. G., Sunarti, S., & Saddhono, K.: Optimization Pillars of Potential Culture and Creative Industry in Banyuwangi, East Java, Indonesia. Psychology and Education, 58 (3), 2025-2032 (2021) \"]","page_idx":"3","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[38,14,314,27]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"E3S Web of Conferences 331, 04015 (2021) ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[38,28,127,40]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ICDMM 2021 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[665,14,959,28]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.1051/e3sconf/202133104015 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[492,975,504,985]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 ","text_format":null,"text_level":null,"type":"page_number"}],"string":"[\n {\n \"bbox\": \"[129,156,919,200]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Empowering community through creative economy as a disaster risk reduction strategy in Indonesia \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[127,219,705,235]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ratna W.D.P $^{1*}$ , Noviansyah Rizal $^{1}$ , Riza B.S $^{1}$ , Fauziyah $^{2}$ , and Muhammad Dimyati $^{3}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[127,244,882,282]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"$^{1}$ Institut Teknologi dan Bisnis Widya Gama, Accounting Department, Lumajang East Java Province, Indonesia \\\",\\\"2 University of Science and Technology Beijing, Department of Computer and Communication Engineering, Beijing, China \\\",\\\"<sup>3</sup> Institute of Economic Science Mandala, Accounting Department, Jember East Java Province, Indonesia \\\"]\",\n \"page_idx\": \"0\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[206,307,826,504]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract. In the 21st century, the world is faced with COVID-19 global pandemic that has had wideranging health, social, food, and economic negative impacts throughout the world and also in Indonesia due to the cessation of the mobility of most people and economic engines. The creative empowers the community based on innovation and creativity proven and is one of the pillars of disaster risk reduction recovering the social and economic impact of COVID-19 in the community. The method used is an ethnographic approach starting from the collection of literature sources and supported by field data. The results show that the concept of the creative economy is creative ideas developed by humans supported by the use of technology to produce innovative products and cultures, having regional imagery but are globally competitive and can ultimately provide value to the national economy. Intellectual actors included are not only academics but also include industry players, pioneers, figures in the arts, culture, and sciences who have a role in providing creative ideas, innovations in the creative industry. The result is a triplex helix model in mitigating the effect of the COVID19 disaster by connecting intellectuals, businesses, and the government in a creative economic structure. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[90,539,230,557]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[87,565,482,715]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Indonesian government has set targets to support an inclusive and sustainable national economy, a full and productive workforce, and decent work for all. The government has designed and implemented policies that support sustainable tourism that enable the creation of jobs while supporting local culture and products by 2030 [1]. However, the spread of COVID-19 has a disastrous impact on local and national economies around the world, many industries and policymakers are researching the impact of COVID-19 on the creative and cultural industries, as well as workers in the sector [2]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[87,717,482,840]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The global recession caused by COVID19 reduce consumption and investment in various countries. In Indonesia in the second quarter of 2020, Indonesia's economic growth was depressed to a negative $5.32\\\\%$ (yoy) compared to the same period last year. This contraction is more than in the previous quarter of $2.97\\\\%$ . Only China started showing positive growth to $3.2\\\\%$ in the second quarter of 2020. Whereas in the first quarter of 2020, economic growth was negative up to $6.8\\\\%$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[87,840,482,896]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The development of a local culture-based creative economy ecosystem, supported by competent business actors, requires support and cooperation from several parties including the government, the private sector, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,539,907,583]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"community, academia, as well as the media. In addition, the development of the sector can also be carried out with training to improve the welfare of business actors ([3]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[515,590,895,745]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 1. 2020's Global Economic Growth (YoY) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0118/619d9fb46e34895993c2b1d162b678b660477a4f2c851a5c41e0896b6e14686e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[510,781,907,892]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The creative economy has been widely developed in major countries such as the US, UK [4], all countries in Australia [5], including in Indonesia [6]. Optimizing the creative economy in Indonesia has become the focus of researchers in recent years [7]. There are challenges concerning creativity and culture that are widely recognized in the academic world. However, it is often invisible to policymakers ([2]. The most populous \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[38,14,312,27]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E3S Web of Conferences 331, 04015 (2021) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[38,28,127,40]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ICDMM 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[665,14,959,27]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.1051/e3sconf/202133104015 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[127,917,462,933]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"* Corresponding Author: pradnyataj@gmail.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[58,969,944,994]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[92,82,478,151]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"population in Indonesia is on the island of Java, along with its potential for cultural diversity and creative economic activities. Local wisdom and cultural events reflect the creativity of the community and can provide economic value for the community [8]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,152,480,329]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The creative economy has started to be discussed in Indonesia and has received proper facilities since 2006. The creative economy has contributed significantly to the economic development in Indonesia. Following the growing business world, the creative economy will also continue to develop along with the increasing creativity of business actors. In the future, it is expected that the creative industry can grow bigger and wider. The government provides various support and assistance for the development of the creative economy and development of this business. The government also holds various important events as a means of exhibiting products created by the community's creativity. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,330,480,451]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Based on the results of research between the Creative Economy Agency (Bekraf) and the Central Statistics Agency (BPS) in 2016, the creative economy contributed to the national economic growth of 922.59 billion rupiahs or 7.44 percent of the national Gross Domestic Product (GDP). Numerous empirical evidence supports that the creative industry has an impact on GDP through the creative industry sub-sector and creates a new form of governance of the cultural industry. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,453,480,737]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There are six main characteristics of the creative economy as quoted from Creative Economy Based on Local Wisdom in 2020. First is the existence of creative economy intellectual creations, emphasizing the importance of creativity. Therefore, creativity and expertise in a sector are a necessity and become one of the main characteristics of the creative economy. Second is the nature of easy to replace. Creations and innovations must continue to be developed under the development of the economic activity. The goal is to be accepted by the market and useful for the consumers. The third is the direct and indirect distribution based on company policies and consumer needs. Fourth, is the requirement of cooperation between all parties, such as between the entrepreneur and the government that regulates the policy. Fifth is based on ideas, that ideas are vital in developing creative industries and will always be related to innovation and creativity. The sixth is the nature of no limitation in product creation which means that innovation and creativity in product creation will always occur and has no definite limits. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,739,478,834]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This research focuses more on innovation and creativity to be developed, which is necessary for the growth of the creative industry in Indonesia. The result shows that the Indonesian economy can compete with other countries and will automatically become an indication that the Indonesian creative economy is one step ahead of its competitors \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,854,183,871]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 Method \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,879,478,922]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This study uses an ethnographic approach. The ethnographic method [9] started with literature sources and was complemented by field data obtained through \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,82,904,259]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"observation, participation, and in-depth interviews with selected respondents. Literature sources are obtained from researches that have been carried out starting in 2017 on culture in Banyuwangi. Informants are determined based on their involvement and role in the development of the creative industry, MSME actors, the community, and local government. The validation of the data is performed using triangulation of sources. Data analysis is carried out thoroughly and continuously starting from the process of data inventory, identification, classification, and data interpretation. Interpretation of data is carried out comprehensively by looking at the overall relationship of the data. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,261,904,384]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There are two approaches used in community development, the direct and indirect approaches [10]. The direct approach is carried out by implementing community empowerment, while the indirect approach is carried out with a process of public awareness of the importance of developing creative industries. The model developed is a participatory cycle model, which includes processes that are carried out simultaneously starting from input, process, output, impact, and result. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,403,749,419]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 Result and Discussion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,430,904,757]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The creative industries in Banyuwangi include several creative industries spread across various regions and contribute significantly to the regional economy. One of the creative industries that are developing is the bamboo handicrafts - crafts that use natural products to be used as household products, household accessories, fashion, musical instruments, and others. The creative industry of bamboo handicrafts has shown an increasing trend, attracting the market and opening up wide business opportunities. In certain areas, such as Papring has its prosperous resources of bamboo raw materials. The bamboo craft equipped with ideas and creativity will be able to produce products that are competitive in the global market. Communities in the Papring have started their activities to develop their potential by utilizing the natural potential in their environment by producing handicrafts from bamboo. Papring area has abundant various types of bamboo such as their unique origin Panggonane Pring, which is the source of bamboo plants. The types of bamboo in this area include stone bamboo, apus, petung, nineg, and various other types. The bamboo crafts industry is self-sufficient in terms of raw materials since it did not require supplies from outside of the region for handicraft production. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,758,904,879]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Triple Helix concept integrates the roles of academia, business, and government into creativity, innovation, and technology activities for the creative industry. The creativity and innovation have a close relationship that supports innovation performance. Carayannis and Campbell (2009) stated the importance of government, university, and industry policies and practices to interact intelligently, effectively, and efficiently. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,881,902,922]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The role of intellectuals is to spread and implement science, art, and technology and to form constructive values for the development of creative industries in \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[38,14,312,27]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E3S Web of Conferences 331, 04015 (2021) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[40,29,127,40]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ICDMM 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[665,14,959,27]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.1051/e3sconf/202133104015 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[492,975,504,986]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[92,84,480,355]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"society. Intellectuals are related to new creating activities that have the bargaining power to the market and the formation of creative people. Academics as part of intellectuals have 3 roles, namely, the role of education aimed at encouraging the birth of future creative generations that support the creation, innovation, and work in the creative industry. The role of research is carried out to provide input on creative industry development policy models and their required instruments, manage the renewal of ideas and creative processes that will produce new products, services, and technology that supports efficient working and use of resources and makes a competitive national creative industry. The role of community service is carried out in forming a society with social institutions and social orders that support the growth of the national creative industry. In realizing the importance of innovation performance in the creative industry, it is necessary to measure the performance of innovation that is appropriate for business actors in the creative industry. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,357,480,425]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Intellectual actors are not only academics but also include culturalists, artists, writers, pioneers, figures in the arts, culture, and sciences whom all have their roles in providing creative ideas, innovations in the creative industry. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,426,480,615]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The business becomes a liaison in the context of economic exchange and the transformation of creativity into economic value. The business function as actors, investors, and creators of new technology, as well as consumers. The creative industry serves as the center of excellence of creators of products, services, and creative culture, as a new market that absorbs products produced and created new jobs for creative individuals or other supporting individuals. The business also plays a role as a community and creative entrepreneur builder which function as a motor that forms a space for thinking, honing creative ideas in doing creative industry business, as well as business coaching or business management training in the creative industry. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,615,480,929]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Government has a major role in the mechanism for providing incentive programs, controlling a conducive business climate, educative directions as well as supporting the development of creative industries for the community and the private sector. The government also acts as a catalyst and facilitator and advocate that provides stimulation, challenges, and encouragement in the development of creative industries so that innovative ideas move to a higher level of competence. This support can be in the form of the government's commitment to using its political power proportionally and by providing good public administration services in addition to financial assistance, incentives, or protection. The government also acts as a regulator that produces policies related to society, industry, institutions, intermediation, and resources and technology. The government can accelerate the development of the creative industry if they were able to make policies that create a conducive business climate for the creative industry. The government as an investor must be able to empower the state's assets to be productive within the scope of the creative industry and to be responsible for industrial infrastructure investment. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,84,904,218]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the creative industry, the government makes a model based on creative individuals with five main pillars, namely: (1) industries involved in the production of creative industries; (2) technology as a supporter of the realization of individual creativity; (3) resources required such as natural resources and land; (4) institutions ranging from norms and values in society, industry associations and supporting communities to the protection of intellectual property; and (5) financial intermediary institutions. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,241,643,255]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 Conclusion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,267,904,388]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Indonesian sustainable tourism policy goals of 2030 which support that enable the creation of jobs while supporting local culture and products have their challenges with the spread of COVID-19. Based on the Triple Helix concept for the creative industry, the roles of academia, business, and government in creativity, innovation, and technology activities to interact intelligently, effectively, and efficiently as required for such goal. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,390,904,661]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"With its nature as being the largest archipelagic country in the world, Indonesia has the potential for a creative economy in terms of unique resources, language, and culture with the support of intellectuals such as academics, culturalists, artists, writers, pioneers, figures in the arts, culture, and sciences whom all have their roles in providing creative ideas, innovations in the creative industry. Academics intellectuals serve their roles in education, research, and community service. The business becomes a liaison in the context of economic exchange and the transformation of creativity into economic value, while the Government has its major role as major support of financial assistance, regulator, business climate, educative directions, catalyst, facilitator, and advocate of creative industries. The development of a local culture-based creative economy ecosystem, supported by competent business actors, requires support and cooperation from several parties including the government, the private sector, community, academia, as well as the media. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,684,626,697]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[515,709,907,926]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. R. B. Sulistyan, & R. W. Paramita., D. Business Location Planning Assistance: Preservation of Traditional Culture of Kampoeng Batara Banyuwangi. Empowerment Society, 4 (1), 17-21. (2021) \\\",\\\"2. R. Comunian, & L. England., Creative and cultural work without filters: Covid-19 and exposed precarity in the creative economy. Cultural Trends, 29 (2), 112-128. (2020) \\\",\\\"3. I. Hilmiana, Soemaryani, & A. R. Prasetyo., Strategic Partnership Model In Developing The Local Art And Culture-Based Creative Economy. Academy of Strategic Management Journal, 20 (1), 1-9(2021) \\\",\\\"4. T. Kemeny, M. Nathan, & D. O'Brien, Creative differences? Measuring creative economy \\\"]\",\n \"page_idx\": \"2\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[40,15,312,27]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E3S Web of Conferences 331, 04015 (2021) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[40,29,127,40]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ICDMM 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[667,16,957,27]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.1051/e3sconf/202133104015 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[494,975,502,985]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[92,83,482,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"employment in the United States and the UK. Regional Studies, 54 (3), 377-387. (2020) \\\",\\\"5. S. Cunningham, M. McCutcheon, G. Hearn, & M. D. Ryan., 'Demand' for culture and 'allied' industries: policy insights from multi-site creative economy research. International Journal of Cultural Policy, 1-14. (2020) \\\",\\\"6. Y. Agustina, A. Winarno, H. Pratikto, B. S. Narmaditya, & F. Filianti. A Creative Economy Development Strategy: The Case of Trenggalek Creative Network for Trenggalek Regency, Indonesia. The Journal of Asian Finance, Economics and Business, 7 (12), 1111-1122.(2020) \\\",\\\"7. S. Madelan, Optimizing Exports of Indonesian Creative Economy Products Towards Increasing Competitiveness. Journal of Business Economics, \\\"]\",\n \"page_idx\": \"3\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[515,83,904,299]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"Communication, and Social Sciences (BECOSS), 2(3), 273-284. (2020) \\\",\\\"8. N. Nurdiani, R. Hendarti, & M. Tedja. Physical Quality of Creative Economic Space on Cultural Tourism Areas in Java Island. IOP Conference Series: Earth and Environmental Science, 452, 012057. (2020) \\\",\\\"9. Spradley, Metode Etnografi. Yogyakarta: PT Tiara J. P.Wacana. (1997) \\\",\\\"10. N. Anoegrajekti, Asrumi, Macaryus, S., Iskandar, I., Attas, S. G., Sunarti, S., & Saddhono, K.: Optimization Pillars of Potential Culture and Creative Industry in Banyuwangi, East Java, Indonesia. Psychology and Education, 58 (3), 2025-2032 (2021) \\\"]\",\n \"page_idx\": \"3\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[38,14,314,27]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E3S Web of Conferences 331, 04015 (2021) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[38,28,127,40]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ICDMM 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[665,14,959,28]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.1051/e3sconf/202133104015 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[492,975,504,985]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.1051/e3sconf/202133104015"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Medicine and Health Sciences"},"sha256":{"kind":"string","value":"20180ba16227f0984fd2f416b303cb0c92521f496fad34dfbd9e42d2c7bcac4c"},"title":{"kind":"string","value":"Empowering community through creative economy as a disaster risk reduction strategy in Indonesia"}}},{"rowIdx":57,"cells":{"abstract":{"kind":"string","value":"\n \nThe exoplanet detection is the most exciting and challenging field of astronomy. The discovery of many exoplanets has revolutionized our understanding of the formation and evolution of planetary systems and has showed new ways to search for extra terrestrial life. In recent years, some primary methods of exoplanet detection like transit, radial velocity, gravitational microlensing, direct imaging and astrometry have played a important role for the discovery of exoplanets. In this paper we explored detection methodologies with all the implications and analytics of comparison between them. Here we also discussed on different machine learning algorithms for exoplanet detection and visualization. Finally, concluded with the significant discoveries made by some missions and their implications on our understanding for the properties, environmental conditions and importance of exoplanets in the universe.\n "},"author":{"kind":"string","value":"Aditee Mattoo|Mahima Kaushik|Ritesh Rastogi"},"content_list":{"kind":"list like","value":[{"bbox":"[80,146,561,167]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Exoplanet Detection : A Detailed Analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[152,193,657,211]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Mahima Kaushik $^{1,*}$ , Aditee Mattoo $^{1}$ and Dr. Ritesh Rastogi $^{2}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[152,225,915,281]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"$^{1}$ Department of M.Tech Integrated (CSE), Noida Institute of Engineering and Technology, Greater Noida, 201310, U.P, India \",\"$^{2}$ Department of CS, Noida Institute of Engineering and Technology, Greater Noida, 201310, U.P, India. \",\"* Corresponding author. E-mail: mahimakaushik333@gmail.com \"]","page_idx":"0","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[152,291,915,422]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract. The exoplanet detection is the most exciting and challenging field of astronomy. The discovery of many exoplanets has revolutionized our understanding of the formation and evolution of planetary systems and has showed new ways to search for extra terrestrial life. In recent years, some primary methods of exoplanet detection like transit, radial velocity, gravitational microlensing, direct imaging and astrometry have played a important role for the discovery of exoplanets. In this paper we explored detection methodologies with all the implications and analytics of comparison between them. Here we also discussed on different machine learning algorithms for exoplanet detection and visualization. Finally, concluded with the significant discoveries made by some missions and their implications on our understanding for the properties, environmental conditions and importance of exoplanets in the universe. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[152,436,667,450]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords. Exoplanet Detection—Transit—Missions—Exoplanets—Gaia. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,464,205,476]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,493,485,910]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Since the discovery of exoplanets began, they have continually surprised us with the hidden secrets of the cosmos. Here on earth, wealth is often associated with the possession of precious rocks and metals. But what about worlds made of these rocks and metals? For instance, 55 Cancri e is an exoplanet made of diamonds and orbits a sun-like star. It was discovered by McArthur McArthur et al. (2004) using the radial velocity method by observing the spectrum of one of the parent stars in a binary system located 40 light-years from the planet Earth. Exoplanets are poised to become a significant part of human civilization in various aspects, such as energy sources, colonization of other planets, transportation, space exploration, employment, and many others. Since the first exoplanet was discovered, astronomers have been working tirelessly to discover other extrasolar planets. Numerous missions have been conducted by several space agencies to study exoplanets, and the results have been truly fulfilling. Till date, approximately $5500+$ exoplanets have been discovered Akeson et al. (2013), with many of them undergoing detailed studies. Exoplanets come in various types such as earth-size, Neptune-size, super-earth, Jupiter-size, sub-earth, super-jupiters, brown dwarfs, and stellar companions. There are also various techniques for detecting exoplanets Wei (2018), including radial velocity, transit method, astrometry, direct imaging, and gravitational microlensing. Each method of ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,464,915,728]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"fers distinct perspectives, strengths, limitations, and other details, often represented in tabular and graphical forms. By analyzing the light that passes through the atmosphere of a planet, scientists can understand its composition. Some exoplanets have been found to have atmospheres containing elements such as carbon dioxide, methane, and even water vapor Perryman (2018). If we compare a piece of wood and a piece of diamond to see which one is more expensive, the correct answer is wood. This is because, so far, earth is the only place in the universe where wood is produced, while diamonds are present on different exoplanets like 55 Cancri e. Gliese $667\\mathrm{Cc}$ is a super-earth exoplanet that lies 22 light-years away from earth. It's a potentially habitable super-earth with a mass of 3.8 times that of earth. Humans are still in the race of hunting for those habitable exoplanets, on which we can grow more plants and expand human colonization beyond Earth. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,734,915,910]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"51 Pegasib, the first exoplanet, orbiting around a star similar to sun, was found using the radial velocity approach by Mayor & Queloz. Several space agencies are running missions specifically for exoplanet detection using different detection methods Shafi (2021). These missions include hubble, spitzer, corot, kepler/K2, gaia, TESS, CHEOPs, webb, and many more, which continuously monitor the universe, and astronomers keep detecting exoplanets with their help. In the future, there will be missions for specific exoplanets to understand their properties in detail and their environmental conditions. However, as of now, it's hard to ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[23,288,60,760]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"arXiv:2404.09143v2 [astro-ph.EP] 17 Apr 2024 ","text_format":null,"text_level":null,"type":"aside_text"},{"bbox":"[78,65,302,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Astrophys. Astr. (0000) 000: ###DOI ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[80,923,270,934]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\langle \\widehat{\\mathbb{C}}\\rangle$ Indian Academy of Sciences ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[907,923,914,933]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[78,101,919,423]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 1. Exoplanet detection methods, shown on a logarithmic mass scale, show current and upcoming measuring possibilities. Open arrows show previous evaluations, whereas solid arrows represent discoveries. Data collected from the NASA Exoplanet Archive (2018) Perryman (2018). \"]","image_footnote":"[]","img_path":"images/0425/e2687fb83dada3ae395e2bdfaa50673f1530edc9b20dc690a3b14c7bc41c7587.jpg","list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[75,487,490,665]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"travel to even nearby exoplanets within a lifetime. Further more, introduced space missions briefly, including details about data formats and other details. Astronomers use various machine learning Kumari et al. (2023) and data cleaning algorithms to work on the data that comes from these missions. Such algorithms include the BLS algorithm, light curve analyzer, CNN, and other algorithms. all exoplanets are formed by normal matter but some scientists assumes that some exoplanets could me formed through dark matter Bai et al. (2023). Introduction about exoplanets with there basic details are going to be discussed in the next section. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,690,226,708]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. The exoplanets ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[75,721,489,914]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Extrasolar planets, also referred to as exoplanets, are planets that circle stars outside of our solar system. From the first exoplanet to be discovered in the early 1990s to the current booming field with thousands of confirmed planets, the growth of exoplanet investigation has been incredible wikipedia (2022). Section 2 discusses the crucial roles that various detection methods have had in this process. Technological developments like adaptive optics, space-based observatories like kepler and TESS, and high-precision spectrographs have made these approaches far more useful Woodrum et al. (2023). The discovery of smaller and farther-off planets has been made feasible by these developments. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[504,487,917,546]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Exoplanet research is essential because it can increase our knowledge of planetary systems and the conditions necessary for habitability and the possibility that life exists elsewhere in the cosmos. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[505,561,721,575]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.1 Stars That Host Planets ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[504,583,917,776]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On average, each star Scalo et al. (2007) is believed to host at least one planet. It's estimated that around $20\\%$ of sun like stars have an \"earth-sized\" planet in their habitable zones. The majority of known exoplanets orbit stars similar to our Sun, specifically those in the main-sequence spectral categories F, G, or K Lee et al. (2012) Perryman (2018) Meunier et al. (2023). Planets detectable by the radial-velocity method are less common around lower mass stars, such as the red dwarfs in the M spectral group Ananyeva et al. (2023). Despite this, the transit method, which excels at detecting smaller planets, has enabled the kepler telescope to identify dozens of planets orbiting near red dwarfs. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[505,790,684,804]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2 Detection methods ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[504,813,917,903]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Various detection methods Rouan & Lagrange (2023) exist for exoplanets, and astronomers continuously work toward developing new ways of detection. Some effective methods include radial velocity, which led to the discovery of the 1st exoplanet in the late '90s; the transit method, which analyzes light curves and peri ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,78,205,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Page 2 of 10 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[668,78,917,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Astrophys. Astr. (0000) 000: ### ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[95,101,473,294]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 2. Diagram showing the habitable zones surrounding certain stars. The green area indicates where water on a planet's surface can stay liquid at the ideal distance from the star. The blue area is too cool, and the red area is too heated. \"]","image_footnote":"[]","img_path":"images/1003/3ea8042c021cfb4334c8c479070f7c2922288958c75471cfabb8f79c49080c26.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[75,379,489,483]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"odograms Bozza et al. (2016); direct imaging, capturing exoplanets through images; gravitational microlensing; and astrometry Perryman (2018). Table 1 shows the total number of exoplanets discovered by various methods of detection till beginning of 2024 NASA (2024). data taken from NASA exoplanet archive Akeson et al. (2013). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[95,536,473,718]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0448/136f5e40222c4287e7151d5dc5a33ec30cca5c8c119c8f5ba91ce971872a21e2.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":"<table><tr><td>Method of discovery</td><td>Number of planets</td></tr><tr><td>Astrometry</td><td>3</td></tr><tr><td>Disk Kinematics</td><td>1</td></tr><tr><td>Radial Velocity</td><td>1071</td></tr><tr><td>Eclipse timing variations</td><td>17</td></tr><tr><td>Orbital brightness modulations</td><td>9</td></tr><tr><td>Transit</td><td>4146</td></tr><tr><td>Transit timing variations</td><td>28</td></tr><tr><td>Pulsar timing variations</td><td>7</td></tr><tr><td>Microlensing</td><td>204</td></tr><tr><td>Imaging</td><td>69</td></tr><tr><td>Pulsion timing variations</td><td>2</td></tr></table>","table_caption":"[\"Table 1. Exoplanets discovered by various methods till date \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[78,747,260,764]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.3 Types of exoplanets ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[75,770,489,904]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As of the conclusion of 2023, the existence of over 5500 planets had been confirmed. In an attempt to better understand these celestial bodies, scientists have undertaken numerous initiatives to categorize them based on various attributes, including their size, mass, and composition Perryman (2018). Table 2 shows the classification of exoplanets based on mass, size and chemical compositions of known gases such as $ZnS$ , $H_2O$ , $CO_2$ & $CH_4$ Forestano et al. (2023). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[564,111,875,419]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 3. planet masses are categorized based on solar system objects, as suggested by Stevens & Gaudi (2013). The solar system's planet masses are displayed on the right, with the size of the circles corresponding to the planets' radii Perryman (2018) Mathur et al. (2021). \"]","image_footnote":"[]","img_path":"images/0342/10cb776c6a7b09b4e5e8bd992b70789aa06107c37af19ea48af10ee8dfda58c8.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[505,511,719,526]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.4 Atmospheric conditions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[504,533,917,741]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finding transiting exoplanets is a significant breakthrough that provides previously unattainable knowledge about the composition and physical properties of these far-off worlds Yakovlev et al. (2022). Transit and eclipse spectroscopy allows the investigation of atmospheric dynamics Walker et al. (2018) and heat transport processes, especially for hot Jupiters, while the resulting densities, derived from detailed measurements of masses and radii, offer important hints regarding internal structures and compositions. These discoveries have been made possible in large part by drawing on knowledge amassed over the previous 50 years from research on the planets and satellites of the solar system Jakka (2023). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[504,745,919,910]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Future developments in photometric and spectroscopic observations, such as those brought about by the 2020 launch of the James Webb Space Telescope (JWST), are very promising. Advanced diagnostics may reveal the precise atmospheric compositions of exoplanets that are located in the \"habitable zone Jagtap et al. (2021),\" which is where liquid water may be present. These compositions may include indications of either primitive or advanced life forms. These developments advance not only the hunt for extraterrestrial life but also help answer the important question of ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,78,322,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Astrophys. Astr. (0000)000: ### ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[791,78,917,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Page 3 of 10 ### ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[78,128,497,568]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0947/8778a248dc935b98261f4f35498f8b8db2deca4e5970dc83500fa11bcdb50300.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":"<table><tr><td>Factor</td><td>Planet type</td><td>Parameter</td></tr><tr><td rowspan=\"3\">Size</td><td>Earth size</td><td>(1.25 - 2R⊕)</td></tr><tr><td>Super Earth size</td><td>(2 - 6R⊕)</td></tr><tr><td>Jupiter size</td><td>(6 - 15R⊕)</td></tr><tr><td rowspan=\"8\">Mass</td><td>Earth</td><td>(0.1M⊕ - 2M⊕)</td></tr><tr><td>Sub Earth</td><td>(10-8M⊕ - 0.1M⊕)</td></tr><tr><td>Super Earth</td><td>(2M⊕ - 10M⊕)</td></tr><tr><td>Neptunes</td><td>(10M⊕ - 100M⊕)</td></tr><tr><td>Jupiters</td><td>(100M⊕ - 103M⊕)</td></tr><tr><td>Super Jupiters</td><td>(103M⊕ - 13MJ)</td></tr><tr><td>Brown Dwarfs</td><td>(13MJ - 0.07M⊕)</td></tr><tr><td>Stellar Companions</td><td>(0.07M⊕ - 1M⊕)</td></tr><tr><td rowspan=\"15\">Composition(stellar flux)</td><td>Hot super Earths</td><td>(187 - 1.12)</td></tr><tr><td>Warm Super Earths</td><td>(1.12 - 0.30)</td></tr><tr><td>Cold super Earth</td><td>(0.30 - 0.0030)</td></tr><tr><td>Hot Rocky</td><td>(182 - 1.0)</td></tr><tr><td>Warm Rocky</td><td>(1.0 - 0.28)</td></tr><tr><td>Cold Rocky</td><td>(0.28 - 0.0035)</td></tr><tr><td>Hot Sub Neptunes</td><td>(188 - 1.15)</td></tr><tr><td>Warm Sub Neptunes</td><td>(1.15 - 0.32)</td></tr><tr><td>Cold Sub Neptunes</td><td>(0.32 - 0.0.0030)</td></tr><tr><td>Hot Sub Jovians</td><td>(220 - 1.65)</td></tr><tr><td>Warm Sub Jovians</td><td>(1.65 - 0.45)</td></tr><tr><td>Cold Sub Jovians</td><td>(0.45 - 0.0030)</td></tr><tr><td>Hot Jovians</td><td>(220 - 1.65)</td></tr><tr><td>Warm Jovians</td><td>(1.65 - 0.40)</td></tr><tr><td>Cold Jovians</td><td>(0.40 - 0.0025)</td></tr></table>","table_caption":"[\"Table 2. Types of exoplanets based on various factors \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[75,590,490,668]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"how common or uncommon life is in the universe Seager & Deming (2010). The dynamic interplay between theoretical models and observational methods is set to propel future research and transform our knowledge of exoplanetary systems. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,694,337,711]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3. Exoplanet detection methods ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[75,724,489,800]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As discussed previously there are various detection methods which played an important role to present this unique field of study. all methods works on various different properties such as light, noise Smith et al. (2008), EMwaves etc. some of them are explained here. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,812,290,828]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.1 Radial Velocity Method ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[75,835,490,910]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The radial velocity technique for identifying exoplanets involves tracking the reflexive movement of a star, which is triggered by the gravitational force of a planet in orbit. This method assesses three visible characteristics of the star that are altered by the planet's ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[505,104,586,117]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"existence: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[504,133,917,311]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Radial Velocity: When a planet orbits a star, both celestial bodies revolve around a shared center of mass, termed the barycenter. The star's motion around this point results in a periodic fluctuation in its radial velocity Hatzes (2016), which signifies its speed moving towards or away from an observer located on Earth. It is discovered by spectroscopic studies that this radial velocity shift exists. The spectral lines of a star change towards the blue end of the spectrum as it travels towards the observer, and to the red end when it goes away (blueshift and redshift, respectively) Bozza et al. (2016) Chen (2023). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[504,325,917,475]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Angular (Astrometric) Position in the Sky: The star moves in a modest circular or elliptical path around the shared center of mass due to the gravitational attraction of an orbiting planet Pham et al. (2024) Bozza et al. (2016). The star's angular location in the sky changes periodically, indicating its motion. Astrometric measures entail monitoring the star's apparent location with accuracy over time. Measurements of radial velocity have been used more frequently than this approach, which is more difficult Perryman (2018) Chen (2023). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[504,488,917,608]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Time of Arrival of Periodic Reference Signal: Variations in the arrival times of periodic signals, such pulsar pulses or eclipse timings in binary star systems, can be caused by the gravitational interaction between a star and its circling planet Bozza et al. (2016). Timing variations in these signals of reference may indicate the existence of an unseen planet and offer additional evidence of its impact Fischer et al. (2015). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[504,621,919,771]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Radial velocity Dumusque et al. (2017) measurements revealed the first credible discovery of an exoplanet in 1995 Perryman (2018). Since then, a number of exoplanets have been found using this method because to technological and observational breakthroughs. Via radial velocity metrics, 1075 exoplanets had been found as of 02/05/2024; some of these planets have been observed to transit their host stars, allowing the star's light to periodically fade as the planet passes in front of it. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[505,784,658,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.2 Transit method ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[504,806,917,910]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Astronomy achieved a major milestone in 1999 with the ground-breaking observation of the first transit of an extrasolar planet Perryman (2018) Chen (2023). The 'hot Jupiter' system HD 209458 was observed photometrically Perryman (2018), which allowed for this observation. A gas giant planet known as a \"hot Jupiter\" has a close orbit around its host star, which ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,78,208,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Page 4 of 10 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[668,78,917,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Astrophys. Astr. (0000) 000: ### ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[80,104,485,369]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"increases the planet's temperature somewhat. other star systems found by radial velocity surveys came into focus after this finding Fischer et al. (2015). After a planet passes in front of its host star, causing a periodic dimming of the star's brightness, researchers kept an eye on these systems in the hopes of finding other transits Yang (2023). Identifying exoplanets became increasingly dependent on this methodology, particularly for those that were not initially found via the radial velocity method Bozza et al. (2016). Dedicated surveys were rapidly launched to perform \"blind hunts,\" initially from space then from ground-based observatories Perryman (2018). The goal of these surveys was to find new planets only by using the periodic transit fingerprints observed in star light curves. With this method, scientists were able to find a wide variety of exoplanets without any prior awareness of their presence Chen (2023). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,385,485,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the field of exoplanet research, transiting planets are especially significant Afanasev (2018). The light curves acquired during transits yield important details on planet sizes since the radii of the planets may be estimated due to the star's periodic dimming of light Perryman (2018) Fischer et al. (2015). Through techniques like radial velocity measurements, astronomers can establish the transiting planet's mass as well as its density, providing preliminary information about the planet's composition Bozza et al. (2016) Chen (2023). Around the end of the year of 2023, exoplanet research had undergone a major transformation. There were now close to 4153 transiting planets known, with the Kepler space observatory detecting nearly 2778 of them as reported by NASA. Kepler began operations in 2009 and has been a key player in the area of transiting exoplanet research, providing an abundance of data that has made significant effects Panahi et al. (2022) Chen (2023). A wide range of exoplanetary systems have been found as a result of the joint efforts of ground-based surveys Perryman (2018), such HAT and WASP, which revealed bright and easily detectable transiting systems, and Kepler's numerous discoveries, particularly those of close-orbiting multi-planet systems Afanasev (2018). These systems contribute to our understanding of the diversity of planetary systems outside of our solar system by displaying a wide range of physical attributes and circumstances. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,813,285,829]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.3 Direct imaging method ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,835,485,910]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"\"Imaging\" in the context of modern exoplanet research is the process of recognizing a planet as an individual source of light Perryman (2018) Chen (2023). This may happen as a result of the planet's own thermal emission or light reflected from the host star. It is impor ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,104,915,398]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"tant that this imaging idea is not the same as reaching the planetary surface's spatial resolution Bozza et al. (2016). Additionally, it is distinct from techniques that detect fluctuations in the total brightness Manfredi et al. (2020) of the planet and star system as a result of things like secondary eclipses, reflected light, or changes in the stellar spectrum Birkby (2018) throughout a transit. In 2005, the first imaging detections were reported, namely the identification of huge, youthful, self-luminous planets in broad orbits Bozza et al. (2016) Perryman (2018). The NASA exoplanet archive listed 44 of these discoveries as at the end of 2017 Perryman (2018). Prominent instances comprise planets discovered in well researched debris disk systems such as $\\beta$ Pic and fomalhaut, in along with multi-planet systems like HR 8799 (which includes four planets) and LkCa 15 (containing two planets) Perryman (2018). There are certain deep learning models are used to enhance the capabilities of detection such as deep PACO Flasseur et al. (2024) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,400,915,813]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Researchers have been able to determine the orbital velocity of planets in systems such as $\\beta$ Pic, HR 8799, and fomalhaut thanks to multi-epoch research. Over the past ten years, developments in imaging instrument technology Daglayan et al. (2023) Hebrard et al. (2018) have led to an emergence of cutting-edge devices specifically intended for exoplanet imaging Beuzit et al. (2006). VLT-SPHERE, Gemini-GPI Galicher et al. (2014), and Subaru-HiCIAO are a few examples Fischer et al. (2015). It is important to remember that these devices can only directly picture massive planets that are young, warm, and selfluminous at this time Perryman (2018). It is expected that with the next 30-40-meter telescopes, direct detection of point source pictures resulting from reflected starlight would become possible Bozza et al. (2016) Perryman (2018). On the other hand, picturing planets similar to earth is still a distant objective Chen (2023). It is crucial to take into account imaging Koechlin et al. (2005) at radio and X-ray wavelengths while studying star-planet systems. These other wavelength ranges may provide light on many facets of the interactions that take place between stars and their planets Flasseur et al. (2023). There is talk about the potential to expand imaging capabilities to X-ray and radio wavelengths, which implies a more thorough investigation of observational methods in exoplanet studies Rosa & do Nascimento Dias (2023) Daglayan et al. (2023). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,829,746,844]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.4 Gravitational Microlensing ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[510,851,915,910]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The general relativity-based observation method known as gravitational microlensing Perryman (2018). Using gravitational microlensing, scientists found 210 exoplanets by the end of 2023. This technique uses ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,78,322,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Astrophys. Astr. (0000)000: ### ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[793,78,917,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Page 5 of 10 ### ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[82,103,356,244]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0282/4680ba2d4a53870d6cb16e757cd68ca136f9d7903e5308c73655ca2653cafb61.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[359,102,636,244]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0537/7fe40bc8d1c8f73e3ca483a61b7de153fc52207c0482cccbea4abf8ff46e4909.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[643,102,914,244]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0202/e0be68e3b8cd67ff4c112ec522c470c9486eb7a4e1880c1cd1512f8a02aada94.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[90,249,357,392]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0750/850e6f8358ecc668b17792713c2638163567f08ec53e944f5ff6609580c0a6cc.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[366,249,636,392]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0065/6cd43de8e5f93bcc668548d7b6515fdb684e5591416e9ce4e1077f9437812a56.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[643,250,914,392]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 4. Exoplanets detection methods. fig 4.1 represents Number of exoplanets detected by Radial velocity method in previous years, fig 4.2 represents Number of exoplanets detected by Transit method in previous years, fig 4.3 represents Number of exoplanets detected by Gravitational Microlensing method in previous years, fig 4.4 represents Number of exoplanets detected by Direct Imaging method in previous years, fig 4.5 represents Number of exoplanets detected by Timing method in previous years, fig 4.6 represents Number of exoplanets detected by Astrometry method in previous years. \"]","image_footnote":"[]","img_path":"images/0135/088d5ad0f07f87f81aa57179c0630216f236a871fcdd2afd36bb45f7ef999a13.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[82,481,492,625]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 5. Representing the periodogram of exoplanet KIC 6922244 \"]","image_footnote":"[]","img_path":"images/0871/c7f122e8409ca5be7f2bbca55c3fc48811ffd4365cdb0339826103bdde9b8e4b.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[80,701,487,908]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"the gravitational potential of a foreground object (the lens) to bend light beams from a distant background object (the source) Bozza et al. (2016) Perryman (2018). Depending on how the source, lens, and observer are aligned, the outcome is a distorted image of the source, sometimes in many images Chen (2023) Beaulieu (2023). The effects of gravitational lensing can be observed, leading to the classification into various domains Jones (2008). Robust lensing, observable at the level of individual objects, is further classified into macro- and microlensing. Whereas microlensing deals with discrete, unresolved multiple visuals, macrolensing generates multiple resolved images, or \"arcs.\" Weak lensing is a different regime that is sta ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,481,915,615]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"tistically noticed without a clear resolution of individual objects. Specific circumstances within gravitational lensing are described by the introduction of specialized words. Mesolensing is the term used to characterize the astrometric displacement that accompanied a high-proper motion foreground star, whereas nanolensing is the lensing caused by planetary-mass objects Gaudi (2012). Relative motion between the source, lens, and observer causes time-varying amplification of pictures ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,629,915,910]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Depending on the characteristics of the radiation source and the lens, this amplification takes place across an assortment of intervals, from hours to years. A more complex light curve may be seen from the background source if the foreground lensing object is gravitationally complex, such a star with planets or a cluster of galaxies Wright & Gaudi (2012). This complexity results from the changing alignment geometry's time-varying magnification. Significant discoveries have been made using gravitational microlensing Tamanini & Danielski (2019) in exoplanet exploration. In 2004, a 4MJ planet was first clearly detected at approximately 4 astronomical units (au) Perryman (2018). A $5\\mathrm{M} \\oplus$ planet was then found in 2006. An observation of a two-planet system in 2008 made it possible to quantify orbital motion during the lensing event. In 2015, the lens mass and microlens parallax were measured. In early 2018, evidence for planets flying freely was proposed. These results validate gravitational microlens ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,78,205,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Page 6 of 10 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[668,78,917,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Astrophys. Astr. (0000) 000: ### ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[80,104,485,148]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ing as an independent and potent probe of exoplanets, especially over an important mass and orbital radius region Bozza et al. (2016). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,162,196,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.5 Astrometry ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,185,485,407]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The exact measurement of the motions and positions of different celestial objects, from bodies of the solar system and stars within our galaxy to, theoretically, entire galaxies and clusters, is known as astrophysicist. Determining stellar parallaxes and proper movements has historically been a major goal of astrometry, as these are essential for comprehending the characteristics of host stars Perryman (2018). In the present development, astrometry plays a new role: it seeks to identify the crosswise element of the displacement of a host star due to an orbiting planet's gravity. Measures of radial velocity, which are sensitive to the concomitant movement of the photocenter along the line-of-sight, are closely associated with this dynamic manifestation in the sky Saffé et al. (2005). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,422,485,865]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It has been difficult to achieve high-accuracy astrometry; hippocarcos and the fine guidance sensors on the hubble space telescope, for example, achieved accuracies of about 1 milliarsecond (mas). Still, this degree of accuracy has only just touched the tip of the iceberg in terms of how much star position displacements caused by planets in orbit can be identified. By the end of 2023, astrometry had only found three exoplanets, two of which were 5MJ and 2.26MJ in mass, and one of which was a substellar partner with a mass 28 times that of jupiter (28MJ). These discoveries were recorded in the NASA exoplanet archive. Since the gaia Ranalli et al. (2018) mission was launched in 2013, the field of astrophysicists has seen a dramatic change Panahi et al. (2022). Over a billion stars with magnitudes up to $\\mathrm{V} = 20 - 21$ are being actively measured by gaia, with accuracies of about 20-25 microarcseconds (μas) for stars with a magnitude of 15 Perryman (2018). Our knowledge of planetary systems is predicted to be completely transformed by this astounding increase in precision Hodgkin et al. (2021) Panahi et al. (2022). Several thousand planetary systems with correct planet masses and absolute orbits free of the sin i ambiguity (the uncertainty surrounding the inclination angle of the planetary orbit) should be found thanks to gaia's large dataset Panahi et al. (2022) Perryman (2018). Moreover, gaia will provide insightful data on planetary systems' co-planarity, illuminating the configuration and behaviour of several planets within a system wikipedia (2022). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,104,870,118]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4. Comparative analysis of detection methods ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,133,915,607]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Count of detected exoplanets by specific method is the proof of its importance. Every method of extrasolar planet detection works on different principles and scientific concepts Perryman (2018) Chen (2023). While transit photometry and radial velocity methods are playing main role for the majority detection. after discovery there are various parameters and properties to know for detecting habitable planet, which is possible through other methods Meunier et al. (2012). hence every detection method have its own advantages and disadvantages Fischer et al. (2015). The transit photometry method covers wide range of stars but its false positive rate is high Tembhare et al. (2023). transit method works for star-planet systems, which have orbits aligned in such a way that, they can be visible from earth. those exoplanets should move in between host star and earth. Radial velocity method is highly prolific and works for many planets but they must be close to the spectator McArthur et al. (2004). Timing method works for limited materiality, but provide accurate data of exoplanets residing far from earth. Direct imaging also works for few stars but straight forward detection of exoplanets. Large orbital radii planets can be detected by gravitational microlensing, but here measurements alignments can't be repeated and questionable Hebrard et al. (2018). The majority of precise searches employ multiple methodologies to collect multifaceted information on a certain planet. In particular exoplanet HD 86266 c alternative name TOI-652 c detected by primary transit method in 2020 and gave radius parameter details. Table 3 shows the comparative analysis of detection methods with important parameters. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,636,766,651]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5. Space missions for exoplanets ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,665,915,858]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There are several missions and are working towards exoplanets and there discoveries. These missions are running by various space agencies like NASA, ESA, JAXA, SETI etc. some missions ended up working and some are currently doing good work. here is the details of some space missions for exoplanets. The landscape of exoplanet exploration has been significantly shaped by a series of pioneering space missions, each contributing unique insights into the vast diversity of planetary systems beyond our solar neighborhood Galicher et al. (2014). Among these missions, TESS (Transiting Exoplanet Survey Satellite) Kane et al. (2020) and GAIA Ranalli et al. (2018) stand as ongoing endeavors, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,863,915,907]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"tirelessly scanning the celestial expanse for transiting exoplanets and providing precise astrometric data for over a billion stars, respectively. There are various ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,78,322,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Astrophys. Astr. (0000)000: ### ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[793,78,917,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Page 7 of 10 ### ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[80,129,963,409]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0894/dd9aafb7c481d0e1c04c3aa31e406f5c40e0883ada14e2ed5a62ea0932cb580a.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":"<table><tr><td>Detection method</td><td>Principle</td><td>Measurements</td><td>Advantages</td><td>speciality</td><td>Drawbacks</td><td>Equipement</td></tr><tr><td>Radial velocity</td><td>Gravitational Wobble</td><td>mp,T,a</td><td>large amount of planets can be observe at once, precision</td><td>large planets, small cool stars</td><td>its for only planets closer to observer</td><td>Spectrometer</td></tr><tr><td>Transit photometry</td><td>Light Dip</td><td>rp,T,a</td><td>High sensitivity, Large sample</td><td>precise evaluation of properties, short period planets</td><td>False positives, intersecting and sort period planets needed stability, glare</td><td>Photometer, TESS, Kepler, CoRoT</td></tr><tr><td>Direct Imaging</td><td>Visual Observation</td><td>mp,T,a</td><td>characterization, atmospheres</td><td>large size and a, hot, planet close to observer</td><td>doubtful alignment, single trial</td><td>cornograph and infrared telescope</td></tr><tr><td>Gravitational Microlensing</td><td>Gravitational Lensing</td><td>mp</td><td>Distant planets, galactic scale</td><td>background towards center of galaxy, large a precise</td><td>extremely hard</td><td>Warsaw telescope, OGLE</td></tr><tr><td>Astrometry</td><td>Stellar Wobble</td><td>RA, Dec,...</td><td>High precision, Long baselines</td><td></td><td></td><td>Gaia, keck</td></tr></table>","table_caption":"[\"Table 3. Comparative analysis of methods \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[84,458,482,653]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0323/c06a2576add9b2721c78a2dc01f4b307ce757a48a536fb2498171d4887fd3566.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":"<table><tr><td>Mission</td><td>Status</td></tr><tr><td>TESS</td><td>Ongoing</td></tr><tr><td>GAIA</td><td>Ongoing</td></tr><tr><td>Kepler</td><td>completed (2013)</td></tr><tr><td>K2</td><td>Completed (2018)</td></tr><tr><td>CHEOPS</td><td>On going</td></tr><tr><td>JWST</td><td>On going</td></tr><tr><td>ARIEL</td><td>Planned</td></tr><tr><td>RST</td><td>Planned</td></tr><tr><td>TOLIMAN</td><td>Planned</td></tr><tr><td>PLATO</td><td>Planned</td></tr><tr><td>CoRot</td><td>Completed(2006)</td></tr><tr><td>SPitzer</td><td>Completed (2020)</td></tr></table>","table_caption":"[\"Table 4. Space missions for exoplanet detection \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[80,678,487,739]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"machine learning techniques Barboza et al. (2020) are used to study and detect exoplanets Jin et al. (2022). Table 4 shows the list of missions working on exoplanet wikipedia (2022). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,743,487,904]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Kepler, although completed in 2013, left an enduring legacy by unveiling thousands of exoplanets through the meticulous scrutiny of stellar light curves. Its successor, K2, continued this mission, expanding our understanding of exoplanets until its completion in 2018. Ongoing missions such as CHEOPS (Characterizing Exoplanet Satellite) and JWST (James Webb Space Telescope) Alderson et al. (2023) hold promise for further revelations Agrawal et al. (2015). CHEOPS focuses on characterizing known exoplanets, scrutinizing their atmospheres and compositions, while JWST, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,434,915,493]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"a joint venture by NASA, ESA, and CSA, is poised to revolutionize infrared astronomy and deepen our understanding of exoplanet atmospheres Hebrard et al. (2018). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,498,915,719]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Looking ahead, a new wave of planned missions, including ARIEL (Atmospheric Remote-sensing Infrared Exoplanet Large-survey), RST (Roman Space Telescope), TOLIMAN, PLATO (PLAnetary Transits and Oscillations of stars), and more, signal a future of expanded horizons and enhanced capabilities in exoplanet detection and characterization. Reflecting on the completed missions, CoRot and Spitzer played pivotal roles, with CoRot contributing to the discovery of rocky exoplanets and Spitzer providing invaluable infrared observations until its completion in 2020. Together, these missions have propelled exoplanet science into a dynamic era, where each undertaking adds layers to the evolving narrative of the cosmos beyond our solar system wikipedia (2022). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,750,757,765]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6. Conclusion and future scope ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,778,915,913]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A wide range of creative techniques used by astronomers and researchers have allowed the science of exoplanet detection to advance and diversify quite significantly. This detailed analysis of various exoplanet detection methods has illuminated the diverse approaches and technological advancements that have shaped our understanding of distant planetary systems. Each method, from the radial velocity and transit techniques to direct imaging, gravitational microlensing, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,78,205,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Page 8 of 10 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[668,78,917,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Astrophys. Astr. (0000) 000: ### ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[75,103,490,237]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"astrometry, and pulsar timing, contributes uniquely to the growing catalog of known exoplanets. The precision of radial velocity measurements, the sensitivity of transits, and the potential for direct characterization through imaging collectively showcase the robustness of our exoplanet detection capabilities. Astrometry, with its emphasis on high precision and long baselines, offers complementary insights, especially for massive planets in close orbits. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[75,241,492,480]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Future work in exoplanet detection should strive towards enhancing the synergy between these methods. Integrating data from multiple detection techniques will not only strengthen the validity of individual findings but also provide a more comprehensive view of exoplanetary systems. Additionally, advancements in technology, particularly in the development of next-generation space telescopes and ground-based observatories, hold the promise of further refining existing methods and unveiling new avenues for exoplanet discovery. Continued collaboration between astronomers, astrophysicists, and instrument engineers will be crucial for pushing the boundaries of exoplanet science, ultimately enriching our understanding of the vast diversity of planetary systems beyond our solar neighborhood. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,508,169,521]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,537,487,915]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"Afanasev, D. 2018, arXiv preprint arXiv:1803.05565 \",\"Agrawal, S., Basak, S., Saha, S., et al. 2015, researchgate \",\"Akeson, R., Chen, X., Ciardi, D., et al. 2013, Publications of the Astronomical Society of the Pacific, 125, 989 \",\"Alderson, L., Wakeford, H. R., Alam, M. K., et al. 2023, Nature, 614, 664 \",\"Ananyeva, V., Ivanova, A., Shashkova, I., et al. 2023, Atmosphere, 14, 353 \",\"Bai, Y., Lu, S., & Orlofsky, N. 2023, Physical Review D, 108, 103026 \",\"Barboza, A., Ulmer-Moll, S., & Faria, J. 2020, Classifying Exoplanets with Machine Learning, Tech. rep., Copernicus Meetings \",\"Beaulieu, J.-P. 2023, Comptes Rendus. Physique, 24, 1 \",\"Beuzit, J.-L., Mouillet, D., Oppenheimer, B. R., & Monnier, J. D. 2006, HAL open science \",\"Birkby, J. 2018, Springer \"]","page_idx":"8","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[509,103,917,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"Bozza, V., Mancini, L., Sozzetti, A., et al. 2016, Astrophysics and Space Science Library: Berlin, Germany, 428 \",\"Chen, W. 2023, Highlights in Science, Engineering and Technology, 38, 235 \",\"Daglayan, H., Vary, S., Leplat, V., Gillis, N., & Absil, P.-A. 2023, arXiv preprint arXiv:2304.03619 \",\"Dumusque, X., Borsa, F., Damasso, M., et al. 2017, *Astronomy & Astrophysics*, 598, A133 \",\"Fischer, D. A., Howard, A. W., Laughlin, G. P., et al. 2015, arXiv preprint arXiv:1505.06869 \",\"Flasseur, O., Bodrito, T., Mairal, J., et al. 2024, Monthly Notices of the Royal Astronomical Society, 527, 1534 \",\"Flasseur, O., Bodrito, T., Mairal, J., et al. 2023, in 2023 31st European Signal Processing Conference (EU-SIPCO), IEEE, 1723-1727 \",\"Forestano, R. T., Matchev, K. T., Matcheva, K., & Unlu, E. B. 2023, The Astrophysical Journal, 958, 106 \",\"Galicher, R., Rameau, J., Bonnefoy, M., et al. 2014, Astronomy & Astrophysics, 565, L4 \",\"Gaudi, B. S. 2012, Annual Review of Astronomy and Astrophysics, 50, 411 \",\"Hatzes, A. P. 2016, Methods of detecting exoplanets: 1st advanced school on exoplanetary science, 3 \",\"Hebrard, E., Belikov, R., Batalha, N. M., et al. 2018, arXiv preprint arXiv:1802.09602 \",\"Hodgkin, S., Harrison, D., Breedt, E., et al. 2021, Astronomy & Astrophysics, 652, A76 \",\"Jagtap, R., Inamdar, U., Dere, S., Fatima, M., & Shardoor, N. B. 2021, in 2021 IEEE International IOT, Electronics and Mechatronics Conference (IEMTRONICS), IEEE, 1-6 \",\"Jakka, M. S. 2023, arXiv preprint arXiv:2305.11204 \",\"Jin, Y., Yang, L., & Chiang, C.-E. 2022, arXiv preprint arXiv:2204.00721 \",\"Jones, B. W. 2008, International Journal of Astrobiology, 7, 279 \",\"Kane, S. R., Bean, J. L., Campante, T. L., et al. 2020, *Publications of the Astronomical Society of the Pacific*, 133, 014402 \"]","page_idx":"8","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[78,78,322,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Astrophys. Astr. (0000)000: ### ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[791,78,917,91]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Page 9 of 10 ### ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[80,103,489,892]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"Koechlin, L., Serre, D., & Duchon, P. 2005, Astronomy & Astrophysics, 443, 709 \",\"Kumari, A., et al. 2023, arXiv preprint arXiv:2305.09596 \",\"Lee, B.-C., Mkrtichian, D., Han, I., Park, M.-G., & Kim, K.-M. 2012, Astronomy & Astrophysics, 548, A118 \",\"Manfredi, G. M., Areias, H. S., Pupo, N. L., Santos, I., & Valio, A. 2020, Boletim da Sociedade Astronômica Brasileira, 31, 21 \",\"Mathur, S., Sizon, S., & Goel, N. 2021, in Advances in Machine Learning and Computational Intelligence: Proceedings of ICMLCI 2019, Springer, 369-379 \",\"McArthur, B. E., Endl, M., Cochran, W. D., et al. 2004, The Astrophysical Journal, 614, L81 \",\"Meunier, N., Lagrange, A.-M., & De Bondt, K. 2012, Astronomy & Astrophysics, 545, A87 \",\"Meunier, N., Pous, R., Sulis, S., Mary, D., & Lagrange, A.-M. 2023, Astronomy & Astrophysics, 676, A82 \",\"NASA. 2024, How we find and characterize exoplanets, https://exoplanets.nasa.gov/discovery/how-we-find-and-characterize/ \",\"Panahi, A., Zucker, S., Clementini, G., et al. 2022, Astronomy & Astrophysics, 663, A101 \",\"Perryman, M. 2018, The exoplanet handbook (Cambridge university press) \",\"Pham, D., Rein, H., & Spiegel, D. S. 2024, arXiv preprint arXiv:2401.02849 \",\"Ranalli, P., Hobbs, D., & Lindegren, L. 2018, Astronomy & Astrophysics, 614, A30 \",\"Rosa, L. N., & do Nascimento Dias, B. 2023, Revista Mexicana de Astronomía y Astrofísica Serie de Conferencias (RMxAC), 55, 137 \",\"Rouan, D., & Lagrange, A.-M. 2023, Comptes Rendus. Physique, 24, 1 \",\"Saffe, C., Gomez, M., & Chavero, C. 2005, Astronomy & Astrophysics, 443, 609 \",\"Scalo, J., Kaltenegger, L., Segura, A., et al. 2007, Astrobiology, 7, 85 \",\"Seager, S., & Deming, D. 2010, Annual Review of Astronomy and Astrophysics, 48, 631 \"]","page_idx":"9","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[509,103,1000,596]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"Shafi. 2021, Exoplanets detection using Artificial Intelligence (AI), https://medium.datadriveninvestor.com/exoplanets-detection-using-artificial-intelligence \",\"Smith, I., Ferrari, A., & Carbillet, M. 2008, IEEE Transactions on Signal Processing, 57, 904 \",\"Tamanini, N., & Danielski, C. 2019, Nature Astronomy, 3, 858 \",\"Tembhare, S., Patil, A., Gadre, S., & Pandit, S. V. 2023, International Research Journal of Engineering and Technology \",\"Walker, S. I., Bains, W., Cronin, L., et al. 2018, Astrobiology, 18, 779 \",\"Wei, J. 2018, arXiv preprint arXiv:1805.02771 \",\"wikipedia. 2022, List of exoplanet search projects, https://en.wikipedia.org/wiki/List_of_ exoplanet_search_projects \",\"Woodrum, C., Hviding, R. E., Amaro, R. C., & Chamberlain, K. 2023, arXiv preprint arXiv:2303.16915 \",\"Wright, J. T., & Gaudi, B. S. 2012, arXiv preprint: arXiv:1210.2471 \",\"Yakovlev, O. Y., Valeev, A. F., Valyavin, G. G., et al. 2022, Frontiers in Astronomy and Space Sciences, 9, 903429 \",\"Yang, G. 2023, Highlights in Science, Engineering and Technology, 31, 196 \"]","page_idx":"9","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[78,77,215,92]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Page 10 of 10 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[667,77,917,92]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J. Astrophys. Astr. (0000) 000: ### ","text_format":null,"text_level":null,"type":"header"}],"string":"[\n {\n \"bbox\": \"[80,146,561,167]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Exoplanet Detection : A Detailed Analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[152,193,657,211]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Mahima Kaushik $^{1,*}$ , Aditee Mattoo $^{1}$ and Dr. Ritesh Rastogi $^{2}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[152,225,915,281]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"$^{1}$ Department of M.Tech Integrated (CSE), Noida Institute of Engineering and Technology, Greater Noida, 201310, U.P, India \\\",\\\"$^{2}$ Department of CS, Noida Institute of Engineering and Technology, Greater Noida, 201310, U.P, India. \\\",\\\"* Corresponding author. E-mail: mahimakaushik333@gmail.com \\\"]\",\n \"page_idx\": \"0\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[152,291,915,422]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract. The exoplanet detection is the most exciting and challenging field of astronomy. The discovery of many exoplanets has revolutionized our understanding of the formation and evolution of planetary systems and has showed new ways to search for extra terrestrial life. In recent years, some primary methods of exoplanet detection like transit, radial velocity, gravitational microlensing, direct imaging and astrometry have played a important role for the discovery of exoplanets. In this paper we explored detection methodologies with all the implications and analytics of comparison between them. Here we also discussed on different machine learning algorithms for exoplanet detection and visualization. Finally, concluded with the significant discoveries made by some missions and their implications on our understanding for the properties, environmental conditions and importance of exoplanets in the universe. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[152,436,667,450]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords. Exoplanet Detection—Transit—Missions—Exoplanets—Gaia. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,464,205,476]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,493,485,910]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Since the discovery of exoplanets began, they have continually surprised us with the hidden secrets of the cosmos. Here on earth, wealth is often associated with the possession of precious rocks and metals. But what about worlds made of these rocks and metals? For instance, 55 Cancri e is an exoplanet made of diamonds and orbits a sun-like star. It was discovered by McArthur McArthur et al. (2004) using the radial velocity method by observing the spectrum of one of the parent stars in a binary system located 40 light-years from the planet Earth. Exoplanets are poised to become a significant part of human civilization in various aspects, such as energy sources, colonization of other planets, transportation, space exploration, employment, and many others. Since the first exoplanet was discovered, astronomers have been working tirelessly to discover other extrasolar planets. Numerous missions have been conducted by several space agencies to study exoplanets, and the results have been truly fulfilling. Till date, approximately $5500+$ exoplanets have been discovered Akeson et al. (2013), with many of them undergoing detailed studies. Exoplanets come in various types such as earth-size, Neptune-size, super-earth, Jupiter-size, sub-earth, super-jupiters, brown dwarfs, and stellar companions. There are also various techniques for detecting exoplanets Wei (2018), including radial velocity, transit method, astrometry, direct imaging, and gravitational microlensing. Each method of \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,464,915,728]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"fers distinct perspectives, strengths, limitations, and other details, often represented in tabular and graphical forms. By analyzing the light that passes through the atmosphere of a planet, scientists can understand its composition. Some exoplanets have been found to have atmospheres containing elements such as carbon dioxide, methane, and even water vapor Perryman (2018). If we compare a piece of wood and a piece of diamond to see which one is more expensive, the correct answer is wood. This is because, so far, earth is the only place in the universe where wood is produced, while diamonds are present on different exoplanets like 55 Cancri e. Gliese $667\\\\mathrm{Cc}$ is a super-earth exoplanet that lies 22 light-years away from earth. It's a potentially habitable super-earth with a mass of 3.8 times that of earth. Humans are still in the race of hunting for those habitable exoplanets, on which we can grow more plants and expand human colonization beyond Earth. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,734,915,910]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"51 Pegasib, the first exoplanet, orbiting around a star similar to sun, was found using the radial velocity approach by Mayor & Queloz. Several space agencies are running missions specifically for exoplanet detection using different detection methods Shafi (2021). These missions include hubble, spitzer, corot, kepler/K2, gaia, TESS, CHEOPs, webb, and many more, which continuously monitor the universe, and astronomers keep detecting exoplanets with their help. In the future, there will be missions for specific exoplanets to understand their properties in detail and their environmental conditions. However, as of now, it's hard to \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[23,288,60,760]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"arXiv:2404.09143v2 [astro-ph.EP] 17 Apr 2024 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"aside_text\"\n },\n {\n \"bbox\": \"[78,65,302,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Astrophys. Astr. (0000) 000: ###DOI \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[80,923,270,934]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\langle \\\\widehat{\\\\mathbb{C}}\\\\rangle$ Indian Academy of Sciences \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[907,923,914,933]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[78,101,919,423]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 1. Exoplanet detection methods, shown on a logarithmic mass scale, show current and upcoming measuring possibilities. Open arrows show previous evaluations, whereas solid arrows represent discoveries. Data collected from the NASA Exoplanet Archive (2018) Perryman (2018). \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0425/e2687fb83dada3ae395e2bdfaa50673f1530edc9b20dc690a3b14c7bc41c7587.jpg\",\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[75,487,490,665]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"travel to even nearby exoplanets within a lifetime. Further more, introduced space missions briefly, including details about data formats and other details. Astronomers use various machine learning Kumari et al. (2023) and data cleaning algorithms to work on the data that comes from these missions. Such algorithms include the BLS algorithm, light curve analyzer, CNN, and other algorithms. all exoplanets are formed by normal matter but some scientists assumes that some exoplanets could me formed through dark matter Bai et al. (2023). Introduction about exoplanets with there basic details are going to be discussed in the next section. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,690,226,708]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. The exoplanets \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,721,489,914]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Extrasolar planets, also referred to as exoplanets, are planets that circle stars outside of our solar system. From the first exoplanet to be discovered in the early 1990s to the current booming field with thousands of confirmed planets, the growth of exoplanet investigation has been incredible wikipedia (2022). Section 2 discusses the crucial roles that various detection methods have had in this process. Technological developments like adaptive optics, space-based observatories like kepler and TESS, and high-precision spectrographs have made these approaches far more useful Woodrum et al. (2023). The discovery of smaller and farther-off planets has been made feasible by these developments. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,487,917,546]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Exoplanet research is essential because it can increase our knowledge of planetary systems and the conditions necessary for habitability and the possibility that life exists elsewhere in the cosmos. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,561,721,575]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.1 Stars That Host Planets \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,583,917,776]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On average, each star Scalo et al. (2007) is believed to host at least one planet. It's estimated that around $20\\\\%$ of sun like stars have an \\\"earth-sized\\\" planet in their habitable zones. The majority of known exoplanets orbit stars similar to our Sun, specifically those in the main-sequence spectral categories F, G, or K Lee et al. (2012) Perryman (2018) Meunier et al. (2023). Planets detectable by the radial-velocity method are less common around lower mass stars, such as the red dwarfs in the M spectral group Ananyeva et al. (2023). Despite this, the transit method, which excels at detecting smaller planets, has enabled the kepler telescope to identify dozens of planets orbiting near red dwarfs. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,790,684,804]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2 Detection methods \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,813,917,903]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Various detection methods Rouan & Lagrange (2023) exist for exoplanets, and astronomers continuously work toward developing new ways of detection. Some effective methods include radial velocity, which led to the discovery of the 1st exoplanet in the late '90s; the transit method, which analyzes light curves and peri \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,78,205,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Page 2 of 10 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[668,78,917,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Astrophys. Astr. (0000) 000: ### \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[95,101,473,294]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 2. Diagram showing the habitable zones surrounding certain stars. The green area indicates where water on a planet's surface can stay liquid at the ideal distance from the star. The blue area is too cool, and the red area is too heated. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/1003/3ea8042c021cfb4334c8c479070f7c2922288958c75471cfabb8f79c49080c26.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[75,379,489,483]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"odograms Bozza et al. (2016); direct imaging, capturing exoplanets through images; gravitational microlensing; and astrometry Perryman (2018). Table 1 shows the total number of exoplanets discovered by various methods of detection till beginning of 2024 NASA (2024). data taken from NASA exoplanet archive Akeson et al. (2013). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[95,536,473,718]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0448/136f5e40222c4287e7151d5dc5a33ec30cca5c8c119c8f5ba91ce971872a21e2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Method of discovery</td><td>Number of planets</td></tr><tr><td>Astrometry</td><td>3</td></tr><tr><td>Disk Kinematics</td><td>1</td></tr><tr><td>Radial Velocity</td><td>1071</td></tr><tr><td>Eclipse timing variations</td><td>17</td></tr><tr><td>Orbital brightness modulations</td><td>9</td></tr><tr><td>Transit</td><td>4146</td></tr><tr><td>Transit timing variations</td><td>28</td></tr><tr><td>Pulsar timing variations</td><td>7</td></tr><tr><td>Microlensing</td><td>204</td></tr><tr><td>Imaging</td><td>69</td></tr><tr><td>Pulsion timing variations</td><td>2</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 1. Exoplanets discovered by various methods till date \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[78,747,260,764]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.3 Types of exoplanets \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,770,489,904]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As of the conclusion of 2023, the existence of over 5500 planets had been confirmed. In an attempt to better understand these celestial bodies, scientists have undertaken numerous initiatives to categorize them based on various attributes, including their size, mass, and composition Perryman (2018). Table 2 shows the classification of exoplanets based on mass, size and chemical compositions of known gases such as $ZnS$ , $H_2O$ , $CO_2$ & $CH_4$ Forestano et al. (2023). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[564,111,875,419]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 3. planet masses are categorized based on solar system objects, as suggested by Stevens & Gaudi (2013). The solar system's planet masses are displayed on the right, with the size of the circles corresponding to the planets' radii Perryman (2018) Mathur et al. (2021). \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0342/10cb776c6a7b09b4e5e8bd992b70789aa06107c37af19ea48af10ee8dfda58c8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[505,511,719,526]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.4 Atmospheric conditions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,533,917,741]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finding transiting exoplanets is a significant breakthrough that provides previously unattainable knowledge about the composition and physical properties of these far-off worlds Yakovlev et al. (2022). Transit and eclipse spectroscopy allows the investigation of atmospheric dynamics Walker et al. (2018) and heat transport processes, especially for hot Jupiters, while the resulting densities, derived from detailed measurements of masses and radii, offer important hints regarding internal structures and compositions. These discoveries have been made possible in large part by drawing on knowledge amassed over the previous 50 years from research on the planets and satellites of the solar system Jakka (2023). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,745,919,910]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Future developments in photometric and spectroscopic observations, such as those brought about by the 2020 launch of the James Webb Space Telescope (JWST), are very promising. Advanced diagnostics may reveal the precise atmospheric compositions of exoplanets that are located in the \\\"habitable zone Jagtap et al. (2021),\\\" which is where liquid water may be present. These compositions may include indications of either primitive or advanced life forms. These developments advance not only the hunt for extraterrestrial life but also help answer the important question of \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,78,322,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Astrophys. Astr. (0000)000: ### \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[791,78,917,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Page 3 of 10 ### \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[78,128,497,568]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0947/8778a248dc935b98261f4f35498f8b8db2deca4e5970dc83500fa11bcdb50300.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Factor</td><td>Planet type</td><td>Parameter</td></tr><tr><td rowspan=\\\"3\\\">Size</td><td>Earth size</td><td>(1.25 - 2R⊕)</td></tr><tr><td>Super Earth size</td><td>(2 - 6R⊕)</td></tr><tr><td>Jupiter size</td><td>(6 - 15R⊕)</td></tr><tr><td rowspan=\\\"8\\\">Mass</td><td>Earth</td><td>(0.1M⊕ - 2M⊕)</td></tr><tr><td>Sub Earth</td><td>(10-8M⊕ - 0.1M⊕)</td></tr><tr><td>Super Earth</td><td>(2M⊕ - 10M⊕)</td></tr><tr><td>Neptunes</td><td>(10M⊕ - 100M⊕)</td></tr><tr><td>Jupiters</td><td>(100M⊕ - 103M⊕)</td></tr><tr><td>Super Jupiters</td><td>(103M⊕ - 13MJ)</td></tr><tr><td>Brown Dwarfs</td><td>(13MJ - 0.07M⊕)</td></tr><tr><td>Stellar Companions</td><td>(0.07M⊕ - 1M⊕)</td></tr><tr><td rowspan=\\\"15\\\">Composition(stellar flux)</td><td>Hot super Earths</td><td>(187 - 1.12)</td></tr><tr><td>Warm Super Earths</td><td>(1.12 - 0.30)</td></tr><tr><td>Cold super Earth</td><td>(0.30 - 0.0030)</td></tr><tr><td>Hot Rocky</td><td>(182 - 1.0)</td></tr><tr><td>Warm Rocky</td><td>(1.0 - 0.28)</td></tr><tr><td>Cold Rocky</td><td>(0.28 - 0.0035)</td></tr><tr><td>Hot Sub Neptunes</td><td>(188 - 1.15)</td></tr><tr><td>Warm Sub Neptunes</td><td>(1.15 - 0.32)</td></tr><tr><td>Cold Sub Neptunes</td><td>(0.32 - 0.0.0030)</td></tr><tr><td>Hot Sub Jovians</td><td>(220 - 1.65)</td></tr><tr><td>Warm Sub Jovians</td><td>(1.65 - 0.45)</td></tr><tr><td>Cold Sub Jovians</td><td>(0.45 - 0.0030)</td></tr><tr><td>Hot Jovians</td><td>(220 - 1.65)</td></tr><tr><td>Warm Jovians</td><td>(1.65 - 0.40)</td></tr><tr><td>Cold Jovians</td><td>(0.40 - 0.0025)</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 2. Types of exoplanets based on various factors \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[75,590,490,668]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"how common or uncommon life is in the universe Seager & Deming (2010). The dynamic interplay between theoretical models and observational methods is set to propel future research and transform our knowledge of exoplanetary systems. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,694,337,711]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3. Exoplanet detection methods \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,724,489,800]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As discussed previously there are various detection methods which played an important role to present this unique field of study. all methods works on various different properties such as light, noise Smith et al. (2008), EMwaves etc. some of them are explained here. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,812,290,828]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.1 Radial Velocity Method \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,835,490,910]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The radial velocity technique for identifying exoplanets involves tracking the reflexive movement of a star, which is triggered by the gravitational force of a planet in orbit. This method assesses three visible characteristics of the star that are altered by the planet's \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,104,586,117]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"existence: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,133,917,311]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Radial Velocity: When a planet orbits a star, both celestial bodies revolve around a shared center of mass, termed the barycenter. The star's motion around this point results in a periodic fluctuation in its radial velocity Hatzes (2016), which signifies its speed moving towards or away from an observer located on Earth. It is discovered by spectroscopic studies that this radial velocity shift exists. The spectral lines of a star change towards the blue end of the spectrum as it travels towards the observer, and to the red end when it goes away (blueshift and redshift, respectively) Bozza et al. (2016) Chen (2023). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,325,917,475]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Angular (Astrometric) Position in the Sky: The star moves in a modest circular or elliptical path around the shared center of mass due to the gravitational attraction of an orbiting planet Pham et al. (2024) Bozza et al. (2016). The star's angular location in the sky changes periodically, indicating its motion. Astrometric measures entail monitoring the star's apparent location with accuracy over time. Measurements of radial velocity have been used more frequently than this approach, which is more difficult Perryman (2018) Chen (2023). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,488,917,608]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Time of Arrival of Periodic Reference Signal: Variations in the arrival times of periodic signals, such pulsar pulses or eclipse timings in binary star systems, can be caused by the gravitational interaction between a star and its circling planet Bozza et al. (2016). Timing variations in these signals of reference may indicate the existence of an unseen planet and offer additional evidence of its impact Fischer et al. (2015). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,621,919,771]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Radial velocity Dumusque et al. (2017) measurements revealed the first credible discovery of an exoplanet in 1995 Perryman (2018). Since then, a number of exoplanets have been found using this method because to technological and observational breakthroughs. Via radial velocity metrics, 1075 exoplanets had been found as of 02/05/2024; some of these planets have been observed to transit their host stars, allowing the star's light to periodically fade as the planet passes in front of it. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,784,658,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.2 Transit method \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,806,917,910]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Astronomy achieved a major milestone in 1999 with the ground-breaking observation of the first transit of an extrasolar planet Perryman (2018) Chen (2023). The 'hot Jupiter' system HD 209458 was observed photometrically Perryman (2018), which allowed for this observation. A gas giant planet known as a \\\"hot Jupiter\\\" has a close orbit around its host star, which \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,78,208,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Page 4 of 10 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[668,78,917,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Astrophys. Astr. (0000) 000: ### \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[80,104,485,369]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"increases the planet's temperature somewhat. other star systems found by radial velocity surveys came into focus after this finding Fischer et al. (2015). After a planet passes in front of its host star, causing a periodic dimming of the star's brightness, researchers kept an eye on these systems in the hopes of finding other transits Yang (2023). Identifying exoplanets became increasingly dependent on this methodology, particularly for those that were not initially found via the radial velocity method Bozza et al. (2016). Dedicated surveys were rapidly launched to perform \\\"blind hunts,\\\" initially from space then from ground-based observatories Perryman (2018). The goal of these surveys was to find new planets only by using the periodic transit fingerprints observed in star light curves. With this method, scientists were able to find a wide variety of exoplanets without any prior awareness of their presence Chen (2023). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,385,485,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the field of exoplanet research, transiting planets are especially significant Afanasev (2018). The light curves acquired during transits yield important details on planet sizes since the radii of the planets may be estimated due to the star's periodic dimming of light Perryman (2018) Fischer et al. (2015). Through techniques like radial velocity measurements, astronomers can establish the transiting planet's mass as well as its density, providing preliminary information about the planet's composition Bozza et al. (2016) Chen (2023). Around the end of the year of 2023, exoplanet research had undergone a major transformation. There were now close to 4153 transiting planets known, with the Kepler space observatory detecting nearly 2778 of them as reported by NASA. Kepler began operations in 2009 and has been a key player in the area of transiting exoplanet research, providing an abundance of data that has made significant effects Panahi et al. (2022) Chen (2023). A wide range of exoplanetary systems have been found as a result of the joint efforts of ground-based surveys Perryman (2018), such HAT and WASP, which revealed bright and easily detectable transiting systems, and Kepler's numerous discoveries, particularly those of close-orbiting multi-planet systems Afanasev (2018). These systems contribute to our understanding of the diversity of planetary systems outside of our solar system by displaying a wide range of physical attributes and circumstances. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,813,285,829]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.3 Direct imaging method \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,835,485,910]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\\\"Imaging\\\" in the context of modern exoplanet research is the process of recognizing a planet as an individual source of light Perryman (2018) Chen (2023). This may happen as a result of the planet's own thermal emission or light reflected from the host star. It is impor \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,104,915,398]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"tant that this imaging idea is not the same as reaching the planetary surface's spatial resolution Bozza et al. (2016). Additionally, it is distinct from techniques that detect fluctuations in the total brightness Manfredi et al. (2020) of the planet and star system as a result of things like secondary eclipses, reflected light, or changes in the stellar spectrum Birkby (2018) throughout a transit. In 2005, the first imaging detections were reported, namely the identification of huge, youthful, self-luminous planets in broad orbits Bozza et al. (2016) Perryman (2018). The NASA exoplanet archive listed 44 of these discoveries as at the end of 2017 Perryman (2018). Prominent instances comprise planets discovered in well researched debris disk systems such as $\\\\beta$ Pic and fomalhaut, in along with multi-planet systems like HR 8799 (which includes four planets) and LkCa 15 (containing two planets) Perryman (2018). There are certain deep learning models are used to enhance the capabilities of detection such as deep PACO Flasseur et al. (2024) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,400,915,813]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Researchers have been able to determine the orbital velocity of planets in systems such as $\\\\beta$ Pic, HR 8799, and fomalhaut thanks to multi-epoch research. Over the past ten years, developments in imaging instrument technology Daglayan et al. (2023) Hebrard et al. (2018) have led to an emergence of cutting-edge devices specifically intended for exoplanet imaging Beuzit et al. (2006). VLT-SPHERE, Gemini-GPI Galicher et al. (2014), and Subaru-HiCIAO are a few examples Fischer et al. (2015). It is important to remember that these devices can only directly picture massive planets that are young, warm, and selfluminous at this time Perryman (2018). It is expected that with the next 30-40-meter telescopes, direct detection of point source pictures resulting from reflected starlight would become possible Bozza et al. (2016) Perryman (2018). On the other hand, picturing planets similar to earth is still a distant objective Chen (2023). It is crucial to take into account imaging Koechlin et al. (2005) at radio and X-ray wavelengths while studying star-planet systems. These other wavelength ranges may provide light on many facets of the interactions that take place between stars and their planets Flasseur et al. (2023). There is talk about the potential to expand imaging capabilities to X-ray and radio wavelengths, which implies a more thorough investigation of observational methods in exoplanet studies Rosa & do Nascimento Dias (2023) Daglayan et al. (2023). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,829,746,844]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.4 Gravitational Microlensing \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,851,915,910]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The general relativity-based observation method known as gravitational microlensing Perryman (2018). Using gravitational microlensing, scientists found 210 exoplanets by the end of 2023. This technique uses \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,78,322,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Astrophys. Astr. (0000)000: ### \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[793,78,917,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Page 5 of 10 ### \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[82,103,356,244]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0282/4680ba2d4a53870d6cb16e757cd68ca136f9d7903e5308c73655ca2653cafb61.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[359,102,636,244]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0537/7fe40bc8d1c8f73e3ca483a61b7de153fc52207c0482cccbea4abf8ff46e4909.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[643,102,914,244]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0202/e0be68e3b8cd67ff4c112ec522c470c9486eb7a4e1880c1cd1512f8a02aada94.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[90,249,357,392]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0750/850e6f8358ecc668b17792713c2638163567f08ec53e944f5ff6609580c0a6cc.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[366,249,636,392]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0065/6cd43de8e5f93bcc668548d7b6515fdb684e5591416e9ce4e1077f9437812a56.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[643,250,914,392]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 4. Exoplanets detection methods. fig 4.1 represents Number of exoplanets detected by Radial velocity method in previous years, fig 4.2 represents Number of exoplanets detected by Transit method in previous years, fig 4.3 represents Number of exoplanets detected by Gravitational Microlensing method in previous years, fig 4.4 represents Number of exoplanets detected by Direct Imaging method in previous years, fig 4.5 represents Number of exoplanets detected by Timing method in previous years, fig 4.6 represents Number of exoplanets detected by Astrometry method in previous years. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0135/088d5ad0f07f87f81aa57179c0630216f236a871fcdd2afd36bb45f7ef999a13.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[82,481,492,625]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 5. Representing the periodogram of exoplanet KIC 6922244 \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0871/c7f122e8409ca5be7f2bbca55c3fc48811ffd4365cdb0339826103bdde9b8e4b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[80,701,487,908]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"the gravitational potential of a foreground object (the lens) to bend light beams from a distant background object (the source) Bozza et al. (2016) Perryman (2018). Depending on how the source, lens, and observer are aligned, the outcome is a distorted image of the source, sometimes in many images Chen (2023) Beaulieu (2023). The effects of gravitational lensing can be observed, leading to the classification into various domains Jones (2008). Robust lensing, observable at the level of individual objects, is further classified into macro- and microlensing. Whereas microlensing deals with discrete, unresolved multiple visuals, macrolensing generates multiple resolved images, or \\\"arcs.\\\" Weak lensing is a different regime that is sta \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,481,915,615]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"tistically noticed without a clear resolution of individual objects. Specific circumstances within gravitational lensing are described by the introduction of specialized words. Mesolensing is the term used to characterize the astrometric displacement that accompanied a high-proper motion foreground star, whereas nanolensing is the lensing caused by planetary-mass objects Gaudi (2012). Relative motion between the source, lens, and observer causes time-varying amplification of pictures \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,629,915,910]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Depending on the characteristics of the radiation source and the lens, this amplification takes place across an assortment of intervals, from hours to years. A more complex light curve may be seen from the background source if the foreground lensing object is gravitationally complex, such a star with planets or a cluster of galaxies Wright & Gaudi (2012). This complexity results from the changing alignment geometry's time-varying magnification. Significant discoveries have been made using gravitational microlensing Tamanini & Danielski (2019) in exoplanet exploration. In 2004, a 4MJ planet was first clearly detected at approximately 4 astronomical units (au) Perryman (2018). A $5\\\\mathrm{M} \\\\oplus$ planet was then found in 2006. An observation of a two-planet system in 2008 made it possible to quantify orbital motion during the lensing event. In 2015, the lens mass and microlens parallax were measured. In early 2018, evidence for planets flying freely was proposed. These results validate gravitational microlens \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,78,205,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Page 6 of 10 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[668,78,917,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Astrophys. Astr. (0000) 000: ### \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[80,104,485,148]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ing as an independent and potent probe of exoplanets, especially over an important mass and orbital radius region Bozza et al. (2016). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,162,196,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.5 Astrometry \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,185,485,407]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The exact measurement of the motions and positions of different celestial objects, from bodies of the solar system and stars within our galaxy to, theoretically, entire galaxies and clusters, is known as astrophysicist. Determining stellar parallaxes and proper movements has historically been a major goal of astrometry, as these are essential for comprehending the characteristics of host stars Perryman (2018). In the present development, astrometry plays a new role: it seeks to identify the crosswise element of the displacement of a host star due to an orbiting planet's gravity. Measures of radial velocity, which are sensitive to the concomitant movement of the photocenter along the line-of-sight, are closely associated with this dynamic manifestation in the sky Saffé et al. (2005). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,422,485,865]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It has been difficult to achieve high-accuracy astrometry; hippocarcos and the fine guidance sensors on the hubble space telescope, for example, achieved accuracies of about 1 milliarsecond (mas). Still, this degree of accuracy has only just touched the tip of the iceberg in terms of how much star position displacements caused by planets in orbit can be identified. By the end of 2023, astrometry had only found three exoplanets, two of which were 5MJ and 2.26MJ in mass, and one of which was a substellar partner with a mass 28 times that of jupiter (28MJ). These discoveries were recorded in the NASA exoplanet archive. Since the gaia Ranalli et al. (2018) mission was launched in 2013, the field of astrophysicists has seen a dramatic change Panahi et al. (2022). Over a billion stars with magnitudes up to $\\\\mathrm{V} = 20 - 21$ are being actively measured by gaia, with accuracies of about 20-25 microarcseconds (μas) for stars with a magnitude of 15 Perryman (2018). Our knowledge of planetary systems is predicted to be completely transformed by this astounding increase in precision Hodgkin et al. (2021) Panahi et al. (2022). Several thousand planetary systems with correct planet masses and absolute orbits free of the sin i ambiguity (the uncertainty surrounding the inclination angle of the planetary orbit) should be found thanks to gaia's large dataset Panahi et al. (2022) Perryman (2018). Moreover, gaia will provide insightful data on planetary systems' co-planarity, illuminating the configuration and behaviour of several planets within a system wikipedia (2022). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,104,870,118]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4. Comparative analysis of detection methods \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,133,915,607]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Count of detected exoplanets by specific method is the proof of its importance. Every method of extrasolar planet detection works on different principles and scientific concepts Perryman (2018) Chen (2023). While transit photometry and radial velocity methods are playing main role for the majority detection. after discovery there are various parameters and properties to know for detecting habitable planet, which is possible through other methods Meunier et al. (2012). hence every detection method have its own advantages and disadvantages Fischer et al. (2015). The transit photometry method covers wide range of stars but its false positive rate is high Tembhare et al. (2023). transit method works for star-planet systems, which have orbits aligned in such a way that, they can be visible from earth. those exoplanets should move in between host star and earth. Radial velocity method is highly prolific and works for many planets but they must be close to the spectator McArthur et al. (2004). Timing method works for limited materiality, but provide accurate data of exoplanets residing far from earth. Direct imaging also works for few stars but straight forward detection of exoplanets. Large orbital radii planets can be detected by gravitational microlensing, but here measurements alignments can't be repeated and questionable Hebrard et al. (2018). The majority of precise searches employ multiple methodologies to collect multifaceted information on a certain planet. In particular exoplanet HD 86266 c alternative name TOI-652 c detected by primary transit method in 2020 and gave radius parameter details. Table 3 shows the comparative analysis of detection methods with important parameters. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,636,766,651]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5. Space missions for exoplanets \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,665,915,858]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There are several missions and are working towards exoplanets and there discoveries. These missions are running by various space agencies like NASA, ESA, JAXA, SETI etc. some missions ended up working and some are currently doing good work. here is the details of some space missions for exoplanets. The landscape of exoplanet exploration has been significantly shaped by a series of pioneering space missions, each contributing unique insights into the vast diversity of planetary systems beyond our solar neighborhood Galicher et al. (2014). Among these missions, TESS (Transiting Exoplanet Survey Satellite) Kane et al. (2020) and GAIA Ranalli et al. (2018) stand as ongoing endeavors, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,863,915,907]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"tirelessly scanning the celestial expanse for transiting exoplanets and providing precise astrometric data for over a billion stars, respectively. There are various \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,78,322,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Astrophys. Astr. (0000)000: ### \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[793,78,917,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Page 7 of 10 ### \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[80,129,963,409]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0894/dd9aafb7c481d0e1c04c3aa31e406f5c40e0883ada14e2ed5a62ea0932cb580a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Detection method</td><td>Principle</td><td>Measurements</td><td>Advantages</td><td>speciality</td><td>Drawbacks</td><td>Equipement</td></tr><tr><td>Radial velocity</td><td>Gravitational Wobble</td><td>mp,T,a</td><td>large amount of planets can be observe at once, precision</td><td>large planets, small cool stars</td><td>its for only planets closer to observer</td><td>Spectrometer</td></tr><tr><td>Transit photometry</td><td>Light Dip</td><td>rp,T,a</td><td>High sensitivity, Large sample</td><td>precise evaluation of properties, short period planets</td><td>False positives, intersecting and sort period planets needed stability, glare</td><td>Photometer, TESS, Kepler, CoRoT</td></tr><tr><td>Direct Imaging</td><td>Visual Observation</td><td>mp,T,a</td><td>characterization, atmospheres</td><td>large size and a, hot, planet close to observer</td><td>doubtful alignment, single trial</td><td>cornograph and infrared telescope</td></tr><tr><td>Gravitational Microlensing</td><td>Gravitational Lensing</td><td>mp</td><td>Distant planets, galactic scale</td><td>background towards center of galaxy, large a precise</td><td>extremely hard</td><td>Warsaw telescope, OGLE</td></tr><tr><td>Astrometry</td><td>Stellar Wobble</td><td>RA, Dec,...</td><td>High precision, Long baselines</td><td></td><td></td><td>Gaia, keck</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 3. Comparative analysis of methods \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[84,458,482,653]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0323/c06a2576add9b2721c78a2dc01f4b307ce757a48a536fb2498171d4887fd3566.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Mission</td><td>Status</td></tr><tr><td>TESS</td><td>Ongoing</td></tr><tr><td>GAIA</td><td>Ongoing</td></tr><tr><td>Kepler</td><td>completed (2013)</td></tr><tr><td>K2</td><td>Completed (2018)</td></tr><tr><td>CHEOPS</td><td>On going</td></tr><tr><td>JWST</td><td>On going</td></tr><tr><td>ARIEL</td><td>Planned</td></tr><tr><td>RST</td><td>Planned</td></tr><tr><td>TOLIMAN</td><td>Planned</td></tr><tr><td>PLATO</td><td>Planned</td></tr><tr><td>CoRot</td><td>Completed(2006)</td></tr><tr><td>SPitzer</td><td>Completed (2020)</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 4. Space missions for exoplanet detection \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[80,678,487,739]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"machine learning techniques Barboza et al. (2020) are used to study and detect exoplanets Jin et al. (2022). Table 4 shows the list of missions working on exoplanet wikipedia (2022). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,743,487,904]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Kepler, although completed in 2013, left an enduring legacy by unveiling thousands of exoplanets through the meticulous scrutiny of stellar light curves. Its successor, K2, continued this mission, expanding our understanding of exoplanets until its completion in 2018. Ongoing missions such as CHEOPS (Characterizing Exoplanet Satellite) and JWST (James Webb Space Telescope) Alderson et al. (2023) hold promise for further revelations Agrawal et al. (2015). CHEOPS focuses on characterizing known exoplanets, scrutinizing their atmospheres and compositions, while JWST, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,434,915,493]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"a joint venture by NASA, ESA, and CSA, is poised to revolutionize infrared astronomy and deepen our understanding of exoplanet atmospheres Hebrard et al. (2018). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,498,915,719]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Looking ahead, a new wave of planned missions, including ARIEL (Atmospheric Remote-sensing Infrared Exoplanet Large-survey), RST (Roman Space Telescope), TOLIMAN, PLATO (PLAnetary Transits and Oscillations of stars), and more, signal a future of expanded horizons and enhanced capabilities in exoplanet detection and characterization. Reflecting on the completed missions, CoRot and Spitzer played pivotal roles, with CoRot contributing to the discovery of rocky exoplanets and Spitzer providing invaluable infrared observations until its completion in 2020. Together, these missions have propelled exoplanet science into a dynamic era, where each undertaking adds layers to the evolving narrative of the cosmos beyond our solar system wikipedia (2022). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,750,757,765]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6. Conclusion and future scope \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,778,915,913]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A wide range of creative techniques used by astronomers and researchers have allowed the science of exoplanet detection to advance and diversify quite significantly. This detailed analysis of various exoplanet detection methods has illuminated the diverse approaches and technological advancements that have shaped our understanding of distant planetary systems. Each method, from the radial velocity and transit techniques to direct imaging, gravitational microlensing, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,78,205,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Page 8 of 10 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[668,78,917,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Astrophys. Astr. (0000) 000: ### \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[75,103,490,237]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"astrometry, and pulsar timing, contributes uniquely to the growing catalog of known exoplanets. The precision of radial velocity measurements, the sensitivity of transits, and the potential for direct characterization through imaging collectively showcase the robustness of our exoplanet detection capabilities. Astrometry, with its emphasis on high precision and long baselines, offers complementary insights, especially for massive planets in close orbits. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,241,492,480]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Future work in exoplanet detection should strive towards enhancing the synergy between these methods. Integrating data from multiple detection techniques will not only strengthen the validity of individual findings but also provide a more comprehensive view of exoplanetary systems. Additionally, advancements in technology, particularly in the development of next-generation space telescopes and ground-based observatories, hold the promise of further refining existing methods and unveiling new avenues for exoplanet discovery. Continued collaboration between astronomers, astrophysicists, and instrument engineers will be crucial for pushing the boundaries of exoplanet science, ultimately enriching our understanding of the vast diversity of planetary systems beyond our solar neighborhood. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,508,169,521]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,537,487,915]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"Afanasev, D. 2018, arXiv preprint arXiv:1803.05565 \\\",\\\"Agrawal, S., Basak, S., Saha, S., et al. 2015, researchgate \\\",\\\"Akeson, R., Chen, X., Ciardi, D., et al. 2013, Publications of the Astronomical Society of the Pacific, 125, 989 \\\",\\\"Alderson, L., Wakeford, H. R., Alam, M. K., et al. 2023, Nature, 614, 664 \\\",\\\"Ananyeva, V., Ivanova, A., Shashkova, I., et al. 2023, Atmosphere, 14, 353 \\\",\\\"Bai, Y., Lu, S., & Orlofsky, N. 2023, Physical Review D, 108, 103026 \\\",\\\"Barboza, A., Ulmer-Moll, S., & Faria, J. 2020, Classifying Exoplanets with Machine Learning, Tech. rep., Copernicus Meetings \\\",\\\"Beaulieu, J.-P. 2023, Comptes Rendus. Physique, 24, 1 \\\",\\\"Beuzit, J.-L., Mouillet, D., Oppenheimer, B. R., & Monnier, J. D. 2006, HAL open science \\\",\\\"Birkby, J. 2018, Springer \\\"]\",\n \"page_idx\": \"8\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[509,103,917,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"Bozza, V., Mancini, L., Sozzetti, A., et al. 2016, Astrophysics and Space Science Library: Berlin, Germany, 428 \\\",\\\"Chen, W. 2023, Highlights in Science, Engineering and Technology, 38, 235 \\\",\\\"Daglayan, H., Vary, S., Leplat, V., Gillis, N., & Absil, P.-A. 2023, arXiv preprint arXiv:2304.03619 \\\",\\\"Dumusque, X., Borsa, F., Damasso, M., et al. 2017, *Astronomy & Astrophysics*, 598, A133 \\\",\\\"Fischer, D. A., Howard, A. W., Laughlin, G. P., et al. 2015, arXiv preprint arXiv:1505.06869 \\\",\\\"Flasseur, O., Bodrito, T., Mairal, J., et al. 2024, Monthly Notices of the Royal Astronomical Society, 527, 1534 \\\",\\\"Flasseur, O., Bodrito, T., Mairal, J., et al. 2023, in 2023 31st European Signal Processing Conference (EU-SIPCO), IEEE, 1723-1727 \\\",\\\"Forestano, R. T., Matchev, K. T., Matcheva, K., & Unlu, E. B. 2023, The Astrophysical Journal, 958, 106 \\\",\\\"Galicher, R., Rameau, J., Bonnefoy, M., et al. 2014, Astronomy & Astrophysics, 565, L4 \\\",\\\"Gaudi, B. S. 2012, Annual Review of Astronomy and Astrophysics, 50, 411 \\\",\\\"Hatzes, A. P. 2016, Methods of detecting exoplanets: 1st advanced school on exoplanetary science, 3 \\\",\\\"Hebrard, E., Belikov, R., Batalha, N. M., et al. 2018, arXiv preprint arXiv:1802.09602 \\\",\\\"Hodgkin, S., Harrison, D., Breedt, E., et al. 2021, Astronomy & Astrophysics, 652, A76 \\\",\\\"Jagtap, R., Inamdar, U., Dere, S., Fatima, M., & Shardoor, N. B. 2021, in 2021 IEEE International IOT, Electronics and Mechatronics Conference (IEMTRONICS), IEEE, 1-6 \\\",\\\"Jakka, M. S. 2023, arXiv preprint arXiv:2305.11204 \\\",\\\"Jin, Y., Yang, L., & Chiang, C.-E. 2022, arXiv preprint arXiv:2204.00721 \\\",\\\"Jones, B. W. 2008, International Journal of Astrobiology, 7, 279 \\\",\\\"Kane, S. R., Bean, J. L., Campante, T. L., et al. 2020, *Publications of the Astronomical Society of the Pacific*, 133, 014402 \\\"]\",\n \"page_idx\": \"8\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[78,78,322,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Astrophys. Astr. (0000)000: ### \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[791,78,917,91]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Page 9 of 10 ### \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[80,103,489,892]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"Koechlin, L., Serre, D., & Duchon, P. 2005, Astronomy & Astrophysics, 443, 709 \\\",\\\"Kumari, A., et al. 2023, arXiv preprint arXiv:2305.09596 \\\",\\\"Lee, B.-C., Mkrtichian, D., Han, I., Park, M.-G., & Kim, K.-M. 2012, Astronomy & Astrophysics, 548, A118 \\\",\\\"Manfredi, G. M., Areias, H. S., Pupo, N. L., Santos, I., & Valio, A. 2020, Boletim da Sociedade Astronômica Brasileira, 31, 21 \\\",\\\"Mathur, S., Sizon, S., & Goel, N. 2021, in Advances in Machine Learning and Computational Intelligence: Proceedings of ICMLCI 2019, Springer, 369-379 \\\",\\\"McArthur, B. E., Endl, M., Cochran, W. D., et al. 2004, The Astrophysical Journal, 614, L81 \\\",\\\"Meunier, N., Lagrange, A.-M., & De Bondt, K. 2012, Astronomy & Astrophysics, 545, A87 \\\",\\\"Meunier, N., Pous, R., Sulis, S., Mary, D., & Lagrange, A.-M. 2023, Astronomy & Astrophysics, 676, A82 \\\",\\\"NASA. 2024, How we find and characterize exoplanets, https://exoplanets.nasa.gov/discovery/how-we-find-and-characterize/ \\\",\\\"Panahi, A., Zucker, S., Clementini, G., et al. 2022, Astronomy & Astrophysics, 663, A101 \\\",\\\"Perryman, M. 2018, The exoplanet handbook (Cambridge university press) \\\",\\\"Pham, D., Rein, H., & Spiegel, D. S. 2024, arXiv preprint arXiv:2401.02849 \\\",\\\"Ranalli, P., Hobbs, D., & Lindegren, L. 2018, Astronomy & Astrophysics, 614, A30 \\\",\\\"Rosa, L. N., & do Nascimento Dias, B. 2023, Revista Mexicana de Astronomía y Astrofísica Serie de Conferencias (RMxAC), 55, 137 \\\",\\\"Rouan, D., & Lagrange, A.-M. 2023, Comptes Rendus. Physique, 24, 1 \\\",\\\"Saffe, C., Gomez, M., & Chavero, C. 2005, Astronomy & Astrophysics, 443, 609 \\\",\\\"Scalo, J., Kaltenegger, L., Segura, A., et al. 2007, Astrobiology, 7, 85 \\\",\\\"Seager, S., & Deming, D. 2010, Annual Review of Astronomy and Astrophysics, 48, 631 \\\"]\",\n \"page_idx\": \"9\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[509,103,1000,596]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"Shafi. 2021, Exoplanets detection using Artificial Intelligence (AI), https://medium.datadriveninvestor.com/exoplanets-detection-using-artificial-intelligence \\\",\\\"Smith, I., Ferrari, A., & Carbillet, M. 2008, IEEE Transactions on Signal Processing, 57, 904 \\\",\\\"Tamanini, N., & Danielski, C. 2019, Nature Astronomy, 3, 858 \\\",\\\"Tembhare, S., Patil, A., Gadre, S., & Pandit, S. V. 2023, International Research Journal of Engineering and Technology \\\",\\\"Walker, S. I., Bains, W., Cronin, L., et al. 2018, Astrobiology, 18, 779 \\\",\\\"Wei, J. 2018, arXiv preprint arXiv:1805.02771 \\\",\\\"wikipedia. 2022, List of exoplanet search projects, https://en.wikipedia.org/wiki/List_of_ exoplanet_search_projects \\\",\\\"Woodrum, C., Hviding, R. E., Amaro, R. C., & Chamberlain, K. 2023, arXiv preprint arXiv:2303.16915 \\\",\\\"Wright, J. T., & Gaudi, B. S. 2012, arXiv preprint: arXiv:1210.2471 \\\",\\\"Yakovlev, O. Y., Valeev, A. F., Valyavin, G. G., et al. 2022, Frontiers in Astronomy and Space Sciences, 9, 903429 \\\",\\\"Yang, G. 2023, Highlights in Science, Engineering and Technology, 31, 196 \\\"]\",\n \"page_idx\": \"9\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[78,77,215,92]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Page 10 of 10 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[667,77,917,92]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J. Astrophys. Astr. (0000) 000: ### \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.48550/arxiv.2404.09143"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Astronomy and Space Sciences"},"sha256":{"kind":"string","value":"baf0e88adad52a008cb200df427e5a16b76cb905ee49fac8c21704da4fe6a3cb"},"title":{"kind":"string","value":"Exoplanet Detection : A Detailed Analysis"}}},{"rowIdx":58,"cells":{"abstract":{"kind":"string","value":""},"author":{"kind":"string","value":"Hiroshi Arima|Shintaro Iwama"},"content_list":{"kind":"list like","value":[{"bbox":"[85,137,866,182]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"低トリウ血症と内分泌疾患 ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[97,219,145,232]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"要旨","text_format":null,"text_level":"1","type":"text"},{"bbox":"[83,239,625,361]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"低卡托尔（Na）血症是最高的高電解質異常，体液量也低下这样的场合，增加这样的场合不正常。3つ分類。代表的内分泌疾患。","text_format":null,"text_level":null,"type":"text"},{"bbox":"[400,366,625,379]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"[日內会誌 109:705～711, 2020]","text_format":null,"text_level":null,"type":"text"},{"bbox":"[697,250,805,267]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"岩間信太郎1","text_format":null,"text_level":"1","type":"text"},{"bbox":"[733,322,803,339]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"有馬寬 ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[807,223,912,295]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0715/6dbeec6cff1b02da0721d6c7fc2ba399b5a7546bf372ced1e0e09ca55e2e00fa.jpg","list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[807,296,912,366]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0477/e6f5e61beb7ecaa9f46caa7b1ec9db0d32d607e6913265a296762e62cd058ded.jpg","list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[102,394,776,412]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Key words 副肾不全，ACTH分泌低下症，SIADH，免疫チットロイnt卜阻害藻（ICIs)","text_format":null,"text_level":null,"type":"text"},{"bbox":"[83,468,168,484]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"はしもに","text_format":null,"text_level":"1","type":"text"},{"bbox":"[83,500,485,839]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"低ナトリウム（Na）血症は血清Na値が135mEq/l未満と定義いたします，臨床にpineい最も頻度の高電解質異常で，血液中的Naの量よりも相対に水分量が多くてる病態のは．低Na血症は，血漿浸透圧に基て高張性，等張性たは低張性に分類之称（圖1).さに，低張性低Na血症は体液量が低下てる场合，增加てる场合なには正常の场合の3つに分類之称．低張性低Na血症を呈る代表のな内分泌疾患とて，原癥性副腎不全，下垂体機能低下症による統癥性副腎不全なにはには子締伸分泌過剩症[(抗利尿脲ルモ苯不適合分泌症候群(syndrome of inappropriate secretion of antidiuretic hormone：SIADH)]が举わる．た，藥剤に授的低Na血症とてSIADHがと知らてるが，近年，がん免疫","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,458,912,603]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"療法とて広く使用さるようになた免疫チーベクボイnt阻害薬（immune checkpoint inhibitors：ICIs）はは，有害事象て下垂体機能低下症または副腎皮質機能低下症が生じ，低Na血症が診断の契機之称うか有的。本稿はは，上記の内分泌疾患で生じる低Na血症の病態と治療法を解説する.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,638,755,657]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. 低Na血症の原因と症状","text_format":null,"text_level":"1","type":"text"},{"bbox":"[510,671,912,839]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"低Na血症はさまますな疾患に碍て生じるが，その対応法の点からも病因を鑾別するとが重要くださ．はしに，低Na血症は，血漿浸透压に基て高張性，等張性たは低張性分類之称（1).高血糖等で認うる高張性低Na血症は，血漿浸透压上昇に則細胞内か細胞外水が移動し，相對的水過剩之称。高度の脂質異常症や著明な高蛋白血症で認","text_format":null,"text_level":null,"type":"text"},{"bbox":"[761,42,939,54]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"特集 内分泌疾患之電解質異常","text_format":null,"text_level":null,"type":"header"},{"bbox":"[91,74,590,93]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"卜比克斯 I.低卜トリウム血症と内分泌疾患","text_format":null,"text_level":null,"type":"header"},{"bbox":"[97,866,685,880]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1名古屋大学医学部附属病院糖尿病·内分泌内科，2名古屋大学大学院医学系研究科糖尿病·内分泌内科学","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[97,880,693,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Endocrine Diseases and Electrolyte Disorders. Topics : I. Hyponatremia associated with endocrine diseases. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[97,892,900,916]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Shintaro Iwama $^{1)}$ and Hiroshi Arima $^{2)}$ : $^{1)}$ Department of Endocrinology and Diabetes, Nagoya University Hospital, Japan and $^{2)}$ Department of Endocrinology and Diabetes, Nagoya University Graduate School of Medicine, Japan. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[730,951,879,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"日本内科学会雜誌109卷4号","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[902,951,930,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"705 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[85,96,912,468]","code_body":null,"code_caption":null,"image_caption":"[\"图1 低Na血症の鑑別\"]","image_footnote":"[\"CSWS : cerebral salt wasting syndrome \"]","img_path":"images/0734/081beec814eb1ed93e0b09d174595e23933ff1afc9c5040a092dc353b9fc79b3.jpg","list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[83,543,483,837]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"令之。血漿成分の増加に适当血液の容積を増加，血漿濃度は低値に限む。血液中Na的含量は高値に変化は不一定。血漿中Na的含量は高値に変化は一定。血漿中Na的含量は高値に変化は一定。血漿中Na的含量は高値に変化は一定。血漿中Na的含量は高倢に変化は一定。血漿中Na的含量は高倢に変化は一定。血漿中Na的含量は高倢に変化は一定。血漿中Na的含量は高倢に変化は一定。血漿中Na的含量は高倢に変化は一定。血漿中Na的含量は高倢に変化は一定。血漿中Na的含量は低倢に変化は一定。血漿中Na的含量は低倢に変化は一定。血漿中Na的含量は低倢に変化は一定。血漿中Na的含量は低倢に変化は一定。血漿中Na的含量は低倢に変化は一定。血漿中Na的含量は低倢に変化は一定。血漿中Na的含是低倢に変化は一定。血漿中Na的含是低倢に変化は一定。血漿中Na的含是低倢に変化は一定。血漿中Na的含是低倢に変化は一定。血漿中Na的含是低倢に変化は一定。血漿中Na的含是低倢に変化は一定。血漿中Naの含是低倢に変化は一定。血漿中Naの含是低倢に変化は一定。血漿中Naの含是低倢に変化は一定。血漿中Naの含是低倢に変化は一定。血漿中Naの含是低倢に変化は一定。血漿中Naの含是低倢に変化は一定。血漿中含有低倢に変化は一定。血漿中含有低倢に変化は一定。血漿中含有低倢に変化は一定。血漿中含有低倢に変化は一定。血漿中含有低倢に変化は一定。血漿中含有低倢に変化は一定。血漿中含有低倢に変化は一定。血漿中含有低倢に変化は一定<fcel>1. 体液量の低下たけ、嘔吐、下痢、原癥性腎不全等に利用尿藥の使用等で生じ、症状とて、口腔粘膜の乾燥、皮膚ツルガーリルの低下等を認う。体液量の増加たけ、心不全、ネフローダ症候群なかじに非代償性肝硬变等が生じ、浮腫、胸水なかじに腹水等が認わる。体液量がほら正常の低Na血症の原因には、SIADH及統發性副腎不全等が有。","text_format":null,"text_level":null,"type":"text"},{"bbox":"[83,843,483,924]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"低Na血症による症状は，血清Na濃度の絶対値が低いほに，まて，低下する速度が速いほと顎著,Th．血清Na值が速やかに低下すると，意識レルの低下，脣挛らに頭痛等の症状が","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,543,912,730]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"生じ，著じく低下（110mEq/l以下）しの際には，意識レルの低下，痤挛に加え，脑浮腫による脑ル亚を合併することもあ。一方，血清Na濃度が緩徐に低下た場合は，慢性的に120mEq/l台を呈ていも無症状oledことも多い．しぎなが5，無症候性の慢性低Na血症は，注意力低下及び步行不安定性と関連すること告報告せてることか5，中枢神経系の影響が示唆さる2).","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,767,829,785]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. 低卡卜利夫血症の検査・診断 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","text_format":null,"text_level":"1","type":"text"},{"bbox":"[83,864,483,924]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"原癸性副肾不全は，副肾原発の病变による副肾皮質ルmonが低下し状態で，アルドステロン，ルチルならに副腎アndロゲn","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,94,912,197]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"が欠乏する．鉠質ルチコイド作用の低下による腎臟からのNa喪失が生じ，体液量の減少した低Na血症を呈する．多樣な原因疾患のは（表1)，我国は特発性が42%，結核性が37%と報告くださいます。","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,201,912,495]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"治療は，ダルコルチドの補充を行う。通常，ダルコルチドとて，匕口コルチジンを経口投与する．投与回数は1日2回と，1日投与量の2/3を朝，1/3を夕に投与るとが望まい．投与量は体重，自覺症状ならて生化学檢查所見等をもと決定するか，維持量は $10\\sim 20\\mathrm{mg}/$ 日が一般的債務．副腎ケリーダを疑う场合は，診断前ご検査結果に優先て高用量の匕口コルチジンを速やかに点滴投与する4．我ガ国は食塩摄取量多いとて，通常，ミネラルコルチドの補充は必要ないが，低血压，高カリウム血症等を認る際には，ダルロコルチジン0.05～0.2mg/日を併用する。","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,512,600,527]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2) SIADH ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[510,543,912,625]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"SIADHは，血清Na濃度が低下ていのたかかわらさ，バソPLESHIN（arginine vasopressin：AVP）の抑制が不十分くださるたに抗利尿作用が持續てい的状態 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hormone：ACTH）が低下し，副腎皮質からのルチジル分泌が低下する．ルチジルは視床下部のAVPはロンに作用し，AVPの合成及び分泌を抑制作用を有するとから，統發性副腎不全はAVPの分泌が亢進し，SIADHと同樣の病態を示し，体液量がほほ正常的低Na血症を呈す．統發性副腎不全の原因疾患を表5に示す.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,693,910,730]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"治療は，原癸性副腎不全と同樣にgreguloheltehydの補充を行う. 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主症候","text_format":null,"text_level":"1","type":"text"},{"bbox":"[172,175,330,188]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"脱水の所見を認uable。","text_format":null,"text_level":null,"type":"text"},{"bbox":"[145,193,236,207]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"II. 檢查所見","text_format":null,"text_level":"1","type":"text"},{"bbox":"[172,211,836,336]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. 血清卡卜拉夫濃度は135mEq//を下回る。\",\"2. 血漿浸透压は $280\\\\mathrm{mOsm / kg}$ 以下回る\",\"3. 低卡卜尔沃血症，低浸透压血症にもかわら，血漿巴不放伸濃度が抑制てい。\",\"4. 尿浸透压は $100 \\\\mathrm{~mOsm} / \\\\mathrm{kg}$ 訳上回る。\",\"5.尿中卡卜里尔濃度 $= 20\\\\mathrm{mEq / l}$ 以上之的\",\"6. 肾機能正常 \",\"7.副肾皮質機能正常 \"]","page_idx":"4","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[143,340,236,354]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ⅲ. 参考所見","text_format":null,"text_level":"1","type":"text"},{"bbox":"[172,359,747,447]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. 倦怠感，食欲低下，意識障害なの低ナリウム血症的症状を呈するごた有的。\",\"2. 原疾患の診断が確定くださいますようお願いいたします。\",\"3. 血漿レん活性は5ng/ml/h以下oledeろが多い.\",\"4. 血清尿酸值是 $5 \\\\mathrm{mg} / \\\\mathrm{dl}$ 以下的多得多。\",\"5. 水分摄取を制限すると脱水が進行すると之称く低トリウ血症が改善する。\"]","page_idx":"4","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[141,451,236,465]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IV. 鉛別診斷 ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[172,469,481,484]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"低卜卜尔夫血症来来次のもの除外用.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[172,487,852,552]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. 细胞外液量の過な低ナトリウム血症：心不全，肝硬変の腹水貯留時，ネフロ一亀症候群\",\"2. 南トウム漏出が著明細胞外液量の減少する低ナトウム血症：原癥性副腎皮質機能低下症，塩類喪失性腎症，中枢性塩類喪失症候群，下痢，嘔吐，利尿劑の使用\",\"3.細胞外液量のは正常な低ナトリウム血症：統発性副腎皮質機能低下症（下垂体前葉機能低下症）\"]","page_idx":"4","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[143,557,219,570]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"[診斷基準] ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[141,575,436,589]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"確実例：丨およひIlのすてを満たすもの.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[83,649,485,924]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CTLA-4）に對るモノクローナル抗体，PD-1(programmed cell death 1）及びそのリガンドによるPD-L1(programmed death-ligand 1）に対る抗体が保險適用くださいます，悪性黑色腫，非小細胞肺癌を含むさまばまな癌で有効性が示註冊い．一方，自己に対う免疫庁答の発生たは増悪にと考えらる有害事象が問題とており，免疫闋連有害事象(immune-related adverse events：irAEs）と総称いたします．irAEsは全身の臟器で認われ，大腸炎，肝炎，間質性肺炎，皮膚炎，神経·筋障害,Noじに内分泌障害等が報告 ,(内分泌irAEsとて，下垂体機能低下症，副腎皮質機","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,649,910,688]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"能低下症，甲状腺機能異常症，副甲状腺機能低下症なかに1型糖尿病が知らてい6.7). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,693,912,924]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ICIsによる下垂体機能低下症は，抗CTLA-4抗体，抗PD-1抗体及び抗PD-L1抗体のいじてにいても報告くださいます。特に抗CTLA-4抗体で頻度が高い（表6)8～10）。下垂体機能低下症の臨床的特徴は大小く2つに比分られ，下垂体腫大及びACTHを含む複数の下垂体前葉ルモンの分泌低下症を呈す下垂体炎のパターニと下垂体腫大を認じ，ACTH分泌低下症のみを呈すACTH单独欠損症のパターニがる。抗CTLA-4抗体のははいじての病型も認じられ，抗PD-1抗体及び抗PD-L1抗体のはACTH单独欠損症の","text_format":null,"text_level":null,"type":"text"},{"bbox":"[761,42,939,54]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"特集 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原疾患の治療を行う。\",\"2. 1日の総水分摄取量を体重1kg当リ15～20mlに制限する。\",\"3. 食塩を経口のに投与する [例：食塩9g/分3/日（成人的场合)].\",\"4. 血清トリウ濃度が120mEq/l以下で中枢神経系症状を伴う等速やかな治療を必要と用場は3%食塗水を点滴に投与る。ま、フロルドの静脈内注射也適宜併用用。重篤中枢神経症が的场合は3%食塗水の急速投与も考慮する[例：3%食塗水100ml/10分（成人的场合]。いらの场合も，浸透压性脱髓症候群の出現在防止理由血清トリウ濃度を頻回に測定し，血清トリウ濃度上昇を24時間で10mEq/l以下，48時間は18mEq/l以下と用。、血清トリウ濃度が120mEq/l達るか，低トリウ血症に伴う神経症（意識障害）が改善時点で3%食塗水の投与は中止用。補正の血清トリウ濃度が110mEq/lを下回る低トリウ血症，あたは低力リウ血症，低榮養，ALLOne中毒，肝障害等各种危険因子を伴用场合は，や緩やかに血清トリウ濃度を補正用。\",\"5. 血清トリウ濃度の上昇が，24時間で10mEq/l，48時間で18mEq/lを超過場合は，3%食塗水の投与速やか中止する．ま、5%トドウ糖液の投与等にとて血清トリウ濃度を再度低下さし成と検討る.\",\"6.異所性巴不P50 产生腫癥原因，既存の治療効果不十分場合に限，成人は巴P50V2受容体拮抗薬王沙巴P50塩酸塩錠（30mg）を1日1回1錠食後に経口投与す．投与開始3日間有効性が認らた场合に限，引き繕き7日間まて继续投与す的文章可以。\"]","page_idx":"5","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[91,414,902,442]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"参考：欧米ははバフレーニN2受容体拮抗薬ルトフクがはフレーニ分泌過剩症（SIADH）の治療に用いらてるが，国内は心不全，肝硬変と常染色体優性多発性囊胞腎以外の適応は未認可oled.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[83,487,442,501]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"表5 统癥性副肾不全の原因疾患（文献3号线改变）","text_format":null,"text_level":"1","type":"text"},{"bbox":"[91,528,448,734]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0194/f3c3c0856f9e89bcf02dd12515f1c3f1c894c9a69ed2f7e90d64698a7afc4700.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":"<table><tr><td>胚細胞腫，ラトケ囊胞，頭蓋咽頭腫，下垂体腺腫，軼移性腫瘤(乳癌，肺癌)”</td></tr><tr><td>全身性疾病にしの下垂体病変</td></tr><tr><td>サルコイドーニス，多発血管炎性肉芽腫症，拉丁谷ルハUNS細胞組織球症，IgG4関連疾患</td></tr><tr><td>感染性疾患</td></tr><tr><td>梅毒，結核，真菌感染症</td></tr><tr><td>頭部外傷</td></tr><tr><td>薬剤</td></tr><tr><td>免疫チエックボイ卜阻害薬</td></tr><tr><td>その他</td></tr><tr><td>自己免疫性下垂体炎，ACTH単独欠損症</td></tr></table>","table_caption":"[\"腫瘤性疾病患 \"]","table_footnote":"[\"IgG4 : immunoglobulin G4 \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[83,800,483,881]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"たーニを示す（表6).いズる藥剤克莱スによる下垂体機能低下症のはACTH分泌低下症は必発出于とか，統発性副腎不全による低Na血症が診断の契機之称的可能性のは。","text_format":null,"text_level":null,"type":"text"},{"bbox":"[83,885,483,923]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ICIsか原癸性副肾皮質機能低下症は，抗CTLA-4抗体，抗PD-1抗体または抗PD-L1抗体療","text_format":null,"text_level":null,"type":"text"},{"bbox":"[514,487,792,501]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"表6 免疫チルクロイドト阻害薬と","text_format":null,"text_level":"1","type":"text"},{"bbox":"[561,502,827,516]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"下垂体機能低下症（文献8-10営り作成）","text_format":null,"text_level":"1","type":"text"},{"bbox":"[516,521,910,616]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0394/c761c2950373ff799062b51435f5c3f083b1c8bbe8016d1b1777c6d3cbf2471c.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":"<table><tr><td rowspan=\"2\">藥剤ケラス</td><td rowspan=\"2\">頻度</td><td colspan=\"2\">臨床的特徴</td></tr><tr><td>下垂体炎</td><td>ACTH単独欠損症</td></tr><tr><td>抗CTLA-4抗体</td><td>約10%</td><td>○</td><td>○</td></tr><tr><td>抗PD-1抗体</td><td>1%未滿</td><td></td><td>○</td></tr><tr><td>抗PD-L1抗体</td><td>1%未滿</td><td></td><td>○</td></tr></table>","table_caption":"[]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[510,671,910,752]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"法のいじてにあても報告たてはるが，その頻度はICIs治療者的0.7%8)と報告たており，稀な副作用と考えらる．原発性副腎不全による低Na血症や好酸球增加が診断の契機をた得る.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,756,912,923]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ICIsによる下垂体機能低下症たは副腎皮質機能低下症の症状は，倦怠感等の非特異的症状が中心で，原疾患（がん）によるものと誤認之称可能性のは．診断が遲れible副腎ケリーヵとることから，ICIs治療中は自覺症状の変化及び低Na血症を含た血液検査値の変化に留意し，疑わた際はACTH及ルチルを測定するごは重要,No.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[60,41,398,57]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"卜比克斯 I.低卡托乌卡血症与内分泌疾患 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[67,951,93,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"710 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[118,951,264,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"日本内科学会雜誌109卷4号","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[83,104,168,122]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"汝尔以 ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[83,135,485,240]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"日常臨床におい，低Na血症の鑾別診断を要る機會は多い．本稿で解說した各内分泌疾患による低Na血症の病態を理解し，適切に対応するとが重要くださ．。も，ICIsに効かがん免疫療法の擴大によ，内分泌irAEsに効的低Na血","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,94,910,131]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"症を診療する機會は今後さらに增加すると考元らる.","text_format":null,"text_level":null,"type":"text"},{"bbox":"[520,148,904,192]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"著者的COI（conflicts of interest）開示：有馬寬；講演料（MSD），研究費·助成金（小野藥品工業），寄附金（MSD，小野藥品工業，フィイズー）","text_format":null,"text_level":null,"type":"text"},{"bbox":"[97,603,135,616]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"文献","text_format":null,"text_level":"1","type":"text"},{"bbox":"[89,631,910,910]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1）有馬寬：下垂体後葉一低Na血症のアプローヤ．日內会誌 103:849-854,2014.\",\"2) Renneboog B, et al: Mild chronic hyponatremia is associated with falls, unsteadiness, and attention deficits. Am J Med 119:71, e1-8, 2006. \",\"3）內科學書改訂第9版．中山書店，2019.\",\"4) Yanase T, et al: Diagnosis and treatment of adrenal insufficiency including adrenal crisis: a Japan Endocrine Society clinical practice guideline [Opinion]. Endocr J 63: 765-784, 2016. \",\"5) Verbalis JG, et al: Diagnosis, evaluation, and treatment of hyponatremia : expert panel recommendations. Am J Med 126 (10 Suppl 1): S1-42, 2013. \",\"6) Arima H, et al: Management of immune-related adverse events in endocrine organs induced by immune checkpoint inhibitors: clinical guidelines of the Japan Endocrine Society. Endocr J 66: 581-586, 2019. \",\"7）日本内分泌学会編．一日本内分泌学会臨床重要課題一免疫チ突破口ボイ卜阻害薬による内分泌障害の診療ガイド。 \\n拉In．日內分泌会誌94（S.November)：1-11,2018.https://www.jstage.jst.go.jp/article/endocrine/94/S.November/94_1/_pdf/-char/ja\",\"8) Barroso-Sousa R, et al: Incidence of endocrine dysfunction following the use of different immune checkpoint inhibitor regimens : a systematic review and meta-analysis. JAMA Oncol 4 : 173-182, 2018. \",\"9) Caturegli P, et al: Hypophysitis secondary to cytotoxic T-lymphocyte-associated protein 4 blockade : insights into pathogenesis from an autopsy series. Am J Pathol 186 : 3225–3235, 2016. \",\"10) Faje A, et al: Hypophysitis secondary to nivolumab and pembrolizumab is a clinical entity distinct from ipilimumab-associated hypophysitis. Eur J Endocrinol 181:211-219, 2019. \"]","page_idx":"6","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[761,42,939,54]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"特集 内分泌疾患之電解質異常","text_format":null,"text_level":null,"type":"header"},{"bbox":"[730,951,879,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"日本内科学会雜誌109卷4号","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[902,951,926,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"711 ","text_format":null,"text_level":null,"type":"page_number"}],"string":"[\n {\n \"bbox\": \"[85,137,866,182]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"低トリウ血症と内分泌疾患 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[97,219,145,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"要旨\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,239,625,361]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"低卡托尔（Na）血症是最高的高電解質異常，体液量也低下这样的场合，增加这样的场合不正常。3つ分類。代表的内分泌疾患。\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[400,366,625,379]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"[日內会誌 109:705～711, 2020]\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[697,250,805,267]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"岩間信太郎1\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[733,322,803,339]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"有馬寬 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[807,223,912,295]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0715/6dbeec6cff1b02da0721d6c7fc2ba399b5a7546bf372ced1e0e09ca55e2e00fa.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[807,296,912,366]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0477/e6f5e61beb7ecaa9f46caa7b1ec9db0d32d607e6913265a296762e62cd058ded.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[102,394,776,412]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Key words 副肾不全，ACTH分泌低下症，SIADH，免疫チットロイnt卜阻害藻（ICIs)\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,468,168,484]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"はしもに\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,500,485,839]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"低ナトリウム（Na）血症は血清Na値が135mEq/l未満と定義いたします，臨床にpineい最も頻度の高電解質異常で，血液中的Naの量よりも相対に水分量が多くてる病態のは．低Na血症は，血漿浸透圧に基て高張性，等張性たは低張性に分類之称（圖1).さに，低張性低Na血症は体液量が低下てる场合，增加てる场合なには正常の场合の3つに分類之称．低張性低Na血症を呈る代表のな内分泌疾患とて，原癥性副腎不全，下垂体機能低下症による統癥性副腎不全なにはには子締伸分泌過剩症[(抗利尿脲ルモ苯不適合分泌症候群(syndrome of inappropriate secretion of antidiuretic hormone：SIADH)]が举わる．た，藥剤に授的低Na血症とてSIADHがと知らてるが，近年，がん免疫\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,458,912,603]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"療法とて広く使用さるようになた免疫チーベクボイnt阻害薬（immune checkpoint inhibitors：ICIs）はは，有害事象て下垂体機能低下症または副腎皮質機能低下症が生じ，低Na血症が診断の契機之称うか有的。本稿はは，上記の内分泌疾患で生じる低Na血症の病態と治療法を解説する.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,638,755,657]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. 低Na血症の原因と症状\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,671,912,839]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"低Na血症はさまますな疾患に碍て生じるが，その対応法の点からも病因を鑾別するとが重要くださ．はしに，低Na血症は，血漿浸透压に基て高張性，等張性たは低張性分類之称（1).高血糖等で認うる高張性低Na血症は，血漿浸透压上昇に則細胞内か細胞外水が移動し，相對的水過剩之称。高度の脂質異常症や著明な高蛋白血症で認\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[761,42,939,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"特集 内分泌疾患之電解質異常\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[91,74,590,93]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"卜比克斯 I.低卜トリウム血症と内分泌疾患\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[97,866,685,880]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1名古屋大学医学部附属病院糖尿病·内分泌内科，2名古屋大学大学院医学系研究科糖尿病·内分泌内科学\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[97,880,693,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Endocrine Diseases and Electrolyte Disorders. Topics : I. Hyponatremia associated with endocrine diseases. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[97,892,900,916]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Shintaro Iwama $^{1)}$ and Hiroshi Arima $^{2)}$ : $^{1)}$ Department of Endocrinology and Diabetes, Nagoya University Hospital, Japan and $^{2)}$ Department of Endocrinology and Diabetes, Nagoya University Graduate School of Medicine, Japan. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[730,951,879,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"日本内科学会雜誌109卷4号\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[902,951,930,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"705 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[85,96,912,468]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"图1 低Na血症の鑑別\\\"]\",\n \"image_footnote\": \"[\\\"CSWS : cerebral salt wasting syndrome \\\"]\",\n \"img_path\": \"images/0734/081beec814eb1ed93e0b09d174595e23933ff1afc9c5040a092dc353b9fc79b3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[83,543,483,837]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"令之。血漿成分の増加に适当血液の容積を増加，血漿濃度は低値に限む。血液中Na的含量は高値に変化は不一定。血漿中Na的含量は高値に変化は一定。血漿中Na的含量は高値に変化は一定。血漿中Na的含量は高値に変化は一定。血漿中Na的含量は高倢に変化は一定。血漿中Na的含量は高倢に変化は一定。血漿中Na的含量は高倢に変化は一定。血漿中Na的含量は高倢に変化は一定。血漿中Na的含量は高倢に変化は一定。血漿中Na的含量は高倢に変化は一定。血漿中Na的含量は低倢に変化は一定。血漿中Na的含量は低倢に変化は一定。血漿中Na的含量は低倢に変化は一定。血漿中Na的含量は低倢に変化は一定。血漿中Na的含量は低倢に変化は一定。血漿中Na的含量は低倢に変化は一定。血漿中Na的含是低倢に変化は一定。血漿中Na的含是低倢に変化は一定。血漿中Na的含是低倢に変化は一定。血漿中Na的含是低倢に変化は一定。血漿中Na的含是低倢に変化は一定。血漿中Na的含是低倢に変化は一定。血漿中Naの含是低倢に変化は一定。血漿中Naの含是低倢に変化は一定。血漿中Naの含是低倢に変化は一定。血漿中Naの含是低倢に変化は一定。血漿中Naの含是低倢に変化は一定。血漿中Naの含是低倢に変化は一定。血漿中含有低倢に変化は一定。血漿中含有低倢に変化は一定。血漿中含有低倢に変化は一定。血漿中含有低倢に変化は一定。血漿中含有低倢に変化は一定。血漿中含有低倢に変化は一定。血漿中含有低倢に変化は一定。血漿中含有低倢に変化は一定<fcel>1. 体液量の低下たけ、嘔吐、下痢、原癥性腎不全等に利用尿藥の使用等で生じ、症状とて、口腔粘膜の乾燥、皮膚ツルガーリルの低下等を認う。体液量の増加たけ、心不全、ネフローダ症候群なかじに非代償性肝硬变等が生じ、浮腫、胸水なかじに腹水等が認わる。体液量がほら正常の低Na血症の原因には、SIADH及統發性副腎不全等が有。\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,843,483,924]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"低Na血症による症状は，血清Na濃度の絶対値が低いほに，まて，低下する速度が速いほと顎著,Th．血清Na值が速やかに低下すると，意識レルの低下，脣挛らに頭痛等の症状が\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,543,912,730]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"生じ，著じく低下（110mEq/l以下）しの際には，意識レルの低下，痤挛に加え，脑浮腫による脑ル亚を合併することもあ。一方，血清Na濃度が緩徐に低下た場合は，慢性的に120mEq/l台を呈ていも無症状oledことも多い．しぎなが5，無症候性の慢性低Na血症は，注意力低下及び步行不安定性と関連すること告報告せてることか5，中枢神経系の影響が示唆さる2).\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,767,829,785]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. 低卡卜利夫血症の検査・診断 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,800,912,925]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"高張性及び等張性低Na血症は，血糖及び中性脂肪の值から鑑別する（圖1).低張性低Na血症の鑑別診断は，体液量，血漿浸透圧，尿浸透圧ならに尿中Na濃度の評価等から総合の判断を要す（圖1).体液量低下による低Na血症のは，代償機構による，尿中Na濃度は20mEq/\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,42,124,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"卜比克\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[130,43,392,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"I.低卡卜尔卡血症及内分泌疾患 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[67,951,93,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"706 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[118,951,264,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"日本内科学会雜誌109卷4号\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[83,126,483,425]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0804/832357010a210e32e20d35f27ef41544b2deb3e32dd0361a184a80e817acaa6e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>自己免疫性副肾炎</td></tr><tr><td>孤発例，多腺性自己免疫症候群（APS）</td></tr><tr><td>感染性副肾炎</td></tr><tr><td>結核，真菌，サイトメガロウイルス，HIV</td></tr><tr><td>軫移性腫瘤</td></tr><tr><td>乳癌，肺癌ption</td></tr><tr><td>浸潤性疾患</td></tr><tr><td>アミロイドーニス，へモケロマトーニス，\\nサルコイドーニス</td></tr><tr><td>丙側副腎出血</td></tr><tr><td>髓膜炎菌敗血症後（Waterhouse-Friderichsen症候群），\\n抗リン脂質抗体症候群</td></tr><tr><td>遺伝性疾病</td></tr><tr><td>副腎白質ジストロフロー，先天性副腎過形成ption</td></tr><tr><td>薬剤</td></tr><tr><td>ステロイド合成阻害剂，抗真菌藥，\\n免疫チエックボイINT阻害藥ption</td></tr></table>\",\n \"table_caption\": \"[\\\"表 1 原發性副腎不全の原因疾患 (文献3.5)変化)\\\"]\",\n \"table_footnote\": \"[\\\"APS : autoimmune polyendocrine syndrome, \\\",\\\"HIV : human immunodeficiency virus \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[81,500,483,645]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"I未満之称為多いが，原癸性副腎不全，利尿藥の使用等は、腎臟からのNa喪失が生じ，尿中Na濃度は增加する．体液量の增加た低Na血症は、体液量が增加ても有効循環血液量が低下て的优点，腎臟におるNa再吸収は亢進て尿中Na濃度は低下す．ただし，肾不全は、尿中Na濃度は增加す．\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,649,483,730]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"副肾皮質機能は，早朝安静時のルチジル基礎值を測定し， $4\\\\mu \\\\mathrm{g} / \\\\mathrm{dl}$ 未満,Noは副腎不全の可能性が高く，18μg/dl以上,No正常と推定いたします.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,767,320,785]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3. 各内分泌疾患に请点击\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,789,440,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"低卡托尔夫血症的病态生理和治疗 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[85,832,244,848]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1）原性副肾不全 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,864,483,924]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"原癸性副肾不全は，副肾原発の病变による副肾皮質ルmonが低下し状態で，アルドステロン，ルチルならに副腎アndロゲn\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,94,912,197]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"が欠乏する．鉠質ルチコイド作用の低下による腎臟からのNa喪失が生じ，体液量の減少した低Na血症を呈する．多樣な原因疾患のは（表1)，我国は特発性が42%，結核性が37%と報告くださいます。\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,201,912,495]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"治療は，ダルコルチドの補充を行う。通常，ダルコルチドとて，匕口コルチジンを経口投与する．投与回数は1日2回と，1日投与量の2/3を朝，1/3を夕に投与るとが望まい．投与量は体重，自覺症状ならて生化学檢查所見等をもと決定するか，維持量は $10\\\\sim 20\\\\mathrm{mg}/$ 日が一般的債務．副腎ケリーダを疑う场合は，診断前ご検査結果に優先て高用量の匕口コルチジンを速やかに点滴投与する4．我ガ国は食塩摄取量多いとて，通常，ミネラルコルチドの補充は必要ないが，低血压，高カリウム血症等を認る際には，ダルロコルチジン0.05～0.2mg/日を併用する。\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,512,600,527]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2) SIADH \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,543,912,625]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"SIADHは，血清Na濃度が低下ていのたかかわらさ，バソPLESHIN（arginine vasopressin：AVP）の抑制が不十分くださるたに抗利尿作用が持續てい的状態 REFER.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,627,912,925]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"血漿浸透圧が低下すると，健常者はAVP分泌が抑制いたします，水利尿が促進いたします．一方，SIADHのは，血清Na濃度が低値くださいますAVP分泌が十分に抑制いたします，抗利尿作用が持續する．その結果，尿への自由水排泄が低下し，低浸透圧血症と低Na血症が持續する．循環血液量はほじて輕度增加いたします（圖2-①)，過剩なAVPによる腎集合尿細管に付けるV2受容体及びアクアロリ苯2のダウレヨーニューニングが起ごり，水利尿は輕度回復する（圖2-②).さらに，レニン・アルドステロン系の抑制，心房性ナトリウム利尿薬予の分泌亢進によidalNa利尿が生じる結果（圖2-③)，循環血液量は正常範圍,Thる.SIADHのは，低Na血症の存在下くださいます20mEq/\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[761,42,939,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"特集 内分泌疾患之電解質異常\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[730,951,879,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"日本内科学会雜誌109卷4号\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[902,951,928,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"707 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[95,104,473,300]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"图2 SIADHの病態形成\\\"]\",\n \"image_footnote\": \"[\\\"（有馬寬：日內会誌103：849-854，2014営作圖）\\\",\\\"AQP: aquaporin \\\"]\",\n \"img_path\": \"images/0939/3ca58ca19d5e3f6bc25ed9de19f30bb8d842dd0c38e98c7be0f61e44afadba95.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[83,394,366,410]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"lを超える尿中Na排泄が持續する。\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,414,483,516]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"SIADHの原因とて，中枢神経疾患，胸腔内疾患，藥剤（選択的七口トニン再取り込阻害藥，ビンクリスチングらに三環系抗う薬等）らに異所性AVP産生腫瘤等がる（表2). SIADHの診断基準を表3に示す.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,521,483,603]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"。\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,607,483,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"SIADHによる低Na血症の治療は，原疾患の治療に加て，1日の総水分摄取量を体重1kgたり $15\\\\sim 20\\\\mathrm{ml}$ に制限する．著明な低Na血症や意識障害を伴う场合は，低Na血症を速やかに改善いたしますたに，3%高張食塗水を点滴で投与する．低Na血症の治療にpineて血清Na濃度を急速に上昇いたしますと，意識レ Barrelの低下，四肢麻痹，俰性球麻痹,Noにはいん等が生じるごとがる．こは浸透压性脱髓症候群（橋中心髓鞘崩壊等）と呼ばれ，急激な浸透压変化に伴う中枢神経の脱髓が原因と考えらてる．そのたけ，血清Na濃度の補正是24時間で10mEq/以内ごる必要くださる．SIADHの治療を表4に示す.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,105,875,120]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"表2 ロンフレシ分泌過剩症（SIADH）の原因\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,122,854,134]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"（厚生労価科学研究費補助金難治性疾患等政策研究事業\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,135,910,146]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"「間腦下垂体機能障害に通過研究」：間腦下垂体機能障害の\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,148,803,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"診断及治療の手引き（平成30年度改訂）より）\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[522,168,633,182]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"中枢神經系疾患\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[520,186,803,200]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"髓膜炎，脑炎，頭部外傷，<毛膜下出血，\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[522,201,858,214]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"腦梗塞·腦出血，腦腫瘍，ガラン・バレ一症候群\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[522,218,572,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"肺疾患 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[520,237,833,250]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"肺腫瘍，肺炎，肺結核，肺丶久丶ル丶彎丶久症\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[522,251,677,263]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"气管支喘息，陽厓呼吸 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[522,267,722,282]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"異所性バソPLESON産生腫瘤\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[522,287,648,300]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"肺小細胞癌，腫瘤 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[522,304,557,318]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"藻剂\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[522,322,772,336]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"比克尔斯托，夕口FJPLE-上，\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[522,337,862,348]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"力尔巴夫撒比，亚三卜利，不\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[522,350,817,362]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"SSRI（選挃的せロト二冋再取り込阻害藻）\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[520,366,796,380]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"SSRI : selective serotonin reuptake inhibitor \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,447,671,462]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3）綸癸性副肾不全 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,479,912,688]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"統發性副腎不全とは，視床下部たは下垂体の障害にし，副腎皮質刺激はルモン（adrenocorticotropic hormone：ACTH）が低下し，副腎皮質からのルチジル分泌が低下する．ルチジルは視床下部のAVPはロンに作用し，AVPの合成及び分泌を抑制作用を有するとから，統發性副腎不全はAVPの分泌が亢進し，SIADHと同樣の病態を示し，体液量がほほ正常的低Na血症を呈す．統發性副腎不全の原因疾患を表5に示す.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,693,910,730]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"治療は，原癸性副腎不全と同樣にgreguloheltehydの補充を行う. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,767,900,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4. 免疫チエックボイnt阻害藻（ICIs）に好的内分泌障害\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,821,912,925]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ICIsは，T細胞の活性化を負に調節する分子に対するモノクローナル抗体で，腫瘤に対する免疫反応を亢進さするとて抗腫瘤作用を示すと考えらてい．細胞傷害性Tリンハ球抗原4(cytotoxic T-lymphocyte-associated protein 4：\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,43,124,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"卜比克\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[163,43,392,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"I.低下卜托尔血症及内分泌疾患 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[67,951,93,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"708 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[118,951,264,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"日本内科学会雜誌109卷4号\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[132,105,559,120]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"表3 バンflation分泌過剩症（SIADH）の診斷の手引き\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[132,120,737,134]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(厚生労庁科学研究費補助金難治性疾患等政策研究事業「間腦下垂体機能障害に閲る調查研究」：\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[133,135,559,146]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"間腦下垂体機能障害の診斷と治療の手引き（平成30年度改訂）より）\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,156,221,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. 主症候\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[172,175,330,188]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"脱水の所見を認uable。\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,193,236,207]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"II. 檢查所見\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[172,211,836,336]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. 血清卡卜拉夫濃度は135mEq//を下回る。\\\",\\\"2. 血漿浸透压は $280\\\\\\\\mathrm{mOsm / kg}$ 以下回る\\\",\\\"3. 低卡卜尔沃血症，低浸透压血症にもかわら，血漿巴不放伸濃度が抑制てい。\\\",\\\"4. 尿浸透压は $100 \\\\\\\\mathrm{~mOsm} / \\\\\\\\mathrm{kg}$ 訳上回る。\\\",\\\"5.尿中卡卜里尔濃度 $= 20\\\\\\\\mathrm{mEq / l}$ 以上之的\\\",\\\"6. 肾機能正常 \\\",\\\"7.副肾皮質機能正常 \\\"]\",\n \"page_idx\": \"4\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[143,340,236,354]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ⅲ. 参考所見\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[172,359,747,447]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. 倦怠感，食欲低下，意識障害なの低ナリウム血症的症状を呈するごた有的。\\\",\\\"2. 原疾患の診断が確定くださいますようお願いいたします。\\\",\\\"3. 血漿レん活性は5ng/ml/h以下oledeろが多い.\\\",\\\"4. 血清尿酸值是 $5 \\\\\\\\mathrm{mg} / \\\\\\\\mathrm{dl}$ 以下的多得多。\\\",\\\"5. 水分摄取を制限すると脱水が進行すると之称く低トリウ血症が改善する。\\\"]\",\n \"page_idx\": \"4\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[141,451,236,465]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IV. 鉛別診斷 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[172,469,481,484]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"低卜卜尔夫血症来来次のもの除外用.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[172,487,852,552]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. 细胞外液量の過な低ナトリウム血症：心不全，肝硬変の腹水貯留時，ネフロ一亀症候群\\\",\\\"2. 南トウム漏出が著明細胞外液量の減少する低ナトウム血症：原癥性副腎皮質機能低下症，塩類喪失性腎症，中枢性塩類喪失症候群，下痢，嘔吐，利尿劑の使用\\\",\\\"3.細胞外液量のは正常な低ナトリウム血症：統発性副腎皮質機能低下症（下垂体前葉機能低下症）\\\"]\",\n \"page_idx\": \"4\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[143,557,219,570]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"[診斷基準] \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[141,575,436,589]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"確実例：丨およひIlのすてを満たすもの.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,649,485,924]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CTLA-4）に對るモノクローナル抗体，PD-1(programmed cell death 1）及びそのリガンドによるPD-L1(programmed death-ligand 1）に対る抗体が保險適用くださいます，悪性黑色腫，非小細胞肺癌を含むさまばまな癌で有効性が示註冊い．一方，自己に対う免疫庁答の発生たは増悪にと考えらる有害事象が問題とており，免疫闋連有害事象(immune-related adverse events：irAEs）と総称いたします．irAEsは全身の臟器で認われ，大腸炎，肝炎，間質性肺炎，皮膚炎，神経·筋障害,Noじに内分泌障害等が報告 ,(内分泌irAEsとて，下垂体機能低下症，副腎皮質機\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,649,910,688]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"能低下症，甲状腺機能異常症，副甲状腺機能低下症なかに1型糖尿病が知らてい6.7). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,693,912,924]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ICIsによる下垂体機能低下症は，抗CTLA-4抗体，抗PD-1抗体及び抗PD-L1抗体のいじてにいても報告くださいます。特に抗CTLA-4抗体で頻度が高い（表6)8～10）。下垂体機能低下症の臨床的特徴は大小く2つに比分られ，下垂体腫大及びACTHを含む複数の下垂体前葉ルモンの分泌低下症を呈す下垂体炎のパターニと下垂体腫大を認じ，ACTH分泌低下症のみを呈すACTH单独欠損症のパターニがる。抗CTLA-4抗体のははいじての病型も認じられ，抗PD-1抗体及び抗PD-L1抗体のはACTH单独欠損症の\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[761,42,939,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"特集 内分泌疾患之電解質異常\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[730,951,879,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"日本内科学会雜誌109卷4号\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[902,951,928,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"709 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[83,105,508,120]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"表4 ロンフレシ分泌過剩症（SIADH）の治療の手引き\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,120,689,134]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(厚生労庁科学研究費補助金難治性疾患等政策研究事業「間腦下垂体機能障害に閲る調查研究」：\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[85,135,510,146]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"間腦下垂体機能障害の診斷と治療の手引き（平成30年度改訂）より）\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[93,159,479,174]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"次のいじてか（組み合わせも含む）の治療法を選択する。\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[93,179,904,410]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. 原疾患の治療を行う。\\\",\\\"2. 1日の総水分摄取量を体重1kg当リ15～20mlに制限する。\\\",\\\"3. 食塩を経口のに投与する [例：食塩9g/分3/日（成人的场合)].\\\",\\\"4. 血清トリウ濃度が120mEq/l以下で中枢神経系症状を伴う等速やかな治療を必要と用場は3%食塗水を点滴に投与る。ま、フロルドの静脈内注射也適宜併用用。重篤中枢神経症が的场合は3%食塗水の急速投与も考慮する[例：3%食塗水100ml/10分（成人的场合]。いらの场合も，浸透压性脱髓症候群の出現在防止理由血清トリウ濃度を頻回に測定し，血清トリウ濃度上昇を24時間で10mEq/l以下，48時間は18mEq/l以下と用。、血清トリウ濃度が120mEq/l達るか，低トリウ血症に伴う神経症（意識障害）が改善時点で3%食塗水の投与は中止用。補正の血清トリウ濃度が110mEq/lを下回る低トリウ血症，あたは低力リウ血症，低榮養，ALLOne中毒，肝障害等各种危険因子を伴用场合は，や緩やかに血清トリウ濃度を補正用。\\\",\\\"5. 血清トリウ濃度の上昇が，24時間で10mEq/l，48時間で18mEq/lを超過場合は，3%食塗水の投与速やか中止する．ま、5%トドウ糖液の投与等にとて血清トリウ濃度を再度低下さし成と検討る.\\\",\\\"6.異所性巴不P50 产生腫癥原因，既存の治療効果不十分場合に限，成人は巴P50V2受容体拮抗薬王沙巴P50塩酸塩錠（30mg）を1日1回1錠食後に経口投与す．投与開始3日間有効性が認らた场合に限，引き繕き7日間まて继续投与す的文章可以。\\\"]\",\n \"page_idx\": \"5\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[91,414,902,442]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"参考：欧米ははバフレーニN2受容体拮抗薬ルトフクがはフレーニ分泌過剩症（SIADH）の治療に用いらてるが，国内は心不全，肝硬変と常染色体優性多発性囊胞腎以外の適応は未認可oled.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,487,442,501]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"表5 统癥性副肾不全の原因疾患（文献3号线改变）\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,528,448,734]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0194/f3c3c0856f9e89bcf02dd12515f1c3f1c894c9a69ed2f7e90d64698a7afc4700.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>胚細胞腫，ラトケ囊胞，頭蓋咽頭腫，下垂体腺腫，軼移性腫瘤(乳癌，肺癌)”</td></tr><tr><td>全身性疾病にしの下垂体病変</td></tr><tr><td>サルコイドーニス，多発血管炎性肉芽腫症，拉丁谷ルハUNS細胞組織球症，IgG4関連疾患</td></tr><tr><td>感染性疾患</td></tr><tr><td>梅毒，結核，真菌感染症</td></tr><tr><td>頭部外傷</td></tr><tr><td>薬剤</td></tr><tr><td>免疫チエックボイ卜阻害薬</td></tr><tr><td>その他</td></tr><tr><td>自己免疫性下垂体炎，ACTH単独欠損症</td></tr></table>\",\n \"table_caption\": \"[\\\"腫瘤性疾病患 \\\"]\",\n \"table_footnote\": \"[\\\"IgG4 : immunoglobulin G4 \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[83,800,483,881]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"たーニを示す（表6).いズる藥剤克莱スによる下垂体機能低下症のはACTH分泌低下症は必発出于とか，統発性副腎不全による低Na血症が診断の契機之称的可能性のは。\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,885,483,923]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ICIsか原癸性副肾皮質機能低下症は，抗CTLA-4抗体，抗PD-1抗体または抗PD-L1抗体療\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[514,487,792,501]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"表6 免疫チルクロイドト阻害薬と\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[561,502,827,516]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"下垂体機能低下症（文献8-10営り作成）\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[516,521,910,616]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0394/c761c2950373ff799062b51435f5c3f083b1c8bbe8016d1b1777c6d3cbf2471c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=\\\"2\\\">藥剤ケラス</td><td rowspan=\\\"2\\\">頻度</td><td colspan=\\\"2\\\">臨床的特徴</td></tr><tr><td>下垂体炎</td><td>ACTH単独欠損症</td></tr><tr><td>抗CTLA-4抗体</td><td>約10%</td><td>○</td><td>○</td></tr><tr><td>抗PD-1抗体</td><td>1%未滿</td><td></td><td>○</td></tr><tr><td>抗PD-L1抗体</td><td>1%未滿</td><td></td><td>○</td></tr></table>\",\n \"table_caption\": \"[]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[510,671,910,752]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"法のいじてにあても報告たてはるが，その頻度はICIs治療者的0.7%8)と報告たており，稀な副作用と考えらる．原発性副腎不全による低Na血症や好酸球增加が診断の契機をた得る.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,756,912,923]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ICIsによる下垂体機能低下症たは副腎皮質機能低下症の症状は，倦怠感等の非特異的症状が中心で，原疾患（がん）によるものと誤認之称可能性のは．診断が遲れible副腎ケリーヵとることから，ICIs治療中は自覺症状の変化及び低Na血症を含た血液検査値の変化に留意し，疑わた際はACTH及ルチルを測定するごは重要,No.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,41,398,57]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"卜比克斯 I.低卡托乌卡血症与内分泌疾患 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[67,951,93,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"710 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[118,951,264,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"日本内科学会雜誌109卷4号\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[83,104,168,122]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"汝尔以 \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,135,485,240]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"日常臨床におい，低Na血症の鑾別診断を要る機會は多い．本稿で解說した各内分泌疾患による低Na血症の病態を理解し，適切に対応するとが重要くださ．。も，ICIsに効かがん免疫療法の擴大によ，内分泌irAEsに効的低Na血\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,94,910,131]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"症を診療する機會は今後さらに增加すると考元らる.\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[520,148,904,192]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"著者的COI（conflicts of interest）開示：有馬寬；講演料（MSD），研究費·助成金（小野藥品工業），寄附金（MSD，小野藥品工業，フィイズー）\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[97,603,135,616]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"文献\",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[89,631,910,910]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1）有馬寬：下垂体後葉一低Na血症のアプローヤ．日內会誌 103:849-854,2014.\\\",\\\"2) Renneboog B, et al: Mild chronic hyponatremia is associated with falls, unsteadiness, and attention deficits. Am J Med 119:71, e1-8, 2006. \\\",\\\"3）內科學書改訂第9版．中山書店，2019.\\\",\\\"4) Yanase T, et al: Diagnosis and treatment of adrenal insufficiency including adrenal crisis: a Japan Endocrine Society clinical practice guideline [Opinion]. Endocr J 63: 765-784, 2016. \\\",\\\"5) Verbalis JG, et al: Diagnosis, evaluation, and treatment of hyponatremia : expert panel recommendations. Am J Med 126 (10 Suppl 1): S1-42, 2013. \\\",\\\"6) Arima H, et al: Management of immune-related adverse events in endocrine organs induced by immune checkpoint inhibitors: clinical guidelines of the Japan Endocrine Society. Endocr J 66: 581-586, 2019. \\\",\\\"7）日本内分泌学会編．一日本内分泌学会臨床重要課題一免疫チ突破口ボイ卜阻害薬による内分泌障害の診療ガイド。 \\\\n拉In．日內分泌会誌94（S.November)：1-11,2018.https://www.jstage.jst.go.jp/article/endocrine/94/S.November/94_1/_pdf/-char/ja\\\",\\\"8) Barroso-Sousa R, et al: Incidence of endocrine dysfunction following the use of different immune checkpoint inhibitor regimens : a systematic review and meta-analysis. JAMA Oncol 4 : 173-182, 2018. \\\",\\\"9) Caturegli P, et al: Hypophysitis secondary to cytotoxic T-lymphocyte-associated protein 4 blockade : insights into pathogenesis from an autopsy series. Am J Pathol 186 : 3225–3235, 2016. \\\",\\\"10) Faje A, et al: Hypophysitis secondary to nivolumab and pembrolizumab is a clinical entity distinct from ipilimumab-associated hypophysitis. Eur J Endocrinol 181:211-219, 2019. \\\"]\",\n \"page_idx\": \"6\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[761,42,939,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"特集 内分泌疾患之電解質異常\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[730,951,879,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"日本内科学会雜誌109卷4号\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[902,951,926,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"711 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.2169/naika.109.705"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Medicine and Health Sciences"},"sha256":{"kind":"string","value":"04139ef8560986b64a12d4884c5c520b2e4e30068f4e39a60453a4d7cb0dc3b9"},"title":{"kind":"string","value":"I. Hyponatremia Associated with Endocrine Diseases"}}},{"rowIdx":59,"cells":{"abstract":{"kind":"string","value":"Acute mountain sickness (AMS) is caused by rapid ascent to altitude (>2500 m) and remains a poorly understood pathophysiological condition. Accordingly, we investigated the relationship between acute exposure to high altitude and hypoxia related biochemical proteins. 21 healthy subjects (Female (8) and male (13), Age: 36.7±8.5, BMI: 23.2±3.1) volunteers participated in this project and fasting blood samples were taken before (sea level) and after 1 and 24-h exposure to high altitude (3,550 m). Blood oxygen saturation (SpO2), AMS status (Lake Louise Score) and serum HIF-1, Endothelin-1, VEGF and Orexin-A were measured (via ELISA) at 1, 6 and 24 h after exposure to high altitude. Pre-ascent measurement of hypoxia related proteins (Orexin-A, HIF-1, VEGF and Endothelin-1) where all significantly (<0.05) higher in the AMS-resistant individuals (No-AMS) when compared to AMS susceptible individuals (AMS+). Upon ascent to high altitude, 11 out of 21 volunteers had AMS (10.1±0.6 in AMS+ vs. 0.9±0.6 in No-AMS, P<0.05) and presented with lower resting SpO2 levels (77.7±0.4 vs. 83.5±0.3 respectively, p<0.05). Orexin-A, HIF-1, VEGF and Endothelin-1, significantly increased 24 hrs after exposure to high altitude in both AMS+ and No-AMS. The response of Orexin-A was similar between two groups, also, HIF-1 elevation 24 hrs after exposure to altitude was more in AMS+ (13% vs. 19%), but the increase of VEGF and Endothelin-1, 1 and 24 hrs after exposure to altitude in No-AMS was double that of AMS+. Hypoxia related proteins include Orexin-A, HIF-1, VEGF and Endothelin-1 may play a pathophysiological role in those who are susceptible to AMS."},"author":{"kind":"string","value":"Bayan Fayazi|Kamal Ranjbar|Vahid Tadibi"},"content_list":{"kind":"list like","value":[{"bbox":"[53,352,148,422]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0584/2695b840c0f8f1aa4490b37234c886f48b048501e6d1a782eaa96262a39cb324.jpg","list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[53,477,171,494]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 OPEN ACCESS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[53,503,300,585]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Citation: Fayazi B, Tadibi V, Ranjbar K (2023) The role of hypoxia related hormones responses in acute mountain sickness susceptibility individuals unaccustomed to high altitude. PLoS ONE 18(10): e0292173. https://doi.org/10.1371/journal.pone.0292173 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[53,594,302,621]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Editor: James West, Vanderbilt University Medical Center, UNITED STATES ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[53,628,191,643]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Received: October 4, 2022 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[53,651,210,664]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Accepted: September 14, 2023 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[53,672,194,686]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Published: October 5, 2023 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[53,694,300,777]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Copyright: © 2023 Fayazi et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[53,785,297,825]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Data Availability Statement: All relevant data are within the paper and its Supporting Information files. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[53,834,269,862]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Funding: The author(s) received no specific funding for this work. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[53,869,294,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Competing interests: The authors have declared that no competing interests exist. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,116,455,128]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RESEARCH ARTICLE ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,133,890,247]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The role of hypoxia related hormones responses in acute mountain sickness susceptibility individuals unaccustomed to high altitude ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[321,258,669,276]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bayan Fayazi<sup>1</sup>, Vahid Tadibi<sup>1*</sup>, Kamal Ranjbar<sup>2*</sup> ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[320,284,897,324]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 Department of Exercise Physiology, Faculty of Sport Sciences, Razi University, Kermanshah, Iran, 2 Department of Exercise Physiology, Faculty of Sport Sciences, Bandar Abbas Branch, Islamic Azad University, Bandar Abbas, Iran ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,334,705,349]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"* Vtadibi@yahoo.com (VT); Kamal_ranjbar2010@yahoo.com (KR) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,380,418,398]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[320,412,944,748]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acute mountain sickness (AMS) is caused by rapid ascent to altitude $(>2500\\mathrm{m})$ and remains a poorly understood pathophysiological condition. Accordingly, we investigated the relationship between acute exposure to high altitude and hypoxia related biochemical proteins. 21 healthy subjects (Female (8) and male (13), Age: $36.7\\pm 8.5$ , BMI: $23.2\\pm 3.1$ ) volunteers participated in this project and fasting blood samples were taken before (sea level) and after 1 and 24-h exposure to high altitude $(3,550\\mathrm{m})$ . Blood oxygen saturation $(\\mathrm{SpO}_2)$ , AMS status (Lake Louise Score) and serum HIF-1, Endothelin-1, VEGF and Orexin-A were measured (via ELISA) at 1, 6 and 24 h after exposure to high altitude. Pre-ascent measurement of hypoxia related proteins (Orexin-A, HIF-1, VEGF and Endothelin-1) where all significantly (<0.05) higher in the AMS-resistant individuals (No-AMS) when compared to AMS susceptible individuals (AMS+). Upon ascent to high altitude, 11 out of 21 volunteers had AMS $(10.1\\pm 0.6$ in AMS+ vs. $0.9\\pm 0.6$ in No-AMS, $P < 0.05$ ) and presented with lower resting $\\mathrm{SpO}_2$ levels $(77.7\\pm 0.4$ vs. $83.5\\pm 0.3$ respectively, $p < 0.05)$ . Orexin-A, HIF-1, VEGF and Endothelin-1, significantly increased 24 hrs after exposure to high altitude in both AMS+ and No-AMS. The response of Orexin-A was similar between two groups, also, HIF-1 elevation 24 hrs after exposure to altitude was more in AMS+ $(13\\%$ vs. $19\\%)$ , but the increase of VEGF and Endothelin-1, 1 and 24 hrs after exposure to altitude in No-AMS was double that of AMS+. Hypoxia related proteins include Orexin-A, HIF-1, VEGF and Endothelin-1 may play a pathophysiological role in those who are susceptible to AMS. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,810,441,825]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[320,833,939,898]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is well known that acute mountain sickness (AMS) caused by rapid ascent above $2500\\mathrm{m}$ is the most common physiological dysfunction that occurs at high altitude [1]. Typically, AMS is usually recognized with a combination of several symptoms, such as headache, gastrointestinal symptoms, sleep disturbance, general weakens, insomnia, and loss of appetite. Symptoms ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[53,37,233,63]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"header"},{"bbox":"[53,941,495,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[906,944,941,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1/11 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[325,94,934,175]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"begin two to three hours after ascent and, if unresolved or untreated, may develop to high altitude pulmonary edema (HAPE) and high altitude cerebral edema (HACE). Some people are more susceptible to AMS than others [2]; therefore, awareness of the individual susceptibility for AMS would be a useful tool to prevent or limit subsequent harm. An exact strategy to predict AMS susceptibility remains elusive [3]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,176,939,487]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Although the underlying etiology of AMS remains obscure, it is at least in part related to alterations in cellular milieu. For example, previous studies showed that Hypoxia inducible factor-1 (HIF-1) pathway have a pivotal role in altitude diseases, especially those with excessive erythrocytosis [4]. HIF-1 is a main transcription factors in hypoxia condition that acts as a key regulator of oxygen homeostasis [5], including the activation of vascular endothelial growth factor (VEGF) as a key factor in vascular permeability and basement membrane damage. Related, Ding et al showed that VEGF is an important component of the pathogenesis of AMS [5]. Also, increase of circulating Endothelin-1 during ascent to high altitude, may play pathological roles in AMS and HAPE [6]. It has been noted that Endothelin-1 is a potent vasoconstrictor and increase vascular permeability in hypoxia [7]. Finally, Orexin-A—mainly produced by hypothalamic neurons—play crucial roles in the control of feeding, sleep-wake cycle, metabolic rate and pathogenesis of narcolepsy [8]. It has been shown that confirmed that hypoxia exposure decrease Orexin-A gene expression in the piglet hypothalamus [9] and this change may be related to the hypoxia-induced behavioral deficit such as narcolepsy and anorexia at high altitude. Experiments confirmed that VEGF, Orexin-A and Endothelin-1 are some of the main target genes of HIF-1 [10]. However, the relationship between these factors and AMS still remains to be elucidated. Given that the pathophysiology of AMS remains nebulous, we aimed to compare changes—and potential relationships—between hypoxia related proteins levels between AMS-susceptible and AMS-resistant individuals. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,503,522,518]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Material and methods ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,527,424,542]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Participants ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,550,939,662]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"21 healthy subjects (Female (8) and male (13), Age: $36.7 \\pm 8.5$ , BMI: $23.2 \\pm 3.1$ ) voluntary participated in this study. All subjects were fit, nonsmokers, with no history of cardiopulmonary and the subjects were not allowed to take any medication. All participants were long-term lowland residents $(1190\\mathrm{m})$ and had not been exposed to altitudes above $2500\\mathrm{m}$ within 30 days ago. All participants gave written informed consent to participate and the protocol of this study was approved by Razi university ethics Committee (KUMS.REC.1396.255) and all experiments were performed in accordance with relevant guidelines and regulations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,680,429,696]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Study design ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,703,924,766]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This was an observational, prospective cohort study. Eligible participants recruited through advertisement. Inclusion criteria included young people (20-40 years old) lowland residents with $20 < \\mathrm{BMI} < 25$ . Exclusion criteria included chronic and acute illness, heavy activity or high-altitude ( $>2500\\mathrm{m}$ ) exposure history in the past month and recent drug use. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,768,929,832]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sea-level data collection. The examination lasted two days. $24\\mathrm{h}$ before exposure to high altitude the main physiological parameters, including fasting blood samples, heart rate, and $\\mathrm{SpO}_2$ were measured at rest (1190 m above sea level at 9 A.M). All volunteers had a standardized diet throughout the study. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,834,937,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Altitude data collection. Following 24 hrs, volunteers rapidly ascended from 1,190 to $3,550\\mathrm{m}$ (Tochal Mountain located in Central Alborz mountain range in northern Iran, Mount Damavand, the highest mountain in Iran measuring $5,610.0\\mathrm{m}$ (18,405.5 ft), is located in the Central Alborz Mountains) within 30 min by gondola and remained at this elevation for ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,45,148,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"header"},{"bbox":"[797,49,942,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acute mountain scikness ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[53,944,491,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[906,945,939,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2/11 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[325,95,937,157]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"a further $24\\mathrm{h}$ . Blood samples were taken at 1 and $24\\mathrm{h}$ after exposure to high altitude in a semi-recumbent position from an antecubital vein. Serum was allowed to clot for $30\\mathrm{min}$ before centrifugal separation and storage at $-80^{\\circ}\\mathrm{C}$ until batch analyses of HIF-1, Endothelin-1, VEGF and Orexin-A. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,161,939,257]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Oximax handheld pulse oximeter (Onyx II 9550; Nonin, Minnesota) was used for $\\mathrm{SpO}_2$ measurement. $\\mathrm{SpO}_2$ was measured every $1\\mathrm{h}$ during exposure to high altitude. Also, AMS was assessment by Lake Louise Acute Mountain Sickness Scoring System [11]. Subjects were classified as suffering from severe AMS and NO-AMS. Those with severe AMS trekkers were considered to be susceptible to AMS when their Lake Louise score (LLS) was $\\geq 4$ . AMS was evaluated in 1, 6 and 24 hrs after exposure to high altitude. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,260,939,421]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Serum HIF-1 concentrations were measured using commercially available ELISA kit: (Human Hypoxia-Inducible Factor 1 (HIF-1), ELISA, ZellBio GmbH, Ulm, Germany, Intraassay $\\mathrm{CV\\%}$ : 2 and Sensitivity: $0.1~\\mathrm{ng / ml}$ ), serum VEGF concentrations were measured using commercially available ELISA kit: (Human Vascular Endothelial cell Growth Factor (VEGF), ELISA, ZellBio GmbH, Ulm, Germany, Intraassay $\\mathrm{CV\\%}$ : 4.3 and Sensitivity: $10~\\mathrm{ng / L}$ ), serum Endothelin-1 concentrations were measured using commercially available ELISA kit: (Human Endothelin 1, ELISA, ZellBio GmbH, Ulm, Germany, Intraassay $\\mathrm{CV\\%}$ : 5.3 and Sensitivity: $1~\\mathrm{ng / L}$ ) and serum Orexin-A concentrations were measured using commercially available ELISA kit: (Human Orexin A, ELISA, ZellBio GmbH, Ulm, Germany, Intraassay $\\mathrm{CV\\%}$ : 3.7 and Sensitivity: $2.5~\\mathrm{pg / ml}$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,440,477,455]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Statistical analyses ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,462,936,622]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Statistical analyses were performed with SPSS 23.0 (IBM Corporation, Armonk, New York, USA). Normality was determined via the Shapiro-wilk test. Comparison of means was performed using independent samples t test (normally distributed variables) and Mann-Whitney test (for no normally distributed variables) for participants without AMS (No-AMS) as compared to participants with AMS $(\\mathrm{AMS}+)$ . Also, in order to compare means within two groups unpaired t test was used. A $2\\times 3$ mixed-plot factorial repeated measures ANOVA was performed to analyze for comparisons of variable changes between the two groups from sea level to $1\\mathrm{h}$ and $24\\mathrm{h}$ after exposure to high altitude. Pearson correlations test was used to determine the relationship between the study variables. Significance was set as $\\mathsf{p}\\leq 0.05$ . Values are given as Mean $\\pm \\mathrm{SD}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,641,395,656]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Results ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,665,939,777]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Of the 21 participants recruited at the beginning of the study, 11 (52%) individuals were AMS + while the others were No-AMS (The average of AMS in three time point: $10.10 \\pm 0.55$ in AMS + vs. $0.9 \\pm 0.59$ in No-AMS). Difficulty sleeping and headache were the most frequent symptoms of AMS (72% and 66%, respectively), followed by gastrointestinal symptoms (34%), fatigue (20%) and dizziness (13%). Lake Louise score in 1, 6 and 24 hrs after exposure to high altitude was shown in Table 1. It should be noted that 2 subjects with AMS, several hours after exposure to hypoxia, due to AMS severity transferred to the sea level. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,795,395,810]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sea level ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,818,939,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The level of Orexin-A in No-AMS group was $1256 \\pm 482 \\, \\mathrm{pg/ml}$ and $432.8 \\pm 51.0 \\, \\mathrm{pg/ml}$ in AMS + group ( $T = -5.6$ , $P = 0.0001$ ). Moreover, HIF-1 level in No-AMS group was higher than AMS + group ( $14.2 \\pm 7.9 \\, \\mathrm{ng/ml}$ vs. $2.7 \\pm 0.3 \\, \\mathrm{ng/ml}$ ; $T = -4.8$ , $P = 0.0001$ ). Similarly, both serum VEGF and Endothelin-1 was higher in the No-AMS group when compared to the AMS+ group (all $P < 0.01$ ). Finally, $\\mathrm{SpO}_2$ was comparable between groups ( $T = 0.5$ , $P = 0.6$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,45,148,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"header"},{"bbox":"[797,49,942,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acute mountain scikness ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[53,944,491,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[906,945,939,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3/11 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[321,109,942,179]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0425/3059f3439724f4b78af0a965205db6a1fd8f341202b434201f0053c3cefb5411.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":"<table><tr><td>After high altitude exposure</td><td>AMS+</td><td>No-AMS</td></tr><tr><td>1 h</td><td>9.09±0.653*</td><td>0.30±0.153</td></tr><tr><td>6 hrs</td><td>13.09±0.513*†</td><td>1.60±0.306†</td></tr><tr><td>24 hrs</td><td>8.11±0.512*</td><td>0.80±0.133</td></tr></table>","table_caption":"[\"Table 1. Lake Louise score after exposure to high altitude (3550 m). \"]","table_footnote":"[\"* Significant difference between AMS and No-AMS groups \",\"† Significant difference compared 1 and 24 hrs after high altitude exposure. Sig≤0.05, values are Mean±SD. \",\"Also, the general characteristic of the volunteers in the two groups are presented in Table 2. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[321,241,573,255]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.1371/journal.pone.0292173.t001 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,272,849,290]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Exposure to Hypoxia (1 and 24 h after exposure to high altitude) ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[321,294,933,358]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Orexin-A. In No-AMS group, Orexin-A level increased $10\\%$ and $35\\%$ at 1 and $24\\mathrm{h}$ after exposure to high altitude (respectively, $\\mathrm{P} = 0.01$ and $\\mathrm{P} = 0.001$ ) in comparison to sea level (Fig 1). The level of Orexin-A at 24 hrs after exposure to high altitude was more than $1\\mathrm{h}$ after exposure to high altitude $(\\mathrm{P} = 0.001)$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,359,924,407]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In $\\mathrm{AMS + }$ group, circulating Orexin-A rose $15\\%$ $(\\mathrm{P} = 0.01)$ and $36\\%$ $(\\mathrm{P} = 0.0001)$ in 1 and $24\\mathrm{h}$ after exposure to altitude in comparison to sea level. Serum Orexin-A was more in $24\\mathrm{h}$ after exposure to altitude compared to $1\\mathrm{h}$ after exposure to altitude $(\\mathrm{P} = 0.0001)$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,409,936,441]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Also, serum Orexin-A was significantly different between $\\mathrm{AMS + }$ and No-AMS at the 1 and $24\\mathrm{h}$ after exposure to high altitude (respectively, $\\mathfrak{p} = 0.0001$ and $\\mathfrak{p} = 0.0001$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,441,939,523]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hypoxia Inducible Factor-1 (HIF-1). As shown in Fig 2, HIF-1 increased $6\\%$ $(P = 0.2)$ and $12\\%$ $(P = 0.03)$ respectively at 1 and $24\\mathrm{h}$ after exposure to altitude in compared to sea level in No-AMS group. Circulating HIF-1 rose $7\\%$ $(P = 0.7)$ and $19\\%$ $(P = 0.01)$ in 1 and $24\\mathrm{h}$ after exposure to altitude in comparison sea level in AMS+ group. Furthermore, there was no significant difference between 1 and $24\\mathrm{h}$ after exposure to high altitude in this group $(P = 0.5)$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,523,941,556]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A comparison between groups showed that the levels of HIF-1 at 1 $(\\mathrm{P} = 0.0001)$ and 24 hrs $(\\mathrm{P} = 0.0001)$ after exposure to high altitude in No-AMS group was more than $\\mathrm{AMS + }$ group (Fig 2). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,556,939,622]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Vascular Endothelial Growth Factor (VEGF). VEGF increased significantly 1 and $24\\mathrm{h}$ after exposure to high altitude in compared to sea level (respectively $17\\%$ and $34\\%$ ) in No-AMS (Fig 3). Furthermore VEGF at $24\\mathrm{h}$ after exposure to high altitude increased $15\\%$ in compared $1\\mathrm{h}$ after exposure to high altitude $(\\mathrm{P} = 0.01)$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,623,934,719]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Serum VEGF increased in 1 and $24\\mathrm{h}$ after exposure to high altitude compared to sea level (respectively, $7\\%$ $(p = 0.07)$ and $28\\%$ $(p = 0.01)$ ) in AMS+ group. VEGF concentration in this group at $24\\mathrm{h}$ after exposure to altitude was more in comparison to $1\\mathrm{h}$ after exposure to high altitude $(P = 0.01)$ . At both time points of hypoxia, VEGF concentration was elevated in those who did not develop AMS as compared with those who did (respectively, $P = 0.006$ and $P = 0.01$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,753,942,857]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0299/4139ae1e6d638b52561bbb18f097ff3913f9e2c0a706dc707402f67ac5682e48.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":"<table><tr><td>Variable</td><td>AMS+ (n = 11)</td><td>No-AMS (n = 10)</td></tr><tr><td>Sex</td><td>Female (4)/male(7)</td><td>Female (4)/male(6)</td></tr><tr><td>Age, years</td><td>36.91±1.69</td><td>36.50±1.58</td></tr><tr><td>Body weight, kg</td><td>70.55±3.95</td><td>67.65±2.21</td></tr><tr><td>Body Height, m</td><td>1.75 ±0.30</td><td>1.72±0.28</td></tr><tr><td>BMI, kg/m2</td><td>23.1±1.4</td><td>23.4±1.09</td></tr></table>","table_caption":"[\"Table 2. Baseline characteristics of the subjects. \"]","table_footnote":"[\"Values are Mean±SD. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[321,888,573,902]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.1371/journal.pone.0292173.t002 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[53,45,151,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"header"},{"bbox":"[797,47,942,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acute mountain scikness ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[53,941,493,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[906,944,939,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4/11 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[312,95,944,465]","code_body":null,"code_caption":null,"image_caption":"[\"Fig 1. Orexin-A response to exposure to high altitude in No-AMS and AMS+ groups. * Significant difference between AMS and No-AMS groups, ** Significant different vs. Sea level, # Significant different vs. 1 h after exposure to high altitude. Sig≤0.05, values are Mean±SD. \"]","image_footnote":"[]","img_path":"images/0164/691494fbc94e2d8199711d3ac2205bb855481094f1324c8091f4641452fb2fca.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[313,513,566,527]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.1371/journal.pone.0292173.g001 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[320,539,941,669]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Endothelin-1. Endothelin-1 increased $22\\%$ $(P = 0.006)$ and $38\\%$ $(P = 0.0001)$ at 1 and 24 h after exposure to altitude in compared to sea level in No-AMS group (Fig 4). Endothelin-1, $24\\mathrm{h}$ after exposure to high altitude increased significantly in compared $1\\mathrm{h}$ after exposure to altitude in this group $(P = 0.03)$ . Although there was a rise in Endothelin-1 at $1\\mathrm{h}$ $(10\\%)$ and $24\\mathrm{h}$ $(20\\%)$ after exposure to high altitude (respectively, $P = 0.03$ and $P = 0.02$ ) in AMS+ group, this was not different between time points $(P = 1.0)$ . Endothelin-1 concentration after exposure to hypoxia in AMS-resistant individuals was greater in compared to subjects who remained apparently healthy $(P < 0.05)$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[320,670,937,752]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\mathbf{SpO}_2$ . The mean of $\\mathrm{SpO}_2$ during exposure to high altitude was $83.5 \\pm 0.3$ in No-AMS group and was $77.66 \\pm 0.4$ in AMS+ group. The mean of $\\mathrm{SpO}_2$ decreased significantly $16\\%$ in No-AMS group and $25\\%$ in AMS+ group after exposure to altitude; this difference was significant (Fig 5). As well as, we found a significant difference for the mean of $\\mathrm{SpO}_2$ between the AMS+ study group and the No-AMS study group ( $T = 10.8$ , $P = 0.0001$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[320,753,939,818]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As shown in Table 3, at baseline, The Lake Louise score inversely correlate with Orexin-A, HIF-1, VEGF and Endothelin-1, as well as, no correlation was found between $\\mathrm{SpO}_2$ at sea level and LLS. In 1 and $24\\mathrm{h}$ after exposure to high altitude Lake Louise score was significant inversely correlation with Orexin-A, VEGF, HIF-1, Endothelin-1 and $\\mathrm{SpO}_2$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,843,433,859]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Discussion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[320,866,939,898]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The aim of the present study was to assess the potential role of the four hypoxia-related proteins in AMS pathogenesis. Taken together, the study's main findings were that serum Orexin-","text_format":null,"text_level":null,"type":"text"},{"bbox":"[53,44,151,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"header"},{"bbox":"[797,47,942,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acute mountain sickness ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[53,941,495,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[906,944,939,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5/11 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[320,95,941,481]","code_body":null,"code_caption":null,"image_caption":"[\"Fig 2. HIF-1 response to exposure to high altitude in No-AMS and $\\\\mathrm{AMS + }$ groups. \\\\* Significant difference between AMS and No-AMS groups, \\\\*\\\\* Significant different vs. Sea level, # Significant different vs. $1\\\\textrm{h}$ after exposure to high altitude. $\\\\mathrm{Sig}\\\\leq 0.05$ , values are Mean $\\\\pm \\\\mathrm{SD}$ \"]","image_footnote":"[]","img_path":"images/0162/1e3e2295d28a82c1f9d3839cd625a9534f9b847d45dd4a98f73c299987bcd729.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[321,530,578,542]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.1371/journal.pone.0292173.g002 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[320,560,934,657]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A, HIF-1, VEGF and Endothelin-1 at sea level was more in AMS-resistant individuals, and all of them increased after exposure to altitude in No-AMS and $\\mathrm{AMS + }$ ; however, the magnitude of the Orexin-A response to the altitude at 1 and $24\\mathrm{h}$ after exposure to hypoxia in individuals with $\\mathrm{AMS + }$ was approximately identical with No-AMS group. Elevation in HIF-1 24 hrs after exposure to altitude in $\\mathrm{AMS + }$ was more than No-AMS. In contrast, the increase in VEGF and Endothelin 1 and 24 hrs after exposure to altitude in No-AMS were more than $\\mathrm{AMS + }$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[320,660,934,773]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this study, difficulty sleeping and headache were the most common symptoms of individual with AMS. Meanwhile, Orexin-A has pivotal role in difficulty sleeping and sleep/wake regulation [12]. In support of our findings, Liu and colleagues confirmed that hypoxia increases the expression of Orexin-A and this change leads to sleep fragmentation after exposure to altitude [13]. Since Orexin-A to altitude was similar between groups, it is possible that other factors that regulate the sleep cycle are participate in difficulty sleeping after exposure to altitude in $\\mathrm{AMS + }$ group. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[320,773,941,888]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"At sea level, 1 and 24 hrs after exposure to altitude, HIF-1 levels in $\\mathrm{AMS + }$ group was less than No-AMS group. In this regards, Kline and coworkers showed that HIF-1 partial deficient impaired ventilatory responses to chronic hypoxia [14]. The ventilatory response to the hypoxic is the main physiological response of the body at altitude, which can be directly related to adaptation to altitude and resistance to AMS. HIF-1 Elevations in HIF-1 24 hrs after exposure to high altitude in $\\mathrm{AMS + }$ was more compared to the No-AMS. It seems likely that HIF-responsive proteins such as VEGF and Endothelin-1 may predispose an individual to AMS. VEGF ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[53,44,151,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"header"},{"bbox":"[797,47,942,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acute mountain sickness ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[53,941,495,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[906,944,939,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6/11 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[320,95,916,478]","code_body":null,"code_caption":null,"image_caption":"[\"Fig 3. VEGF response to exposure to high altitude in No-AMS and $\\\\mathrm{AMS + }$ groups. \\\\* Significant difference between AMS and No-AMS groups, \\\\*\\\\* Significant different vs. Sea level, # Significant different vs. $1\\\\textrm{h}$ after exposure to high altitude. Sig $\\\\leq 0.05$ , values are Mean $\\\\pm \\\\mathrm{SD}$ \"]","image_footnote":"[]","img_path":"images/0127/7a02807f3f3699a013bfef46b8e3f1b2d760807c304ab743a1e7b3194618165b.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[321,525,578,539]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.1371/journal.pone.0292173.g003 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[320,551,936,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"upregulated by hypoxia and demonstrated as an increase in vascular permeability [15]. There is an increasing body of evidence suggesting that VEGF is a potent permeability factor, although its role in altitude disease is obscure. In agreement with previous investigations, VEGF increased 1 and $24\\mathrm{h}$ after exposure to hypoxia in $\\mathrm{AMS + }$ and No-AMS groups, but this change was more in No-AMS group (respectively $7\\%$ vs. $18\\%$ and $16\\%$ vs. $35\\%$ ). In contrast of our findings Harrison et al. showed that serum VEGF at baseline was not different between $\\mathrm{AMS + }$ and No-AMS group; however, they did report that serum VEGF elevation was more in $\\mathrm{AMS + }$ group in comparison to No-AMS group at 3 days after exposure to altitude ( $43\\%$ vs. $26\\%$ ) [16]. A probable explanation for the difference between the results of this study and the findings of Harrison related to the individual variability and different ascent profile, including the altitude at which measurements were made, duration of high altitude stay and the altitude of residency of the subjects. The mechanism behind VEGF elevation in response to hypoxia is unclear; however, it remains possible that less serum VEGF in subjects more susceptible to AMS may in part be due to enhanced renal VEGF clearance and fluid retention. For example, Maloney et al. showed that decrease in circulating VEGF after exposure to altitude may indicate protection against the development of VEGF-mediated capillary leak in organs such as the brain [17]. Furthermore, Kendall et al. showed that soluble receptors act to sequester unbound VEGF from the circulation, therefore preventing its interaction with the endothelium-bound receptors that mediate its effects upon endothelial function and vascular permeability [18]. While some studies have reported a relation between serum VEGF and AMS susceptibility, the majority did not [17, 19, 20]. It should be noted that MRI brain scans ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[53,45,151,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"header"},{"bbox":"[797,47,942,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acute mountain scikness ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[53,941,495,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[906,944,941,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7/11 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[320,95,916,459]","code_body":null,"code_caption":null,"image_caption":"[\"Fig 4. Endothelin-1 response to exposure to high altitude in No-AMS and $\\\\mathrm{AMS + }$ groups.\\\\* Significant difference between AMS and No-AMS groups, \\\\*\\\\* Significant different vs. Sea level, # Significant different vs. $1\\\\textrm{h}$ after exposure to high altitude. $\\\\mathrm{Sig}\\\\leq 0.05$ , values are Mean±SD. \"]","image_footnote":"[]","img_path":"images/0468/ac215c9f364624dbfe2c0576ba310cb30a1797fcba23a5809c3728e9cf4c3ec7.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[321,508,578,522]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.1371/journal.pone.0292173.g004 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[251,537,942,851]","code_body":null,"code_caption":null,"image_caption":"[\"Fig 5. $\\\\mathrm{SpO}_2$ changes in response to exposure to high altitude in No-AMS and $\\\\mathrm{AMS + }$ $^*$ show significant difference between $\\\\mathrm{AMS + }$ and No-AMS groups. $\\\\mathrm{Sig}\\\\leq 0.05$ , values are Mean $\\\\pm \\\\mathrm{SD}$ \"]","image_footnote":"[]","img_path":"images/0162/663a6305e55e45b2c15fc1dc5a6a1ff71fb075f1f8f6884a81f318b32783618e.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[253,887,509,901]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.1371/journal.pone.0292173.g005 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[53,44,151,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"header"},{"bbox":"[797,47,942,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acute mountain scikness ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[53,941,493,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[906,944,939,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"8/11 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[321,119,942,425]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0508/05a871f3c5933c24cc3ab9540ed9a90a9610d785326d370d41da53e2647a18cb.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":"<table><tr><td>Factors</td><td>Pearson Correlation</td><td>P value</td></tr><tr><td>Sea level</td><td></td><td></td></tr><tr><td>Orexin-A</td><td>-0.831</td><td>0.0001</td></tr><tr><td>HIF-1</td><td>-0.789</td><td>0.0001</td></tr><tr><td>VEGF</td><td>-0.774</td><td>0.0001</td></tr><tr><td>Endothelin-1</td><td>-0.691</td><td>0.001</td></tr><tr><td>SpO2</td><td>-0.181</td><td>0.43</td></tr><tr><td>1 h after exposure to altitude</td><td></td><td></td></tr><tr><td>Orexin-A</td><td>-0.817</td><td>0.0001</td></tr><tr><td>HIF-1</td><td>-0.803</td><td>0.0001</td></tr><tr><td>VEGF</td><td>-0.769</td><td>0.0001</td></tr><tr><td>Endothelin-1</td><td>-0.710</td><td>0.0001</td></tr><tr><td>SpO2</td><td>-0.697</td><td>0.0001</td></tr><tr><td>24 h after exposure to altitude</td><td></td><td></td></tr><tr><td>Orexin-A</td><td>-0.774</td><td>0.0001</td></tr><tr><td>HIF-1</td><td>-0.818</td><td>0.0001</td></tr><tr><td>VEGF</td><td>-0.753</td><td>0.0001</td></tr><tr><td>Endothelin-1</td><td>-0.770</td><td>0.0001</td></tr><tr><td>SpO2 mean</td><td>-0.884</td><td>0.0001</td></tr></table>","table_caption":"[\"Table 3. The correlation between hypoxia related proteins with LLS at sea level and 1 and 24 hrs after exposure to high altitude. \"]","table_footnote":"[\"https://doi.org/10.1371/journal.pone.0292173.t003 \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[325,457,941,537]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"following hypoxic exposure showed that brain swelling was not different between subjects who developed AMS and those who did not (An indirect argument against a role of VEGF in AMS) [19, 21]. Also, the concentration of VEGF in the cerebrospinal fluid is not different between subjects with and without AMS [22]. These findings demonstrated that vasogenic edema is not involved in the pathophysiology of AMS. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,539,939,654]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Previous experimental studies suggests that increased brain volume with hypobaric hypoxia elevates intracranial pressure and, when accompanied by impaired or diminished intracranial buffering capacity, contributes to the development of the symptoms that define AMS (tight-fit hypothesis). On the other hand, Julian and coworker for the first time showed that resistance to AMS may be driven, in part, by the ability to mount an adequate \"defensive\" anti-inflammatory or anti-permeability response after exposure to altitude [23]. These findings lend deep insight into the pathophysiology of AMS. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,654,939,719]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The increase of Endothelin-1 after exposure to hypoxia in both groups are in line with previous studies [7, 24]. In contrast to these studies and our findings, Barker et al. showed that Endothelin-1 was not different between those with and without AMS susceptibility [6]. It was concluded that Endothelin -1 have a pathological role in HAPE instead of AMS [6]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,720,929,785]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The mean $\\mathrm{SpO}_2$ of the AMS+ group after exposure to altitude was significantly lower than that of the No-AMS group. In contrast out findings Wagner et al. showed that there was no significant difference in the mean oxygen saturations at summit or average altitude between the AMS+ and No-AMS groups [25]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,785,934,883]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"AMS symptoms do not appear immediately after exposure to altitude, it may last up to 12 hrs; therefore, if AMS can be predicted before it occurs, there will be enough time to prevent its possible adverse consequences. In fact, hypoxia related proteins (Orexin-A, HIF-1, VEGF and Endothelin-1) evaluation at sea level could be consider as a predictive indices to identify acute mountain sickness susceptibility individuals unaccustomed to high altitude. It should to be noted that the most important limitation of this study is that the sample size is not big enough. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[53,45,151,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"header"},{"bbox":"[797,47,942,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acute mountain scikness ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[53,944,493,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[906,944,939,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"9/11 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[325,94,433,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conclusion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,118,936,248]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The results of this study in summary showed that hypoxia related proteins include Orexin-A, HIF-1, VEGF and Endothelin-1 were higher in the AMS-resistant individuals when compared to AMS susceptible individuals at sea level and these proteins did not show consistent changes after exposure to high altitude. Susceptibility to AMS may be associated with a high response HIF-1 and more $\\mathrm{SpO}_2$ reduction to hypoxia. To better determine the underlying factors in AMS etiology, future work in this area should focus on genetic and local measurement of other factors. Further studies are needed to validate these measurements, but they suggest an alternative approach to obtain a more accurate simple method of predicting AMS. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,272,542,289]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Supporting information ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,295,913,342]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"S1 Data. ELISA data set of Orexin, HIF-1, VEGF and Endothelin-1. Abbreviation: AMS: Acute Mountain Sickness. (XLS) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,375,498,392]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acknowledgments ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,398,926,429]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The authors thank Prof Philip Ainslie (University of British Columbia, Canada) for feedback and edit of the paper. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,454,522,469]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Author Contributions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,478,516,492]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Data curation: Bayan Fayazi. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,500,511,515]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Methodology: Bayan Fayazi. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,523,503,537]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Supervision: Vahid Tadibi. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,545,596,560]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Writing - original draft: Kamal Ranjbar. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,568,616,583]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Writing - review & editing: Kamal Ranjbar. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,614,429,628]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[330,633,939,895]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. Liu B, Huang H, Wu G, Xu G, Sun B-D, Zhang E-L, et al. A signature of circulating microRNAs predicts the susceptibility of acute mountain sickness. Frontiers in physiology. 2017; 8. https://doi.org/10.3389/fphys.2017.00055 PMID: 28228730 \",\"2. MacInnis MJ, Wang P, Koehle MS, Rupert JL. The genetics of altitude tolerance: the evidence for inherited susceptibility to acute mountain sickness. Journal of occupational and environmental medicine. 2011; 53(2):159-68. https://doi.org/10.1097/JOM.0b013e318206b112 PMID: 21270658 \",\"3. Chen H-C, Lin W-L, Wu J-Y, Wang S-H, Chiu T-F, Weng Y-M, et al. Change in oxygen saturation does not predict acute mountain sickness on Jade Mountain. Wilderness & environmental medicine. 2012; 23(2):122-7. \",\"4. Painschab MS, Malpartida GE, Dávila-Roman VG, Gilman RH, Kolb TM, León-Velarde F, et al. Association between serum concentrations of hypoxia inducible factor responsive proteins and excessive Erythrocytosis in high altitude Peru. High altitude medicine & biology. 2015; 16(1):26-33. https://doi.org/10.1089/ham.2014.1086 PMID: 25760230 \",\"5. Ding H, Liu Q, Hua M, Ding M, Du H, Zhang W, et al. Polymorphisms of hypoxia-related genes in subjects susceptible to acute mountain sickness. Respiration. 2011; 81(3):236-41. https://doi.org/10.1159/000322850 PMID: 21242666 \",\"6. Barker KR, Conroy AL, Hawkes M, Murphy H, Pandey P, Kain KC. Biomarkers of hypoxia, endothelial and circulatory dysfunction among climbers in Nepal with AMS and HAPE: a prospective case-control study. Journal of travel medicine. 2016; 23(3):taw005. https://doi.org/10.1093/jtm/taw005 PMID: 26984355 \"]","page_idx":"9","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[53,45,151,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"header"},{"bbox":"[797,47,942,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acute mountain scikness ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[53,944,493,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[901,944,939,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10/11 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[325,93,942,858]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"7. Boos CJ, Woods DR, Varias A, Biscocho S, Heseltine P, Mellor AJ. High altitude and acute mountain sickness and changes in circulating endothelin-1, interleukin-6, and interleukin-17a. High altitude medicine & biology. 2016; 17(1):25-31. https://doi.org/10.1089/ham.2015.0098 PMID: 26680502 \",\"8. Xu T-R, Yang Y, Ward R, Gao L, Liu Y. Orexin receptors: multi-functional therapeutic targets for sleeping disorders, eating disorders, drug addiction, cancers and other physiological disorders. Cellular signalling. 2013; 25(12):2413-23. https://doi.org/10.1016/j.cellsig.2013.07.025 PMID: 23917208 \",\"9. Du MK, Hunt NJ, Waters KA, Machaalani R. Cumulative effects of repetitive intermittent hypercapnic hypoxia on orexin in the developing piglet hypothalamus. International Journal of Developmental Neuroscience. 2016; 48:1-8. https://doi.org/10.1016/j.ijdevneu.2015.10.007 PMID: 26548856 \",\"10. Wang L, Zhou Y, Li M, Zhu Y. Expression of hypoxia-inducible factor-1α, endothelin-1 and adrenomedullin in newborn rats with hypoxia-induced pulmonary hypertension. Experimental and therapeutic medicine. 2014; 8(1):335-9. \",\"11. Bp R, Hackett P, Oelz O, editors. The Lake Louise acute mountain sickness scoring system. Hypoxia and Mountain Medicine: proceeding of the International hypoxia Symposium; 1993. \",\"12. Sokolowska P, Urbańska A, Biegańska K, Wagner W, Ciszewski W, Namiecińska M, et al. Orexins protect neuronal cell cultures against hypoxic stress: an involvement of Akt signaling. Journal of Molecular Neuroscience. 2014; 52(1):48-55. https://doi.org/10.1007/s12031-013-0165-7 PMID: 24243084 \",\"13. Liu Z, Jiang L, Zhu F, Fu C, Lu S, Zhou J, et al. Chronic intermittent hypoxia and the expression of orexin and its receptors in the brains of rats. Sleep and Biological Rhythms. 2014; 12(1):22-9. \",\"14. Kline DD, Peng Y-J, Manalo DJ, Semenza GL, Prabhakar NR. Defective carotid body function and impaired ventilatory responses to chronic hypoxia in mice partially deficient for hypoxia-inducible factor 1α. Proceedings of the National Academy of Sciences. 2002; 99(2):821-6. \",\"15. Schoch HJ, Fischer S, Marti HH. Hypoxia-induced vascular endothelial growth factor expression causes vascular leakage in the brain. Brain. 2002; 125(11):2549-57. https://doi.org/10.1093/brain/awf257 PMID: 12390979 \",\"16. Harrison MF, Anderson PJ, Miller AD, O'Malley KA, Richert ML, Johnson JB, et al. Physiological variables associated with the development of acute mountain sickness at the South Pole. BMJ open. 2013; 3(7):e003064. https://doi.org/10.1136/bmjopen-2013-003064 PMID: 23869103 \",\"17. Maloney J, Wang D, Duncan T, Voelkel N, Ruoss S. Plasma vascular endothelial growth factor in acute mountain sickness. Chest Journal. 2000; 118(1):47-52. https://doi.org/10.1378/chest.118.1.47 PMID: 10893358 \",\"18. Kendall RL, Wang G, Thomas KA. Identification of a natural soluble form of the vascular endothelial growth factor receptor, FLT-1, and its heterodimerization with KDR. Biochemical and biophysical research communications. 1996; 226(2):324-8. https://doi.org/10.1006/bbrc.1996.1355 PMID: 8806634 \",\"19. Schommer K, Wiesegart N, Dehnert C, Mairbäurl H, Bärsch P. No correlation between plasma levels of vascular endothelial growth factor or its soluble receptor and acute mountain sickness. High altitude medicine & biology. 2011; 12(4):323-7. https://doi.org/10.1089/ham.2011.1020 PMID: 22206557 \",\"20. Ding H, Liu Q, Hua M, Ding M, Du H, Zhang W, et al. Associations between vascular endothelial growth factor gene polymorphisms and susceptibility to acute mountain sickness. Journal of International Medical Research. 2012; 40(6):2135-44. https://doi.org/10.1177/030006051204000611 PMID: 23321170 \",\"21. Schoonman GG, Sander PS, Nirkko AC, Lange T, Jaermann T, Dydak U, et al. Hypoxia-induced acute mountain sickness is associated with intracellular cerebral edema: a 3 T magnetic resonance imaging study. Journal of Cerebral Blood Flow & Metabolism. 2008; 28(1):198-206. https://doi.org/10.1038/sj.jcbfm.9600513 PMID: 17519973 \",\"22. Bailey DM, Roukens R, Knauth M, Kallenberg K, Christ S, Mohr A, et al. Free radical-mediated damage to barrier function is not associated with altered brain morphology in high-altitude headache. Journal of Cerebral Blood Flow & Metabolism. 2006; 26(1):99-111. \",\"23. Julian CG, Subudhi AW, Wilson MJ, Dimmen AC, Pecha T, Roach RC. Acute mountain sickness, inflammation, and permeability: new insights from a blood biomarker study. Journal of Applied Physiology. 2011; 111(2):392-9. https://doi.org/10.1152/japplphysiol.00391.2011 PMID: 21636566 \",\"24. Sartori C, Vollenweider L, Löffler B-M, Delabays A, Nicod P, Bärtsch P, et al. Exaggerated endothelin release in high-altitude pulmonary edema. Circulation. 1999; 99(20):2665-8. https://doi.org/10.1161/01.cir.99.20.2665 PMID: 10338460 \",\"25. Wagner DR, Knott JR, Fry JP. Oximetry fails to predict acute mountain sickness or summit success during a rapid ascent to 5640 meters. Wilderness & environmental medicine. 2012; 23(2):114-21. https://doi.org/10.1016/j.wem.2012.02.015 PMID: 22656656 \"]","page_idx":"10","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[53,45,151,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"header"},{"bbox":"[797,47,942,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acute mountain scikness ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[53,941,493,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[901,944,939,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"11/11 ","text_format":null,"text_level":null,"type":"page_number"}],"string":"[\n {\n \"bbox\": \"[53,352,148,422]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0584/2695b840c0f8f1aa4490b37234c886f48b048501e6d1a782eaa96262a39cb324.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[53,477,171,494]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 OPEN ACCESS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,503,300,585]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Citation: Fayazi B, Tadibi V, Ranjbar K (2023) The role of hypoxia related hormones responses in acute mountain sickness susceptibility individuals unaccustomed to high altitude. PLoS ONE 18(10): e0292173. https://doi.org/10.1371/journal.pone.0292173 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,594,302,621]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Editor: James West, Vanderbilt University Medical Center, UNITED STATES \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,628,191,643]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Received: October 4, 2022 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,651,210,664]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Accepted: September 14, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,672,194,686]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Published: October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,694,300,777]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Copyright: © 2023 Fayazi et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,785,297,825]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data Availability Statement: All relevant data are within the paper and its Supporting Information files. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,834,269,862]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Funding: The author(s) received no specific funding for this work. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,869,294,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Competing interests: The authors have declared that no competing interests exist. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,116,455,128]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RESEARCH ARTICLE \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,133,890,247]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The role of hypoxia related hormones responses in acute mountain sickness susceptibility individuals unaccustomed to high altitude \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,258,669,276]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bayan Fayazi<sup>1</sup>, Vahid Tadibi<sup>1*</sup>, Kamal Ranjbar<sup>2*</sup> \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[320,284,897,324]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 Department of Exercise Physiology, Faculty of Sport Sciences, Razi University, Kermanshah, Iran, 2 Department of Exercise Physiology, Faculty of Sport Sciences, Bandar Abbas Branch, Islamic Azad University, Bandar Abbas, Iran \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,334,705,349]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"* Vtadibi@yahoo.com (VT); Kamal_ranjbar2010@yahoo.com (KR) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,380,418,398]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[320,412,944,748]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acute mountain sickness (AMS) is caused by rapid ascent to altitude $(>2500\\\\mathrm{m})$ and remains a poorly understood pathophysiological condition. Accordingly, we investigated the relationship between acute exposure to high altitude and hypoxia related biochemical proteins. 21 healthy subjects (Female (8) and male (13), Age: $36.7\\\\pm 8.5$ , BMI: $23.2\\\\pm 3.1$ ) volunteers participated in this project and fasting blood samples were taken before (sea level) and after 1 and 24-h exposure to high altitude $(3,550\\\\mathrm{m})$ . Blood oxygen saturation $(\\\\mathrm{SpO}_2)$ , AMS status (Lake Louise Score) and serum HIF-1, Endothelin-1, VEGF and Orexin-A were measured (via ELISA) at 1, 6 and 24 h after exposure to high altitude. Pre-ascent measurement of hypoxia related proteins (Orexin-A, HIF-1, VEGF and Endothelin-1) where all significantly (<0.05) higher in the AMS-resistant individuals (No-AMS) when compared to AMS susceptible individuals (AMS+). Upon ascent to high altitude, 11 out of 21 volunteers had AMS $(10.1\\\\pm 0.6$ in AMS+ vs. $0.9\\\\pm 0.6$ in No-AMS, $P < 0.05$ ) and presented with lower resting $\\\\mathrm{SpO}_2$ levels $(77.7\\\\pm 0.4$ vs. $83.5\\\\pm 0.3$ respectively, $p < 0.05)$ . Orexin-A, HIF-1, VEGF and Endothelin-1, significantly increased 24 hrs after exposure to high altitude in both AMS+ and No-AMS. The response of Orexin-A was similar between two groups, also, HIF-1 elevation 24 hrs after exposure to altitude was more in AMS+ $(13\\\\%$ vs. $19\\\\%)$ , but the increase of VEGF and Endothelin-1, 1 and 24 hrs after exposure to altitude in No-AMS was double that of AMS+. Hypoxia related proteins include Orexin-A, HIF-1, VEGF and Endothelin-1 may play a pathophysiological role in those who are susceptible to AMS. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,810,441,825]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[320,833,939,898]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is well known that acute mountain sickness (AMS) caused by rapid ascent above $2500\\\\mathrm{m}$ is the most common physiological dysfunction that occurs at high altitude [1]. Typically, AMS is usually recognized with a combination of several symptoms, such as headache, gastrointestinal symptoms, sleep disturbance, general weakens, insomnia, and loss of appetite. Symptoms \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,37,233,63]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[53,941,495,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[906,944,941,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1/11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[325,94,934,175]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"begin two to three hours after ascent and, if unresolved or untreated, may develop to high altitude pulmonary edema (HAPE) and high altitude cerebral edema (HACE). Some people are more susceptible to AMS than others [2]; therefore, awareness of the individual susceptibility for AMS would be a useful tool to prevent or limit subsequent harm. An exact strategy to predict AMS susceptibility remains elusive [3]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,176,939,487]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Although the underlying etiology of AMS remains obscure, it is at least in part related to alterations in cellular milieu. For example, previous studies showed that Hypoxia inducible factor-1 (HIF-1) pathway have a pivotal role in altitude diseases, especially those with excessive erythrocytosis [4]. HIF-1 is a main transcription factors in hypoxia condition that acts as a key regulator of oxygen homeostasis [5], including the activation of vascular endothelial growth factor (VEGF) as a key factor in vascular permeability and basement membrane damage. Related, Ding et al showed that VEGF is an important component of the pathogenesis of AMS [5]. Also, increase of circulating Endothelin-1 during ascent to high altitude, may play pathological roles in AMS and HAPE [6]. It has been noted that Endothelin-1 is a potent vasoconstrictor and increase vascular permeability in hypoxia [7]. Finally, Orexin-A—mainly produced by hypothalamic neurons—play crucial roles in the control of feeding, sleep-wake cycle, metabolic rate and pathogenesis of narcolepsy [8]. It has been shown that confirmed that hypoxia exposure decrease Orexin-A gene expression in the piglet hypothalamus [9] and this change may be related to the hypoxia-induced behavioral deficit such as narcolepsy and anorexia at high altitude. Experiments confirmed that VEGF, Orexin-A and Endothelin-1 are some of the main target genes of HIF-1 [10]. However, the relationship between these factors and AMS still remains to be elucidated. Given that the pathophysiology of AMS remains nebulous, we aimed to compare changes—and potential relationships—between hypoxia related proteins levels between AMS-susceptible and AMS-resistant individuals. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,503,522,518]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Material and methods \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,527,424,542]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Participants \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,550,939,662]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"21 healthy subjects (Female (8) and male (13), Age: $36.7 \\\\pm 8.5$ , BMI: $23.2 \\\\pm 3.1$ ) voluntary participated in this study. All subjects were fit, nonsmokers, with no history of cardiopulmonary and the subjects were not allowed to take any medication. All participants were long-term lowland residents $(1190\\\\mathrm{m})$ and had not been exposed to altitudes above $2500\\\\mathrm{m}$ within 30 days ago. All participants gave written informed consent to participate and the protocol of this study was approved by Razi university ethics Committee (KUMS.REC.1396.255) and all experiments were performed in accordance with relevant guidelines and regulations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,680,429,696]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Study design \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,703,924,766]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This was an observational, prospective cohort study. Eligible participants recruited through advertisement. Inclusion criteria included young people (20-40 years old) lowland residents with $20 < \\\\mathrm{BMI} < 25$ . Exclusion criteria included chronic and acute illness, heavy activity or high-altitude ( $>2500\\\\mathrm{m}$ ) exposure history in the past month and recent drug use. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,768,929,832]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sea-level data collection. The examination lasted two days. $24\\\\mathrm{h}$ before exposure to high altitude the main physiological parameters, including fasting blood samples, heart rate, and $\\\\mathrm{SpO}_2$ were measured at rest (1190 m above sea level at 9 A.M). All volunteers had a standardized diet throughout the study. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,834,937,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Altitude data collection. Following 24 hrs, volunteers rapidly ascended from 1,190 to $3,550\\\\mathrm{m}$ (Tochal Mountain located in Central Alborz mountain range in northern Iran, Mount Damavand, the highest mountain in Iran measuring $5,610.0\\\\mathrm{m}$ (18,405.5 ft), is located in the Central Alborz Mountains) within 30 min by gondola and remained at this elevation for \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,45,148,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[797,49,942,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acute mountain scikness \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[53,944,491,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[906,945,939,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2/11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[325,95,937,157]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"a further $24\\\\mathrm{h}$ . Blood samples were taken at 1 and $24\\\\mathrm{h}$ after exposure to high altitude in a semi-recumbent position from an antecubital vein. Serum was allowed to clot for $30\\\\mathrm{min}$ before centrifugal separation and storage at $-80^{\\\\circ}\\\\mathrm{C}$ until batch analyses of HIF-1, Endothelin-1, VEGF and Orexin-A. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,161,939,257]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Oximax handheld pulse oximeter (Onyx II 9550; Nonin, Minnesota) was used for $\\\\mathrm{SpO}_2$ measurement. $\\\\mathrm{SpO}_2$ was measured every $1\\\\mathrm{h}$ during exposure to high altitude. Also, AMS was assessment by Lake Louise Acute Mountain Sickness Scoring System [11]. Subjects were classified as suffering from severe AMS and NO-AMS. Those with severe AMS trekkers were considered to be susceptible to AMS when their Lake Louise score (LLS) was $\\\\geq 4$ . AMS was evaluated in 1, 6 and 24 hrs after exposure to high altitude. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,260,939,421]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Serum HIF-1 concentrations were measured using commercially available ELISA kit: (Human Hypoxia-Inducible Factor 1 (HIF-1), ELISA, ZellBio GmbH, Ulm, Germany, Intraassay $\\\\mathrm{CV\\\\%}$ : 2 and Sensitivity: $0.1~\\\\mathrm{ng / ml}$ ), serum VEGF concentrations were measured using commercially available ELISA kit: (Human Vascular Endothelial cell Growth Factor (VEGF), ELISA, ZellBio GmbH, Ulm, Germany, Intraassay $\\\\mathrm{CV\\\\%}$ : 4.3 and Sensitivity: $10~\\\\mathrm{ng / L}$ ), serum Endothelin-1 concentrations were measured using commercially available ELISA kit: (Human Endothelin 1, ELISA, ZellBio GmbH, Ulm, Germany, Intraassay $\\\\mathrm{CV\\\\%}$ : 5.3 and Sensitivity: $1~\\\\mathrm{ng / L}$ ) and serum Orexin-A concentrations were measured using commercially available ELISA kit: (Human Orexin A, ELISA, ZellBio GmbH, Ulm, Germany, Intraassay $\\\\mathrm{CV\\\\%}$ : 3.7 and Sensitivity: $2.5~\\\\mathrm{pg / ml}$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,440,477,455]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Statistical analyses \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,462,936,622]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Statistical analyses were performed with SPSS 23.0 (IBM Corporation, Armonk, New York, USA). Normality was determined via the Shapiro-wilk test. Comparison of means was performed using independent samples t test (normally distributed variables) and Mann-Whitney test (for no normally distributed variables) for participants without AMS (No-AMS) as compared to participants with AMS $(\\\\mathrm{AMS}+)$ . Also, in order to compare means within two groups unpaired t test was used. A $2\\\\times 3$ mixed-plot factorial repeated measures ANOVA was performed to analyze for comparisons of variable changes between the two groups from sea level to $1\\\\mathrm{h}$ and $24\\\\mathrm{h}$ after exposure to high altitude. Pearson correlations test was used to determine the relationship between the study variables. Significance was set as $\\\\mathsf{p}\\\\leq 0.05$ . Values are given as Mean $\\\\pm \\\\mathrm{SD}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,641,395,656]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Results \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,665,939,777]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Of the 21 participants recruited at the beginning of the study, 11 (52%) individuals were AMS + while the others were No-AMS (The average of AMS in three time point: $10.10 \\\\pm 0.55$ in AMS + vs. $0.9 \\\\pm 0.59$ in No-AMS). Difficulty sleeping and headache were the most frequent symptoms of AMS (72% and 66%, respectively), followed by gastrointestinal symptoms (34%), fatigue (20%) and dizziness (13%). Lake Louise score in 1, 6 and 24 hrs after exposure to high altitude was shown in Table 1. It should be noted that 2 subjects with AMS, several hours after exposure to hypoxia, due to AMS severity transferred to the sea level. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,795,395,810]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sea level \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,818,939,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The level of Orexin-A in No-AMS group was $1256 \\\\pm 482 \\\\, \\\\mathrm{pg/ml}$ and $432.8 \\\\pm 51.0 \\\\, \\\\mathrm{pg/ml}$ in AMS + group ( $T = -5.6$ , $P = 0.0001$ ). Moreover, HIF-1 level in No-AMS group was higher than AMS + group ( $14.2 \\\\pm 7.9 \\\\, \\\\mathrm{ng/ml}$ vs. $2.7 \\\\pm 0.3 \\\\, \\\\mathrm{ng/ml}$ ; $T = -4.8$ , $P = 0.0001$ ). Similarly, both serum VEGF and Endothelin-1 was higher in the No-AMS group when compared to the AMS+ group (all $P < 0.01$ ). Finally, $\\\\mathrm{SpO}_2$ was comparable between groups ( $T = 0.5$ , $P = 0.6$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,45,148,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[797,49,942,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acute mountain scikness \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[53,944,491,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[906,945,939,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3/11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[321,109,942,179]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0425/3059f3439724f4b78af0a965205db6a1fd8f341202b434201f0053c3cefb5411.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>After high altitude exposure</td><td>AMS+</td><td>No-AMS</td></tr><tr><td>1 h</td><td>9.09±0.653*</td><td>0.30±0.153</td></tr><tr><td>6 hrs</td><td>13.09±0.513*†</td><td>1.60±0.306†</td></tr><tr><td>24 hrs</td><td>8.11±0.512*</td><td>0.80±0.133</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 1. Lake Louise score after exposure to high altitude (3550 m). \\\"]\",\n \"table_footnote\": \"[\\\"* Significant difference between AMS and No-AMS groups \\\",\\\"† Significant difference compared 1 and 24 hrs after high altitude exposure. Sig≤0.05, values are Mean±SD. \\\",\\\"Also, the general characteristic of the volunteers in the two groups are presented in Table 2. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[321,241,573,255]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.1371/journal.pone.0292173.t001 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,272,849,290]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Exposure to Hypoxia (1 and 24 h after exposure to high altitude) \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,294,933,358]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Orexin-A. In No-AMS group, Orexin-A level increased $10\\\\%$ and $35\\\\%$ at 1 and $24\\\\mathrm{h}$ after exposure to high altitude (respectively, $\\\\mathrm{P} = 0.01$ and $\\\\mathrm{P} = 0.001$ ) in comparison to sea level (Fig 1). The level of Orexin-A at 24 hrs after exposure to high altitude was more than $1\\\\mathrm{h}$ after exposure to high altitude $(\\\\mathrm{P} = 0.001)$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,359,924,407]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In $\\\\mathrm{AMS + }$ group, circulating Orexin-A rose $15\\\\%$ $(\\\\mathrm{P} = 0.01)$ and $36\\\\%$ $(\\\\mathrm{P} = 0.0001)$ in 1 and $24\\\\mathrm{h}$ after exposure to altitude in comparison to sea level. Serum Orexin-A was more in $24\\\\mathrm{h}$ after exposure to altitude compared to $1\\\\mathrm{h}$ after exposure to altitude $(\\\\mathrm{P} = 0.0001)$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,409,936,441]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Also, serum Orexin-A was significantly different between $\\\\mathrm{AMS + }$ and No-AMS at the 1 and $24\\\\mathrm{h}$ after exposure to high altitude (respectively, $\\\\mathfrak{p} = 0.0001$ and $\\\\mathfrak{p} = 0.0001$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,441,939,523]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hypoxia Inducible Factor-1 (HIF-1). As shown in Fig 2, HIF-1 increased $6\\\\%$ $(P = 0.2)$ and $12\\\\%$ $(P = 0.03)$ respectively at 1 and $24\\\\mathrm{h}$ after exposure to altitude in compared to sea level in No-AMS group. Circulating HIF-1 rose $7\\\\%$ $(P = 0.7)$ and $19\\\\%$ $(P = 0.01)$ in 1 and $24\\\\mathrm{h}$ after exposure to altitude in comparison sea level in AMS+ group. Furthermore, there was no significant difference between 1 and $24\\\\mathrm{h}$ after exposure to high altitude in this group $(P = 0.5)$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,523,941,556]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A comparison between groups showed that the levels of HIF-1 at 1 $(\\\\mathrm{P} = 0.0001)$ and 24 hrs $(\\\\mathrm{P} = 0.0001)$ after exposure to high altitude in No-AMS group was more than $\\\\mathrm{AMS + }$ group (Fig 2). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,556,939,622]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Vascular Endothelial Growth Factor (VEGF). VEGF increased significantly 1 and $24\\\\mathrm{h}$ after exposure to high altitude in compared to sea level (respectively $17\\\\%$ and $34\\\\%$ ) in No-AMS (Fig 3). Furthermore VEGF at $24\\\\mathrm{h}$ after exposure to high altitude increased $15\\\\%$ in compared $1\\\\mathrm{h}$ after exposure to high altitude $(\\\\mathrm{P} = 0.01)$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,623,934,719]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Serum VEGF increased in 1 and $24\\\\mathrm{h}$ after exposure to high altitude compared to sea level (respectively, $7\\\\%$ $(p = 0.07)$ and $28\\\\%$ $(p = 0.01)$ ) in AMS+ group. VEGF concentration in this group at $24\\\\mathrm{h}$ after exposure to altitude was more in comparison to $1\\\\mathrm{h}$ after exposure to high altitude $(P = 0.01)$ . At both time points of hypoxia, VEGF concentration was elevated in those who did not develop AMS as compared with those who did (respectively, $P = 0.006$ and $P = 0.01$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,753,942,857]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0299/4139ae1e6d638b52561bbb18f097ff3913f9e2c0a706dc707402f67ac5682e48.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Variable</td><td>AMS+ (n = 11)</td><td>No-AMS (n = 10)</td></tr><tr><td>Sex</td><td>Female (4)/male(7)</td><td>Female (4)/male(6)</td></tr><tr><td>Age, years</td><td>36.91±1.69</td><td>36.50±1.58</td></tr><tr><td>Body weight, kg</td><td>70.55±3.95</td><td>67.65±2.21</td></tr><tr><td>Body Height, m</td><td>1.75 ±0.30</td><td>1.72±0.28</td></tr><tr><td>BMI, kg/m2</td><td>23.1±1.4</td><td>23.4±1.09</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 2. Baseline characteristics of the subjects. \\\"]\",\n \"table_footnote\": \"[\\\"Values are Mean±SD. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[321,888,573,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.1371/journal.pone.0292173.t002 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,45,151,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[797,47,942,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acute mountain scikness \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[53,941,493,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[906,944,939,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4/11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[312,95,944,465]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig 1. Orexin-A response to exposure to high altitude in No-AMS and AMS+ groups. * Significant difference between AMS and No-AMS groups, ** Significant different vs. Sea level, # Significant different vs. 1 h after exposure to high altitude. Sig≤0.05, values are Mean±SD. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0164/691494fbc94e2d8199711d3ac2205bb855481094f1324c8091f4641452fb2fca.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[313,513,566,527]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.1371/journal.pone.0292173.g001 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[320,539,941,669]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Endothelin-1. Endothelin-1 increased $22\\\\%$ $(P = 0.006)$ and $38\\\\%$ $(P = 0.0001)$ at 1 and 24 h after exposure to altitude in compared to sea level in No-AMS group (Fig 4). Endothelin-1, $24\\\\mathrm{h}$ after exposure to high altitude increased significantly in compared $1\\\\mathrm{h}$ after exposure to altitude in this group $(P = 0.03)$ . Although there was a rise in Endothelin-1 at $1\\\\mathrm{h}$ $(10\\\\%)$ and $24\\\\mathrm{h}$ $(20\\\\%)$ after exposure to high altitude (respectively, $P = 0.03$ and $P = 0.02$ ) in AMS+ group, this was not different between time points $(P = 1.0)$ . Endothelin-1 concentration after exposure to hypoxia in AMS-resistant individuals was greater in compared to subjects who remained apparently healthy $(P < 0.05)$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[320,670,937,752]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\mathbf{SpO}_2$ . The mean of $\\\\mathrm{SpO}_2$ during exposure to high altitude was $83.5 \\\\pm 0.3$ in No-AMS group and was $77.66 \\\\pm 0.4$ in AMS+ group. The mean of $\\\\mathrm{SpO}_2$ decreased significantly $16\\\\%$ in No-AMS group and $25\\\\%$ in AMS+ group after exposure to altitude; this difference was significant (Fig 5). As well as, we found a significant difference for the mean of $\\\\mathrm{SpO}_2$ between the AMS+ study group and the No-AMS study group ( $T = 10.8$ , $P = 0.0001$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[320,753,939,818]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As shown in Table 3, at baseline, The Lake Louise score inversely correlate with Orexin-A, HIF-1, VEGF and Endothelin-1, as well as, no correlation was found between $\\\\mathrm{SpO}_2$ at sea level and LLS. In 1 and $24\\\\mathrm{h}$ after exposure to high altitude Lake Louise score was significant inversely correlation with Orexin-A, VEGF, HIF-1, Endothelin-1 and $\\\\mathrm{SpO}_2$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,843,433,859]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Discussion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[320,866,939,898]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The aim of the present study was to assess the potential role of the four hypoxia-related proteins in AMS pathogenesis. Taken together, the study's main findings were that serum Orexin-\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,44,151,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[797,47,942,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acute mountain sickness \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[53,941,495,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[906,944,939,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5/11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[320,95,941,481]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig 2. HIF-1 response to exposure to high altitude in No-AMS and $\\\\\\\\mathrm{AMS + }$ groups. \\\\\\\\* Significant difference between AMS and No-AMS groups, \\\\\\\\*\\\\\\\\* Significant different vs. Sea level, # Significant different vs. $1\\\\\\\\textrm{h}$ after exposure to high altitude. $\\\\\\\\mathrm{Sig}\\\\\\\\leq 0.05$ , values are Mean $\\\\\\\\pm \\\\\\\\mathrm{SD}$ \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0162/1e3e2295d28a82c1f9d3839cd625a9534f9b847d45dd4a98f73c299987bcd729.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[321,530,578,542]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.1371/journal.pone.0292173.g002 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[320,560,934,657]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A, HIF-1, VEGF and Endothelin-1 at sea level was more in AMS-resistant individuals, and all of them increased after exposure to altitude in No-AMS and $\\\\mathrm{AMS + }$ ; however, the magnitude of the Orexin-A response to the altitude at 1 and $24\\\\mathrm{h}$ after exposure to hypoxia in individuals with $\\\\mathrm{AMS + }$ was approximately identical with No-AMS group. Elevation in HIF-1 24 hrs after exposure to altitude in $\\\\mathrm{AMS + }$ was more than No-AMS. In contrast, the increase in VEGF and Endothelin 1 and 24 hrs after exposure to altitude in No-AMS were more than $\\\\mathrm{AMS + }$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[320,660,934,773]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this study, difficulty sleeping and headache were the most common symptoms of individual with AMS. Meanwhile, Orexin-A has pivotal role in difficulty sleeping and sleep/wake regulation [12]. In support of our findings, Liu and colleagues confirmed that hypoxia increases the expression of Orexin-A and this change leads to sleep fragmentation after exposure to altitude [13]. Since Orexin-A to altitude was similar between groups, it is possible that other factors that regulate the sleep cycle are participate in difficulty sleeping after exposure to altitude in $\\\\mathrm{AMS + }$ group. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[320,773,941,888]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"At sea level, 1 and 24 hrs after exposure to altitude, HIF-1 levels in $\\\\mathrm{AMS + }$ group was less than No-AMS group. In this regards, Kline and coworkers showed that HIF-1 partial deficient impaired ventilatory responses to chronic hypoxia [14]. The ventilatory response to the hypoxic is the main physiological response of the body at altitude, which can be directly related to adaptation to altitude and resistance to AMS. HIF-1 Elevations in HIF-1 24 hrs after exposure to high altitude in $\\\\mathrm{AMS + }$ was more compared to the No-AMS. It seems likely that HIF-responsive proteins such as VEGF and Endothelin-1 may predispose an individual to AMS. VEGF \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,44,151,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[797,47,942,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acute mountain sickness \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[53,941,495,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[906,944,939,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6/11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[320,95,916,478]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig 3. VEGF response to exposure to high altitude in No-AMS and $\\\\\\\\mathrm{AMS + }$ groups. \\\\\\\\* Significant difference between AMS and No-AMS groups, \\\\\\\\*\\\\\\\\* Significant different vs. Sea level, # Significant different vs. $1\\\\\\\\textrm{h}$ after exposure to high altitude. Sig $\\\\\\\\leq 0.05$ , values are Mean $\\\\\\\\pm \\\\\\\\mathrm{SD}$ \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0127/7a02807f3f3699a013bfef46b8e3f1b2d760807c304ab743a1e7b3194618165b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[321,525,578,539]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.1371/journal.pone.0292173.g003 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[320,551,936,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"upregulated by hypoxia and demonstrated as an increase in vascular permeability [15]. There is an increasing body of evidence suggesting that VEGF is a potent permeability factor, although its role in altitude disease is obscure. In agreement with previous investigations, VEGF increased 1 and $24\\\\mathrm{h}$ after exposure to hypoxia in $\\\\mathrm{AMS + }$ and No-AMS groups, but this change was more in No-AMS group (respectively $7\\\\%$ vs. $18\\\\%$ and $16\\\\%$ vs. $35\\\\%$ ). In contrast of our findings Harrison et al. showed that serum VEGF at baseline was not different between $\\\\mathrm{AMS + }$ and No-AMS group; however, they did report that serum VEGF elevation was more in $\\\\mathrm{AMS + }$ group in comparison to No-AMS group at 3 days after exposure to altitude ( $43\\\\%$ vs. $26\\\\%$ ) [16]. A probable explanation for the difference between the results of this study and the findings of Harrison related to the individual variability and different ascent profile, including the altitude at which measurements were made, duration of high altitude stay and the altitude of residency of the subjects. The mechanism behind VEGF elevation in response to hypoxia is unclear; however, it remains possible that less serum VEGF in subjects more susceptible to AMS may in part be due to enhanced renal VEGF clearance and fluid retention. For example, Maloney et al. showed that decrease in circulating VEGF after exposure to altitude may indicate protection against the development of VEGF-mediated capillary leak in organs such as the brain [17]. Furthermore, Kendall et al. showed that soluble receptors act to sequester unbound VEGF from the circulation, therefore preventing its interaction with the endothelium-bound receptors that mediate its effects upon endothelial function and vascular permeability [18]. While some studies have reported a relation between serum VEGF and AMS susceptibility, the majority did not [17, 19, 20]. It should be noted that MRI brain scans \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,45,151,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[797,47,942,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acute mountain scikness \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[53,941,495,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[906,944,941,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7/11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[320,95,916,459]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig 4. Endothelin-1 response to exposure to high altitude in No-AMS and $\\\\\\\\mathrm{AMS + }$ groups.\\\\\\\\* Significant difference between AMS and No-AMS groups, \\\\\\\\*\\\\\\\\* Significant different vs. Sea level, # Significant different vs. $1\\\\\\\\textrm{h}$ after exposure to high altitude. $\\\\\\\\mathrm{Sig}\\\\\\\\leq 0.05$ , values are Mean±SD. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0468/ac215c9f364624dbfe2c0576ba310cb30a1797fcba23a5809c3728e9cf4c3ec7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[321,508,578,522]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.1371/journal.pone.0292173.g004 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[251,537,942,851]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig 5. $\\\\\\\\mathrm{SpO}_2$ changes in response to exposure to high altitude in No-AMS and $\\\\\\\\mathrm{AMS + }$ $^*$ show significant difference between $\\\\\\\\mathrm{AMS + }$ and No-AMS groups. $\\\\\\\\mathrm{Sig}\\\\\\\\leq 0.05$ , values are Mean $\\\\\\\\pm \\\\\\\\mathrm{SD}$ \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0162/663a6305e55e45b2c15fc1dc5a6a1ff71fb075f1f8f6884a81f318b32783618e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[253,887,509,901]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.1371/journal.pone.0292173.g005 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,44,151,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[797,47,942,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acute mountain scikness \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[53,941,493,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[906,944,939,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"8/11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[321,119,942,425]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0508/05a871f3c5933c24cc3ab9540ed9a90a9610d785326d370d41da53e2647a18cb.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Factors</td><td>Pearson Correlation</td><td>P value</td></tr><tr><td>Sea level</td><td></td><td></td></tr><tr><td>Orexin-A</td><td>-0.831</td><td>0.0001</td></tr><tr><td>HIF-1</td><td>-0.789</td><td>0.0001</td></tr><tr><td>VEGF</td><td>-0.774</td><td>0.0001</td></tr><tr><td>Endothelin-1</td><td>-0.691</td><td>0.001</td></tr><tr><td>SpO2</td><td>-0.181</td><td>0.43</td></tr><tr><td>1 h after exposure to altitude</td><td></td><td></td></tr><tr><td>Orexin-A</td><td>-0.817</td><td>0.0001</td></tr><tr><td>HIF-1</td><td>-0.803</td><td>0.0001</td></tr><tr><td>VEGF</td><td>-0.769</td><td>0.0001</td></tr><tr><td>Endothelin-1</td><td>-0.710</td><td>0.0001</td></tr><tr><td>SpO2</td><td>-0.697</td><td>0.0001</td></tr><tr><td>24 h after exposure to altitude</td><td></td><td></td></tr><tr><td>Orexin-A</td><td>-0.774</td><td>0.0001</td></tr><tr><td>HIF-1</td><td>-0.818</td><td>0.0001</td></tr><tr><td>VEGF</td><td>-0.753</td><td>0.0001</td></tr><tr><td>Endothelin-1</td><td>-0.770</td><td>0.0001</td></tr><tr><td>SpO2 mean</td><td>-0.884</td><td>0.0001</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 3. The correlation between hypoxia related proteins with LLS at sea level and 1 and 24 hrs after exposure to high altitude. \\\"]\",\n \"table_footnote\": \"[\\\"https://doi.org/10.1371/journal.pone.0292173.t003 \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[325,457,941,537]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"following hypoxic exposure showed that brain swelling was not different between subjects who developed AMS and those who did not (An indirect argument against a role of VEGF in AMS) [19, 21]. Also, the concentration of VEGF in the cerebrospinal fluid is not different between subjects with and without AMS [22]. These findings demonstrated that vasogenic edema is not involved in the pathophysiology of AMS. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,539,939,654]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Previous experimental studies suggests that increased brain volume with hypobaric hypoxia elevates intracranial pressure and, when accompanied by impaired or diminished intracranial buffering capacity, contributes to the development of the symptoms that define AMS (tight-fit hypothesis). On the other hand, Julian and coworker for the first time showed that resistance to AMS may be driven, in part, by the ability to mount an adequate \\\"defensive\\\" anti-inflammatory or anti-permeability response after exposure to altitude [23]. These findings lend deep insight into the pathophysiology of AMS. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,654,939,719]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The increase of Endothelin-1 after exposure to hypoxia in both groups are in line with previous studies [7, 24]. In contrast to these studies and our findings, Barker et al. showed that Endothelin-1 was not different between those with and without AMS susceptibility [6]. It was concluded that Endothelin -1 have a pathological role in HAPE instead of AMS [6]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,720,929,785]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The mean $\\\\mathrm{SpO}_2$ of the AMS+ group after exposure to altitude was significantly lower than that of the No-AMS group. In contrast out findings Wagner et al. showed that there was no significant difference in the mean oxygen saturations at summit or average altitude between the AMS+ and No-AMS groups [25]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,785,934,883]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"AMS symptoms do not appear immediately after exposure to altitude, it may last up to 12 hrs; therefore, if AMS can be predicted before it occurs, there will be enough time to prevent its possible adverse consequences. In fact, hypoxia related proteins (Orexin-A, HIF-1, VEGF and Endothelin-1) evaluation at sea level could be consider as a predictive indices to identify acute mountain sickness susceptibility individuals unaccustomed to high altitude. It should to be noted that the most important limitation of this study is that the sample size is not big enough. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,45,151,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[797,47,942,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acute mountain scikness \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[53,944,493,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[906,944,939,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"9/11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[325,94,433,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conclusion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,118,936,248]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The results of this study in summary showed that hypoxia related proteins include Orexin-A, HIF-1, VEGF and Endothelin-1 were higher in the AMS-resistant individuals when compared to AMS susceptible individuals at sea level and these proteins did not show consistent changes after exposure to high altitude. Susceptibility to AMS may be associated with a high response HIF-1 and more $\\\\mathrm{SpO}_2$ reduction to hypoxia. To better determine the underlying factors in AMS etiology, future work in this area should focus on genetic and local measurement of other factors. Further studies are needed to validate these measurements, but they suggest an alternative approach to obtain a more accurate simple method of predicting AMS. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,272,542,289]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Supporting information \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,295,913,342]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"S1 Data. ELISA data set of Orexin, HIF-1, VEGF and Endothelin-1. Abbreviation: AMS: Acute Mountain Sickness. (XLS) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,375,498,392]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acknowledgments \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,398,926,429]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The authors thank Prof Philip Ainslie (University of British Columbia, Canada) for feedback and edit of the paper. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,454,522,469]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Author Contributions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,478,516,492]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data curation: Bayan Fayazi. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,500,511,515]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Methodology: Bayan Fayazi. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,523,503,537]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Supervision: Vahid Tadibi. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,545,596,560]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Writing - original draft: Kamal Ranjbar. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,568,616,583]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Writing - review & editing: Kamal Ranjbar. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,614,429,628]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[330,633,939,895]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. Liu B, Huang H, Wu G, Xu G, Sun B-D, Zhang E-L, et al. A signature of circulating microRNAs predicts the susceptibility of acute mountain sickness. Frontiers in physiology. 2017; 8. https://doi.org/10.3389/fphys.2017.00055 PMID: 28228730 \\\",\\\"2. MacInnis MJ, Wang P, Koehle MS, Rupert JL. The genetics of altitude tolerance: the evidence for inherited susceptibility to acute mountain sickness. Journal of occupational and environmental medicine. 2011; 53(2):159-68. https://doi.org/10.1097/JOM.0b013e318206b112 PMID: 21270658 \\\",\\\"3. Chen H-C, Lin W-L, Wu J-Y, Wang S-H, Chiu T-F, Weng Y-M, et al. Change in oxygen saturation does not predict acute mountain sickness on Jade Mountain. Wilderness & environmental medicine. 2012; 23(2):122-7. \\\",\\\"4. Painschab MS, Malpartida GE, Dávila-Roman VG, Gilman RH, Kolb TM, León-Velarde F, et al. Association between serum concentrations of hypoxia inducible factor responsive proteins and excessive Erythrocytosis in high altitude Peru. High altitude medicine & biology. 2015; 16(1):26-33. https://doi.org/10.1089/ham.2014.1086 PMID: 25760230 \\\",\\\"5. Ding H, Liu Q, Hua M, Ding M, Du H, Zhang W, et al. Polymorphisms of hypoxia-related genes in subjects susceptible to acute mountain sickness. Respiration. 2011; 81(3):236-41. https://doi.org/10.1159/000322850 PMID: 21242666 \\\",\\\"6. Barker KR, Conroy AL, Hawkes M, Murphy H, Pandey P, Kain KC. Biomarkers of hypoxia, endothelial and circulatory dysfunction among climbers in Nepal with AMS and HAPE: a prospective case-control study. Journal of travel medicine. 2016; 23(3):taw005. https://doi.org/10.1093/jtm/taw005 PMID: 26984355 \\\"]\",\n \"page_idx\": \"9\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[53,45,151,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[797,47,942,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acute mountain scikness \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[53,944,493,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[901,944,939,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10/11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[325,93,942,858]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"7. Boos CJ, Woods DR, Varias A, Biscocho S, Heseltine P, Mellor AJ. High altitude and acute mountain sickness and changes in circulating endothelin-1, interleukin-6, and interleukin-17a. High altitude medicine & biology. 2016; 17(1):25-31. https://doi.org/10.1089/ham.2015.0098 PMID: 26680502 \\\",\\\"8. Xu T-R, Yang Y, Ward R, Gao L, Liu Y. Orexin receptors: multi-functional therapeutic targets for sleeping disorders, eating disorders, drug addiction, cancers and other physiological disorders. Cellular signalling. 2013; 25(12):2413-23. https://doi.org/10.1016/j.cellsig.2013.07.025 PMID: 23917208 \\\",\\\"9. Du MK, Hunt NJ, Waters KA, Machaalani R. Cumulative effects of repetitive intermittent hypercapnic hypoxia on orexin in the developing piglet hypothalamus. International Journal of Developmental Neuroscience. 2016; 48:1-8. https://doi.org/10.1016/j.ijdevneu.2015.10.007 PMID: 26548856 \\\",\\\"10. Wang L, Zhou Y, Li M, Zhu Y. Expression of hypoxia-inducible factor-1α, endothelin-1 and adrenomedullin in newborn rats with hypoxia-induced pulmonary hypertension. Experimental and therapeutic medicine. 2014; 8(1):335-9. \\\",\\\"11. Bp R, Hackett P, Oelz O, editors. The Lake Louise acute mountain sickness scoring system. Hypoxia and Mountain Medicine: proceeding of the International hypoxia Symposium; 1993. \\\",\\\"12. Sokolowska P, Urbańska A, Biegańska K, Wagner W, Ciszewski W, Namiecińska M, et al. Orexins protect neuronal cell cultures against hypoxic stress: an involvement of Akt signaling. Journal of Molecular Neuroscience. 2014; 52(1):48-55. https://doi.org/10.1007/s12031-013-0165-7 PMID: 24243084 \\\",\\\"13. Liu Z, Jiang L, Zhu F, Fu C, Lu S, Zhou J, et al. Chronic intermittent hypoxia and the expression of orexin and its receptors in the brains of rats. Sleep and Biological Rhythms. 2014; 12(1):22-9. \\\",\\\"14. Kline DD, Peng Y-J, Manalo DJ, Semenza GL, Prabhakar NR. Defective carotid body function and impaired ventilatory responses to chronic hypoxia in mice partially deficient for hypoxia-inducible factor 1α. Proceedings of the National Academy of Sciences. 2002; 99(2):821-6. \\\",\\\"15. Schoch HJ, Fischer S, Marti HH. Hypoxia-induced vascular endothelial growth factor expression causes vascular leakage in the brain. Brain. 2002; 125(11):2549-57. https://doi.org/10.1093/brain/awf257 PMID: 12390979 \\\",\\\"16. Harrison MF, Anderson PJ, Miller AD, O'Malley KA, Richert ML, Johnson JB, et al. Physiological variables associated with the development of acute mountain sickness at the South Pole. BMJ open. 2013; 3(7):e003064. https://doi.org/10.1136/bmjopen-2013-003064 PMID: 23869103 \\\",\\\"17. Maloney J, Wang D, Duncan T, Voelkel N, Ruoss S. Plasma vascular endothelial growth factor in acute mountain sickness. Chest Journal. 2000; 118(1):47-52. https://doi.org/10.1378/chest.118.1.47 PMID: 10893358 \\\",\\\"18. Kendall RL, Wang G, Thomas KA. Identification of a natural soluble form of the vascular endothelial growth factor receptor, FLT-1, and its heterodimerization with KDR. Biochemical and biophysical research communications. 1996; 226(2):324-8. https://doi.org/10.1006/bbrc.1996.1355 PMID: 8806634 \\\",\\\"19. Schommer K, Wiesegart N, Dehnert C, Mairbäurl H, Bärsch P. No correlation between plasma levels of vascular endothelial growth factor or its soluble receptor and acute mountain sickness. High altitude medicine & biology. 2011; 12(4):323-7. https://doi.org/10.1089/ham.2011.1020 PMID: 22206557 \\\",\\\"20. Ding H, Liu Q, Hua M, Ding M, Du H, Zhang W, et al. Associations between vascular endothelial growth factor gene polymorphisms and susceptibility to acute mountain sickness. Journal of International Medical Research. 2012; 40(6):2135-44. https://doi.org/10.1177/030006051204000611 PMID: 23321170 \\\",\\\"21. Schoonman GG, Sander PS, Nirkko AC, Lange T, Jaermann T, Dydak U, et al. Hypoxia-induced acute mountain sickness is associated with intracellular cerebral edema: a 3 T magnetic resonance imaging study. Journal of Cerebral Blood Flow & Metabolism. 2008; 28(1):198-206. https://doi.org/10.1038/sj.jcbfm.9600513 PMID: 17519973 \\\",\\\"22. Bailey DM, Roukens R, Knauth M, Kallenberg K, Christ S, Mohr A, et al. Free radical-mediated damage to barrier function is not associated with altered brain morphology in high-altitude headache. Journal of Cerebral Blood Flow & Metabolism. 2006; 26(1):99-111. \\\",\\\"23. Julian CG, Subudhi AW, Wilson MJ, Dimmen AC, Pecha T, Roach RC. Acute mountain sickness, inflammation, and permeability: new insights from a blood biomarker study. Journal of Applied Physiology. 2011; 111(2):392-9. https://doi.org/10.1152/japplphysiol.00391.2011 PMID: 21636566 \\\",\\\"24. Sartori C, Vollenweider L, Löffler B-M, Delabays A, Nicod P, Bärtsch P, et al. Exaggerated endothelin release in high-altitude pulmonary edema. Circulation. 1999; 99(20):2665-8. https://doi.org/10.1161/01.cir.99.20.2665 PMID: 10338460 \\\",\\\"25. Wagner DR, Knott JR, Fry JP. Oximetry fails to predict acute mountain sickness or summit success during a rapid ascent to 5640 meters. Wilderness & environmental medicine. 2012; 23(2):114-21. https://doi.org/10.1016/j.wem.2012.02.015 PMID: 22656656 \\\"]\",\n \"page_idx\": \"10\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[53,45,151,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[797,47,942,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acute mountain scikness \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[53,941,493,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0292173 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[901,944,939,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"11/11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.1371/journal.pone.0292173"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Life Sciences"},"sha256":{"kind":"string","value":"34cbb6715a8d26986a47357bc56300d5aa490763f16894ac2e9d92b1abcbcbd6"},"title":{"kind":"string","value":"The role of hypoxia related hormones responses in acute mountain sickness susceptibility individuals unaccustomed to high altitude"}}},{"rowIdx":60,"cells":{"abstract":{"kind":"string","value":"This paper aims to map the daily functional characteristics of people diagnosed with essential tremor (ET) based on their subjective self-reports. In addition, we provide objective measurements of a cup-drinking task. This study involved 20 participants diagnosed with ET who completed the Columbia University Assessment of Disability in Essential Tremor (CADET) questionnaire that included five additional tasks related to digital equipment operation we wrote. Participants also described task-performance modifications they implemented. To create objective personal performance profiles, they performed a cup-drinking task while being monitored using a sensor measurement system. The CADET's subjective self-report results indicate that the most prevalent tasks participants reported as having difficulty with or requiring modifications were writing, threading a needle, carrying a cup, using a spoon, pouring, and taking a photo or video on a mobile phone. Analysis of participants' modifications revealed that holding the object with two hands or with one hand supporting the other were the most prevalent types. No significant correlation was found between the CADET total scores and the cup drinking objective measures. Capturing patients' perspectives on their functional disability, alongside objective performance measures, is envisioned to contribute to the development of custom-tailored interventions aligned with individual profiles, i.e., patient-based/smart healthcare."},"author":{"kind":"string","value":"Rosenblum, Sara|Salih, Adham|Roth, Navit"},"content_list":{"kind":"list like","value":[{"bbox":"[53,122,107,136]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Article ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,137,932,184]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Subjective and Objective Day-to-Day Performance Measures of People with Essential Tremor ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[53,197,519,212]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Navit Roth $^{1,2}$ , Adham Salih $^{2}$ and Sara Rosenblum $^{1,*}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,237,939,303]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1 The Laboratory of Complex Human Activity and Participation (CHAP), Department of Occupational Therapy, University of Haifa, Haifa 3498838, Israel; navroth@braude.ac.il \",\"$^{2}$ Department of Mechanical Engineering, Braude College of Engineering, Karmiel 2161002, Israel; adhamsalih@braude.ac.il \",\"* Correspondence: rosens@research.haifa.ac.il \"]","page_idx":"0","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[272,319,942,565]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract: This paper aims to map the daily functional characteristics of people diagnosed with essential tremor (ET) based on their subjective self-reports. In addition, we provide objective measurements of a cup-drinking task. This study involved 20 participants diagnosed with ET who completed the Columbia University Assessment of Disability in Essential Tremor (CADET) questionnaire that included five additional tasks related to digital equipment operation we wrote. Participants also described task-performance modifications they implemented. To create objective personal performance profiles, they performed a cup-drinking task while being monitored using a sensor measurement system. The CADET's subjective self-report results indicate that the most prevalent tasks participants reported as having difficulty with or requiring modifications were writing, threading a needle, carrying a cup, using a spoon, pouring, and taking a photo or video on a mobile phone. Analysis of participants' modifications revealed that holding the object with two hands or with one hand supporting the other were the most prevalent types. No significant correlation was found between the CADET total scores and the cup drinking objective measures. Capturing patients' perspectives on their functional disability, alongside objective performance measures, is envisioned to contribute to the development of custom-tailored interventions aligned with individual profiles, i.e., patient-based/smart healthcare. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,554,146,576]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0250/7afbd7e38300f9234205385dabb4b9d13fa7e343c9bffac2d0984886a4ee544e.jpg","list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[53,581,243,593]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Citation: Roth, N.; Salih, A. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,595,213,607]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Rosenblum, S. Subjective and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,609,242,621]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Objective Day-to-Day Performance ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,624,236,636]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Measures of People with Essential ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,638,215,650]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Tremor. Sensors 2024, 24, 4854. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,652,245,664]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.3390/s24154854 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,674,250,686]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Academic Editors: Chan Hwang See, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,688,226,699]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Arfan Ghani and Simeon Keates ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,709,178,722]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Received: 26 June 2024 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,724,169,736]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Revised: 19 July 2024 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,738,176,750]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Accepted: 24 July 2024 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,753,181,764]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Published: 26 July 2024 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,777,154,803]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0703/9f582f766f8b36313e136c55d535147a8408637b65bc5adbdeb775175d8998a7.jpg","list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[53,808,257,919]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,580,744,595]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords: essential tremor; functional disability; ADL; modifications ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[275,639,394,653]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[272,657,939,791]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Essential tremor (ET), one of the most common movement disorders, is found in a wide age range, with a prevalence of about $5\\%$ in people over 65 years [1]. The disorder is characterized mainly as an action tremor (postural or kinetic) but can also appear while at rest [2-4]. The frequency of the tremor in ET may range between 4 and $12\\mathrm{Hz}$ and decreases with time [5]. Although ET may affect assorted body parts, most often upper limbs are affected with a slight asymmetry [6,7]. Most people with ET do not seek medical diagnosis [1,6,8,9]. About $85\\%$ report that the tremor causes functional disability [10] in the activities of daily living (ADL)—such as eating, drinking, and using tools and self-care instruments that require fine and gross motor skills [10] and in work performance [6]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,791,941,912]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Previous studies indicated that stress and fatigue may also affect the tremor, whereas non-motoric symptoms, including changes in cognition, depression, and anxiety, may also be involved among patients with ET [6,9,11,12]. The potential for ET to cause embarrassment and social consequences [6] seems to influence the patients' decisions to seek medical treatment [13]. Embarrassment exists in almost $50\\%$ of those with mild ET (even without head tremor) [13], and the social effects may result in patients avoiding social meetings and tasks that will expose the tremor. These studies implied that ET and its associated depression and anxiety may cause a decline in quality of life among people with ET [14-16]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,57,115,98]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"# ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[121,68,253,87]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"sensors ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[863,64,937,99]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"MDPI ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[55,964,403,977]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sensors 2024, 24, 4854. https://doi.org/10.3390/s24154854 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[687,964,939,977]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://www.mdpi.com/journal/sensors ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[278,102,939,177]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Currently, ET is clinically diagnosed using a combination of a neurological examination and a medical history review [17]. Assessment of tremor severity may include scales that are not ET-specific, such as the Fahn-Tolosa-Marin Clinical Tremor Rating Scale [18] and scales developed specifically for ET such as the Tremor Research Group Essential Tremor Rating Assessment Scale [19]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,178,939,370]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Usually, functional disability due to tremors is assessed with self-report questionnaires and performance-based tests [20] that may be part of a severity-assessment scale. Self-report questionnaires provide assessors with the perspective of the person coping with the tremor by asking respondents whether they have difficulty or disability in ADL tasks and how difficult those tasks are to perform. Such instruments are the Bain and Findley ADL scale [21] and the Columbia University Assessment of Disability in Essential Tremor (CADET) questionnaire [22]. The CADET, which was implemented in the current study, is a valid and reliable 31-item self-report questionnaire. It focuses on the respondent's difficulty/disability while performing ADL tasks and whether the respondent made modifications (or had a change in efficiency) [22,23]. Eliciting this information is especially valuable because people who make modifications to perform ADL do not necessarily report having a disability [24]. Modifications include changing the environmental, psychosocial, behavioral, psychological, or physical method of performing a specific task [25]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,372,939,489]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In previous studies, the highest percentage of people with ET who were surveyed reported the physical tasks of using a spoon, drinking from, holding, or carrying a cup, pouring a drink, and of writing were difficult to perform, required modification, or carried out less efficiently than previously [6,10]. To the best of our knowledge, however, there is no detailed documentation on the modification types/profiles they implemented in various daily tasks. Thus, this current study focuses on physical modifications. In addition, because we did not find detailed documentation on the reported disability while performing diverse tasks of digital and smartphone operation, we made these tasks a secondary focus of our research. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,491,939,579]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Another aim of this paper is to demonstrate the need for performance-based tests that supply objective measures of day-to-day performance. Measurements—to quantify tremors in various ADL tasks—may be acquired using digital-graphic boards, accelerometers and gyroscopes, image processing, and electromyography [20,26-28]. The resulting metrics may be implemented as reciprocal validation [29] of neurology assessments and/or self-report scales [20], providing support for the individual's functional health status evaluation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,580,941,699]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this study, the performance-based objective measures were gathered for the cup-drinking task as an example of a functional daily task that challenges people with ET [6,21,22]. This particular task was chosen as more than $65\\%$ of participants with ET reported difficulties/disabilities or needed modification/efficiency changes while performing the task [10,27]. In addition, this daily task comprises multiple phases with different kinematic features [30], during which tremor characteristics, which may differ with posture and force levels, may affect tremor amplitude [31,32]. The cup-drinking task was used by the Glass scale [33] as one task for assessing tremor severity among people with ET. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,699,941,788]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In previous studies, objective measures of the tremor were obtained while performing cup tasks using hand or cup-attached accelerometers in drinking and holding-a-cup tasks [26,34,35] or by measuring spilled water volume in the holding-a-cup task [27]. To the best of our knowledge, however, the correlation between the objective drinking task measures and the total score of a self-report disability questionnaire was not reported and thus was examined in the current study. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,789,941,907]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In line with the personal medicine view [36], understanding patients' perspectives and the ways they perform daily tasks may aid in the pre-clinical phase, as well as in assessment and treatment processes, toward improved physical and emotional health and quality of life [25]. Thus, this study also aims to map activity performance features of people who are diagnosed with ET but who may not be followed routinely or medically. To this end, we combine the individual patient's responses on a self-report disability questionnaire and objective performance-based quantitative measures for the common ADL task of drinking to create an individual profile. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,57,184,68]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sensors 2024, 24, 4854 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[897,60,939,70]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 of 13 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[275,103,364,116]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. Methods ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[275,118,391,133]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.1. Participants ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[273,137,941,256]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We recruited participants by advertising among the general population (e.g., via social media, bulletin boards, and senior culture clubs). Inclusion criteria were being aged 20 or more and having a written physician's diagnosis of ET. Exclusion criteria were another disability that could affect the participants' tremors based on their self-reports (previously also described) [37]. Twenty people $(N = 20)$ met the inclusion and exclusion criteria and participated in the study. This study was conducted according to the guidelines of the Declaration of Helsinki; the University of Haifa Ethics Committee approved the study (018/19). All participants signed informed consent forms. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[275,268,393,282]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2. Instruments ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[273,286,941,348]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Participants who met the criteria signed an informed consent form and completed a demographic questionnaire, a Mini-Mental State Examination [38], and an expanded and translated CADET questionnaire, including questions on extra digital tasks [22]. In addition, they performed an ADL task of drinking from a cup using three grip types. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[275,357,532,373]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2.1. Demographic Questionnaire ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[273,376,941,437]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The demographic questionnaire included information on the participants' age, education (years), dominant hand, tremor-specific medication, general medication, duration (years) since noticing the tremor, duration (years) since diagnosis with ET, and family members diagnosed with ET. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[275,447,917,462]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2.2. CADET Questionnaire with Further Questions on Extra Tasks and Modifications ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[273,466,941,541]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To address disability in daily tasks due to the tremor, a Hebrew translation of the validated CADET questionnaire was administrated after approval by the corresponding author [23]. The 31 tasks were rated as 0 (no disability, need to modify activities, or no loss of efficiency), 1 (no disability but a need to modify activities or a loss of efficiency), or 2 (disability or both a need to modify and loss of efficiency) [10]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,541,941,690]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Questions related to the five digital equipment operation tasks were added for this study: (1) connect to digital accessories (e.g., connect a phone charger, headphones, and a USB drive); (2) select an object with a computer mouse (e.g., select and click on an object from a toolbar or a recipient from an e-mail list); (3) type digits or a text message with fingers on a mobile phone; (4) take a photo or video using a mobile phone; and (5) use a finger to choose an object on a mobile phone screen (e.g., an application icon). The CADET outcome measures comprised two mean scores: one for the questionnaire's original 31 items and the other for the five questions related to the extra tasks. Participants also listed up to five tasks in which the tremor bothered them the most and elaborated on modifications in the task performance they made due to the tremor. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[275,701,796,715]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2.3. The Cup Drinking Task: Objective Performance-Based Measures ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[273,720,941,869]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As described in the introduction section, the ADL task of drinking has been chosen as a representative ADL task for the objective measurements. This task included lifting a cup from the table, moving the cup toward the mouth, performing a drinking-like movement, and returning the cup to the table. Participants were seated in front of a table with the cup positioned on it at about $25\\mathrm{cm}$ from the edge in front of the participant's dominant hand. This layout was applied to enable the task simulation of picking up the cup and drinking in their regular manner during a meal. Participants were instructed to perform the task in their regular manner as close as possible to their real-life daily performance but using three grip types: (i) around-the-cup grip (without using the handle), (ii) below-the-handle grip (leaning the handle on the fingers), and (iii) on-the-handle grip (holding only the handle). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,870,941,914]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The task was characterized and divided into six states describing the different positions of the cup during the task. This task, which addressed the effect of the tremor on the cup acceleration, was performed using a specially built, customized measurement system, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,57,184,68]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sensors 2024, 24, 4854 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[897,60,939,70]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 of 13 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[272,102,944,209]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"which included a triaxial accelerometer (ADXL335 (Module by SparkFun, Niwot, CO, USA, Sensor by Analog Devices, Wilmington, MA, USA) with a range of $\\pm 3\\mathrm{g}$ ) attached to the cup. Data were sampled at $100\\mathrm{Hz}$ using an acquisition component (NI-6008) and a computer. The cup system included an $80\\%$ volume-filled cup of water covered by a lid. The system was previously described by Roth and Rosenblum in their study [35]. The axes system used with the accelerometer is shown in Figure 1, whereas the cup position states appear in Figure 2. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[284,224,379,313]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 1. Cup accelerometer axes layout. \"]","image_footnote":"[]","img_path":"images/0124/31140ab2299d4e3ddb70c466aa9dce4be892d2c2f2175f4b1b3f2b08dae11f72.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[115,351,374,470]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0085/0ba9cacf92fbbc5499e252de363753c2efbbf0fa91c1f808b5dcfc43a1feb9e9.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[371,351,630,470]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0033/32093ab236449ac9363c0965b89ee068af23f51d9541aaf690ada85b9f0c0b5e.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[628,351,885,470]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0563/2409cd039013f7442e32f87df1f09bb72d4dab87bd675c5dd4008c807f452bf0.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[115,480,374,598]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/1001/482449f89b658158455fd1cca66005c466b2d282025a3c5368744b57deb52316.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[371,480,630,598]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0017/36576ad29befc1a6d95523434bf27c97d68ba512bef1bd9cfbf6b677f0d025e3.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[628,480,885,598]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 2. Cup position states: State 1—On the table, start position; State 2—Mid-duration between States 2 and 3; State 3—Near the mouth, before drinking; State 4—Near the mouth, after drinking, start of returning the cup to the table movement; State 5—Mid-duration between States 4 and 6; State 6—Contact with the table, end of returning movement. \"]","image_footnote":"[]","img_path":"images/0993/50f41767219b2a91302af67dfb08de0530b4fa9cd003785d439e84230b2bc60d.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[272,682,944,829]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To evaluate each participant's ability to perform the drinking task, two performance measures were developed. The first measure, which is hereby termed T46, was defined as the duration between States 4 and 6. The recorded videos showed that 'reaching the mouth' movements, i.e., going from State 1 to State 4, were performed differently by each participant. For example, some participants held the cup near their mouth for several seconds, whereas other participants started the 'return to the table' movement immediately after reaching State 3. In addition, some participants tilted the cup or vertically raised the cup to mouth level at the beginning of the movement and then began moving the cup toward their mouth. Therefore, to ensure consistency among participants when analyzing the performance characteristics, we decided to consider only the time between States 4 and 6. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,831,944,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The second measure aimed to evaluate the overall \"error\" caused by the tremor. To calculate this measure, first, we filtered out the signals that are associated with ET (4-12 Hz). Then, the mean-squared error (MSE) was calculated with reference to constant zero. This procedure was applied to each axis (X, Y, and Z). In addition, the overall MSE, denoted by MSE-T and constitutes the sum of the MSE components, was also calculated. Figure 3 illustrates the calculation steps for the X-axis. The black curve shows the voltage as recorded by the sensor. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,55,184,68]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sensors 2024, 24, 4854 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[897,59,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 of 13 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[273,102,941,133]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The filtered data are represented by the blue curve, whereas the red curves represent the data trend (by applying a moving average window). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,149,784,444]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 3. MSE Calculation. \"]","image_footnote":"[]","img_path":"images/0091/11b76787c7694fb07fba7ddfbf2a1e072a61d8de1b1c48366454e189aafeea52.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[272,480,941,555]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As shown in Figure 3, as expected, the filtered data trend is approximately zero. This observation suggests that the filtered data provide a good approximation of the tremor effect. It should be noted that according to the limitations of the experimental setup regarding the acceleration measurement without a gyro element, as reported in [35], we decided to analyze and report the MSE values only for States 5 to 6. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,565,445,581]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3. Statistical Analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[272,583,941,674]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Statistical analysis of the demographic data was conducted using the SPSS software, Ver 25, frequency and descriptive tools. Prevalence was calculated for CADET tasks scored one or more by participants and for the tasks they noted as bothering them most. The internal reliability of the five extra digital tasks was calculated using Cronbach's alpha. Nonparametric correlates using the Spearman test were calculated among CADET mean scores, age, and duration since noticing the tremor. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,674,941,734]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Analysis of modification data included modification prevalence in each task (for scored tasks) and modification methods. The percentages of participants who reported performing at least one task slower than had been the case prior to ET presenting, avoiding a task, and (although not asked) embarrassment and feelings of stress were also calculated. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,734,941,778]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For the objective performance-based measures, nonparametric correlates using the Spearman test were calculated among CADET mean scores and each objective performance-based measure. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,790,354,804]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4. Results ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[273,806,408,822]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4.1. Demographics ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[272,824,941,898]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The participants' demographic details were described in the methods section. The participants' ages ranged from 23 to 81 years ( $M = 64.7$ , $SD = 15.7$ ). Males comprised half of the sample ( $n = 10$ ) and $75\\%$ of the participants ( $n = 15$ ) had a family history of tremors based on their self-reports. Their education ranged from 12 to 27 years ( $M = 17.2$ , $SD = 3.4$ ) (see [37] for further details). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,55,184,68]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sensors 2024, 24, 4854 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[897,59,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 of 13 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[273,102,840,118]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4.2. CADET Scores and Item's Frequency Distribution, Including the Added Tasks ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[272,120,941,193]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The internal reliability of the five extra digital-operation tasks was $\\alpha = 0.58$ , and significant correlations were found between their mean scores and the original 31 tasks on the CADET ( $r = 0.61$ , $p < 0.01$ ). No significant correlations, however, were found between mean scores (either of the 31 CADET tasks or the five extra tasks) and age or tremor duration. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,196,941,255]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The 31 CADET task scores ranged from 5.77 to 66.67 ( $M = 32.29$ , $SD = 17.43$ ) (previously also described in [37]). The extra digital task scores ranged from 0 to 80.00 ( $M = 32.50$ , $SD = 20.74$ ). Most $(85\\%)$ participants reported avoiding or preferring someone else to perform at least one task. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,255,941,360]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Table 1 details the results for the CADET and extra tasks in which more than $50\\%$ of participants scored one or higher (indicating disability, need for modifications, or changed efficiency). The table also shows tasks in which $100\\%$ of participants scored zero and tasks more than $25\\%$ avoided. Regarding digital tasks not displayed (i.e., $<50\\%$ ), $45.0\\%$ of participants scored one or higher for the task \"Select an object with the computer mouse\", and $30.0\\%$ for \"Connect to digital accessories and choose an object on a mobile phone screen with your finger\". ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,397,989,783]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0860/03f4a041a05ecd19d13e31a22b94f88128cb93f0628440e96d1a80972e1c369f.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":"<table><tr><td>Item #</td><td>Description</td><td>Prevalence</td></tr><tr><td colspan=\"3\">CADET Questionnairea</td></tr><tr><td>2</td><td>Writing a letter, postcard, thank you card, or chequeb</td><td>100.0%</td></tr><tr><td>26</td><td>Threading a needleb</td><td>92.9%</td></tr><tr><td>7</td><td>Carrying a cup of coffee</td><td>90.0%</td></tr><tr><td>8</td><td>Using a spoon to drink soup</td><td>90.0%</td></tr><tr><td>6</td><td>Pouring milk or juice from a bottle</td><td>80.0%</td></tr><tr><td>10</td><td>Eating in a restaurant</td><td>75.0%</td></tr><tr><td>5</td><td>Drinking from a glass</td><td>70.0%</td></tr><tr><td>1</td><td>Signing your name</td><td>60.0%</td></tr><tr><td>24</td><td>Reading a book, magazine, or newspaper</td><td>60.0%</td></tr><tr><td>30</td><td>Tying your necktie (males) or putting on your lipstick (females)b</td><td>58.3%</td></tr><tr><td>31</td><td>Shaving (males) or putting on your eyeliner (females)b</td><td>57.1%</td></tr><tr><td>25</td><td>Unlocking door with a key</td><td>55.0%</td></tr><tr><td>9</td><td>Carrying a tray of foodb</td><td>52.6%</td></tr><tr><td colspan=\"3\">Five extra digital equipment operation tasksa</td></tr><tr><td>4</td><td>Taking a photo or video with a mobile phone</td><td>80.0%</td></tr><tr><td>3</td><td>Typing digits or texting messages with fingers on a mobile phone</td><td>50.0%</td></tr><tr><td colspan=\"3\">Minimally difficult tasksc</td></tr><tr><td>21\n(CADET)</td><td>Putting on your watch</td><td></td></tr><tr><td colspan=\"3\">Avoided tasksd</td></tr><tr><td>4\n(five extra digital tasks)</td><td>Taking a photo or video with a mobile phone</td><td>40.0%</td></tr><tr><td>2 (CADET)</td><td>Writing a letter, postcard, thank you card, or cheque</td><td>31.6%</td></tr></table>","table_caption":"[\"Table 1. Task Prevalence. \"]","table_footnote":"[\"Note. CADET = Columbia University Assessment of Disability in Essential Tremor [22]. a Tasks for which at least $50\\\\%$ of participants scored one or more (have difficulty, need modification, loss of efficiency); b $N = 20$ , but percentages are of participants reporting on specific tasks: $n = 19$ for Items 9 and 2 (CADET), $n = 12$ for Items 30 and 21, and $n = 14$ for Items 26 and 31 (spelling as in original questionnaire); c $100\\\\%$ of participants scored zero (no difficulty, need for modification, or loss of efficiency); d tasks at least $25\\\\%$ of participants reported they avoided or preferred others to perform it for them. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[272,872,941,917]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is important to note that $65\\%$ of participants reported other tasks that bothered them the most, ones that were not included in the CADET questionnaire. Such tasks were preparing food, eating (not just in a restaurant but also when using a fork at home), and performing work or ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,55,184,68]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sensors 2024, 24, 4854 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[895,59,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 of 13 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[272,102,941,133]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"hobby tasks, such as painting. Tasks reported with a prevalence of $35\\%$ or greater were eating $(55.0\\%)$ , carrying or drinking from a cup $(50.0\\%)$ , and writing $(35.0\\%)$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,143,470,159]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4.3. Reported Modifications ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[272,162,941,223]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"All participants reported implementing at least one modification, and $65.0\\%$ reported performing at least one task slower because of the tremor. These modifications are listed below. Table 2 details the tasks for which at least $50\\%$ of participants made modifications. The highest percentage of modifications were made to carrying a cup. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,260,942,435]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0172/609251ab0c0b52acbc0b0c2500ad17dbb4498cb3af1f8dae6a26148c8fc51381.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":"<table><tr><td>Item #</td><td>Description</td><td>Prevalence (% Participants Using Modification)</td></tr><tr><td>CADET</td><td></td><td></td></tr><tr><td>7</td><td>Carrying a cup of coffee</td><td>80.0</td></tr><tr><td>6</td><td>Pouring milk or juice from a bottle</td><td>65.0</td></tr><tr><td>2</td><td>Writing a letter, postcard, thank you card, or cheque</td><td>63.2</td></tr><tr><td>5</td><td>Drinking from a glass</td><td>60.0</td></tr><tr><td>24</td><td>Reading a book, magazine, or newspaper</td><td>55.0</td></tr><tr><td>8</td><td>Using a spoon to drink soup</td><td>50.0</td></tr><tr><td>Five digital tasks</td><td></td><td></td></tr><tr><td>4</td><td>Taking a photo or video with a mobile phonea</td><td>50.0</td></tr></table>","table_caption":"[\"Table 2. Tasks for Which at Least ${50}\\\\%$ of Participants Reported Using Modifications. \"]","table_footnote":"[\"Note. $N = 20$ . CADET = Columbia University Assessment of Disability in Essential Tremor [22]. a Included taking a picture or video with regular camera modifications. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[309,478,875,493]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Participants reported various modifications to perform different daily tasks: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,495,848,631]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"Performing the task with the other hand. \",\"- Holding the object with two hands/supporting one hand with the other. \",\"- Changing their hand posture. \",\"- Changing their grip force. \",\"- Leaning their body (e.g., forearm) against the table/another surface. \",\"- Holding the object in a different place. \",\"Utilizing an alternative instrument (e.g., using a spoon to eat a salad). \",\"Using an assistive device. \",\"- Altering system settings (e.g., computer mouse sensitivity). \"]","page_idx":"6","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[272,634,941,799]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Table 3 shows the prevalence of participants making modifications in at least one task. Stabilizing one hand with the other or holding an object with two hands was the most common modification type. Although the questionnaire did not specify this task (i.e., it specified only eating in a restaurant), participants reported making modifications in eating (including at home), such as using a spoon in place of a fork $(45.0\\%)$ or drinking soup instead of using a spoon. Some participants mentioned trying to relax or change eye focus or head tilt to suppress the tremor effect. The writing and carrying-a-cup tasks were among those with the highest prevalence having scores of one or more, and results show the most prevalent modification type for these tasks was holding the object with two hands/supporting one hand with the other hand $(33.3\\%)$ while writing and $50\\%$ while carrying a cup). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,55,184,68]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sensors 2024, 24, 4854 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[895,59,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7 of 13 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[57,124,942,298]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0703/f037c771d78ce421aa527d8246939ef8f296e5840b63d80cbc984593b56ddcb9.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":"<table><tr><td>Modification Type</td><td>Prevalence (% Participants Using Modification in at Least One Task)</td></tr><tr><td>Holding the object with two hands/supporting one hand with the other</td><td>85.0</td></tr><tr><td>Leaning their body against the table/another surface</td><td>65.0</td></tr><tr><td>Changing the grip force</td><td>45.0</td></tr><tr><td>Changing their hand posture</td><td>40.0</td></tr><tr><td>Holding the object in a different place</td><td>35.0</td></tr><tr><td>Performing the task with the other hand</td><td>30.0</td></tr><tr><td>Utilizing an alternative instrument</td><td>30.0</td></tr><tr><td>Using an assistive device</td><td>30.0</td></tr><tr><td>Altering system settings</td><td>30.0</td></tr></table>","table_caption":"[\"Table 3. Modification Types Used in at Least One Task $\\\\left( {N = {20}}\\\\right)$ . \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[273,313,554,329]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4.4. Additional Characteristics Reported ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[272,332,942,451]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Although the study instruments did not directly ask about other affected body parts, stress, or embarrassment, some participants added information regarding these issues. For instance, $25\\%$ reported head tremors and $15.0\\%$ said physical activity increased the tremor. Furthermore, $65\\%$ mentioned that excitement, stress, the presence of other people (not just strangers), or the anticipation of experiencing tremors when meeting people might enhance the tremor. Overall, $55.0\\%$ reported that tremors caused embarrassment or discomfort in the presence of other people. Of these, $45.5\\%$ brought up embarrassment or discomfort during the CADET interview relative to the \"eating in a restaurant\" task. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,463,620,479]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4.5. Objective Measures of the Cup Drinking Task ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[272,481,942,571]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this study, 18 of the 20 participants reported difficulty in performing drinking-related tasks (Task 5 \"Drinking from a glass\" and/or Task 7 \"Carrying a cup\" of coffee in the CADET questionnaire). Figure 4 shows the MSE-T measure vs. the participants' overall CADET score among these participants. Each of the figure's subplots addresses a different grip type. Based on the results shown in Figure 4, no correlation is found between the CADET score and the MSE-T measure. This result was evident for all performance measures. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[285,587,847,684]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0276/ec65eb5b81d68e4fc68ac66645d28f393cbeee3313b030faf9114c6279ac2d80.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[278,688,847,785]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0445/168a424e3896bbc2ad2f1eaef45c1301755bb3dfadedc188ee688d894b5d521e.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[278,790,847,887]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 4. MSE-T 5-6 Measure vs. CADET Score. \"]","image_footnote":"[]","img_path":"images/0142/2fdcee6c54caefda2a39553085a7015609bb93842e5c5ac7092e22ec18bdac1a.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[55,55,184,68]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sensors 2024, 24, 4854 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[897,59,941,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"8 of 13 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[275,102,421,118]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4.6. Personal Profiles ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[273,120,941,285]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To further explore this result, we focused on four participants between the ages of 68 and 70. These participants were selected based on their reported difficulties in performing drinking-related tasks (5 and/or 7) on the CADET questionnaire. Among them, two participants, A (68 years old) and B (69 years old), had similar total CADET scores and reported difficulties in performing drinking-related tasks. Participant C (68 years old), who had the lowest total CADET score, reported no difficulties in performing drinking tasks. Finally, participant D (69 years old), who reported difficulties in drinking-related tasks, had the highest total CADET score. Table 4 outlines the obtained performance outcome measures of these four participants. We also added, from the CADET questionnaire, their reported scores for Tasks 5 and 7, as well as their overall CADET scores. It should be noted that a lower CADET score represents a lower level of reported difficulties. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,323,939,535]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0849/21d8a7c510643d73db0c50069d68c87fe074f00a39657981f7b249eeaf1c02e6.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":"<table><tr><td rowspan=\"2\">Subject</td><td rowspan=\"2\">Grip Type</td><td rowspan=\"2\">T46 (s)</td><td rowspan=\"2\">MSE-X (10-6)</td><td rowspan=\"2\">MSE-Y (10-6)</td><td rowspan=\"2\">MSE-Z (10-6)</td><td rowspan=\"2\">MSE-T (10-6)</td><td colspan=\"2\">CADET Specific Task</td><td rowspan=\"2\">CADET Total Score</td></tr><tr><td>Drinking from a Glass—Task 5</td><td>Carrying a Cup—Task 7</td></tr><tr><td>A</td><td>i</td><td>1.82</td><td>11,500</td><td>2440</td><td>1790</td><td>15,800</td><td>0</td><td>1</td><td>17.85</td></tr><tr><td>A</td><td>ii</td><td>1.75</td><td>536</td><td>1030</td><td>759</td><td>2320</td><td>0</td><td>1</td><td>17.85</td></tr><tr><td>A</td><td>iii</td><td>1.94</td><td>692</td><td>657</td><td>838</td><td>2190</td><td>0</td><td>1</td><td>17.85</td></tr><tr><td>B</td><td>i</td><td>1.87</td><td>92</td><td>14</td><td>144</td><td>250</td><td>0</td><td>1</td><td>22.22</td></tr><tr><td>B</td><td>ii</td><td>3.62</td><td>58</td><td>138</td><td>420</td><td>616</td><td>0</td><td>1</td><td>22.22</td></tr><tr><td>B</td><td>iii</td><td>3.1</td><td>181</td><td>122</td><td>476</td><td>780</td><td>0</td><td>1</td><td>22.22</td></tr><tr><td>C</td><td>i</td><td>3.9</td><td>85</td><td>128</td><td>40</td><td>253</td><td>0</td><td>0</td><td>5.77</td></tr><tr><td>C</td><td>ii</td><td>4.7</td><td>70</td><td>36</td><td>23</td><td>129</td><td>0</td><td>0</td><td>5.77</td></tr><tr><td>C</td><td>iii</td><td>3.8</td><td>39</td><td>47</td><td>21</td><td>107</td><td>0</td><td>0</td><td>5.77</td></tr><tr><td>D</td><td>i</td><td>1.68</td><td>2520</td><td>6370</td><td>627</td><td>9510</td><td>2</td><td>2</td><td>56</td></tr><tr><td>D</td><td>ii</td><td>1.31</td><td>1280</td><td>811</td><td>746</td><td>2840</td><td>2</td><td>2</td><td>56</td></tr><tr><td>D</td><td>iii</td><td>1.45</td><td>3500</td><td>355</td><td>3690</td><td>7450</td><td>2</td><td>2</td><td>56</td></tr></table>","table_caption":"[\"Table 4. Performance Measures and CADET. \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[273,555,941,659]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Results of the comparison between the CADET-related columns of Participant A with those of Participant B suggest that these two participants should have similar objective measurement values. The difference between their objective performance measures, however, suggests that they perform the drinking task differently. On the other hand, for participants C and D, the difference is evident in both the CADET-related and the performance-related columns. These results were also verified by watching the recorded videos of these participants. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[275,671,381,684]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5. Discussion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[273,689,941,793]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This study focuses on characterizing daily functional difficulties and modifications that people with ET use, based on their self-reports. The CADET mean scores were similar to those of the community cases presented in [10]. Other studies showed that over time, tremor amplitude and frequency change and, from the patient's perspective, worsen [5,39]. In our study, scores were collected at only one time point, and we found no significant correlation between self-reported disability and the duration of the tremor since it was first noticed. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,793,941,914]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As in previous works that used the Bain and Findley ADL scale [6] and CADET questionnaire [10], we found writing, carrying/holding a cup, using a spoon, and pouring among the six tasks with the highest percentages of participants reporting difficulty (or modification or changed efficiency). Participants also reported difficulty with digital equipment operation tasks, such as taking photos with a mobile phone. The five digital operation tasks included in the current study involved gross as well as fine motor tasks and postural/static, as well as movement/dynamic tasks (e.g., taking a photo or connecting a cable/plug to a digital accessory). The task of taking photos with a mobile phone is ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,57,184,68]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sensors 2024, 24, 4854 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[897,60,939,70]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"9 of 13 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[278,102,939,250]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"usually performed while trying to stabilize the hand/s, holding the mobile phone in the air in a desired posture against gravity. In contrast to this task, the other four tasks usually involve stabilizing the hands or the objects on the table/body. Among the five extra tasks, the \"Taking a Photo\" task was the most prevalent task with which participants reported having difficulty, needing to implement modifications, or losing efficiency. Thus, because digital equipment has become an inseparable tool in people's daily functioning, results indicating that people with ET have difficulty maintaining stability when attempting to use the equipment should be considered in evaluation and intervention processes. Further, such difficulties might be considered when developing embedded digital solutions, enabling people with ET the essential daily use of these instruments. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,252,939,430]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The diverse tasks in which participants scored one or more indicate that tremor does not manifest equally across the tasks addressed in the questionnaire. These results align with previous studies that used performance-based measures and showed that tremor amplitude differed among tasks [27,40]. When listing tasks in which the tremor bothered them most, participants did not always choose those included in the questionnaire. This suggests that although, for example, drinking and writing are essential, these are not necessarily the tasks that most bother people with ET. Because the prevalence of ET rises in people over the age of 65, the daily activities of people with ET may include more hobbies and home-based activities than in younger groups. Thus, in line with the personalized medicine trend, it is important to incorporate the patients' perspectives on their daily coping with ET [39] and consider widening the scope of self-report questionnaires with tasks relevant to patients' perspectives, such as hobby-related activities. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,431,939,653]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Task-performance modification may be a way for people with functional deficiencies to compensate for their difficulties and optimize the performance of the task [24,25,41]. In this study, the most commonly reported physical modification was holding the object with two hands/supporting one hand with the other hand. Although not asked this question about each task, the participants' self-reports indicated that not all tried to perform the task differently or implement different modification types. Some participants seemed unaware of the various modification types they could try for each task or did not remember which modifications they had implemented due to the tremor. Thus, participants may have implemented modifications beyond those they were able to report via the questionnaire. In addition, it seems that in some cases, participants did not feel comfortable implementing modifications in public. For example, 16 participants reported implementing modifications when carrying a cup (a more private, home-based task), whereas only six participants reported modifications when eating in a restaurant (more public). At the same time, it is important to point out that not all modification types were relevant for all tasks. For example, leaning on the table would not be a relevant modification for the carrying-a-tray task. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,653,939,713]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A better understanding and mapping of the task's related modifications and their effect on daily activities may help characterize future support guides and lists of possible modifications that occupational therapists offer people with ET [42]. Analyzing these modifications may also help in developing mechanical tremor-suppression devices. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,714,939,803]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Although not part of the data planned to be gathered during the interviews, about half of the participants addressed the issue of discomfort or embarrassment relevant to daily functions, such as carrying a cup or eating in a restaurant. Some described how just the notion of other people's presence enhances their tremors. Thus, we recommend that therapists relate not only to the tremor's impact on the task but also to the task-performance environment and whether the task is performed in front of other people. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[278,804,939,907]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Self-report disability questionnaires are an important tool as they complement the clinical and severity tests performed by neurologists. Nevertheless, this tool may not provide detailed information per specific task when addressing multiple ADL tasks. The drinking task was chosen for the primary objective measures as this is an ADL task comprising multiple phases with different kinematic features and reported difficulty by people with ET. Results from the objective performance-based measurements of the drinking task from the presented work show that the total score of the questionnaire provides a general view ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,57,184,68]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sensors 2024, 24, 4854 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[890,60,939,70]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10 of 13 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[272,102,942,159]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"of the disability in ADL rather than specific task characteristics. This was emphasized in the analysis of the results shown in Table 4. Hence, when creating an individual profile, it is important to combine objective measurements with the responses on (subjective) self-report questionnaires. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,162,942,222]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This study aimed to map activity performance features of people who are diagnosed with ET but who may not be followed routinely or medically. Moreover, it combined a self-report disability questionnaire and performance-based quantitative measures in a common ADL task of drinking to create an individual profile of each study participant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,222,942,326]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This work had two main limitations. First, the limited sample size can be seen as a shortcoming. We were unable to recruit a larger group because of the difficulty finding participants who met the inclusion and exclusion criteria when advertising to the general population, as most people with ET do not seek medical diagnosis or treatment [1,2]. Second, $25\\%$ of participants reported head tremors, which may have affected the disability and modification types reported. In addition, as this percentage relies on reported tremors, it could be that a larger percentage of participants had head tremors. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,338,391,351]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6. Conclusions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[272,356,942,491]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This study's results enable the mapping of individuals' daily function capabilities from their perspective. We recommend including digital equipment operation, as well as more hobby-related and home-based tasks, in ET disability self-report questionnaires. Further, we suggest that therapists should raise patients' awareness of possible modifications to improve task performance and thereby improve the patient's quality of life. Furthermore, it is suggested to combine specific performance-based tests using measurement systems to design tailor-made solutions for patient-based healthcare. Finally, to provide a general understanding of the difficulties faced by people with ET when performing ADL tasks, it is recommended that future research include performance-based analyses concerning various tasks. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,504,941,560]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Author Contributions: S.R. and N.R. performed the study's conceptualization and design. All authors performed the data analysis. N.R. was further responsible for data collection and writing the manuscript drafts. All authors worked on the final manuscript. All authors have read and agreed to the published version of the manuscript. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,567,633,581]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Funding: This research received no external funding. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,588,941,615]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Institutional Review Board Statement: This study was conducted according to the guidelines of the Declaration of Helsinki and the University of Haifa Ethics Committee approved the study (018/19). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,623,890,637]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Informed Consent Statement: All participants in this study signed informed consent forms. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,644,853,658]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Data Availability Statement: Data is unavailable due to privacy or ethical restrictions. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,665,942,693]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conflicts of Interest: The authors declare that they have neither financial nor non-financial competing interests. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[273,708,389,721]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abbreviations ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[272,737,512,751]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ADL Activities of daily living ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,751,791,764]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CADET Columbia University Assessment of Disability in Essential Tremor ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[272,764,462,777]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ET Essential tremor ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,791,144,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[53,810,941,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. Louis, E.D.; McCreary, M. How common is essential tremor? Update on the worldwide prevalence of essential tremor. Tremor Other Hyperkinetic Mov. 2021, 11, 28. [CrossRef] [PubMed] \",\"2. Elble, R.J. What is Essential Tremor? Curr. Neurol. Neurosci. Rep. 2013, 13, 353. [CrossRef] [PubMed] \",\"3. Louis, E.D.; Hernandez, N.; Michalec, M. Prevalence and correlates of rest tremor in essential tremor: Cross-sectional survey of 831 patients across four distinct cohorts. Eur. J. Neurol. 2015, 22, 927-932. [CrossRef] [PubMed] \",\"4. Brennan, K.C.; Jurewicz, E.C.; Ford, B.; Pullman, S.L.; Louis, E.D. Is essenmtial tremor predominantly a kinetic or a postural tremor? A clinical and electrophysiological study. Mov. Disord. 2002, 17, 313-316. [CrossRef] [PubMed] \",\"5. Elble, R.J. Essential tremor frequency decreases with time. Neurology 2000, 55, 1547-1551. [CrossRef] [PubMed] \"]","page_idx":"10","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[55,55,184,68]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sensors 2024, 24, 4854 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[890,59,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"11 of 13 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[53,102,941,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"6. Bain, P.G.; Findley, L.J.; Thompson, P.D.; Gresty, M.A.; Rothwell, J.C.; Harding, A.E.; Marsden, C.D. A study of hereditary essential tremor. Brain 1994, 117, 805-824. [CrossRef] [PubMed] \",\"7. Louis, E.D.; Wendt, K.J.; Pullman, S.L.; Ford, B. Is Essential Tremor Symmetric? Arch. Neurol. 1998, 55, 1553-1559. [CrossRef] [PubMed] \",\"8. Dogu, O.; Louis, E.D.; Sevim, S.; Kaleagasi, H.; Aral, M. Clinical characteristics of essential tremor in Mersin, Turkey: A population-based door-to-door study. J. Neurol. 2005, 252, 516-520. [CrossRef] [PubMed] \",\"9. Louis, E.D.; Okun, M.S. It is time to remove the \\\"benign\\\" from the essential tremor label. Park. Relat. Disord. 2011, 17. [CrossRef] \",\"10. Louis, E.D.; Barnes, L.; Albert, S.M.; Cote, L.; Schneier, F.R.; Pulman, S.L.; Yu, Q. Correlates of functional disability in essential tremor. Mov. Disord. 2001, 16, 914-920. [CrossRef] \",\"11. Louis, E.D.; Benito-León, J.; Bermejo-Pareja, F. Self-reported depression and anti-depressant medication use in essential tremor: Cross-sectional and prospective analyses in a population-based study. Eur. J. Neurol. 2007, 14, 1138-1146. [CrossRef] \",\"12. Louis, E.D. Functional correlates of lower cognitive test scores in essential tremor. Mov. Disord. 2010, 25, 481-485. [CrossRef] \",\"13. Louis, E.D.; Rios, E. Embarrassment in essential tremor: Prevalence, clinical correlates and therapeutic implications. Parkinsonism Relat. Disord. 2009, 15, 535-538. [CrossRef] [PubMed] \",\"14. Louis, E.D.; Machado, D.G. Tremor-related quality of life: A comparison of essential tremor vs. Parkinson's disease patients. Parkinsonism Relat. Disord. 2015, 21, 729-735. [CrossRef] \",\"15. Nguyen, H.V.; Ngian, V.; Cordato, D.; Shen, Q.; Chan, D.K.Y. Quality of life in a random sample of community dwelling older patients with essential tremor. Acta Neurol. Scand. 2007, 116, 289-292. [CrossRef] [PubMed] \",\"16. Chandran, V.; Pal, P.K. Quality of life and its determinants in essential tremor. Parkinsonism Relat. Disord. 2013, 19, 62-65. [CrossRef] \",\"17. Benito-León, J.; Louis, E.D. Clinical update: Diagnosis and treatment of essential tremor. Lancet 2007, 369, 1152-1154. [CrossRef] \",\"18. Fahn, S.; Tolosa, E.; Marin, C. Clinical Rating Scale for Tremor. In Parkinson's Disease and Movement Disorders; Urban & Schwarzenberg: Baltimore, MD, USA; Munich, Germany, 1988. \",\"19. Elble, R.; Comella, C.; Fahn, S.; Hallett, M.; Jankovic, J.; Juncos, J.L.; LeWitt, P.; Lyons, K.; Ondo, W.; Pahwa, R.; et al. Reliability of a new scale for essential tremor. Mov. Disord. 2012, 27, 1567-1569. [CrossRef] \",\"20. Louis, E.D.; Wendt, K.; Albert, S.M.; Pullman, S.L.; Yu, Q.; Andrews, H. Validity of a Performance-Based Test of function in essential tremor. Arch. Neurol. 1999, 56, 841-846. [CrossRef] [PubMed] \",\"21. Bain, P.G.; Findley, L.J.; Atchison, P.; Behari, M.; Vidailhet, M.; Gresty, M.; Rothwell, J.C.; Thompson, P.D.; Marsden, C.D. Assessing tremor severity. J. Neurol. Neurosurg. Psychiatry 1993, 56, 868-873. [CrossRef] \",\"22. Wendt, K.J.; Albert, S.M.; Pullman, S.L.; Schneier, F.R.; Louis, E.D. Columbia University Assessment of Disability in Essential Tremor (CADET): Methodological issues in essential tremor research. *Parkinsonism Relat. Disord.* 2000, 6, 17-23. [CrossRef] [PubMed] \",\"23. Louis, E.D.; Barnes, L.F.; Wendt, K.J.; Albert, S.M.; Pullman, S.L.; Yu, Q.; Schneier, F.R. Validity and test-retest reliability of a disability questionnaire for essential tremor. Mov. Disord. 2000, 15, 516-523. [CrossRef] [PubMed] \",\"24. Fried, L.P.; Bandeen-Roche, K.; Williamson, J.D. Functional decline in older adults: Expanding methods of ascertainment. J. Gerontol. Ser. A Biol. Sci. Med. Sci. 1996, 51, M206-M214. [CrossRef] [PubMed] \",\"25. Fried, L.P.; Herdman, S.J.; Kuhn, K.E.; Rubin, G.; Turano, K. Preclinical Disability: Hypotheses About the Bottom of the Iceberg. J. Aging Health 1991, 3, 285-300. [CrossRef] \",\"26. Heldman, D.A.; Jankovic, J.; Vaillancourt, D.E.; Prodoehl, J.; Elble, R.J.; Giuffrida, J.P. Essential tremor quantification during activities of daily living. Parkinsonism Relat. Disord. 2011, 17, 537-542. [CrossRef] [PubMed] \",\"27. Bain, P.G.; Mally, J.; Gresty, M.; Findley, L.J. Assessing the impact of essential tremor on upper limb function. J. Neurol. 1993, 241, 54-61. [CrossRef] [PubMed] \",\"28. Schuhmayer, N.; Weber, C.; Kieler, M.; Voller, B.; Pirker, W.; Auff, E.; Haubenberger, D. Task-dependent variability of Essential Tremor. Parkinsonism Relat. Disord. 2017, 41, 79-85. [CrossRef] [PubMed] \",\"29. Mostile, G.; Giuffrida, J.P.; Adam, O.R.; Davidson, A.; Jankovic, J. Correlation between kinesia system assessments and clinical tremor scores in patients with essential tremor. Mov. Disord. 2010, 25, 1938-1943. [CrossRef] [PubMed] \",\"30. Murphy, M.A.; Sinnerhagen, K.S.; Johnels, B.; Willén, C. Three-dimensional kinematic motion analysis of a daily activity drinking from a glass: A pilot study. J. Neuroeng. Rehabil. 2006, 3, 18. [CrossRef] \",\"31. Kavanagh, J.J.; Keogh, J.W.L. Correlates between Force and Postural Tremor in Older Individuals with Essential Tremor. Cerebellum 2016, 15, 688-695. [CrossRef] \",\"32. Héroux, M.E.; Pari, G.; Norman, K.E. The effect of inertial loading on wrist postural tremor in essential tremor. Clin. Neurophysiol. 2009, 120, 1020-1029. [CrossRef] [PubMed] \",\"33. Gironell, A.; Martínez-Corral, M.; Pagonabarraga, J.; Kulisevsky, J. The Glass scale: A simple tool to determine severity in essential tremor. Parkinsonism Relat. Disord. 2010, 16, 412-414. [CrossRef] [PubMed] \",\"34. Golan, D.; Giladi, N.; Thorne, R.; Korczyn, A.D.; Simon, E.S. Subdivision of essential tremor patients according to physiologic characteristics. Acta Neurol. Scand. 2004, 109, 393-397. [CrossRef] [PubMed] \",\"35. Roth, N.; Rosenblum, S. Does cup-grip type affect tremor among people with essential tremor? Sensors 2021, 21, 7797. [CrossRef] [PubMed] \"]","page_idx":"11","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[55,57,184,68]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sensors 2024, 24, 4854 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[890,59,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"12 of 13 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[53,103,941,282]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"36. Goetz, L.H.; Schork, N.J. Personalized medicine: Motivation, challenges, and progress. Fertil. Steril. 2018, 109, 952-963. [CrossRef] [PubMed] \",\"37. Roth, N.; Braun-Benyamin, O.; Rosenblum, S. Drawing direction effect on a task's performance characteristics among people with essential tremor. Sensors 2021, 21, 5814. [CrossRef] [PubMed] \",\"38. Folstein, M.F.; Folstein, S.E.; McHugh, P.R. \\\"Mini-mental state\\\": A practical method for grading the cognitive state of patients for the clinician. J. Psychiatr. Res. 1975, 12, 189-198. [CrossRef] [PubMed] \",\"39. Gutierrez, J.; Park, J.; Badejo, O.; Louis, E.D. Worse and worse and worse: Essential tremor patients' longitudinal perspectives on their condition. Front. Neurol. 2016, 7, 175. [CrossRef] [PubMed] \",\"40. Elble, R.J.; Brilliant, M.; Leffler, K.; Higgins, C. Quantification of essential tremor in writing and drawing. Mov. Disord. 1996, 11, 70-78. [CrossRef] \",\"41. Moore, J.; Eccles, F.J.R.; Simpson, J. Post-diagnostic lived experiences of individuals with essential tremor. Disabil. Rehabil. 2020, 42, 2894-2902. [CrossRef] \",\"42. Hopfner, F.; Deuschl, G. Managing Essential Tremor. Neurotherapeutics 2020, 17, 1603-1621. [CrossRef] [PubMed] \"]","page_idx":"12","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[53,298,942,341]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Disclaimer/Publisher's Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,57,184,68]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sensors 2024, 24, 4854 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[890,59,939,71]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"12","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"13 of 13 ","text_format":null,"text_level":null,"type":"page_number"}],"string":"[\n {\n \"bbox\": \"[53,122,107,136]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Article \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,137,932,184]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Subjective and Objective Day-to-Day Performance Measures of People with Essential Tremor \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,197,519,212]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Navit Roth $^{1,2}$ , Adham Salih $^{2}$ and Sara Rosenblum $^{1,*}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,237,939,303]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1 The Laboratory of Complex Human Activity and Participation (CHAP), Department of Occupational Therapy, University of Haifa, Haifa 3498838, Israel; navroth@braude.ac.il \\\",\\\"$^{2}$ Department of Mechanical Engineering, Braude College of Engineering, Karmiel 2161002, Israel; adhamsalih@braude.ac.il \\\",\\\"* Correspondence: rosens@research.haifa.ac.il \\\"]\",\n \"page_idx\": \"0\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[272,319,942,565]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract: This paper aims to map the daily functional characteristics of people diagnosed with essential tremor (ET) based on their subjective self-reports. In addition, we provide objective measurements of a cup-drinking task. This study involved 20 participants diagnosed with ET who completed the Columbia University Assessment of Disability in Essential Tremor (CADET) questionnaire that included five additional tasks related to digital equipment operation we wrote. Participants also described task-performance modifications they implemented. To create objective personal performance profiles, they performed a cup-drinking task while being monitored using a sensor measurement system. The CADET's subjective self-report results indicate that the most prevalent tasks participants reported as having difficulty with or requiring modifications were writing, threading a needle, carrying a cup, using a spoon, pouring, and taking a photo or video on a mobile phone. Analysis of participants' modifications revealed that holding the object with two hands or with one hand supporting the other were the most prevalent types. No significant correlation was found between the CADET total scores and the cup drinking objective measures. Capturing patients' perspectives on their functional disability, alongside objective performance measures, is envisioned to contribute to the development of custom-tailored interventions aligned with individual profiles, i.e., patient-based/smart healthcare. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,554,146,576]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0250/7afbd7e38300f9234205385dabb4b9d13fa7e343c9bffac2d0984886a4ee544e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[53,581,243,593]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Citation: Roth, N.; Salih, A. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,595,213,607]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Rosenblum, S. Subjective and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,609,242,621]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Objective Day-to-Day Performance \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,624,236,636]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Measures of People with Essential \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,638,215,650]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Tremor. Sensors 2024, 24, 4854. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,652,245,664]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.3390/s24154854 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,674,250,686]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Academic Editors: Chan Hwang See, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,688,226,699]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Arfan Ghani and Simeon Keates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,709,178,722]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Received: 26 June 2024 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,724,169,736]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Revised: 19 July 2024 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,738,176,750]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Accepted: 24 July 2024 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,753,181,764]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Published: 26 July 2024 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,777,154,803]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0703/9f582f766f8b36313e136c55d535147a8408637b65bc5adbdeb775175d8998a7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[53,808,257,919]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,580,744,595]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords: essential tremor; functional disability; ADL; modifications \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[275,639,394,653]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,657,939,791]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Essential tremor (ET), one of the most common movement disorders, is found in a wide age range, with a prevalence of about $5\\\\%$ in people over 65 years [1]. The disorder is characterized mainly as an action tremor (postural or kinetic) but can also appear while at rest [2-4]. The frequency of the tremor in ET may range between 4 and $12\\\\mathrm{Hz}$ and decreases with time [5]. Although ET may affect assorted body parts, most often upper limbs are affected with a slight asymmetry [6,7]. Most people with ET do not seek medical diagnosis [1,6,8,9]. About $85\\\\%$ report that the tremor causes functional disability [10] in the activities of daily living (ADL)—such as eating, drinking, and using tools and self-care instruments that require fine and gross motor skills [10] and in work performance [6]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,791,941,912]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Previous studies indicated that stress and fatigue may also affect the tremor, whereas non-motoric symptoms, including changes in cognition, depression, and anxiety, may also be involved among patients with ET [6,9,11,12]. The potential for ET to cause embarrassment and social consequences [6] seems to influence the patients' decisions to seek medical treatment [13]. Embarrassment exists in almost $50\\\\%$ of those with mild ET (even without head tremor) [13], and the social effects may result in patients avoiding social meetings and tasks that will expose the tremor. These studies implied that ET and its associated depression and anxiety may cause a decline in quality of life among people with ET [14-16]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,57,115,98]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"# \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[121,68,253,87]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"sensors \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[863,64,937,99]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"MDPI \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[55,964,403,977]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sensors 2024, 24, 4854. https://doi.org/10.3390/s24154854 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[687,964,939,977]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://www.mdpi.com/journal/sensors \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[278,102,939,177]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Currently, ET is clinically diagnosed using a combination of a neurological examination and a medical history review [17]. Assessment of tremor severity may include scales that are not ET-specific, such as the Fahn-Tolosa-Marin Clinical Tremor Rating Scale [18] and scales developed specifically for ET such as the Tremor Research Group Essential Tremor Rating Assessment Scale [19]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,178,939,370]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Usually, functional disability due to tremors is assessed with self-report questionnaires and performance-based tests [20] that may be part of a severity-assessment scale. Self-report questionnaires provide assessors with the perspective of the person coping with the tremor by asking respondents whether they have difficulty or disability in ADL tasks and how difficult those tasks are to perform. Such instruments are the Bain and Findley ADL scale [21] and the Columbia University Assessment of Disability in Essential Tremor (CADET) questionnaire [22]. The CADET, which was implemented in the current study, is a valid and reliable 31-item self-report questionnaire. It focuses on the respondent's difficulty/disability while performing ADL tasks and whether the respondent made modifications (or had a change in efficiency) [22,23]. Eliciting this information is especially valuable because people who make modifications to perform ADL do not necessarily report having a disability [24]. Modifications include changing the environmental, psychosocial, behavioral, psychological, or physical method of performing a specific task [25]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,372,939,489]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In previous studies, the highest percentage of people with ET who were surveyed reported the physical tasks of using a spoon, drinking from, holding, or carrying a cup, pouring a drink, and of writing were difficult to perform, required modification, or carried out less efficiently than previously [6,10]. To the best of our knowledge, however, there is no detailed documentation on the modification types/profiles they implemented in various daily tasks. Thus, this current study focuses on physical modifications. In addition, because we did not find detailed documentation on the reported disability while performing diverse tasks of digital and smartphone operation, we made these tasks a secondary focus of our research. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,491,939,579]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Another aim of this paper is to demonstrate the need for performance-based tests that supply objective measures of day-to-day performance. Measurements—to quantify tremors in various ADL tasks—may be acquired using digital-graphic boards, accelerometers and gyroscopes, image processing, and electromyography [20,26-28]. The resulting metrics may be implemented as reciprocal validation [29] of neurology assessments and/or self-report scales [20], providing support for the individual's functional health status evaluation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,580,941,699]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this study, the performance-based objective measures were gathered for the cup-drinking task as an example of a functional daily task that challenges people with ET [6,21,22]. This particular task was chosen as more than $65\\\\%$ of participants with ET reported difficulties/disabilities or needed modification/efficiency changes while performing the task [10,27]. In addition, this daily task comprises multiple phases with different kinematic features [30], during which tremor characteristics, which may differ with posture and force levels, may affect tremor amplitude [31,32]. The cup-drinking task was used by the Glass scale [33] as one task for assessing tremor severity among people with ET. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,699,941,788]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In previous studies, objective measures of the tremor were obtained while performing cup tasks using hand or cup-attached accelerometers in drinking and holding-a-cup tasks [26,34,35] or by measuring spilled water volume in the holding-a-cup task [27]. To the best of our knowledge, however, the correlation between the objective drinking task measures and the total score of a self-report disability questionnaire was not reported and thus was examined in the current study. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,789,941,907]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In line with the personal medicine view [36], understanding patients' perspectives and the ways they perform daily tasks may aid in the pre-clinical phase, as well as in assessment and treatment processes, toward improved physical and emotional health and quality of life [25]. Thus, this study also aims to map activity performance features of people who are diagnosed with ET but who may not be followed routinely or medically. To this end, we combine the individual patient's responses on a self-report disability questionnaire and objective performance-based quantitative measures for the common ADL task of drinking to create an individual profile. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,57,184,68]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sensors 2024, 24, 4854 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[897,60,939,70]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 of 13 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[275,103,364,116]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. Methods \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[275,118,391,133]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.1. Participants \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,137,941,256]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We recruited participants by advertising among the general population (e.g., via social media, bulletin boards, and senior culture clubs). Inclusion criteria were being aged 20 or more and having a written physician's diagnosis of ET. Exclusion criteria were another disability that could affect the participants' tremors based on their self-reports (previously also described) [37]. Twenty people $(N = 20)$ met the inclusion and exclusion criteria and participated in the study. This study was conducted according to the guidelines of the Declaration of Helsinki; the University of Haifa Ethics Committee approved the study (018/19). All participants signed informed consent forms. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[275,268,393,282]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2. Instruments \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,286,941,348]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Participants who met the criteria signed an informed consent form and completed a demographic questionnaire, a Mini-Mental State Examination [38], and an expanded and translated CADET questionnaire, including questions on extra digital tasks [22]. In addition, they performed an ADL task of drinking from a cup using three grip types. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[275,357,532,373]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2.1. Demographic Questionnaire \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,376,941,437]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The demographic questionnaire included information on the participants' age, education (years), dominant hand, tremor-specific medication, general medication, duration (years) since noticing the tremor, duration (years) since diagnosis with ET, and family members diagnosed with ET. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[275,447,917,462]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2.2. CADET Questionnaire with Further Questions on Extra Tasks and Modifications \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,466,941,541]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To address disability in daily tasks due to the tremor, a Hebrew translation of the validated CADET questionnaire was administrated after approval by the corresponding author [23]. The 31 tasks were rated as 0 (no disability, need to modify activities, or no loss of efficiency), 1 (no disability but a need to modify activities or a loss of efficiency), or 2 (disability or both a need to modify and loss of efficiency) [10]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,541,941,690]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Questions related to the five digital equipment operation tasks were added for this study: (1) connect to digital accessories (e.g., connect a phone charger, headphones, and a USB drive); (2) select an object with a computer mouse (e.g., select and click on an object from a toolbar or a recipient from an e-mail list); (3) type digits or a text message with fingers on a mobile phone; (4) take a photo or video using a mobile phone; and (5) use a finger to choose an object on a mobile phone screen (e.g., an application icon). The CADET outcome measures comprised two mean scores: one for the questionnaire's original 31 items and the other for the five questions related to the extra tasks. Participants also listed up to five tasks in which the tremor bothered them the most and elaborated on modifications in the task performance they made due to the tremor. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[275,701,796,715]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2.2.3. The Cup Drinking Task: Objective Performance-Based Measures \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,720,941,869]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As described in the introduction section, the ADL task of drinking has been chosen as a representative ADL task for the objective measurements. This task included lifting a cup from the table, moving the cup toward the mouth, performing a drinking-like movement, and returning the cup to the table. Participants were seated in front of a table with the cup positioned on it at about $25\\\\mathrm{cm}$ from the edge in front of the participant's dominant hand. This layout was applied to enable the task simulation of picking up the cup and drinking in their regular manner during a meal. Participants were instructed to perform the task in their regular manner as close as possible to their real-life daily performance but using three grip types: (i) around-the-cup grip (without using the handle), (ii) below-the-handle grip (leaning the handle on the fingers), and (iii) on-the-handle grip (holding only the handle). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,870,941,914]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The task was characterized and divided into six states describing the different positions of the cup during the task. This task, which addressed the effect of the tremor on the cup acceleration, was performed using a specially built, customized measurement system, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,57,184,68]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sensors 2024, 24, 4854 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[897,60,939,70]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 of 13 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[272,102,944,209]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"which included a triaxial accelerometer (ADXL335 (Module by SparkFun, Niwot, CO, USA, Sensor by Analog Devices, Wilmington, MA, USA) with a range of $\\\\pm 3\\\\mathrm{g}$ ) attached to the cup. Data were sampled at $100\\\\mathrm{Hz}$ using an acquisition component (NI-6008) and a computer. The cup system included an $80\\\\%$ volume-filled cup of water covered by a lid. The system was previously described by Roth and Rosenblum in their study [35]. The axes system used with the accelerometer is shown in Figure 1, whereas the cup position states appear in Figure 2. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[284,224,379,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 1. Cup accelerometer axes layout. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0124/31140ab2299d4e3ddb70c466aa9dce4be892d2c2f2175f4b1b3f2b08dae11f72.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[115,351,374,470]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0085/0ba9cacf92fbbc5499e252de363753c2efbbf0fa91c1f808b5dcfc43a1feb9e9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[371,351,630,470]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0033/32093ab236449ac9363c0965b89ee068af23f51d9541aaf690ada85b9f0c0b5e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[628,351,885,470]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0563/2409cd039013f7442e32f87df1f09bb72d4dab87bd675c5dd4008c807f452bf0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[115,480,374,598]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/1001/482449f89b658158455fd1cca66005c466b2d282025a3c5368744b57deb52316.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[371,480,630,598]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0017/36576ad29befc1a6d95523434bf27c97d68ba512bef1bd9cfbf6b677f0d025e3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[628,480,885,598]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 2. Cup position states: State 1—On the table, start position; State 2—Mid-duration between States 2 and 3; State 3—Near the mouth, before drinking; State 4—Near the mouth, after drinking, start of returning the cup to the table movement; State 5—Mid-duration between States 4 and 6; State 6—Contact with the table, end of returning movement. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0993/50f41767219b2a91302af67dfb08de0530b4fa9cd003785d439e84230b2bc60d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[272,682,944,829]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To evaluate each participant's ability to perform the drinking task, two performance measures were developed. The first measure, which is hereby termed T46, was defined as the duration between States 4 and 6. The recorded videos showed that 'reaching the mouth' movements, i.e., going from State 1 to State 4, were performed differently by each participant. For example, some participants held the cup near their mouth for several seconds, whereas other participants started the 'return to the table' movement immediately after reaching State 3. In addition, some participants tilted the cup or vertically raised the cup to mouth level at the beginning of the movement and then began moving the cup toward their mouth. Therefore, to ensure consistency among participants when analyzing the performance characteristics, we decided to consider only the time between States 4 and 6. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,831,944,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The second measure aimed to evaluate the overall \\\"error\\\" caused by the tremor. To calculate this measure, first, we filtered out the signals that are associated with ET (4-12 Hz). Then, the mean-squared error (MSE) was calculated with reference to constant zero. This procedure was applied to each axis (X, Y, and Z). In addition, the overall MSE, denoted by MSE-T and constitutes the sum of the MSE components, was also calculated. Figure 3 illustrates the calculation steps for the X-axis. The black curve shows the voltage as recorded by the sensor. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,55,184,68]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sensors 2024, 24, 4854 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[897,59,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 of 13 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[273,102,941,133]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The filtered data are represented by the blue curve, whereas the red curves represent the data trend (by applying a moving average window). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,149,784,444]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 3. MSE Calculation. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0091/11b76787c7694fb07fba7ddfbf2a1e072a61d8de1b1c48366454e189aafeea52.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[272,480,941,555]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As shown in Figure 3, as expected, the filtered data trend is approximately zero. This observation suggests that the filtered data provide a good approximation of the tremor effect. It should be noted that according to the limitations of the experimental setup regarding the acceleration measurement without a gyro element, as reported in [35], we decided to analyze and report the MSE values only for States 5 to 6. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,565,445,581]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3. Statistical Analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,583,941,674]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Statistical analysis of the demographic data was conducted using the SPSS software, Ver 25, frequency and descriptive tools. Prevalence was calculated for CADET tasks scored one or more by participants and for the tasks they noted as bothering them most. The internal reliability of the five extra digital tasks was calculated using Cronbach's alpha. Nonparametric correlates using the Spearman test were calculated among CADET mean scores, age, and duration since noticing the tremor. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,674,941,734]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Analysis of modification data included modification prevalence in each task (for scored tasks) and modification methods. The percentages of participants who reported performing at least one task slower than had been the case prior to ET presenting, avoiding a task, and (although not asked) embarrassment and feelings of stress were also calculated. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,734,941,778]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For the objective performance-based measures, nonparametric correlates using the Spearman test were calculated among CADET mean scores and each objective performance-based measure. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,790,354,804]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4. Results \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,806,408,822]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4.1. Demographics \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,824,941,898]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The participants' demographic details were described in the methods section. The participants' ages ranged from 23 to 81 years ( $M = 64.7$ , $SD = 15.7$ ). Males comprised half of the sample ( $n = 10$ ) and $75\\\\%$ of the participants ( $n = 15$ ) had a family history of tremors based on their self-reports. Their education ranged from 12 to 27 years ( $M = 17.2$ , $SD = 3.4$ ) (see [37] for further details). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,55,184,68]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sensors 2024, 24, 4854 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[897,59,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 of 13 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[273,102,840,118]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4.2. CADET Scores and Item's Frequency Distribution, Including the Added Tasks \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,120,941,193]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The internal reliability of the five extra digital-operation tasks was $\\\\alpha = 0.58$ , and significant correlations were found between their mean scores and the original 31 tasks on the CADET ( $r = 0.61$ , $p < 0.01$ ). No significant correlations, however, were found between mean scores (either of the 31 CADET tasks or the five extra tasks) and age or tremor duration. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,196,941,255]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The 31 CADET task scores ranged from 5.77 to 66.67 ( $M = 32.29$ , $SD = 17.43$ ) (previously also described in [37]). The extra digital task scores ranged from 0 to 80.00 ( $M = 32.50$ , $SD = 20.74$ ). Most $(85\\\\%)$ participants reported avoiding or preferring someone else to perform at least one task. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,255,941,360]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Table 1 details the results for the CADET and extra tasks in which more than $50\\\\%$ of participants scored one or higher (indicating disability, need for modifications, or changed efficiency). The table also shows tasks in which $100\\\\%$ of participants scored zero and tasks more than $25\\\\%$ avoided. Regarding digital tasks not displayed (i.e., $<50\\\\%$ ), $45.0\\\\%$ of participants scored one or higher for the task \\\"Select an object with the computer mouse\\\", and $30.0\\\\%$ for \\\"Connect to digital accessories and choose an object on a mobile phone screen with your finger\\\". \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,397,989,783]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0860/03f4a041a05ecd19d13e31a22b94f88128cb93f0628440e96d1a80972e1c369f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Item #</td><td>Description</td><td>Prevalence</td></tr><tr><td colspan=\\\"3\\\">CADET Questionnairea</td></tr><tr><td>2</td><td>Writing a letter, postcard, thank you card, or chequeb</td><td>100.0%</td></tr><tr><td>26</td><td>Threading a needleb</td><td>92.9%</td></tr><tr><td>7</td><td>Carrying a cup of coffee</td><td>90.0%</td></tr><tr><td>8</td><td>Using a spoon to drink soup</td><td>90.0%</td></tr><tr><td>6</td><td>Pouring milk or juice from a bottle</td><td>80.0%</td></tr><tr><td>10</td><td>Eating in a restaurant</td><td>75.0%</td></tr><tr><td>5</td><td>Drinking from a glass</td><td>70.0%</td></tr><tr><td>1</td><td>Signing your name</td><td>60.0%</td></tr><tr><td>24</td><td>Reading a book, magazine, or newspaper</td><td>60.0%</td></tr><tr><td>30</td><td>Tying your necktie (males) or putting on your lipstick (females)b</td><td>58.3%</td></tr><tr><td>31</td><td>Shaving (males) or putting on your eyeliner (females)b</td><td>57.1%</td></tr><tr><td>25</td><td>Unlocking door with a key</td><td>55.0%</td></tr><tr><td>9</td><td>Carrying a tray of foodb</td><td>52.6%</td></tr><tr><td colspan=\\\"3\\\">Five extra digital equipment operation tasksa</td></tr><tr><td>4</td><td>Taking a photo or video with a mobile phone</td><td>80.0%</td></tr><tr><td>3</td><td>Typing digits or texting messages with fingers on a mobile phone</td><td>50.0%</td></tr><tr><td colspan=\\\"3\\\">Minimally difficult tasksc</td></tr><tr><td>21\\n(CADET)</td><td>Putting on your watch</td><td></td></tr><tr><td colspan=\\\"3\\\">Avoided tasksd</td></tr><tr><td>4\\n(five extra digital tasks)</td><td>Taking a photo or video with a mobile phone</td><td>40.0%</td></tr><tr><td>2 (CADET)</td><td>Writing a letter, postcard, thank you card, or cheque</td><td>31.6%</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 1. Task Prevalence. \\\"]\",\n \"table_footnote\": \"[\\\"Note. CADET = Columbia University Assessment of Disability in Essential Tremor [22]. a Tasks for which at least $50\\\\\\\\%$ of participants scored one or more (have difficulty, need modification, loss of efficiency); b $N = 20$ , but percentages are of participants reporting on specific tasks: $n = 19$ for Items 9 and 2 (CADET), $n = 12$ for Items 30 and 21, and $n = 14$ for Items 26 and 31 (spelling as in original questionnaire); c $100\\\\\\\\%$ of participants scored zero (no difficulty, need for modification, or loss of efficiency); d tasks at least $25\\\\\\\\%$ of participants reported they avoided or preferred others to perform it for them. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[272,872,941,917]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is important to note that $65\\\\%$ of participants reported other tasks that bothered them the most, ones that were not included in the CADET questionnaire. Such tasks were preparing food, eating (not just in a restaurant but also when using a fork at home), and performing work or \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,55,184,68]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sensors 2024, 24, 4854 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[895,59,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 of 13 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[272,102,941,133]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"hobby tasks, such as painting. Tasks reported with a prevalence of $35\\\\%$ or greater were eating $(55.0\\\\%)$ , carrying or drinking from a cup $(50.0\\\\%)$ , and writing $(35.0\\\\%)$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,143,470,159]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4.3. Reported Modifications \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,162,941,223]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"All participants reported implementing at least one modification, and $65.0\\\\%$ reported performing at least one task slower because of the tremor. These modifications are listed below. Table 2 details the tasks for which at least $50\\\\%$ of participants made modifications. The highest percentage of modifications were made to carrying a cup. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,260,942,435]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0172/609251ab0c0b52acbc0b0c2500ad17dbb4498cb3af1f8dae6a26148c8fc51381.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Item #</td><td>Description</td><td>Prevalence (% Participants Using Modification)</td></tr><tr><td>CADET</td><td></td><td></td></tr><tr><td>7</td><td>Carrying a cup of coffee</td><td>80.0</td></tr><tr><td>6</td><td>Pouring milk or juice from a bottle</td><td>65.0</td></tr><tr><td>2</td><td>Writing a letter, postcard, thank you card, or cheque</td><td>63.2</td></tr><tr><td>5</td><td>Drinking from a glass</td><td>60.0</td></tr><tr><td>24</td><td>Reading a book, magazine, or newspaper</td><td>55.0</td></tr><tr><td>8</td><td>Using a spoon to drink soup</td><td>50.0</td></tr><tr><td>Five digital tasks</td><td></td><td></td></tr><tr><td>4</td><td>Taking a photo or video with a mobile phonea</td><td>50.0</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 2. Tasks for Which at Least ${50}\\\\\\\\%$ of Participants Reported Using Modifications. \\\"]\",\n \"table_footnote\": \"[\\\"Note. $N = 20$ . CADET = Columbia University Assessment of Disability in Essential Tremor [22]. a Included taking a picture or video with regular camera modifications. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[309,478,875,493]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Participants reported various modifications to perform different daily tasks: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,495,848,631]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"Performing the task with the other hand. \\\",\\\"- Holding the object with two hands/supporting one hand with the other. \\\",\\\"- Changing their hand posture. \\\",\\\"- Changing their grip force. \\\",\\\"- Leaning their body (e.g., forearm) against the table/another surface. \\\",\\\"- Holding the object in a different place. \\\",\\\"Utilizing an alternative instrument (e.g., using a spoon to eat a salad). \\\",\\\"Using an assistive device. \\\",\\\"- Altering system settings (e.g., computer mouse sensitivity). \\\"]\",\n \"page_idx\": \"6\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[272,634,941,799]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Table 3 shows the prevalence of participants making modifications in at least one task. Stabilizing one hand with the other or holding an object with two hands was the most common modification type. Although the questionnaire did not specify this task (i.e., it specified only eating in a restaurant), participants reported making modifications in eating (including at home), such as using a spoon in place of a fork $(45.0\\\\%)$ or drinking soup instead of using a spoon. Some participants mentioned trying to relax or change eye focus or head tilt to suppress the tremor effect. The writing and carrying-a-cup tasks were among those with the highest prevalence having scores of one or more, and results show the most prevalent modification type for these tasks was holding the object with two hands/supporting one hand with the other hand $(33.3\\\\%)$ while writing and $50\\\\%$ while carrying a cup). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,55,184,68]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sensors 2024, 24, 4854 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[895,59,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7 of 13 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[57,124,942,298]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0703/f037c771d78ce421aa527d8246939ef8f296e5840b63d80cbc984593b56ddcb9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Modification Type</td><td>Prevalence (% Participants Using Modification in at Least One Task)</td></tr><tr><td>Holding the object with two hands/supporting one hand with the other</td><td>85.0</td></tr><tr><td>Leaning their body against the table/another surface</td><td>65.0</td></tr><tr><td>Changing the grip force</td><td>45.0</td></tr><tr><td>Changing their hand posture</td><td>40.0</td></tr><tr><td>Holding the object in a different place</td><td>35.0</td></tr><tr><td>Performing the task with the other hand</td><td>30.0</td></tr><tr><td>Utilizing an alternative instrument</td><td>30.0</td></tr><tr><td>Using an assistive device</td><td>30.0</td></tr><tr><td>Altering system settings</td><td>30.0</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 3. Modification Types Used in at Least One Task $\\\\\\\\left( {N = {20}}\\\\\\\\right)$ . \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[273,313,554,329]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4.4. Additional Characteristics Reported \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,332,942,451]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Although the study instruments did not directly ask about other affected body parts, stress, or embarrassment, some participants added information regarding these issues. For instance, $25\\\\%$ reported head tremors and $15.0\\\\%$ said physical activity increased the tremor. Furthermore, $65\\\\%$ mentioned that excitement, stress, the presence of other people (not just strangers), or the anticipation of experiencing tremors when meeting people might enhance the tremor. Overall, $55.0\\\\%$ reported that tremors caused embarrassment or discomfort in the presence of other people. Of these, $45.5\\\\%$ brought up embarrassment or discomfort during the CADET interview relative to the \\\"eating in a restaurant\\\" task. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,463,620,479]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4.5. Objective Measures of the Cup Drinking Task \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,481,942,571]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this study, 18 of the 20 participants reported difficulty in performing drinking-related tasks (Task 5 \\\"Drinking from a glass\\\" and/or Task 7 \\\"Carrying a cup\\\" of coffee in the CADET questionnaire). Figure 4 shows the MSE-T measure vs. the participants' overall CADET score among these participants. Each of the figure's subplots addresses a different grip type. Based on the results shown in Figure 4, no correlation is found between the CADET score and the MSE-T measure. This result was evident for all performance measures. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[285,587,847,684]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0276/ec65eb5b81d68e4fc68ac66645d28f393cbeee3313b030faf9114c6279ac2d80.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[278,688,847,785]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0445/168a424e3896bbc2ad2f1eaef45c1301755bb3dfadedc188ee688d894b5d521e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[278,790,847,887]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 4. MSE-T 5-6 Measure vs. CADET Score. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0142/2fdcee6c54caefda2a39553085a7015609bb93842e5c5ac7092e22ec18bdac1a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[55,55,184,68]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sensors 2024, 24, 4854 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[897,59,941,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"8 of 13 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[275,102,421,118]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4.6. Personal Profiles \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,120,941,285]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To further explore this result, we focused on four participants between the ages of 68 and 70. These participants were selected based on their reported difficulties in performing drinking-related tasks (5 and/or 7) on the CADET questionnaire. Among them, two participants, A (68 years old) and B (69 years old), had similar total CADET scores and reported difficulties in performing drinking-related tasks. Participant C (68 years old), who had the lowest total CADET score, reported no difficulties in performing drinking tasks. Finally, participant D (69 years old), who reported difficulties in drinking-related tasks, had the highest total CADET score. Table 4 outlines the obtained performance outcome measures of these four participants. We also added, from the CADET questionnaire, their reported scores for Tasks 5 and 7, as well as their overall CADET scores. It should be noted that a lower CADET score represents a lower level of reported difficulties. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,323,939,535]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0849/21d8a7c510643d73db0c50069d68c87fe074f00a39657981f7b249eeaf1c02e6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=\\\"2\\\">Subject</td><td rowspan=\\\"2\\\">Grip Type</td><td rowspan=\\\"2\\\">T46 (s)</td><td rowspan=\\\"2\\\">MSE-X (10-6)</td><td rowspan=\\\"2\\\">MSE-Y (10-6)</td><td rowspan=\\\"2\\\">MSE-Z (10-6)</td><td rowspan=\\\"2\\\">MSE-T (10-6)</td><td colspan=\\\"2\\\">CADET Specific Task</td><td rowspan=\\\"2\\\">CADET Total Score</td></tr><tr><td>Drinking from a Glass—Task 5</td><td>Carrying a Cup—Task 7</td></tr><tr><td>A</td><td>i</td><td>1.82</td><td>11,500</td><td>2440</td><td>1790</td><td>15,800</td><td>0</td><td>1</td><td>17.85</td></tr><tr><td>A</td><td>ii</td><td>1.75</td><td>536</td><td>1030</td><td>759</td><td>2320</td><td>0</td><td>1</td><td>17.85</td></tr><tr><td>A</td><td>iii</td><td>1.94</td><td>692</td><td>657</td><td>838</td><td>2190</td><td>0</td><td>1</td><td>17.85</td></tr><tr><td>B</td><td>i</td><td>1.87</td><td>92</td><td>14</td><td>144</td><td>250</td><td>0</td><td>1</td><td>22.22</td></tr><tr><td>B</td><td>ii</td><td>3.62</td><td>58</td><td>138</td><td>420</td><td>616</td><td>0</td><td>1</td><td>22.22</td></tr><tr><td>B</td><td>iii</td><td>3.1</td><td>181</td><td>122</td><td>476</td><td>780</td><td>0</td><td>1</td><td>22.22</td></tr><tr><td>C</td><td>i</td><td>3.9</td><td>85</td><td>128</td><td>40</td><td>253</td><td>0</td><td>0</td><td>5.77</td></tr><tr><td>C</td><td>ii</td><td>4.7</td><td>70</td><td>36</td><td>23</td><td>129</td><td>0</td><td>0</td><td>5.77</td></tr><tr><td>C</td><td>iii</td><td>3.8</td><td>39</td><td>47</td><td>21</td><td>107</td><td>0</td><td>0</td><td>5.77</td></tr><tr><td>D</td><td>i</td><td>1.68</td><td>2520</td><td>6370</td><td>627</td><td>9510</td><td>2</td><td>2</td><td>56</td></tr><tr><td>D</td><td>ii</td><td>1.31</td><td>1280</td><td>811</td><td>746</td><td>2840</td><td>2</td><td>2</td><td>56</td></tr><tr><td>D</td><td>iii</td><td>1.45</td><td>3500</td><td>355</td><td>3690</td><td>7450</td><td>2</td><td>2</td><td>56</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 4. Performance Measures and CADET. \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[273,555,941,659]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Results of the comparison between the CADET-related columns of Participant A with those of Participant B suggest that these two participants should have similar objective measurement values. The difference between their objective performance measures, however, suggests that they perform the drinking task differently. On the other hand, for participants C and D, the difference is evident in both the CADET-related and the performance-related columns. These results were also verified by watching the recorded videos of these participants. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[275,671,381,684]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5. Discussion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,689,941,793]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This study focuses on characterizing daily functional difficulties and modifications that people with ET use, based on their self-reports. The CADET mean scores were similar to those of the community cases presented in [10]. Other studies showed that over time, tremor amplitude and frequency change and, from the patient's perspective, worsen [5,39]. In our study, scores were collected at only one time point, and we found no significant correlation between self-reported disability and the duration of the tremor since it was first noticed. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,793,941,914]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As in previous works that used the Bain and Findley ADL scale [6] and CADET questionnaire [10], we found writing, carrying/holding a cup, using a spoon, and pouring among the six tasks with the highest percentages of participants reporting difficulty (or modification or changed efficiency). Participants also reported difficulty with digital equipment operation tasks, such as taking photos with a mobile phone. The five digital operation tasks included in the current study involved gross as well as fine motor tasks and postural/static, as well as movement/dynamic tasks (e.g., taking a photo or connecting a cable/plug to a digital accessory). The task of taking photos with a mobile phone is \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,57,184,68]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sensors 2024, 24, 4854 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[897,60,939,70]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"9 of 13 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[278,102,939,250]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"usually performed while trying to stabilize the hand/s, holding the mobile phone in the air in a desired posture against gravity. In contrast to this task, the other four tasks usually involve stabilizing the hands or the objects on the table/body. Among the five extra tasks, the \\\"Taking a Photo\\\" task was the most prevalent task with which participants reported having difficulty, needing to implement modifications, or losing efficiency. Thus, because digital equipment has become an inseparable tool in people's daily functioning, results indicating that people with ET have difficulty maintaining stability when attempting to use the equipment should be considered in evaluation and intervention processes. Further, such difficulties might be considered when developing embedded digital solutions, enabling people with ET the essential daily use of these instruments. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,252,939,430]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The diverse tasks in which participants scored one or more indicate that tremor does not manifest equally across the tasks addressed in the questionnaire. These results align with previous studies that used performance-based measures and showed that tremor amplitude differed among tasks [27,40]. When listing tasks in which the tremor bothered them most, participants did not always choose those included in the questionnaire. This suggests that although, for example, drinking and writing are essential, these are not necessarily the tasks that most bother people with ET. Because the prevalence of ET rises in people over the age of 65, the daily activities of people with ET may include more hobbies and home-based activities than in younger groups. Thus, in line with the personalized medicine trend, it is important to incorporate the patients' perspectives on their daily coping with ET [39] and consider widening the scope of self-report questionnaires with tasks relevant to patients' perspectives, such as hobby-related activities. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,431,939,653]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Task-performance modification may be a way for people with functional deficiencies to compensate for their difficulties and optimize the performance of the task [24,25,41]. In this study, the most commonly reported physical modification was holding the object with two hands/supporting one hand with the other hand. Although not asked this question about each task, the participants' self-reports indicated that not all tried to perform the task differently or implement different modification types. Some participants seemed unaware of the various modification types they could try for each task or did not remember which modifications they had implemented due to the tremor. Thus, participants may have implemented modifications beyond those they were able to report via the questionnaire. In addition, it seems that in some cases, participants did not feel comfortable implementing modifications in public. For example, 16 participants reported implementing modifications when carrying a cup (a more private, home-based task), whereas only six participants reported modifications when eating in a restaurant (more public). At the same time, it is important to point out that not all modification types were relevant for all tasks. For example, leaning on the table would not be a relevant modification for the carrying-a-tray task. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,653,939,713]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A better understanding and mapping of the task's related modifications and their effect on daily activities may help characterize future support guides and lists of possible modifications that occupational therapists offer people with ET [42]. Analyzing these modifications may also help in developing mechanical tremor-suppression devices. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,714,939,803]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Although not part of the data planned to be gathered during the interviews, about half of the participants addressed the issue of discomfort or embarrassment relevant to daily functions, such as carrying a cup or eating in a restaurant. Some described how just the notion of other people's presence enhances their tremors. Thus, we recommend that therapists relate not only to the tremor's impact on the task but also to the task-performance environment and whether the task is performed in front of other people. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[278,804,939,907]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Self-report disability questionnaires are an important tool as they complement the clinical and severity tests performed by neurologists. Nevertheless, this tool may not provide detailed information per specific task when addressing multiple ADL tasks. The drinking task was chosen for the primary objective measures as this is an ADL task comprising multiple phases with different kinematic features and reported difficulty by people with ET. Results from the objective performance-based measurements of the drinking task from the presented work show that the total score of the questionnaire provides a general view \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,57,184,68]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sensors 2024, 24, 4854 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[890,60,939,70]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10 of 13 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[272,102,942,159]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"of the disability in ADL rather than specific task characteristics. This was emphasized in the analysis of the results shown in Table 4. Hence, when creating an individual profile, it is important to combine objective measurements with the responses on (subjective) self-report questionnaires. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,162,942,222]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This study aimed to map activity performance features of people who are diagnosed with ET but who may not be followed routinely or medically. Moreover, it combined a self-report disability questionnaire and performance-based quantitative measures in a common ADL task of drinking to create an individual profile of each study participant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,222,942,326]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This work had two main limitations. First, the limited sample size can be seen as a shortcoming. We were unable to recruit a larger group because of the difficulty finding participants who met the inclusion and exclusion criteria when advertising to the general population, as most people with ET do not seek medical diagnosis or treatment [1,2]. Second, $25\\\\%$ of participants reported head tremors, which may have affected the disability and modification types reported. In addition, as this percentage relies on reported tremors, it could be that a larger percentage of participants had head tremors. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,338,391,351]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6. Conclusions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,356,942,491]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This study's results enable the mapping of individuals' daily function capabilities from their perspective. We recommend including digital equipment operation, as well as more hobby-related and home-based tasks, in ET disability self-report questionnaires. Further, we suggest that therapists should raise patients' awareness of possible modifications to improve task performance and thereby improve the patient's quality of life. Furthermore, it is suggested to combine specific performance-based tests using measurement systems to design tailor-made solutions for patient-based healthcare. Finally, to provide a general understanding of the difficulties faced by people with ET when performing ADL tasks, it is recommended that future research include performance-based analyses concerning various tasks. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,504,941,560]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Author Contributions: S.R. and N.R. performed the study's conceptualization and design. All authors performed the data analysis. N.R. was further responsible for data collection and writing the manuscript drafts. All authors worked on the final manuscript. All authors have read and agreed to the published version of the manuscript. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,567,633,581]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Funding: This research received no external funding. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,588,941,615]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Institutional Review Board Statement: This study was conducted according to the guidelines of the Declaration of Helsinki and the University of Haifa Ethics Committee approved the study (018/19). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,623,890,637]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Informed Consent Statement: All participants in this study signed informed consent forms. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,644,853,658]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data Availability Statement: Data is unavailable due to privacy or ethical restrictions. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,665,942,693]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conflicts of Interest: The authors declare that they have neither financial nor non-financial competing interests. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[273,708,389,721]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abbreviations \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,737,512,751]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ADL Activities of daily living \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,751,791,764]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CADET Columbia University Assessment of Disability in Essential Tremor \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[272,764,462,777]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ET Essential tremor \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,791,144,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[53,810,941,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. Louis, E.D.; McCreary, M. How common is essential tremor? Update on the worldwide prevalence of essential tremor. Tremor Other Hyperkinetic Mov. 2021, 11, 28. [CrossRef] [PubMed] \\\",\\\"2. Elble, R.J. What is Essential Tremor? Curr. Neurol. Neurosci. Rep. 2013, 13, 353. [CrossRef] [PubMed] \\\",\\\"3. Louis, E.D.; Hernandez, N.; Michalec, M. Prevalence and correlates of rest tremor in essential tremor: Cross-sectional survey of 831 patients across four distinct cohorts. Eur. J. Neurol. 2015, 22, 927-932. [CrossRef] [PubMed] \\\",\\\"4. Brennan, K.C.; Jurewicz, E.C.; Ford, B.; Pullman, S.L.; Louis, E.D. Is essenmtial tremor predominantly a kinetic or a postural tremor? A clinical and electrophysiological study. Mov. Disord. 2002, 17, 313-316. [CrossRef] [PubMed] \\\",\\\"5. Elble, R.J. Essential tremor frequency decreases with time. Neurology 2000, 55, 1547-1551. [CrossRef] [PubMed] \\\"]\",\n \"page_idx\": \"10\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[55,55,184,68]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sensors 2024, 24, 4854 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[890,59,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"11 of 13 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[53,102,941,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"6. Bain, P.G.; Findley, L.J.; Thompson, P.D.; Gresty, M.A.; Rothwell, J.C.; Harding, A.E.; Marsden, C.D. A study of hereditary essential tremor. Brain 1994, 117, 805-824. [CrossRef] [PubMed] \\\",\\\"7. Louis, E.D.; Wendt, K.J.; Pullman, S.L.; Ford, B. Is Essential Tremor Symmetric? Arch. Neurol. 1998, 55, 1553-1559. [CrossRef] [PubMed] \\\",\\\"8. Dogu, O.; Louis, E.D.; Sevim, S.; Kaleagasi, H.; Aral, M. Clinical characteristics of essential tremor in Mersin, Turkey: A population-based door-to-door study. J. Neurol. 2005, 252, 516-520. [CrossRef] [PubMed] \\\",\\\"9. Louis, E.D.; Okun, M.S. It is time to remove the \\\\\\\"benign\\\\\\\" from the essential tremor label. Park. Relat. Disord. 2011, 17. [CrossRef] \\\",\\\"10. Louis, E.D.; Barnes, L.; Albert, S.M.; Cote, L.; Schneier, F.R.; Pulman, S.L.; Yu, Q. Correlates of functional disability in essential tremor. Mov. Disord. 2001, 16, 914-920. [CrossRef] \\\",\\\"11. Louis, E.D.; Benito-León, J.; Bermejo-Pareja, F. Self-reported depression and anti-depressant medication use in essential tremor: Cross-sectional and prospective analyses in a population-based study. Eur. J. Neurol. 2007, 14, 1138-1146. [CrossRef] \\\",\\\"12. Louis, E.D. Functional correlates of lower cognitive test scores in essential tremor. Mov. Disord. 2010, 25, 481-485. [CrossRef] \\\",\\\"13. Louis, E.D.; Rios, E. Embarrassment in essential tremor: Prevalence, clinical correlates and therapeutic implications. Parkinsonism Relat. Disord. 2009, 15, 535-538. [CrossRef] [PubMed] \\\",\\\"14. Louis, E.D.; Machado, D.G. Tremor-related quality of life: A comparison of essential tremor vs. Parkinson's disease patients. Parkinsonism Relat. Disord. 2015, 21, 729-735. [CrossRef] \\\",\\\"15. Nguyen, H.V.; Ngian, V.; Cordato, D.; Shen, Q.; Chan, D.K.Y. Quality of life in a random sample of community dwelling older patients with essential tremor. Acta Neurol. Scand. 2007, 116, 289-292. [CrossRef] [PubMed] \\\",\\\"16. Chandran, V.; Pal, P.K. Quality of life and its determinants in essential tremor. Parkinsonism Relat. Disord. 2013, 19, 62-65. [CrossRef] \\\",\\\"17. Benito-León, J.; Louis, E.D. Clinical update: Diagnosis and treatment of essential tremor. Lancet 2007, 369, 1152-1154. [CrossRef] \\\",\\\"18. Fahn, S.; Tolosa, E.; Marin, C. Clinical Rating Scale for Tremor. In Parkinson's Disease and Movement Disorders; Urban & Schwarzenberg: Baltimore, MD, USA; Munich, Germany, 1988. \\\",\\\"19. Elble, R.; Comella, C.; Fahn, S.; Hallett, M.; Jankovic, J.; Juncos, J.L.; LeWitt, P.; Lyons, K.; Ondo, W.; Pahwa, R.; et al. Reliability of a new scale for essential tremor. Mov. Disord. 2012, 27, 1567-1569. [CrossRef] \\\",\\\"20. Louis, E.D.; Wendt, K.; Albert, S.M.; Pullman, S.L.; Yu, Q.; Andrews, H. Validity of a Performance-Based Test of function in essential tremor. Arch. Neurol. 1999, 56, 841-846. [CrossRef] [PubMed] \\\",\\\"21. Bain, P.G.; Findley, L.J.; Atchison, P.; Behari, M.; Vidailhet, M.; Gresty, M.; Rothwell, J.C.; Thompson, P.D.; Marsden, C.D. Assessing tremor severity. J. Neurol. Neurosurg. Psychiatry 1993, 56, 868-873. [CrossRef] \\\",\\\"22. Wendt, K.J.; Albert, S.M.; Pullman, S.L.; Schneier, F.R.; Louis, E.D. Columbia University Assessment of Disability in Essential Tremor (CADET): Methodological issues in essential tremor research. *Parkinsonism Relat. Disord.* 2000, 6, 17-23. [CrossRef] [PubMed] \\\",\\\"23. Louis, E.D.; Barnes, L.F.; Wendt, K.J.; Albert, S.M.; Pullman, S.L.; Yu, Q.; Schneier, F.R. Validity and test-retest reliability of a disability questionnaire for essential tremor. Mov. Disord. 2000, 15, 516-523. [CrossRef] [PubMed] \\\",\\\"24. Fried, L.P.; Bandeen-Roche, K.; Williamson, J.D. Functional decline in older adults: Expanding methods of ascertainment. J. Gerontol. Ser. A Biol. Sci. Med. Sci. 1996, 51, M206-M214. [CrossRef] [PubMed] \\\",\\\"25. Fried, L.P.; Herdman, S.J.; Kuhn, K.E.; Rubin, G.; Turano, K. Preclinical Disability: Hypotheses About the Bottom of the Iceberg. J. Aging Health 1991, 3, 285-300. [CrossRef] \\\",\\\"26. Heldman, D.A.; Jankovic, J.; Vaillancourt, D.E.; Prodoehl, J.; Elble, R.J.; Giuffrida, J.P. Essential tremor quantification during activities of daily living. Parkinsonism Relat. Disord. 2011, 17, 537-542. [CrossRef] [PubMed] \\\",\\\"27. Bain, P.G.; Mally, J.; Gresty, M.; Findley, L.J. Assessing the impact of essential tremor on upper limb function. J. Neurol. 1993, 241, 54-61. [CrossRef] [PubMed] \\\",\\\"28. Schuhmayer, N.; Weber, C.; Kieler, M.; Voller, B.; Pirker, W.; Auff, E.; Haubenberger, D. Task-dependent variability of Essential Tremor. Parkinsonism Relat. Disord. 2017, 41, 79-85. [CrossRef] [PubMed] \\\",\\\"29. Mostile, G.; Giuffrida, J.P.; Adam, O.R.; Davidson, A.; Jankovic, J. Correlation between kinesia system assessments and clinical tremor scores in patients with essential tremor. Mov. Disord. 2010, 25, 1938-1943. [CrossRef] [PubMed] \\\",\\\"30. Murphy, M.A.; Sinnerhagen, K.S.; Johnels, B.; Willén, C. Three-dimensional kinematic motion analysis of a daily activity drinking from a glass: A pilot study. J. Neuroeng. Rehabil. 2006, 3, 18. [CrossRef] \\\",\\\"31. Kavanagh, J.J.; Keogh, J.W.L. Correlates between Force and Postural Tremor in Older Individuals with Essential Tremor. Cerebellum 2016, 15, 688-695. [CrossRef] \\\",\\\"32. Héroux, M.E.; Pari, G.; Norman, K.E. The effect of inertial loading on wrist postural tremor in essential tremor. Clin. Neurophysiol. 2009, 120, 1020-1029. [CrossRef] [PubMed] \\\",\\\"33. Gironell, A.; Martínez-Corral, M.; Pagonabarraga, J.; Kulisevsky, J. The Glass scale: A simple tool to determine severity in essential tremor. Parkinsonism Relat. Disord. 2010, 16, 412-414. [CrossRef] [PubMed] \\\",\\\"34. Golan, D.; Giladi, N.; Thorne, R.; Korczyn, A.D.; Simon, E.S. Subdivision of essential tremor patients according to physiologic characteristics. Acta Neurol. Scand. 2004, 109, 393-397. [CrossRef] [PubMed] \\\",\\\"35. Roth, N.; Rosenblum, S. Does cup-grip type affect tremor among people with essential tremor? Sensors 2021, 21, 7797. [CrossRef] [PubMed] \\\"]\",\n \"page_idx\": \"11\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[55,57,184,68]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sensors 2024, 24, 4854 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[890,59,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"12 of 13 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[53,103,941,282]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"36. Goetz, L.H.; Schork, N.J. Personalized medicine: Motivation, challenges, and progress. Fertil. Steril. 2018, 109, 952-963. [CrossRef] [PubMed] \\\",\\\"37. Roth, N.; Braun-Benyamin, O.; Rosenblum, S. Drawing direction effect on a task's performance characteristics among people with essential tremor. Sensors 2021, 21, 5814. [CrossRef] [PubMed] \\\",\\\"38. Folstein, M.F.; Folstein, S.E.; McHugh, P.R. \\\\\\\"Mini-mental state\\\\\\\": A practical method for grading the cognitive state of patients for the clinician. J. Psychiatr. Res. 1975, 12, 189-198. [CrossRef] [PubMed] \\\",\\\"39. Gutierrez, J.; Park, J.; Badejo, O.; Louis, E.D. Worse and worse and worse: Essential tremor patients' longitudinal perspectives on their condition. Front. Neurol. 2016, 7, 175. [CrossRef] [PubMed] \\\",\\\"40. Elble, R.J.; Brilliant, M.; Leffler, K.; Higgins, C. Quantification of essential tremor in writing and drawing. Mov. Disord. 1996, 11, 70-78. [CrossRef] \\\",\\\"41. Moore, J.; Eccles, F.J.R.; Simpson, J. Post-diagnostic lived experiences of individuals with essential tremor. Disabil. Rehabil. 2020, 42, 2894-2902. [CrossRef] \\\",\\\"42. Hopfner, F.; Deuschl, G. Managing Essential Tremor. Neurotherapeutics 2020, 17, 1603-1621. [CrossRef] [PubMed] \\\"]\",\n \"page_idx\": \"12\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[53,298,942,341]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Disclaimer/Publisher's Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,57,184,68]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sensors 2024, 24, 4854 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[890,59,939,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"12\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"13 of 13 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.3390/s24154854"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Medicine and Health Sciences"},"sha256":{"kind":"string","value":"e8dbf18411e9cb7a9f311ac0b7235b19f2e117d7fab2ed676a45c0a7781768f7"},"title":{"kind":"string","value":"Subjective and Objective Day-to-Day Performance Measures of People with Essential Tremor"}}},{"rowIdx":61,"cells":{"abstract":{"kind":"string","value":""},"author":{"kind":"string","value":""},"content_list":{"kind":"list like","value":[{"bbox":"[312,85,685,112]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Supporting Information for ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[115,181,880,272]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Enhanced Triboelectric Nanogenerator Based on Tungsten Disulfide via Thiolated Ligand Conjugation ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[112,317,806,338]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Tae In Kim, $^{\\dagger}$ Ick-Joon Park, $^{\\ddagger}$ Sumin Kang, $^{\\S}$ Taek-Soo Kim $^{\\S}$ and Sung-Yool Choi $^{*,\\dagger}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,376,882,568]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"†School of Electrical Engineering, Graphene/2D Materials Research Center, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, Korea \",\"$^{\\\\ddagger}$ Department of Electrical and Electronic Engineering, Joongbu University, Goyang 10279, Korea \",\"$^{\\\\S}$ Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, Korea 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Characterization of chemically exfoliated $\\\\mathrm{WS}_2$ nanosheets. (a) AFM image of chemically exfoliated $\\\\mathrm{WS}_2$ nanosheets via lithium-intercalated exfoliation. (b) Raman spectrum of chemically exfoliated $\\\\mathrm{WS}_2$ nanosheets. (c) UV-vis spectrum of chemically exfoliated $\\\\mathrm{WS}_2$ nanosheets. The insets show a schematic illustration of $\\\\mathrm{WS}_2$ structure (left) and a photograph of $\\\\mathrm{WS}_2$ nanosheet suspension (right). \"]","image_footnote":"[]","img_path":"images/0528/b29c96d65df6fbe32673da129063d82067971aaa1511399b6d579d44caf4b90e.jpg","list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[492,926,505,938]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[329,86,678,225]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0593/e47bc3dc70f3f33b6aff9ef8d7bbb82f5c64485e727f48e66a72bd99913e330e.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[327,225,678,363]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/1023/48566e5b4f661795ae6daff601cf6c05666232341bee8437b8938b0138c95daf.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[329,363,678,502]","code_body":null,"code_caption":null,"image_caption":"[\"Figure S2. ATR-IR spectra of unconjugated ligand and ligand-conjugated $\\\\mathrm{WS}_2$ . ATR-IR spectra of (a) MPA molecule and $\\\\mathrm{MPA - WS}_2$ , (b) MHA molecule and MHA- $\\\\mathrm{WS}_2$ , and (c) MOA molecule and MOA- $\\\\mathrm{WS}_2$ samples, exhibiting the binding between thiol containing ligands and $\\\\mathrm{WS}_2$ nanosheets. \"]","image_footnote":"[]","img_path":"images/0640/2eea0e1edd7067b23c809b45e30fd18031956c510ba065b28ca414acf4fd0673.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[492,926,505,938]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[337,85,660,366]","code_body":null,"code_caption":null,"image_caption":"[\"Figure S3. Deconvoluted W 4f peaks of the pristine $\\\\mathrm{WS}_2$ and ligand-conjugated $\\\\mathrm{WS}_2$ . \"]","image_footnote":"[]","img_path":"images/0379/c88bcf3131e7c4fc84312489d52d98aa313c61fa5830f92ca15f4e297f854f9d.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[492,927,505,938]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[309,87,687,432]","code_body":null,"code_caption":null,"image_caption":"[\"Figure S4. UPS spectra of the pristine $\\\\mathrm{WS}_2$ and ligand-conjugated $\\\\mathrm{WS}_2$ with MPA, MHA, MOA, and MUA molecules. \"]","image_footnote":"[]","img_path":"images/0817/9b47564a2bb5f92876bbc9a762aba7ab109551f285e9dd690520cbfba99fdd91.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[492,926,505,939]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[247,80,751,244]","code_body":null,"code_caption":null,"image_caption":"[\"Figure S5. Triboelectric output current of the pristine and ligand-conjugated $\\\\mathrm{WS}_2$ TENG devices. \"]","image_footnote":"[]","img_path":"images/0131/c77d4717fd0a28aa5d045c15871f4006a97441f5081bdcc8648c32618dfa4dfb.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[492,927,505,938]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[122,82,267,186]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0905/f84d877db898475fff4e9e500793e32c598ece4c0def7c2d727e2c2726e8ac6b.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[273,82,418,186]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0940/ae39ca7a1f1bd682ace416d2599d2ada81b5e1f7f298bf8003b70c3ae10f02a7.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[426,82,573,186]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0630/1838cf539667f21bf8d2ad0ba12f48da72b154197d2022227ea16f16233a0da2.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[581,82,727,186]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0720/6b1c5c14a3b465d53282c7e1193da45b232ba78249ced23003c13c354f7055fd.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[734,82,880,186]","code_body":null,"code_caption":null,"image_caption":"[\"Figure S6. AFM images of the pristine $\\\\mathrm{WS}_2$ and ligand-conjugated $\\\\mathrm{WS}_2$ surfaces. The surface \"]","image_footnote":"[\"topographies of the (a) pristine $\\\\mathrm{WS}_2$ , (b) MPA- $\\\\mathrm{WS}_2$ , (c) MHA- $\\\\mathrm{WS}_2$ , (d) MOA- $\\\\mathrm{WS}_2$ , and (e) \",\"$\\\\mathrm{MUA - WS_2}$ surfaces, having surface roughness values of 1.5, 2.3, 2.6, 3.3, and $4.6~\\\\mathrm{nm}$ , respectively. \"]","img_path":"images/0609/38f86d86ae27395db902918133d8cbda0182cf0ba13d1b06a9287fba90746a51.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[492,926,505,938]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[260,83,739,249]","code_body":null,"code_caption":null,"image_caption":"[\"Figure S7. Triboelectric output performance of the MUA-WS $_2$ TENG with the concentration of MUA treated with WS $_2$ . \"]","image_footnote":"[]","img_path":"images/0637/c360616e7024866207a873343b4b745686287677cdfe7295ee0cafb144715cd4.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[492,927,505,938]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"8 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[122,84,240,168]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0436/e0364f5ed25ce557a0372ed0acda18453cbbf6bf214cc2943384740e2115bcba.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[245,84,366,168]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0448/206e61780563ed17ab91e9fce1e334405a5a5168d514347af180084b54492a07.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[371,84,492,168]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0106/5edfcf48561076428672f742f9321bd62c0f71e7eed65a928920b51c4b170f9f.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[497,83,618,168]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0373/692916f6a7cf898a2e90ea9c270d484afd16450670fda08bf177397e57bbdc23.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[621,83,742,168]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0071/95e24d5669ef48ccc2fecc6dbe27873d265faabac76df115ddad928e3c16e5f7.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[747,83,868,168]","code_body":null,"code_caption":null,"image_caption":"[\"Figure S8. AFM images of the pristine $\\\\mathrm{WS}_2$ and ligand-conjugated $\\\\mathrm{WS}_2$ surfaces functionalized with MUA having different concentrations. The surface topographies of the (a) pristine $\\\\mathrm{WS}_2$ , and ligand-conjugated $\\\\mathrm{WS}_2$ treated with MUA having a concentration of (b) 5, (c) 25, (d) 50, (e) 100, and (f) $500~\\\\mathrm{mM}$ , respectively. The surface roughness values of the pristine $\\\\mathrm{WS}_2$ and ligand-conjugated $\\\\mathrm{WS}_2$ surfaces treated with MUA having different concentrations are 1.5, 1.8, 2.5, 2.8, 3.8, and $4.6~\\\\mathrm{nm}$ , respectively. \"]","image_footnote":"[]","img_path":"images/0602/c805ce606985db6b191ae411e23cda2d951f5107cf415fbcda3b527111bf1388.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[492,926,505,938]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"9 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[312,86,684,272]","code_body":null,"code_caption":null,"image_caption":"[\"Figure S9. Charge density of the pristine and ligand-conjugated TMD TENG devices calculated by integrating the short-circuit current over time. Error bars represent standard deviations. \"]","image_footnote":"[]","img_path":"images/0767/27cc2044202dab0a17bf252aca0785139a8a51318873172da138026eca2192ee.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[487,925,510,939]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[220,85,519,236]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0440/2d7040bf6a8894bcd70470d2bdb8097d3c978ce31ee4ad54a000b225901421e8.jpg","list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[522,86,786,237]","code_body":null,"code_caption":null,"image_caption":"[\"Figure S10. Output voltages of the ligand-conjugated $\\\\mathrm{WS}_2$ TENG devices with MUA molecule after repeated pressing and releasing operation. (a) The output performance of MUA- $\\\\mathrm{WS}_2$ TENG devices after the repeated operation over 10, 100, 1000, and 10000 cycles, exhibiting the stable triboelectric operation of the ligand-conjugated TMD TENG devices. (b) The plot of output voltage of MUA- $\\\\mathrm{WS}_2$ TENG devices after the repeated operation cycles in average. \"]","image_footnote":"[]","img_path":"images/0443/77ae8c4496b411abe976c95ed06f7e4b8bc74297b9724aa53b1895de99404edb.jpg","list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[487,925,509,939]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"11 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[154,87,500,299]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0131/8e299926ffdee90a6c092d38f2b1076e002feb2a61cfe3388715c4a664b3a382.jpg","list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[507,87,845,299]","code_body":null,"code_caption":null,"image_caption":"[\"Figure S11. Long-term chemical stability of the pristine $\\\\mathrm{WS}_2$ and MUA- $\\\\mathrm{WS}_2$ samples. \"]","image_footnote":"[]","img_path":"images/0715/39b38678a52958f6c692c9a719729ba6e39d36dc211ea571d349391775223658.jpg","list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[112,348,882,398]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Deconvoluted S 2p peaks of (a) pristine $\\mathrm{WS}_2$ and (b) MUA- $\\mathrm{WS}_2$ samples as-prepared, and after 56 and 98 days. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[487,925,510,939]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"12 ","text_format":null,"text_level":null,"type":"page_number"}],"string":"[\n {\n \"bbox\": \"[312,85,685,112]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Supporting Information for \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,181,880,272]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Enhanced Triboelectric Nanogenerator Based on Tungsten Disulfide via Thiolated Ligand Conjugation \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,317,806,338]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Tae In Kim, $^{\\\\dagger}$ Ick-Joon Park, $^{\\\\ddagger}$ Sumin Kang, $^{\\\\S}$ Taek-Soo Kim $^{\\\\S}$ and Sung-Yool Choi $^{*,\\\\dagger}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,376,882,568]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"†School of Electrical Engineering, Graphene/2D Materials Research Center, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, Korea \\\",\\\"$^{\\\\\\\\ddagger}$ Department of Electrical and Electronic Engineering, Joongbu University, Goyang 10279, Korea \\\",\\\"$^{\\\\\\\\S}$ Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, Korea \\\"]\",\n \"page_idx\": \"0\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[114,590,423,608]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"* E-mail: sungyool.choi@kaist.ac.kr \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[492,926,505,939]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[188,105,354,219]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(a) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0821/1414b30d98b01b9d3973115708dc58a163efe42ee4814ceee6391ea9ef4e3dd4.jpg\",\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[356,102,594,239]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0436/c7e487787f680902572772fcfd24ca3829eb017f8522040f354feb21a6dd2aca.jpg\",\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[596,102,836,239]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure S1. Characterization of chemically exfoliated $\\\\\\\\mathrm{WS}_2$ nanosheets. (a) AFM image of chemically exfoliated $\\\\\\\\mathrm{WS}_2$ nanosheets via lithium-intercalated exfoliation. (b) Raman spectrum of chemically exfoliated $\\\\\\\\mathrm{WS}_2$ nanosheets. (c) UV-vis spectrum of chemically exfoliated $\\\\\\\\mathrm{WS}_2$ nanosheets. The insets show a schematic illustration of $\\\\\\\\mathrm{WS}_2$ structure (left) and a photograph of $\\\\\\\\mathrm{WS}_2$ nanosheet suspension (right). \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0528/b29c96d65df6fbe32673da129063d82067971aaa1511399b6d579d44caf4b90e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[492,926,505,938]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[329,86,678,225]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0593/e47bc3dc70f3f33b6aff9ef8d7bbb82f5c64485e727f48e66a72bd99913e330e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[327,225,678,363]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/1023/48566e5b4f661795ae6daff601cf6c05666232341bee8437b8938b0138c95daf.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[329,363,678,502]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure S2. ATR-IR spectra of unconjugated ligand and ligand-conjugated $\\\\\\\\mathrm{WS}_2$ . ATR-IR spectra of (a) MPA molecule and $\\\\\\\\mathrm{MPA - WS}_2$ , (b) MHA molecule and MHA- $\\\\\\\\mathrm{WS}_2$ , and (c) MOA molecule and MOA- $\\\\\\\\mathrm{WS}_2$ samples, exhibiting the binding between thiol containing ligands and $\\\\\\\\mathrm{WS}_2$ nanosheets. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0640/2eea0e1edd7067b23c809b45e30fd18031956c510ba065b28ca414acf4fd0673.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[492,926,505,938]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[337,85,660,366]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure S3. Deconvoluted W 4f peaks of the pristine $\\\\\\\\mathrm{WS}_2$ and ligand-conjugated $\\\\\\\\mathrm{WS}_2$ . \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0379/c88bcf3131e7c4fc84312489d52d98aa313c61fa5830f92ca15f4e297f854f9d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[492,927,505,938]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[309,87,687,432]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure S4. UPS spectra of the pristine $\\\\\\\\mathrm{WS}_2$ and ligand-conjugated $\\\\\\\\mathrm{WS}_2$ with MPA, MHA, MOA, and MUA molecules. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0817/9b47564a2bb5f92876bbc9a762aba7ab109551f285e9dd690520cbfba99fdd91.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[492,926,505,939]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[247,80,751,244]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure S5. Triboelectric output current of the pristine and ligand-conjugated $\\\\\\\\mathrm{WS}_2$ TENG devices. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0131/c77d4717fd0a28aa5d045c15871f4006a97441f5081bdcc8648c32618dfa4dfb.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[492,927,505,938]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[122,82,267,186]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0905/f84d877db898475fff4e9e500793e32c598ece4c0def7c2d727e2c2726e8ac6b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[273,82,418,186]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0940/ae39ca7a1f1bd682ace416d2599d2ada81b5e1f7f298bf8003b70c3ae10f02a7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[426,82,573,186]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0630/1838cf539667f21bf8d2ad0ba12f48da72b154197d2022227ea16f16233a0da2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[581,82,727,186]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0720/6b1c5c14a3b465d53282c7e1193da45b232ba78249ced23003c13c354f7055fd.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[734,82,880,186]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure S6. AFM images of the pristine $\\\\\\\\mathrm{WS}_2$ and ligand-conjugated $\\\\\\\\mathrm{WS}_2$ surfaces. The surface \\\"]\",\n \"image_footnote\": \"[\\\"topographies of the (a) pristine $\\\\\\\\mathrm{WS}_2$ , (b) MPA- $\\\\\\\\mathrm{WS}_2$ , (c) MHA- $\\\\\\\\mathrm{WS}_2$ , (d) MOA- $\\\\\\\\mathrm{WS}_2$ , and (e) \\\",\\\"$\\\\\\\\mathrm{MUA - WS_2}$ surfaces, having surface roughness values of 1.5, 2.3, 2.6, 3.3, and $4.6~\\\\\\\\mathrm{nm}$ , respectively. \\\"]\",\n \"img_path\": \"images/0609/38f86d86ae27395db902918133d8cbda0182cf0ba13d1b06a9287fba90746a51.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[492,926,505,938]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[260,83,739,249]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure S7. Triboelectric output performance of the MUA-WS $_2$ TENG with the concentration of MUA treated with WS $_2$ . \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0637/c360616e7024866207a873343b4b745686287677cdfe7295ee0cafb144715cd4.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[492,927,505,938]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"8 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[122,84,240,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0436/e0364f5ed25ce557a0372ed0acda18453cbbf6bf214cc2943384740e2115bcba.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[245,84,366,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0448/206e61780563ed17ab91e9fce1e334405a5a5168d514347af180084b54492a07.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[371,84,492,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0106/5edfcf48561076428672f742f9321bd62c0f71e7eed65a928920b51c4b170f9f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[497,83,618,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0373/692916f6a7cf898a2e90ea9c270d484afd16450670fda08bf177397e57bbdc23.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[621,83,742,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0071/95e24d5669ef48ccc2fecc6dbe27873d265faabac76df115ddad928e3c16e5f7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[747,83,868,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure S8. AFM images of the pristine $\\\\\\\\mathrm{WS}_2$ and ligand-conjugated $\\\\\\\\mathrm{WS}_2$ surfaces functionalized with MUA having different concentrations. The surface topographies of the (a) pristine $\\\\\\\\mathrm{WS}_2$ , and ligand-conjugated $\\\\\\\\mathrm{WS}_2$ treated with MUA having a concentration of (b) 5, (c) 25, (d) 50, (e) 100, and (f) $500~\\\\\\\\mathrm{mM}$ , respectively. The surface roughness values of the pristine $\\\\\\\\mathrm{WS}_2$ and ligand-conjugated $\\\\\\\\mathrm{WS}_2$ surfaces treated with MUA having different concentrations are 1.5, 1.8, 2.5, 2.8, 3.8, and $4.6~\\\\\\\\mathrm{nm}$ , respectively. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0602/c805ce606985db6b191ae411e23cda2d951f5107cf415fbcda3b527111bf1388.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[492,926,505,938]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"9 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[312,86,684,272]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure S9. Charge density of the pristine and ligand-conjugated TMD TENG devices calculated by integrating the short-circuit current over time. Error bars represent standard deviations. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0767/27cc2044202dab0a17bf252aca0785139a8a51318873172da138026eca2192ee.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[487,925,510,939]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[220,85,519,236]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0440/2d7040bf6a8894bcd70470d2bdb8097d3c978ce31ee4ad54a000b225901421e8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[522,86,786,237]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure S10. Output voltages of the ligand-conjugated $\\\\\\\\mathrm{WS}_2$ TENG devices with MUA molecule after repeated pressing and releasing operation. (a) The output performance of MUA- $\\\\\\\\mathrm{WS}_2$ TENG devices after the repeated operation over 10, 100, 1000, and 10000 cycles, exhibiting the stable triboelectric operation of the ligand-conjugated TMD TENG devices. (b) The plot of output voltage of MUA- $\\\\\\\\mathrm{WS}_2$ TENG devices after the repeated operation cycles in average. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0443/77ae8c4496b411abe976c95ed06f7e4b8bc74297b9724aa53b1895de99404edb.jpg\",\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[487,925,509,939]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"11 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[154,87,500,299]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0131/8e299926ffdee90a6c092d38f2b1076e002feb2a61cfe3388715c4a664b3a382.jpg\",\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[507,87,845,299]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure S11. Long-term chemical stability of the pristine $\\\\\\\\mathrm{WS}_2$ and MUA- $\\\\\\\\mathrm{WS}_2$ samples. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0715/39b38678a52958f6c692c9a719729ba6e39d36dc211ea571d349391775223658.jpg\",\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[112,348,882,398]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Deconvoluted S 2p peaks of (a) pristine $\\\\mathrm{WS}_2$ and (b) MUA- $\\\\mathrm{WS}_2$ samples as-prepared, and after 56 and 98 days. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[487,925,510,939]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"12 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.1021/acsami.1c02562.s001"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Medicine and Health Sciences/Engineering and Manufacturing Science"},"sha256":{"kind":"string","value":"e71e476234fb054ec68b409108f23ae14375acdb07d704e0f41a07a0a3a00d2b"},"title":{"kind":"string","value":"Enhanced\\nTriboelectric Nanogenerator Based on Tungsten\\nDisulfide via Thiolated Ligand Conjugation"}}},{"rowIdx":62,"cells":{"abstract":{"kind":"string","value":"Background Axial spondyloarthritis (AxSpA) produces structural changes that cause alterations in body functions. One tissue that seems to have a predictive role in the etiology and progression of the disease is the soft tissue, particularly the fascia. However, little is known about the use of myofascial induction in people with AxSpA, and clinical evidence from physiotherapy regarding potential strategies is limited. Objective To evaluate the efficacy of myofascial induction compared with its simulation on joint amplitude in people with AxSpA. Methods In this randomized controlled parallel superiority clinical trial, 84 people with an AxSpA diagnosis confirmed by a rheumatologist will be randomly assigned to groups: the experimental group or the control group. The experimental group will receive myofascial induction, and the control group will undergo a simulation of the technique. Both groups will receive an examination session and six intervention sessions twice per week for three weeks. A baseline follow-up will be performed immediately after the intervention and four weeks after treatment. Conclusion The results of this study may contribute to a better understanding of the efficacy of myofascial induction for joint mobility in people with AxSpA. The implications of these results have a potential transformative effect on the understanding, analysis, evaluation, and physiotherapeutic treatment of this health condition."},"author":{"kind":"string","value":"Sanchez Vera, Maria Alejandra|Jaimes Fernandez, Diego Alejandro|Schleip, Robert"},"content_list":{"kind":"list like","value":[{"bbox":"[325,116,446,130]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"STUDY PROTOCOL ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,135,931,246]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Efficacy of myofascial induction compared with its simulation on joint amplitude in people with axial spondylarthritis: Protocol of a randomized controlled clinical trial ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,260,833,289]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Marı´a Alejandra Sa´nchez Vera $\\mathbb { P } ^ { 1 \\bullet \\ast }$ , Diego Alejandro Jaimes Ferna´ndez2☯, Robert SchleipID3☯ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,297,916,337]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 Faculty of Nursing and Rehabilitation, Physical Therapy Program, Universidad de La Sabana, Chı´a, Colombia, 2 School of Medicine, Universidad de La Sabana, Chı´a, Colombia, 3 Department of Sport and Health Sciences, Technical University of Munich, Munich, Germany ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,349,599,375]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"☯ These authors contributed equally to this work. \\* Mariasanver@unisabana.edu.co ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[75,435,169,446]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"OPEN ACCESS ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,457,302,551]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Citation: Sa´nchez Vera MA, Jaimes Ferna´ndez DA, Schleip R (2023) Efficacy of myofascial induction compared with its simulation on joint amplitude in people with axial spondylarthritis: Protocol of a randomized controlled clinical trial. PLoS ONE 18(10): e0286885. https://doi.org/10.1371/journal. pone.0286885 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[58,560,282,587]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Editor: Malgorzata Wojcik, Pozan University of Physical Education, POLAND ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[58,597,196,608]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Received: January 24, 2023 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,618,178,630]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Accepted: May 12, 2023 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,640,194,651]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Published: October 5, 2023 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,660,299,742]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Copyright: $\\circledcirc$ 2023 Sa´nchez Vera et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,752,303,805]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Data Availability Statement: No datasets were generated or analysed during the current study. All relevant data from this study will be made available upon study completion. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,815,305,896]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Funding: MASV received the funding as part of a process for awarding internal grants for research at the Universidad de La Sabana. https://www. unisabana.edu.co/ The study is supported by the Universidad de La Sabana, Chı´a, Colombia. The funders had no role in study design, data collection ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,406,416,426]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,460,436,477]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Background ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,483,936,569]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Axial spondyloarthritis (AxSpA) produces structural changes that cause alterations in body functions. One tissue that seems to have a predictive role in the etiology and progression of the disease is the soft tissue, particularly the fascia. However, little is known about the use of myofascial induction in people with AxSpA, and clinical evidence from physiotherapy regarding potential strategies is limited. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,593,411,611]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Objective ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,616,933,648]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To evaluate the efficacy of myofascial induction compared with its simulation on joint amplitude in people with AxSpA. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,674,403,689]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Methods ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,695,937,816]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this randomized controlled parallel superiority clinical trial, 84 people with an AxSpA diagnosis confirmed by a rheumatologist will be randomly assigned to groups: the experimental group or the control group. The experimental group will receive myofascial induction, and the control group will undergo a simulation of the technique. Both groups will receive an examination session and six intervention sessions twice per week for three weeks. A baseline follow-up will be performed immediately after the intervention and four weeks after treatment. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,842,428,858]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conclusion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[321,866,939,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The results of this study may contribute to a better understanding of the efficacy of myofascial induction for joint mobility in people with AxSpA. The implications of these results have a ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,39,230,63]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[55,944,493,957]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0286885 October 5, 2023 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[910,945,941,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 / 12 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[57,304,148,375]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"111111111 updates ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[57,429,73,446]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[58,94,300,121]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and analysis, decision to publish, or preparation of the manuscript. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,130,294,156]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Competing interests: The authors have declared that no competing interests exist. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[323,93,931,126]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"potential transformative effect on the understanding, analysis, evaluation, and physiotherapeutic treatment of this health condition. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[326,145,472,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Trial registration ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,170,725,184]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ClinicalTrials.gov NCT04424589. Registered 11 June 2020. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,251,439,268]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,275,937,421]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Axial spondyloarthritis (AxSpA) is a chronic and disabling disease that mainly affects young people and causes clear mobility and functional capacity limitations [1]. According to Londoño et al., in Colombia, its prevalence is $0 . 1 1 \\%$ among the population over 18 years of age, with a 3:1 male–female ratio [2]. Although AxSpA is not among the most prevalent rheumatic conditions in the country, it is regarded as an inflammatory disease with higher rates of functional limitation and disability for affected people [3]. Swinnen [4] showed that a combination of personal factors such as depression, impotence due to disease progression, and coping capacity explained approximately $2 4 \\%$ of the variation in activity limitations in AxSpA in contrast to biological factors, which represented only $1 0 \\%$ of the variation [4]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,422,936,602]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Given the above information, the management of this condition requires solid interdisciplinary actions that meet the needs of individuals with AxSpA [3, 5]. Although pharmacological therapy is the basis of the treatment of this condition [6], nonpharmacological management is an indispensable complement, since it seeks to improve physical and psychological symptoms, readjust the circumstances of patients according to their conditions, and restore bodily movement [5, 7]. Different studies have shown that pharmacological therapy in conjunction with physical rehabilitation is more effective for managing severe symptoms, improving quality of life, fostering independence in activities of daily living, and optimizing anthropometric measurements related to the transverse muscle diameter and fat percentage than biological therapy, which has been demonstrated to not be equally effective when used alone [4, 5, 8–10]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,604,937,717]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Based on physiotherapy, various effective strategies have been developed for the management and control of AxSpA [11, 12]. However, orthopedic manual therapy (OMT) techniques focusing on soft tissue management have not been fully studied; therefore, their efficacy is uncertain in this population [13, 14]. Fascia, which is considered a predictor of joint mobility and axial structure and whose basal tone is not dependent on the central nervous system but on the innate resistance of its composition, appears to have a determining role in the etiology of this disease [15, 16]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,719,941,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Masi et al. stated that the increase in this tone at rest can predispose an individual to developing AxSpA [17], because a greater tension load in the fascia limits the transmission of forces from the enthesis to the spine, resulting in inflammatory changes and local mechanical stress [16, 18]. This anomalous tension causes alterations in the composition of the fascial tissue with an increase in myofibroblasts and a decrease in surrounding water, hyaluronic acid, and proteoglycans [17, 19]. These biochemical responses result in an increase in myofascial thickness and superposition of dehydrated collagen layers, triggering clinical pictures consistent with nonspecific and inflammatory pain, functional limitation, and loss of muscle elasticity among people with AxSpA [20, 21]. These signs become the therapeutic targets of myofascial induction, which is understood as a technique involving guided mechanical forces of low load and a long duration. Its effectiveness has been demonstrated in different mechanical and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,45,148,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[522,47,941,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Myofascial induction on joint amplitude in people with axial spondylarthritis ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[55,944,493,957]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0286885 October 5, 2023 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[908,945,941,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 / 12 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[325,94,934,159]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"inflammatory pathologies, and its mechanism of action allows reorganization of fibroblasts and collagen layers, contributing to transformation of the fundamental substance from a densified state to a more fluid state, which results in clinically better patterns of movement in patients [10, 13, 22–24]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,178,472,194]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Primary objective ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[323,200,936,232]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The objective of this study will be to evaluate the efficacy of myofascial induction compared to its simulation on joint amplitude in people diagnosed with AxSpA. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,251,488,268]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Secondary objective ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,273,923,305]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The secondary objective will be to describe changes in the quality of life and functionality of people with AxSpA between the groups analyzed. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,324,419,342]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hypothesis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,348,941,380]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The myofascial induction group will have better results than the simulation group with respect to joint mobility among people with AxSpA. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,397,535,415]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Materials and methods ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,421,383,439]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Design ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,444,941,574]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This will be a randomized, superiority, double-blind, parallel clinical trial. The trial was designed according to the recommendations for intervention trials (SPIRIT) [25] and the CONSORT guidelines [26]. This protocol complies with the Helsinki guidelines for research involving humans and has been approved by the research and ethics subcommittee of the Faculty of Nursing and Rehabilitation of the Universidad de La Sabana (021-18-2020.) All participants will receive an informed consent form with all the procedures and the risks and benefits of the study. To protect the identity of the participants, each of them will be assigned a numerical code. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,593,485,611]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Study environment ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[323,617,941,647]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This study will be conducted in an outpatient service center for people with rheumatic diseases in the city of Bogota´, Colombia. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,666,467,684]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Inclusion criteria ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,698,936,803]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"• People of both sexes over 18 years of age. \n• People with a rheumatology-confirmed diagnosis of AxSpA regardless of the level of disease activity. \n• People with the cognitive ability to follow orders. \n• People who agree to participate in the study. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,828,470,844]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Exclusion criteria ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[323,858,699,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"• People receiving oral or parenteral coagulation therapy. \n• Pregnant women. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[524,47,941,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Myofascial induction on joint amplitude in people with axial spondylarthritis ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[55,45,148,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[55,944,493,957]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0286885 October 5, 2023 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[908,945,941,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 / 12 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[325,94,514,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"• People with kinesiophobia. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,118,707,133]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"• People with previous physical therapy in the last 15 days. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,142,862,157]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"• The presence of active cancer or current treatment with chemo- or radiotherapy. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,174,429,190]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Intervention ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,196,941,294]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The study interventions are based on the myofascial induction techniques proposed by Andrezj Pilat [21]. The physiotherapist (PT) who performs this procedure in the experimental group will be trained and certified in this technique by the author in question. All techniques will be performed in all planned treatment sessions. 6 myofascial induction and simulation sessions will be done, two per week, every other day, for 3 weeks. Patients will be followed up one month after the last session with final evaluation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[326,318,596,335]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Experimental group intervention ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,340,937,472]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J stroke. After a palpatory examination of the fascia, with the participant in prone position, the PT will apply slight superficial traction with his nondominant hand, which helps maintain tension on the myofascial tissue of the dorsolumbar spine. With his dominant hand, he will move his third finger up to the second level of the distal phalanx while the forearm is pronated, and then he will perform the “J” stroke, slowly sliding with nonpainful superficial depth, and rapid supination in the shape of a “J” on the site of the myofascial restriction. For each of the sites perceived as adhered or restricted, an average of seven J strokes should be performed. This technique does not require any lubricant or contact medium. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,473,941,619]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Release of the paravertebral fascia. The participant will be in a sitting position with the back uncovered, the contacts of the PT will be the knuckles of the second, third, and fourth fingers, the participant will be asked to flex the head, and the contacts will be moved up to the second thoracic vertebra. Once the movement is stopped, the therapist will bring back one of his lower extremities. At this point, he will continue the maneuver, keeping the elbows in extension and leaning on his body mechanics to gain depth. Then, he will ask the participant to slowly flex forward while keeping the legs in abduction while the PT slides the contacts caudally until reaching the high lumbar vertebrae. At each site where a movement restriction is perceived, he will stop without increasing the pressure for seven seconds. This process should be repeated three times. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,621,937,733]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Technique for releasing the quadratus lumborum. The participant will be in the lateral decubitus position with knee and hip flexion of $9 0 ^ { \\circ }$ . The PT will remain bipedal in front of the patient at the height of the torso, the cranial hand will fix the ribs in an ascending direction, and the distal hand will be positioned with the ulnar edge of the hand in the space between the iliac crest and the last rib. The PT will seek to gain depth in this space until resistance is felt. Once he reaches the maximum allowed tissue barrier, he will perform a rhythmic, low-amplitude, flexion and extension movement of the elbow for 15 cycles in three repetitions. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,734,941,815]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In a following moment, in the same position, the patient will slightly drop the legs outside the table, the PT will accompany this tensioning of the quadratus lumborum, locating the pads of the fingers within the common paravertebral mass, and the PT will perform a slide his fingers in the transverse direction without generating flexion of the carpus. He will perform 13 cycles in three repetitions. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,818,941,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This technique will end with the patient in the lateral decubitus position, the legs will fall slightly outside the table, and the PT will have his hands crossed, with one on the iliac crest and another on the rib cage. He will begin by applying light traction on the skin, and then with light pressure, he will try to overcome the barriers of the tissue by facilitating movement of the fascia with sustained pressure for five minutes. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,45,148,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[522,47,941,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Myofascial induction on joint amplitude in people with axial spondylarthritis ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[55,944,493,957]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0286885 October 5, 2023 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[908,945,941,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 / 12 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[325,94,941,174]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Release of the sacroiliac fascia. The patient will remain prone with knee flexion at $9 0 ^ { \\circ }$ , the PT will remain bipedal next to the limb to be worked on, and the PT’s cranial hand will rest on the sacrum in the direction of the iliac crests while maintaining internal rotation of this limb with the caudal hand. He will maintain the pressure until it surpasses between three and six tissue barriers, which should be sustained for four minutes. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,178,941,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sustained pressure technique in the upper trapezius. For the neck, the myofascial release technique will be used in the upper trapezius, the patient will be supine, and the PT will position his hands at the head of the table while seated as follows: His nondominant hand will support the patient. The dominant hand with $9 0 ^ { \\circ }$ flexion will be at the level of the metacarpophalangeal joints, and the initial contact with the upper trapezius and angular muscles of the scapula will begin gently and progressively with the tips of the fingers, increasing the pressure and depth of the position of the hands. The PT will feel each of the tissue barriers and slowly yield to the pressure. This position will be maintained for five minutes. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,308,942,469]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cross-hand technique in thoracolumbar fascia. The intervention continues with the crosshand technique; at this time, the PT will be located on one side of the table, and the patient will have the back uncovered and be in the prone position. The PT will synchronize his breathing with that of the patient. Once he has taken two deep inhalations, the therapist will place the palms of his hands completely on the patient’s back at the level of the lumbosacral region with the fingers in slight abduction, and he will begin applying light traction on the skin, bringing the tissue to a state of pretension; slight pressure will be maintained for five minutes. Once the technique is complete, the pressure will be slowly released, and before removing the hands from the segment, manual contact will be maintained for 10 seconds. Once the hands are removed, the patient will be allowed to rest on the table for approximately two minutes to avoid vasovagal responses. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,472,936,553]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Control group intervention. The simulation intervention in the control group will suggest modification of basic parameters that does not alter the visual demonstration of the technique. The PT assigned to this group will omit the basic principles of myofascial induction: traction, sustained pressure, and tissue depth. All the proposed strategies for the experimental group will be carried out. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,554,934,651]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Discontinuation of participation. During the course of the study, any participant can choose to withdraw from the trial at any time. Participants can choose to suspend treatment and/or evaluations in the study but can remain in the follow-up if they wish; however, participants can also withdraw their consent, which implies that they wish to completely withdraw from the study, completely restricting the researchers’ access to the data obtained to date and those participants’ clinical histories. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,671,410,686]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Outcomes ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,694,936,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The primary outcome will be joint mobility measured by the Bath Ankylosing Spondylitis Metrology Index (BASMI). The secondary outcomes will be the Bath Ankylosing Spondylitis Functional Index (BASFI), Bath Ankylosing Spondylitis Disease Activity Index (BASDAI), and the Ankylosing Spondylitis Quality of Life Questionnaire (AsQol) [27, 28]. Each of these variables will be measured before the intervention, after the intervention, and four weeks after the intervention. The participants will undergo an initial evaluation before randomization. The results will be collected at each time point according to (Figs 1 and 2). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,827,387,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"BASMI ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,851,937,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The index that groups four axial measurements (cervical rotation, Schober test, lateral torso flexion, tragus-to-wall distance) also includes a measurement of peripheral mobility (intermalleolar distance) [29]. These variables will be measured comparatively at least twice to obtain ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,45,148,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[524,47,941,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Myofascial induction on joint amplitude in people with axial spondylarthritis ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[55,944,493,957]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0286885 October 5, 2023 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[908,945,941,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 / 12 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[120,97,941,535]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0019/046e25a2359b1de4651c1a4bb4dcc2d55e9cb9b30b44b07f7d64d426ff73ac28.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":"<table><tr><td rowspan=1 colspan=11>STUDY PERIOD</td></tr><tr><td rowspan=1 colspan=1>TIMEPOINT</td><td rowspan=1 colspan=1>W1</td><td rowspan=1 colspan=1>W2</td><td rowspan=1 colspan=1>W3</td><td rowspan=1 colspan=1>W4</td><td rowspan=1 colspan=1>W5</td><td rowspan=1 colspan=1>W6</td><td rowspan=1 colspan=1>W7</td><td rowspan=1 colspan=1>W9</td><td rowspan=1 colspan=1>W10</td><td rowspan=1 colspan=1>W11</td></tr><tr><td rowspan=1 colspan=1>ENROLMENT:</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Eligibility screen</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Informed consent</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Allocation</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Initial assessment(BASMI,FI,BASDAI, AsQoL)</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>INTERVENTIONS:</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=1>Myofascialinduction</td><td rowspan=2 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=2 colspan=1>一</td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=1> Sham</td><td rowspan=2 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=2 colspan=1>一</td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>ASSESSMENTS:</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Final assessment(BASMI, BASFI,BASDAI, AsQoL)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>FOLLOWUP</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>×</td></tr></table>","table_caption":"[\"W: Weeks. Fig 1. Standard Protocol Items: Recommendations for Interventional Trials (SPIRIT) timeline of measurements. Time represented by weeks (WK). \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[117,589,366,601]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.1371/journal.pone.0286885.g001 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,625,939,672]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"the mean measurements to calculate the final score based on a table of 11 points [25]. This index is evaluated by the examiner and conveys spinal and peripheral mobility findings in centimeters and degrees [27, 30]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,690,380,705]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"BASFI ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,713,934,827]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The BASFI questionnaire is self-administered and consists of 10 questions investigating the functional status of the person diagnosed with AxSpA. Each question can be scored from 0 to 10, with 0 corresponding to no limitation in carrying out the proposed activity and 10 corresponding to the absolute impossibility of carrying it out. This scoring system employs a visual analog scale scored on a horizontal metric line [31]. The final score is obtained from the sum of the results of the questions divided by 10. This instrument is validated for use in Spanish in Colombia [32, 33]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,843,398,858]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"BASDAI ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[323,866,906,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The BASDAI evaluation is self-administered and consists of six questions inquiring about issues related to the five main symptoms of AxSpA: fatigue, axial pain, joint pain and/or ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,45,148,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[522,47,941,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Myofascial induction on joint amplitude in people with axial spondylarthritis ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[55,944,493,957]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0286885 October 5, 2023 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[908,945,941,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 / 12 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[331,94,771,737]","code_body":null,"code_caption":null,"image_caption":"[\"Fig 2. Study flow diagram, adapted from CONSORT. \"]","image_footnote":"[]","img_path":"images/0632/8880b24250d259ffb6bd5f90072d1a38fbd469459b1ec2b6cb8b043b24bf89ad.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[325,761,576,773]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.1371/journal.pone.0286885.g002 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,799,941,881]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"edema, enthesis pain, and morning stiffness [34]. Similar to the BASFI, it is based on a visual analog scale measuring 10 centimeters in a horizontal line. It is scored by adding the scores for questions 1 to 4; for questions 5 and 6, the mean is calculated, and finally, the scores of all questions are added and divided by 5. A higher BASDAI score suggests more active disease [27]. This questionnaire is also validated in Spanish [34]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,944,493,957]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE | https://doi.org/10.1371/journal.pone.0286885 October 5, 2023 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[522,47,941,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Myofascial induction on joint amplitude in people with axial spondylarthritis ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[908,945,941,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7 / 12 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[55,45,150,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PLOS ONE","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[326,94,383,111]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"AsQoL ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,117,936,214]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The AsQoL is self-administered and validated in Spanish in Colombia. It consists of 18 questions with the possibility of dichotomous answers (YES—NO). The affirmative answers are scored as 1, and the negative answers are scored as 0. The overall score is the sum of the scores for each question and can be between 0 (better health-related quality of life) and 18 (worse health-related quality of life). If more than three questions are unanswered, the overall score cannot be calculated [35]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,231,429,247]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recruitment ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,253,939,351]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The people who will be eligible for the study are those receiving some medical or therapeutic service in the clinic selected for the study. A research assistant will review the medical records of people with an AxSpA diagnosis confirmed by a rheumatologist. Individuals who are potentially eligible will be contacted by telephone, and the intention and scope of the study will be briefly explained. After this communication, if a person verbally expresses their desire to be part of the research process, a face-to-face interview will be scheduled. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,367,421,385]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sample size ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,391,937,488]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The sample size was calculated in the program R Studio version 3.5.3, package of superiority clinical trials, based on a mean difference of 4.0 in the BASMI with a standard deviation of 2, an r (ratio) between groups of 1, and assuming a loss of $2 0 \\%$ . An alpha one sided of $0 . 0 5 \\%$ and a power of 0.80 were estimated based on mean difference test corresponding to a clinical trial of superiority (repeated measures ANOVA). The total number of people was 82 (42 per arm). Non probabilistic purposive sampling will be used. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[326,503,596,521]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Randomization and concealment ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,527,941,608]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this study, a simple randomization process will be used by assigning random numbers without repetition in Excel. For randomization concealment, opaque, sealed, consecutively numbered envelopes will be used and kept by a single person. An external person will prepare the envelopes. An envelope will be opened in sequence once the participant has completed the baseline physical examination and the corresponding measurements. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,625,400,642]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Blinding ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,647,941,777]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The people who will perform the baseline and final evaluations will not know the randomization sequences, nor will they be involved in the execution of the treatments in the control group or the experimental group. The participants will visually receive the same procedures since the techniques for the control and experimental groups appear the same; accordingly, they will also be blinded. The PTs who will perform the interventions cannot be blinded due to the need to know the principles of the technique and the form of execution to guarantee correct application of the treatment or simulation. The data analysis will be performed by an external evaluator in the intervention and evaluation process. 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All documents will be safely stored confidentially. Participants will be identified by a specific number or unique test code. The name and any other identifying details will not be included in any part of the database. 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A statistical analysis will be performed based on the purposive analysis principles. The assumption of normality for the quantitative variables will be verified with the Shapiro–Wilk test, followed by a descriptive analysis; for the continuous variables, measures of central tendency or dispersion according to the data will be applied. For categorical variables, the proportions and frequencies will be calculated. Differences between the groups in measurements for the primary and secondary objectives will be analyzed by repeated measures ANOVA the variables with a normal distribution. Nonparametric tests will be performed if the data do not comply with the assumption of normality, interaction of arm by time effect will not be tested. Qualitative variables will be evaluated with the chi-square test of independence or Fisher’s exact test, as appropriate. For all tests, a significance level of p $< 0 . 0 5$ will be employed. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,367,393,383]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Damage ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,390,931,503]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"All adverse events that are found during the execution of the clinical trial should be reported in a timely manner, all the data will be collected according to Common Terminology Criteria for Adverse Events (CTCAE) version 5.0. This information will be recorded in the report format that includes description of the event, start date and end date, relationship with physiotherapeutic treatment and measures taken on the event. Follow-up information will be provided as needed. Each of these formats will be recorded with the serial code assigned to each patient [29, 37–39]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,523,421,540]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Monitoring ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,546,942,626]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Strict compliance with the protocol and accuracy in relation to the source documents will be evaluated. Following the standard operating procedures declared in this document, an external monitor or referee who is not involved in the research will randomly verify that the clinical trial is carried out within the approved and established parameters and that the data generated are documented and reported in accordance with the protocol every two weeks. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,645,429,662]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Discussion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,669,941,784]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Physiotherapeutic intervention is undoubtedly the first-choice treatment for the management of patients with AxSpA; however, studies based on manual therapy with an emphasis on soft tissue are still insufficient [7, 9, 40]. The results of this research can contribute to the understanding and formulation of new treatment paradigms with rapid therapeutic action that complement the strategies that have been shown to be more effective, such as aerobic exercise and resistance and neuromuscular training [1, 5, 7, 41]. Likewise, it will clarify the role of soft tissue in joint mobility in patients with AxSpA [5, 19]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,785,937,864]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The results of this research seek to initially understand the influence of the fascia on the spinal mobility of patients with this health condition. Fascial hypertonicity, as proposed by Masi et al. [18] may be a \"missing link\" in therapeutic understanding of AxSpA. The modulation of fascial and soft tissue tone can become a treatment target from the physiotherapist’s actions to achieve hard declines such as function, quality of life and pain. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[321,866,924,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This study is formulated with high methodological quality, allowing its reproducibility in any clinical scenario. The detailed description of the intervention strategies allows any PT to accurately perform a myofascial induction intervention in this type of patient. Additionally, to date, this is the first study focusing exclusively on evaluating the efficacy of manual therapy techniques, particularly myofascial induction, in patients with AxSpA. 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SPIRIT 2013 checklist: Recommended items to address in a clinical trial protocol and related documents\\*. \n(DOCX) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,343,442,375]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"S1 File. Protocol. (DOCX) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,383,383,415]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"S2 File. (DOCX) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,444,522,462]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Author Contributions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[325,468,661,483]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conceptualization: Marı´a Alejandra Sa´nchez Vera. 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","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,598,848,613]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Writing – review & editing: Diego Alejandro Jaimes Ferna´ndez, Robert Schleip. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[325,642,429,657]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[326,662,941,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Flo´rez Garcı´a MT, Carmona L, Almodo´var R, Ferna´ndez de Las Peñas C, Garcı´a Pe´rez F, Pe´rez Manzanero MA´ et al. Recommendations for the prescription of physical exercise for patients with spondyloarthritis. Reumatol Clin. abril de 2019; 15(2):77–83. \n2. Londoño J, Pela´ez Ballestas I, Cuervo F, Angarita I, Giraldo R, Rueda JC, et al. Prevalencia de la enfermedad reuma´tica en Colombia, segu´n estrategia COPCORD-Asociacio´n Colombiana de Reumatologı´a. Estudio de prevalencia de enfermedad reuma´tica en poblacio´n colombiana mayor de 18 años. Revista Colombiana de Reumatologia. 1 de octubre de 2018; 25(4):245–245-56. \n3. Romanowski MW, Sˇ piritović M, Rutkowski R, Dudek A, Samborski W, Straburzyńska-Lupa A. Comparison of Deep Tissue Massage and Therapeutic Massage for Lower Back Pain, Disease Activity, and Functional Capacity of Ankylosing Spondylitis Patients: A Randomized Clinical Pilot Study. Evid Based Complement Alternat Med. 2017; 2017:9894128. https://doi.org/10.1155/2017/9894128 PMID: 28845185 \n4. Swinnen TW, Westhovens R, De Vlam K, Vlaeyen JWS, Dankaerts W. Activity limitations in patients with axial spondyloarthritis: A role for fear of movement and (Re)injury beliefs. Journal of Rheumatology. 1 de marzo de 2018; 45(3):357-357–66. https://doi.org/10.3899/jrheum.170318 PMID: 29142031 \n5. Perrotta FM, Musto A, Lubrano E. New Insights in Physical Therapy and Rehabilitation in Axial Spondyloarthritis: A Review. Rheumatol Ther. diciembre de 2019; 6(4):479–86. https://doi.org/10.1007/ s40744-019-00170-x PMID: 31410786 \n6. Danve A, Deodhar A. Treatment of axial spondyloarthritis: an update. Nat Rev Rheumatol. abril de 2022; 18(4):205–16. https://doi.org/10.1038/s41584-022-00761-z PMID: 35273385 7. Martey C, Sengupta R. Physical therapy in axial spondyloarthritis: guidelines, evidence and clinical practice. Curr Opin Rheumatol. julio de 2020; 32(4):365–70. https://doi.org/10.1097/BOR. 0000000000000714 PMID: 32453037 \n8. Arguisuelas MD, Liso´n JF, Dome´nech-Ferna´ndez J, Martı´nez-Hurtado I, Coloma PS, Sa´nchez-Zuriaga D. Effects of myofascial release in erector spinae myoelectric activity and lumbar spine kinematics in non-specific chronic low back pain: Randomized controlled trial. Clinical Biomechanics [Internet]. 13 de febrero de 2019 [citado 13 de marzo de 2019]; Disponible en: http://search.ebscohost.com/login.aspx? direct=true&db=edselp&AN=S0268003318300330&site=eds-live \n9. Dagfinrud H, Hagen KB, Kvien TK. The cochrane review of physiotherapy interventions for ankylosing spondylitis. Journal of Rheumatology. 1 de octubre de 2005; 32(10):1899-1899-906. PMID: 16206344 \n10. Parravicini G, Bergna A. Biological effects of direct and indirect manipulation of the fascial system. Narrative review. J Bodyw Mov Ther. abril de 2017; 21(2):435–45. \n11. Masiero S, Bonaldo L, Pigatto M, Lo Nigro A, Ramonda R, Punzi L. Rehabilitation Treatment in Patients with Ankylosing Spondylitis Stabilized with Tumor Necrosis Factor Inhibitor Therapy. A Randomized Controlled Trial. JOURNAL OF RHEUMATOLOGY. 1 de julio de 2011; 38(7):1335-1335–42. \n12. Kyllikki W, Hossein K, Ingia¨ld H. Self- and manual mobilization improves spine mobility in men with ankylosing spondylitis—a randomized study. Clinical Rehabilitation. 1 de julio de 2009; 23(7):599-599– 600. \n13. Cathcart E, McSweeney T, Johnston R, Young H, Edwards DJ. Myofascial pain and treatment: Randomized Controlled Trial: Immediate biomechanical, systemic, and interoceptive effects of myofascial release on the thoracic spine: A randomised controlled trial. Journal of Bodywork & Movement Therapies. 1 de enero de 2019; 23(1):74-74–81. \n14. Ceca D, Elvira L, Guzma´n JF, Pablos A. Benefits of a self-myofascial release program on health-related quality of life in people with fibromyalgia: a randomized controlled trial. J Sports Med Phys Fitness. agosto de 2017; 57(7–8):993–1002. https://doi.org/10.23736/S0022-4707.17.07025-6 PMID: 28139112 \n15. Hanson A, Brown MA. Genetics and the Causes of Ankylosing Spondylitis. Rheum Dis Clin North Am. agosto de 2017; 43(3):401–14. https://doi.org/10.1016/j.rdc.2017.04.006 PMID: 28711142 \n16. Klingler W, Velders M, Hoppe K, Pedro M, Schleip R. Clinical relevance of fascial tissue and dysfunctions. Curr Pain Headache Rep. 2014; 18(8):439. https://doi.org/10.1007/s11916-014-0439-y PMID: 24962403 \n17. Masi AT. Might axial myofascial properties and biomechanical mechanisms be relevant to ankylosing spondylitis and axial spondyloarthritis? Arthritis Res Ther. 2014; 16(2):107. https://doi.org/10.1186/ ar4532 PMID: 25166000 \n18. Masi AT, Nair K, Andonian BJ, Prus KM, Kelly J, Sanchez JR, et al. Integrative structural biomechanical concepts of ankylosing spondylitis. Arthritis. 2011; 2011:205904. https://doi.org/10.1155/2011/205904 PMID: 22216409 \n19. White A, Abbott H, Masi AT, Henderson J, Nair K. Biomechanical properties of low back myofascial tissue in younger adult ankylosing spondylitis patients and matched healthy control subjects. Clin Biomech (Bristol, Avon). agosto de 2018; 57:67–73. https://doi.org/10.1016/j.clinbiomech.2018.06.006 PMID: 29936315 \n20. Andonian BJ, Masi AT, Aldag JC, Barry AJ, Coates BA, Emrich K, et al. Greater Resting Lumbar Extensor Myofascial Stiffness in Younger Ankylosing Spondylitis Patients Than Age-Comparable Healthy Volunteers Quantified by Myotonometry. Arch Phys Med Rehabil. noviembre de 2015; 96(11):2041–7. https://doi.org/10.1016/j.apmr.2015.07.014 PMID: 26254947 \n21. Pilat A, Gonzalez Nieto JL, Batuecas Sua´rez A. Terapias miofaciales: induccio´n miofascial. Ase´ctos teo´ricos y aplicaciones clı´nicas / Andrzej Pilat; revisio´n te´cnica Alicia Batuecas Sua´rez, Jose´ Luis Gonza´lez Nieto. [Internet]. 2003 [citado 27 de marzo de 2019]. Disponible en: http://unisabana22.gsl.com. mx:80/F?func=service&doc_library=CNA01&local_base=CNA01&doc_number= 000065058&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type= MEDIA \n22. Schleip R, Mu¨ller DG. Training principles for fascial connective tissues: scientific foundation and suggested practical applications. J Bodyw Mov Ther. enero de 2013; 17(1):103–15. https://doi.org/10. 1016/j.jbmt.2012.06.007 PMID: 23294691 \n23. Martı´nez Rodrı´guez R, Gala´n del Rı´o F. Mechanistic basis of manual therapy in myofascial injuries. Sonoelastographic evolution control. J Bodyw Mov Ther. abril de 2013; 17(2):221–34. https://doi.org/ 10.1016/j.jbmt.2012.08.006 PMID: 23561871 \n24. Cathcart E, McSweeney T, Johnston R, Young H, Edwards DJ. Immediate biomechanical, systemic, and interoceptive effects of myofascial release on the thoracic spine: A randomised controlled trial. J Bodyw Mov Ther. enero de 2019; 23(1):74–81. https://doi.org/10.1016/j.jbmt.2018.10.006 PMID: 30691766 \n25. Chan AW, Tetzlaff JM, Altman DG, Laupacis A, Gøtzsche PC, Krlezˇa-Jerić K, et al. Declaracio´n SPIRIT 2013: definicio´n de los elementos esta´ndares del protocolo de un ensayo clı´nico. Rev Panam Salud Publica. diciembre de 2015; 38(6):506–14. \n26. Consort—Welcome to the CONSORT Website [Internet]. [citado 13 de junio de 2022]. Disponible en: http://www.consort-statement.org/ \n27. Castro Villegas M del C, Batlle Gualda E. Metrolog´ıa en espondiloartritis. Reumatol Clin. 1 de marzo de 2010; 6:11–7. \n28. Popescu C, Trandafir M, Bădică A, Morar F, Predeţeanu D. Ankylosing spondylitis functional and activity indices in clinical practice. J Med Life. 15 de marzo de 2014; 7(1):78–83. PMID: 24653763 \n29. Martı´n RS, Perera AE, Novo JP, Me´ndez BMG, Annia RG. Clinimetrı´a en las espondiloartritis y sus ´ındices de medidas. Rev Cub de Reu [Internet]. 2013 [citado 13 de junio de 2022]; 15(1). Disponible en: https://www.medigraphic.com/cgi-bin/new/resumen.cgi?IDARTICULO=51080 \n30. Bath Indices [Internet]. National Axial Spondyloarthritis Society. [citado 14 de junio de 2022]. Disponible en: https://nass.co.uk/homepage/health-professionals/resources-for-health-professionals/bath-indices/ \n31. Wariaghli G, Allali F, Berrada K, Idrissi Z, Hmamouchi I, Abouqal R, et al. Normative values for the bath ankylosing spondylitis functional index in the general population compared with ankylosing spondylitis patients in Morocco. BMC Musculoskeletal Disorders. 21 de marzo de 2012; 13(1):40. https://doi.org/ 10.1186/1471-2474-13-40 PMID: 22436379 \n32. Rauseo Vera M, Gutie´rrez-Gonza´lez LA, Maldonado I, Al Snih S. Validacio´n al idioma español de Venezuela de los ´ındices de actividad de enfermedad y el ´ındice funcional para espondilitis anquilosante. Reumatol Clin. 1 de julio de 2019; 15(4):223–8. \n33. Calin A, Jones SD, Garrett SL, Kennedy LG. Bath Ankylosing Spondylitis Functional Index. Br J Rheumatol. agosto de 1995; 34(8):793–4. https://doi.org/10.1093/rheumatology/34.8.793 PMID: 7551671 \n34. Sarabia FN, Ariza RA, Cruz BH. La versio´n española del BASDAI es fiable y se correlaciona con la actividad de la enfermedad en pacientes con espondilitis anquilosante. Revista española de reumatologı´a: O´ rgano Oficial de la Sociedad Española de Reumatologı´a. 2004; 31(6):372. \n35. Ariza-Ariza R, Herna´ndez-Cruz B, Lo´pez-Antequera G, Toyos FJ, Navarro-Sarabia F. Adaptacio´n transcultural y validacio´n de una versio´n en español de un instrumento especı´fico para medir calidad de vida relacionada con la salud en pacientes con espondilitis anquilosante: el ASQoL. Reumatol Clin. 1 de marzo de 2006; 2(2):64–9. \n36. REDCap [Internet]. [citado 13 de junio de 2022]. Disponible en: https://www.project-redcap.org/ \n37. Carnes D, Mars TS, Mullinger B, Froud R, Underwood M. Adverse events and manual therapy: a systematic review. Man Ther. agosto de 2010; 15(4):355–63. https://doi.org/10.1016/j.math.2009.12.006 PMID: 20097115 \n38. Paanalahti K, Holm LW, Nordin M, Asker M, Lyander J, Skillgate E. Adverse events after manual therapy among patients seeking care for neck and/or back pain: a randomized controlled trial. BMC Musculoskelet Disord. 12 de marzo de 2014; 15:77. https://doi.org/10.1186/1471-2474-15-77 PMID: 24618345 \n39. World Medical Association. World Medical Association Declaration of Helsinki: Ethical Principles for Medical Research Involving Human Subjects. JAMA. 27 de noviembre de 2013; 310(20):2191–4. https://doi.org/10.1001/jama.2013.281053 PMID: 24141714 \n40. Kasapoglu Aksoy M, Birtane M, Taştekin N, Ekuklu G. The Effectiveness of Structured Group Education on Ankylosing Spondylitis Patients. J Clin Rheumatol. abril de 2017; 23(3):138–43. https://doi.org/10. 1097/RHU.0000000000000499 PMID: 28248799 \n41. Fernandez-de-las-Penas C, Alonso-Blanco C, Alguacil-Diego IM, Miangolarra-Page JC. One-Year Follow-up of Two Exercise Interventions for the Management of Patients with Ankylosing Spondylitis: A Randomized Controlled Trial [Internet]. Vol. 85. 2006 [citado 27 de marzo de 2019]. 559 p. 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\"[325,349,599,375]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"☯ These authors contributed equally to this work. \\\\* Mariasanver@unisabana.edu.co \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,435,169,446]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"OPEN ACCESS \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,457,302,551]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Citation: Sa´nchez Vera MA, Jaimes Ferna´ndez DA, Schleip R (2023) Efficacy of myofascial induction compared with its simulation on joint amplitude in people with axial spondylarthritis: Protocol of a randomized controlled clinical trial. PLoS ONE 18(10): e0286885. https://doi.org/10.1371/journal. pone.0286885 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[58,560,282,587]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Editor: Malgorzata Wojcik, Pozan University of Physical Education, POLAND \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[58,597,196,608]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Received: January 24, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,618,178,630]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Accepted: May 12, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,640,194,651]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Published: October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,660,299,742]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Copyright: $\\\\circledcirc$ 2023 Sa´nchez Vera et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,752,303,805]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data Availability Statement: No datasets were generated or analysed during the current study. All relevant data from this study will be made available upon study completion. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,815,305,896]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Funding: MASV received the funding as part of a process for awarding internal grants for research at the Universidad de La Sabana. https://www. unisabana.edu.co/ The study is supported by the Universidad de La Sabana, Chı´a, Colombia. The funders had no role in study design, data collection \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,406,416,426]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,460,436,477]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Background \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,483,936,569]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Axial spondyloarthritis (AxSpA) produces structural changes that cause alterations in body functions. One tissue that seems to have a predictive role in the etiology and progression of the disease is the soft tissue, particularly the fascia. However, little is known about the use of myofascial induction in people with AxSpA, and clinical evidence from physiotherapy regarding potential strategies is limited. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,593,411,611]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Objective \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,616,933,648]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To evaluate the efficacy of myofascial induction compared with its simulation on joint amplitude in people with AxSpA. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,674,403,689]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Methods \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,695,937,816]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this randomized controlled parallel superiority clinical trial, 84 people with an AxSpA diagnosis confirmed by a rheumatologist will be randomly assigned to groups: the experimental group or the control group. The experimental group will receive myofascial induction, and the control group will undergo a simulation of the technique. Both groups will receive an examination session and six intervention sessions twice per week for three weeks. A baseline follow-up will be performed immediately after the intervention and four weeks after treatment. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,842,428,858]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conclusion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,866,939,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The results of this study may contribute to a better understanding of the efficacy of myofascial induction for joint mobility in people with AxSpA. The implications of these results have a \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,39,230,63]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[55,944,493,957]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0286885 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[910,945,941,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 / 12 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[57,304,148,375]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"111111111 updates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[57,429,73,446]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[58,94,300,121]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and analysis, decision to publish, or preparation of the manuscript. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,130,294,156]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Competing interests: The authors have declared that no competing interests exist. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[323,93,931,126]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"potential transformative effect on the understanding, analysis, evaluation, and physiotherapeutic treatment of this health condition. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[326,145,472,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Trial registration \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,170,725,184]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ClinicalTrials.gov NCT04424589. Registered 11 June 2020. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,251,439,268]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,275,937,421]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Axial spondyloarthritis (AxSpA) is a chronic and disabling disease that mainly affects young people and causes clear mobility and functional capacity limitations [1]. According to Londoño et al., in Colombia, its prevalence is $0 . 1 1 \\\\%$ among the population over 18 years of age, with a 3:1 male–female ratio [2]. Although AxSpA is not among the most prevalent rheumatic conditions in the country, it is regarded as an inflammatory disease with higher rates of functional limitation and disability for affected people [3]. Swinnen [4] showed that a combination of personal factors such as depression, impotence due to disease progression, and coping capacity explained approximately $2 4 \\\\%$ of the variation in activity limitations in AxSpA in contrast to biological factors, which represented only $1 0 \\\\%$ of the variation [4]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,422,936,602]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Given the above information, the management of this condition requires solid interdisciplinary actions that meet the needs of individuals with AxSpA [3, 5]. Although pharmacological therapy is the basis of the treatment of this condition [6], nonpharmacological management is an indispensable complement, since it seeks to improve physical and psychological symptoms, readjust the circumstances of patients according to their conditions, and restore bodily movement [5, 7]. Different studies have shown that pharmacological therapy in conjunction with physical rehabilitation is more effective for managing severe symptoms, improving quality of life, fostering independence in activities of daily living, and optimizing anthropometric measurements related to the transverse muscle diameter and fat percentage than biological therapy, which has been demonstrated to not be equally effective when used alone [4, 5, 8–10]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,604,937,717]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Based on physiotherapy, various effective strategies have been developed for the management and control of AxSpA [11, 12]. However, orthopedic manual therapy (OMT) techniques focusing on soft tissue management have not been fully studied; therefore, their efficacy is uncertain in this population [13, 14]. Fascia, which is considered a predictor of joint mobility and axial structure and whose basal tone is not dependent on the central nervous system but on the innate resistance of its composition, appears to have a determining role in the etiology of this disease [15, 16]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,719,941,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Masi et al. stated that the increase in this tone at rest can predispose an individual to developing AxSpA [17], because a greater tension load in the fascia limits the transmission of forces from the enthesis to the spine, resulting in inflammatory changes and local mechanical stress [16, 18]. This anomalous tension causes alterations in the composition of the fascial tissue with an increase in myofibroblasts and a decrease in surrounding water, hyaluronic acid, and proteoglycans [17, 19]. These biochemical responses result in an increase in myofascial thickness and superposition of dehydrated collagen layers, triggering clinical pictures consistent with nonspecific and inflammatory pain, functional limitation, and loss of muscle elasticity among people with AxSpA [20, 21]. These signs become the therapeutic targets of myofascial induction, which is understood as a technique involving guided mechanical forces of low load and a long duration. Its effectiveness has been demonstrated in different mechanical and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,45,148,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[522,47,941,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Myofascial induction on joint amplitude in people with axial spondylarthritis \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[55,944,493,957]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0286885 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[908,945,941,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 / 12 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[325,94,934,159]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"inflammatory pathologies, and its mechanism of action allows reorganization of fibroblasts and collagen layers, contributing to transformation of the fundamental substance from a densified state to a more fluid state, which results in clinically better patterns of movement in patients [10, 13, 22–24]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,178,472,194]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Primary objective \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[323,200,936,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The objective of this study will be to evaluate the efficacy of myofascial induction compared to its simulation on joint amplitude in people diagnosed with AxSpA. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,251,488,268]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Secondary objective \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,273,923,305]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The secondary objective will be to describe changes in the quality of life and functionality of people with AxSpA between the groups analyzed. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,324,419,342]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hypothesis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,348,941,380]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The myofascial induction group will have better results than the simulation group with respect to joint mobility among people with AxSpA. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,397,535,415]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Materials and methods \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,421,383,439]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Design \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,444,941,574]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This will be a randomized, superiority, double-blind, parallel clinical trial. The trial was designed according to the recommendations for intervention trials (SPIRIT) [25] and the CONSORT guidelines [26]. This protocol complies with the Helsinki guidelines for research involving humans and has been approved by the research and ethics subcommittee of the Faculty of Nursing and Rehabilitation of the Universidad de La Sabana (021-18-2020.) All participants will receive an informed consent form with all the procedures and the risks and benefits of the study. To protect the identity of the participants, each of them will be assigned a numerical code. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,593,485,611]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Study environment \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[323,617,941,647]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This study will be conducted in an outpatient service center for people with rheumatic diseases in the city of Bogota´, Colombia. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,666,467,684]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Inclusion criteria \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,698,936,803]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"• People of both sexes over 18 years of age. \\n• People with a rheumatology-confirmed diagnosis of AxSpA regardless of the level of disease activity. \\n• People with the cognitive ability to follow orders. \\n• People who agree to participate in the study. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,828,470,844]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Exclusion criteria \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[323,858,699,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"• People receiving oral or parenteral coagulation therapy. \\n• Pregnant women. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[524,47,941,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Myofascial induction on joint amplitude in people with axial spondylarthritis \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[55,45,148,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[55,944,493,957]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0286885 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[908,945,941,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 / 12 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[325,94,514,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"• People with kinesiophobia. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,118,707,133]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"• People with previous physical therapy in the last 15 days. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,142,862,157]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"• The presence of active cancer or current treatment with chemo- or radiotherapy. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,174,429,190]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Intervention \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,196,941,294]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The study interventions are based on the myofascial induction techniques proposed by Andrezj Pilat [21]. The physiotherapist (PT) who performs this procedure in the experimental group will be trained and certified in this technique by the author in question. All techniques will be performed in all planned treatment sessions. 6 myofascial induction and simulation sessions will be done, two per week, every other day, for 3 weeks. Patients will be followed up one month after the last session with final evaluation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[326,318,596,335]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Experimental group intervention \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,340,937,472]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J stroke. After a palpatory examination of the fascia, with the participant in prone position, the PT will apply slight superficial traction with his nondominant hand, which helps maintain tension on the myofascial tissue of the dorsolumbar spine. With his dominant hand, he will move his third finger up to the second level of the distal phalanx while the forearm is pronated, and then he will perform the “J” stroke, slowly sliding with nonpainful superficial depth, and rapid supination in the shape of a “J” on the site of the myofascial restriction. For each of the sites perceived as adhered or restricted, an average of seven J strokes should be performed. This technique does not require any lubricant or contact medium. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,473,941,619]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Release of the paravertebral fascia. The participant will be in a sitting position with the back uncovered, the contacts of the PT will be the knuckles of the second, third, and fourth fingers, the participant will be asked to flex the head, and the contacts will be moved up to the second thoracic vertebra. Once the movement is stopped, the therapist will bring back one of his lower extremities. At this point, he will continue the maneuver, keeping the elbows in extension and leaning on his body mechanics to gain depth. Then, he will ask the participant to slowly flex forward while keeping the legs in abduction while the PT slides the contacts caudally until reaching the high lumbar vertebrae. At each site where a movement restriction is perceived, he will stop without increasing the pressure for seven seconds. This process should be repeated three times. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,621,937,733]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Technique for releasing the quadratus lumborum. The participant will be in the lateral decubitus position with knee and hip flexion of $9 0 ^ { \\\\circ }$ . The PT will remain bipedal in front of the patient at the height of the torso, the cranial hand will fix the ribs in an ascending direction, and the distal hand will be positioned with the ulnar edge of the hand in the space between the iliac crest and the last rib. The PT will seek to gain depth in this space until resistance is felt. Once he reaches the maximum allowed tissue barrier, he will perform a rhythmic, low-amplitude, flexion and extension movement of the elbow for 15 cycles in three repetitions. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,734,941,815]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In a following moment, in the same position, the patient will slightly drop the legs outside the table, the PT will accompany this tensioning of the quadratus lumborum, locating the pads of the fingers within the common paravertebral mass, and the PT will perform a slide his fingers in the transverse direction without generating flexion of the carpus. He will perform 13 cycles in three repetitions. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,818,941,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This technique will end with the patient in the lateral decubitus position, the legs will fall slightly outside the table, and the PT will have his hands crossed, with one on the iliac crest and another on the rib cage. He will begin by applying light traction on the skin, and then with light pressure, he will try to overcome the barriers of the tissue by facilitating movement of the fascia with sustained pressure for five minutes. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,45,148,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[522,47,941,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Myofascial induction on joint amplitude in people with axial spondylarthritis \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[55,944,493,957]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0286885 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[908,945,941,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 / 12 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[325,94,941,174]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Release of the sacroiliac fascia. The patient will remain prone with knee flexion at $9 0 ^ { \\\\circ }$ , the PT will remain bipedal next to the limb to be worked on, and the PT’s cranial hand will rest on the sacrum in the direction of the iliac crests while maintaining internal rotation of this limb with the caudal hand. He will maintain the pressure until it surpasses between three and six tissue barriers, which should be sustained for four minutes. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,178,941,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sustained pressure technique in the upper trapezius. For the neck, the myofascial release technique will be used in the upper trapezius, the patient will be supine, and the PT will position his hands at the head of the table while seated as follows: His nondominant hand will support the patient. The dominant hand with $9 0 ^ { \\\\circ }$ flexion will be at the level of the metacarpophalangeal joints, and the initial contact with the upper trapezius and angular muscles of the scapula will begin gently and progressively with the tips of the fingers, increasing the pressure and depth of the position of the hands. The PT will feel each of the tissue barriers and slowly yield to the pressure. This position will be maintained for five minutes. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,308,942,469]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cross-hand technique in thoracolumbar fascia. The intervention continues with the crosshand technique; at this time, the PT will be located on one side of the table, and the patient will have the back uncovered and be in the prone position. The PT will synchronize his breathing with that of the patient. Once he has taken two deep inhalations, the therapist will place the palms of his hands completely on the patient’s back at the level of the lumbosacral region with the fingers in slight abduction, and he will begin applying light traction on the skin, bringing the tissue to a state of pretension; slight pressure will be maintained for five minutes. Once the technique is complete, the pressure will be slowly released, and before removing the hands from the segment, manual contact will be maintained for 10 seconds. Once the hands are removed, the patient will be allowed to rest on the table for approximately two minutes to avoid vasovagal responses. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,472,936,553]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Control group intervention. The simulation intervention in the control group will suggest modification of basic parameters that does not alter the visual demonstration of the technique. The PT assigned to this group will omit the basic principles of myofascial induction: traction, sustained pressure, and tissue depth. All the proposed strategies for the experimental group will be carried out. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,554,934,651]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Discontinuation of participation. During the course of the study, any participant can choose to withdraw from the trial at any time. Participants can choose to suspend treatment and/or evaluations in the study but can remain in the follow-up if they wish; however, participants can also withdraw their consent, which implies that they wish to completely withdraw from the study, completely restricting the researchers’ access to the data obtained to date and those participants’ clinical histories. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,671,410,686]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Outcomes \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,694,936,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The primary outcome will be joint mobility measured by the Bath Ankylosing Spondylitis Metrology Index (BASMI). The secondary outcomes will be the Bath Ankylosing Spondylitis Functional Index (BASFI), Bath Ankylosing Spondylitis Disease Activity Index (BASDAI), and the Ankylosing Spondylitis Quality of Life Questionnaire (AsQol) [27, 28]. Each of these variables will be measured before the intervention, after the intervention, and four weeks after the intervention. The participants will undergo an initial evaluation before randomization. The results will be collected at each time point according to (Figs 1 and 2). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,827,387,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"BASMI \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,851,937,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The index that groups four axial measurements (cervical rotation, Schober test, lateral torso flexion, tragus-to-wall distance) also includes a measurement of peripheral mobility (intermalleolar distance) [29]. These variables will be measured comparatively at least twice to obtain \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,45,148,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[524,47,941,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Myofascial induction on joint amplitude in people with axial spondylarthritis \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[55,944,493,957]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0286885 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[908,945,941,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 / 12 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[120,97,941,535]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0019/046e25a2359b1de4651c1a4bb4dcc2d55e9cb9b30b44b07f7d64d426ff73ac28.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=1 colspan=11>STUDY PERIOD</td></tr><tr><td rowspan=1 colspan=1>TIMEPOINT</td><td rowspan=1 colspan=1>W1</td><td rowspan=1 colspan=1>W2</td><td rowspan=1 colspan=1>W3</td><td rowspan=1 colspan=1>W4</td><td rowspan=1 colspan=1>W5</td><td rowspan=1 colspan=1>W6</td><td rowspan=1 colspan=1>W7</td><td rowspan=1 colspan=1>W9</td><td rowspan=1 colspan=1>W10</td><td rowspan=1 colspan=1>W11</td></tr><tr><td rowspan=1 colspan=1>ENROLMENT:</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Eligibility screen</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Informed consent</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Allocation</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Initial assessment(BASMI,FI,BASDAI, AsQoL)</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>INTERVENTIONS:</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=1>Myofascialinduction</td><td rowspan=2 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=2 colspan=1>一</td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=1> Sham</td><td rowspan=2 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=2 colspan=1>一</td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>ASSESSMENTS:</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Final assessment(BASMI, BASFI,BASDAI, AsQoL)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>FOLLOWUP</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>×</td></tr></table>\",\n \"table_caption\": \"[\\\"W: Weeks. Fig 1. Standard Protocol Items: Recommendations for Interventional Trials (SPIRIT) timeline of measurements. Time represented by weeks (WK). \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[117,589,366,601]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.1371/journal.pone.0286885.g001 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,625,939,672]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"the mean measurements to calculate the final score based on a table of 11 points [25]. This index is evaluated by the examiner and conveys spinal and peripheral mobility findings in centimeters and degrees [27, 30]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,690,380,705]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"BASFI \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,713,934,827]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The BASFI questionnaire is self-administered and consists of 10 questions investigating the functional status of the person diagnosed with AxSpA. Each question can be scored from 0 to 10, with 0 corresponding to no limitation in carrying out the proposed activity and 10 corresponding to the absolute impossibility of carrying it out. This scoring system employs a visual analog scale scored on a horizontal metric line [31]. The final score is obtained from the sum of the results of the questions divided by 10. This instrument is validated for use in Spanish in Colombia [32, 33]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,843,398,858]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"BASDAI \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[323,866,906,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The BASDAI evaluation is self-administered and consists of six questions inquiring about issues related to the five main symptoms of AxSpA: fatigue, axial pain, joint pain and/or \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,45,148,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[522,47,941,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Myofascial induction on joint amplitude in people with axial spondylarthritis \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[55,944,493,957]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0286885 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[908,945,941,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 / 12 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[331,94,771,737]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig 2. Study flow diagram, adapted from CONSORT. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0632/8880b24250d259ffb6bd5f90072d1a38fbd469459b1ec2b6cb8b043b24bf89ad.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[325,761,576,773]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.1371/journal.pone.0286885.g002 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,799,941,881]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"edema, enthesis pain, and morning stiffness [34]. Similar to the BASFI, it is based on a visual analog scale measuring 10 centimeters in a horizontal line. It is scored by adding the scores for questions 1 to 4; for questions 5 and 6, the mean is calculated, and finally, the scores of all questions are added and divided by 5. A higher BASDAI score suggests more active disease [27]. This questionnaire is also validated in Spanish [34]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,944,493,957]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0286885 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[522,47,941,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Myofascial induction on joint amplitude in people with axial spondylarthritis \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[908,945,941,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7 / 12 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[55,45,150,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[326,94,383,111]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"AsQoL \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,117,936,214]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The AsQoL is self-administered and validated in Spanish in Colombia. It consists of 18 questions with the possibility of dichotomous answers (YES—NO). The affirmative answers are scored as 1, and the negative answers are scored as 0. The overall score is the sum of the scores for each question and can be between 0 (better health-related quality of life) and 18 (worse health-related quality of life). If more than three questions are unanswered, the overall score cannot be calculated [35]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,231,429,247]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recruitment \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,253,939,351]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The people who will be eligible for the study are those receiving some medical or therapeutic service in the clinic selected for the study. A research assistant will review the medical records of people with an AxSpA diagnosis confirmed by a rheumatologist. Individuals who are potentially eligible will be contacted by telephone, and the intention and scope of the study will be briefly explained. After this communication, if a person verbally expresses their desire to be part of the research process, a face-to-face interview will be scheduled. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,367,421,385]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sample size \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,391,937,488]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The sample size was calculated in the program R Studio version 3.5.3, package of superiority clinical trials, based on a mean difference of 4.0 in the BASMI with a standard deviation of 2, an r (ratio) between groups of 1, and assuming a loss of $2 0 \\\\%$ . An alpha one sided of $0 . 0 5 \\\\%$ and a power of 0.80 were estimated based on mean difference test corresponding to a clinical trial of superiority (repeated measures ANOVA). The total number of people was 82 (42 per arm). Non probabilistic purposive sampling will be used. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[326,503,596,521]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Randomization and concealment \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,527,941,608]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this study, a simple randomization process will be used by assigning random numbers without repetition in Excel. For randomization concealment, opaque, sealed, consecutively numbered envelopes will be used and kept by a single person. An external person will prepare the envelopes. An envelope will be opened in sequence once the participant has completed the baseline physical examination and the corresponding measurements. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,625,400,642]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Blinding \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,647,941,777]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The people who will perform the baseline and final evaluations will not know the randomization sequences, nor will they be involved in the execution of the treatments in the control group or the experimental group. The participants will visually receive the same procedures since the techniques for the control and experimental groups appear the same; accordingly, they will also be blinded. The PTs who will perform the interventions cannot be blinded due to the need to know the principles of the technique and the form of execution to guarantee correct application of the treatment or simulation. The data analysis will be performed by an external evaluator in the intervention and evaluation process. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,795,477,810]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data management \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,816,937,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The data obtained from the participants will be recorded in virtual record format on the RED-Cap platform [36] by means of a double digitization process to avoid errors in data recording. This platform requires a username and password that will be known only to the research coordinator. In the case of any eventuality, the primary source will be contacted, and the value will be corrected. All documents will be safely stored confidentially. Participants will be identified by a specific number or unique test code. The name and any other identifying details will not be included in any part of the database. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,45,148,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[524,47,941,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Myofascial induction on joint amplitude in people with axial spondylarthritis \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[55,944,493,957]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0286885 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[908,945,941,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"8 / 12 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[321,94,929,125]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,145,475,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Statistical analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,169,941,348]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The data will be analyzed in the latest version of the R Studio program. A statistical analysis will be performed based on the purposive analysis principles. The assumption of normality for the quantitative variables will be verified with the Shapiro–Wilk test, followed by a descriptive analysis; for the continuous variables, measures of central tendency or dispersion according to the data will be applied. For categorical variables, the proportions and frequencies will be calculated. Differences between the groups in measurements for the primary and secondary objectives will be analyzed by repeated measures ANOVA the variables with a normal distribution. Nonparametric tests will be performed if the data do not comply with the assumption of normality, interaction of arm by time effect will not be tested. Qualitative variables will be evaluated with the chi-square test of independence or Fisher’s exact test, as appropriate. For all tests, a significance level of p $< 0 . 0 5$ will be employed. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,367,393,383]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Damage \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,390,931,503]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"All adverse events that are found during the execution of the clinical trial should be reported in a timely manner, all the data will be collected according to Common Terminology Criteria for Adverse Events (CTCAE) version 5.0. This information will be recorded in the report format that includes description of the event, start date and end date, relationship with physiotherapeutic treatment and measures taken on the event. Follow-up information will be provided as needed. Each of these formats will be recorded with the serial code assigned to each patient [29, 37–39]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,523,421,540]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Monitoring \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,546,942,626]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Strict compliance with the protocol and accuracy in relation to the source documents will be evaluated. Following the standard operating procedures declared in this document, an external monitor or referee who is not involved in the research will randomly verify that the clinical trial is carried out within the approved and established parameters and that the data generated are documented and reported in accordance with the protocol every two weeks. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,645,429,662]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Discussion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,669,941,784]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Physiotherapeutic intervention is undoubtedly the first-choice treatment for the management of patients with AxSpA; however, studies based on manual therapy with an emphasis on soft tissue are still insufficient [7, 9, 40]. The results of this research can contribute to the understanding and formulation of new treatment paradigms with rapid therapeutic action that complement the strategies that have been shown to be more effective, such as aerobic exercise and resistance and neuromuscular training [1, 5, 7, 41]. Likewise, it will clarify the role of soft tissue in joint mobility in patients with AxSpA [5, 19]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,785,937,864]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The results of this research seek to initially understand the influence of the fascia on the spinal mobility of patients with this health condition. Fascial hypertonicity, as proposed by Masi et al. [18] may be a \\\"missing link\\\" in therapeutic understanding of AxSpA. The modulation of fascial and soft tissue tone can become a treatment target from the physiotherapist’s actions to achieve hard declines such as function, quality of life and pain. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[321,866,924,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This study is formulated with high methodological quality, allowing its reproducibility in any clinical scenario. The detailed description of the intervention strategies allows any PT to accurately perform a myofascial induction intervention in this type of patient. Additionally, to date, this is the first study focusing exclusively on evaluating the efficacy of manual therapy techniques, particularly myofascial induction, in patients with AxSpA. The outcomes in this study are intended to elucidate changes in mobility and function; however, these can lead to more research questions with a clinical, biochemical, or biomechanical focus, contributing to the formulation of new disciplinary approaches. Unfortunately, for this study, blinding of the therapists performing the sessions will not be possible, and given the strict eligibility conditions proposed, the results will not be generalizable to another type of population, even a population with a similar rheumatic disease. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,45,148,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[524,47,941,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Myofascial induction on joint amplitude in people with axial spondylarthritis \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[55,944,493,957]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PLOS ONE | https://doi.org/10.1371/journal.pone.0286885 October 5, 2023 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[908,945,941,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"9 / 12 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[325,94,939,239]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,262,544,280]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Supporting information \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[323,286,937,333]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"S1 Checklist. SPIRIT 2013 checklist: Recommended items to address in a clinical trial protocol and related documents\\\\*. \\n(DOCX) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,343,442,375]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"S1 File. Protocol. (DOCX) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,383,383,415]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"S2 File. (DOCX) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,444,522,462]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Author Contributions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,468,661,483]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conceptualization: Marı´a Alejandra Sa´nchez Vera. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,491,627,506]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Investigation: Marı´a Alejandra Sa´nchez Vera. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[320,513,911,545]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Methodology: Marı´a Alejandra Sa´nchez Vera, Diego Alejandro Jaimes Ferna´ndez, Robert Schleip. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[323,553,851,568]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Supervision: Marı´a Alejandra Sa´nchez Vera, Diego Alejandro Jaimes Ferna´ndez. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,575,699,590]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Writing – original draft: Marı´a Alejandra Sa´nchez Vera. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,598,848,613]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Writing – review & editing: Diego Alejandro Jaimes Ferna´ndez, Robert Schleip. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[325,642,429,657]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[326,662,941,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Flo´rez Garcı´a MT, Carmona L, Almodo´var R, Ferna´ndez de Las Peñas C, Garcı´a Pe´rez F, Pe´rez Manzanero MA´ et al. Recommendations for the prescription of physical exercise for patients with spondyloarthritis. Reumatol Clin. abril de 2019; 15(2):77–83. \\n2. Londoño J, Pela´ez Ballestas I, Cuervo F, Angarita I, Giraldo R, Rueda JC, et al. Prevalencia de la enfermedad reuma´tica en Colombia, segu´n estrategia COPCORD-Asociacio´n Colombiana de Reumatologı´a. Estudio de prevalencia de enfermedad reuma´tica en poblacio´n colombiana mayor de 18 años. Revista Colombiana de Reumatologia. 1 de octubre de 2018; 25(4):245–245-56. \\n3. Romanowski MW, Sˇ piritović M, Rutkowski R, Dudek A, Samborski W, Straburzyńska-Lupa A. Comparison of Deep Tissue Massage and Therapeutic Massage for Lower Back Pain, Disease Activity, and Functional Capacity of Ankylosing Spondylitis Patients: A Randomized Clinical Pilot Study. Evid Based Complement Alternat Med. 2017; 2017:9894128. https://doi.org/10.1155/2017/9894128 PMID: 28845185 \\n4. Swinnen TW, Westhovens R, De Vlam K, Vlaeyen JWS, Dankaerts W. Activity limitations in patients with axial spondyloarthritis: A role for fear of movement and (Re)injury beliefs. Journal of Rheumatology. 1 de marzo de 2018; 45(3):357-357–66. https://doi.org/10.3899/jrheum.170318 PMID: 29142031 \\n5. Perrotta FM, Musto A, Lubrano E. New Insights in Physical Therapy and Rehabilitation in Axial Spondyloarthritis: A Review. Rheumatol Ther. diciembre de 2019; 6(4):479–86. https://doi.org/10.1007/ s40744-019-00170-x PMID: 31410786 \\n6. Danve A, Deodhar A. Treatment of axial spondyloarthritis: an update. Nat Rev Rheumatol. abril de 2022; 18(4):205–16. https://doi.org/10.1038/s41584-022-00761-z PMID: 35273385 7. Martey C, Sengupta R. Physical therapy in axial spondyloarthritis: guidelines, evidence and clinical practice. Curr Opin Rheumatol. julio de 2020; 32(4):365–70. https://doi.org/10.1097/BOR. 0000000000000714 PMID: 32453037 \\n8. Arguisuelas MD, Liso´n JF, Dome´nech-Ferna´ndez J, Martı´nez-Hurtado I, Coloma PS, Sa´nchez-Zuriaga D. Effects of myofascial release in erector spinae myoelectric activity and lumbar spine kinematics in non-specific chronic low back pain: Randomized controlled trial. Clinical Biomechanics [Internet]. 13 de febrero de 2019 [citado 13 de marzo de 2019]; Disponible en: http://search.ebscohost.com/login.aspx? direct=true&db=edselp&AN=S0268003318300330&site=eds-live \\n9. Dagfinrud H, Hagen KB, Kvien TK. The cochrane review of physiotherapy interventions for ankylosing spondylitis. Journal of Rheumatology. 1 de octubre de 2005; 32(10):1899-1899-906. PMID: 16206344 \\n10. Parravicini G, Bergna A. Biological effects of direct and indirect manipulation of the fascial system. Narrative review. J Bodyw Mov Ther. abril de 2017; 21(2):435–45. \\n11. Masiero S, Bonaldo L, Pigatto M, Lo Nigro A, Ramonda R, Punzi L. Rehabilitation Treatment in Patients with Ankylosing Spondylitis Stabilized with Tumor Necrosis Factor Inhibitor Therapy. A Randomized Controlled Trial. JOURNAL OF RHEUMATOLOGY. 1 de julio de 2011; 38(7):1335-1335–42. \\n12. Kyllikki W, Hossein K, Ingia¨ld H. Self- and manual mobilization improves spine mobility in men with ankylosing spondylitis—a randomized study. Clinical Rehabilitation. 1 de julio de 2009; 23(7):599-599– 600. \\n13. Cathcart E, McSweeney T, Johnston R, Young H, Edwards DJ. Myofascial pain and treatment: Randomized Controlled Trial: Immediate biomechanical, systemic, and interoceptive effects of myofascial release on the thoracic spine: A randomised controlled trial. Journal of Bodywork & Movement Therapies. 1 de enero de 2019; 23(1):74-74–81. \\n14. Ceca D, Elvira L, Guzma´n JF, Pablos A. Benefits of a self-myofascial release program on health-related quality of life in people with fibromyalgia: a randomized controlled trial. J Sports Med Phys Fitness. agosto de 2017; 57(7–8):993–1002. https://doi.org/10.23736/S0022-4707.17.07025-6 PMID: 28139112 \\n15. Hanson A, Brown MA. Genetics and the Causes of Ankylosing Spondylitis. Rheum Dis Clin North Am. agosto de 2017; 43(3):401–14. https://doi.org/10.1016/j.rdc.2017.04.006 PMID: 28711142 \\n16. Klingler W, Velders M, Hoppe K, Pedro M, Schleip R. Clinical relevance of fascial tissue and dysfunctions. Curr Pain Headache Rep. 2014; 18(8):439. https://doi.org/10.1007/s11916-014-0439-y PMID: 24962403 \\n17. Masi AT. Might axial myofascial properties and biomechanical mechanisms be relevant to ankylosing spondylitis and axial spondyloarthritis? Arthritis Res Ther. 2014; 16(2):107. https://doi.org/10.1186/ ar4532 PMID: 25166000 \\n18. Masi AT, Nair K, Andonian BJ, Prus KM, Kelly J, Sanchez JR, et al. Integrative structural biomechanical concepts of ankylosing spondylitis. Arthritis. 2011; 2011:205904. https://doi.org/10.1155/2011/205904 PMID: 22216409 \\n19. White A, Abbott H, Masi AT, Henderson J, Nair K. Biomechanical properties of low back myofascial tissue in younger adult ankylosing spondylitis patients and matched healthy control subjects. Clin Biomech (Bristol, Avon). agosto de 2018; 57:67–73. https://doi.org/10.1016/j.clinbiomech.2018.06.006 PMID: 29936315 \\n20. Andonian BJ, Masi AT, Aldag JC, Barry AJ, Coates BA, Emrich K, et al. Greater Resting Lumbar Extensor Myofascial Stiffness in Younger Ankylosing Spondylitis Patients Than Age-Comparable Healthy Volunteers Quantified by Myotonometry. Arch Phys Med Rehabil. noviembre de 2015; 96(11):2041–7. https://doi.org/10.1016/j.apmr.2015.07.014 PMID: 26254947 \\n21. Pilat A, Gonzalez Nieto JL, Batuecas Sua´rez A. Terapias miofaciales: induccio´n miofascial. Ase´ctos teo´ricos y aplicaciones clı´nicas / Andrzej Pilat; revisio´n te´cnica Alicia Batuecas Sua´rez, Jose´ Luis Gonza´lez Nieto. [Internet]. 2003 [citado 27 de marzo de 2019]. Disponible en: http://unisabana22.gsl.com. mx:80/F?func=service&doc_library=CNA01&local_base=CNA01&doc_number= 000065058&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type= MEDIA \\n22. Schleip R, Mu¨ller DG. Training principles for fascial connective tissues: scientific foundation and suggested practical applications. J Bodyw Mov Ther. enero de 2013; 17(1):103–15. https://doi.org/10. 1016/j.jbmt.2012.06.007 PMID: 23294691 \\n23. Martı´nez Rodrı´guez R, Gala´n del Rı´o F. Mechanistic basis of manual therapy in myofascial injuries. Sonoelastographic evolution control. J Bodyw Mov Ther. abril de 2013; 17(2):221–34. https://doi.org/ 10.1016/j.jbmt.2012.08.006 PMID: 23561871 \\n24. Cathcart E, McSweeney T, Johnston R, Young H, Edwards DJ. Immediate biomechanical, systemic, and interoceptive effects of myofascial release on the thoracic spine: A randomised controlled trial. J Bodyw Mov Ther. enero de 2019; 23(1):74–81. https://doi.org/10.1016/j.jbmt.2018.10.006 PMID: 30691766 \\n25. Chan AW, Tetzlaff JM, Altman DG, Laupacis A, Gøtzsche PC, Krlezˇa-Jerić K, et al. Declaracio´n SPIRIT 2013: definicio´n de los elementos esta´ndares del protocolo de un ensayo clı´nico. Rev Panam Salud Publica. diciembre de 2015; 38(6):506–14. \\n26. Consort—Welcome to the CONSORT Website [Internet]. [citado 13 de junio de 2022]. Disponible en: http://www.consort-statement.org/ \\n27. Castro Villegas M del C, Batlle Gualda E. Metrolog´ıa en espondiloartritis. Reumatol Clin. 1 de marzo de 2010; 6:11–7. \\n28. Popescu C, Trandafir M, Bădică A, Morar F, Predeţeanu D. Ankylosing spondylitis functional and activity indices in clinical practice. J Med Life. 15 de marzo de 2014; 7(1):78–83. PMID: 24653763 \\n29. Martı´n RS, Perera AE, Novo JP, Me´ndez BMG, Annia RG. Clinimetrı´a en las espondiloartritis y sus ´ındices de medidas. Rev Cub de Reu [Internet]. 2013 [citado 13 de junio de 2022]; 15(1). Disponible en: https://www.medigraphic.com/cgi-bin/new/resumen.cgi?IDARTICULO=51080 \\n30. Bath Indices [Internet]. National Axial Spondyloarthritis Society. [citado 14 de junio de 2022]. Disponible en: https://nass.co.uk/homepage/health-professionals/resources-for-health-professionals/bath-indices/ \\n31. Wariaghli G, Allali F, Berrada K, Idrissi Z, Hmamouchi I, Abouqal R, et al. Normative values for the bath ankylosing spondylitis functional index in the general population compared with ankylosing spondylitis patients in Morocco. BMC Musculoskeletal Disorders. 21 de marzo de 2012; 13(1):40. https://doi.org/ 10.1186/1471-2474-13-40 PMID: 22436379 \\n32. Rauseo Vera M, Gutie´rrez-Gonza´lez LA, Maldonado I, Al Snih S. Validacio´n al idioma español de Venezuela de los ´ındices de actividad de enfermedad y el ´ındice funcional para espondilitis anquilosante. Reumatol Clin. 1 de julio de 2019; 15(4):223–8. \\n33. Calin A, Jones SD, Garrett SL, Kennedy LG. Bath Ankylosing Spondylitis Functional Index. Br J Rheumatol. agosto de 1995; 34(8):793–4. https://doi.org/10.1093/rheumatology/34.8.793 PMID: 7551671 \\n34. Sarabia FN, Ariza RA, Cruz BH. La versio´n española del BASDAI es fiable y se correlaciona con la actividad de la enfermedad en pacientes con espondilitis anquilosante. Revista española de reumatologı´a: O´ rgano Oficial de la Sociedad Española de Reumatologı´a. 2004; 31(6):372. \\n35. Ariza-Ariza R, Herna´ndez-Cruz B, Lo´pez-Antequera G, Toyos FJ, Navarro-Sarabia F. Adaptacio´n transcultural y validacio´n de una versio´n en español de un instrumento especı´fico para medir calidad de vida relacionada con la salud en pacientes con espondilitis anquilosante: el ASQoL. Reumatol Clin. 1 de marzo de 2006; 2(2):64–9. \\n36. REDCap [Internet]. [citado 13 de junio de 2022]. Disponible en: https://www.project-redcap.org/ \\n37. Carnes D, Mars TS, Mullinger B, Froud R, Underwood M. Adverse events and manual therapy: a systematic review. Man Ther. agosto de 2010; 15(4):355–63. https://doi.org/10.1016/j.math.2009.12.006 PMID: 20097115 \\n38. Paanalahti K, Holm LW, Nordin M, Asker M, Lyander J, Skillgate E. Adverse events after manual therapy among patients seeking care for neck and/or back pain: a randomized controlled trial. BMC Musculoskelet Disord. 12 de marzo de 2014; 15:77. https://doi.org/10.1186/1471-2474-15-77 PMID: 24618345 \\n39. World Medical Association. World Medical Association Declaration of Helsinki: Ethical Principles for Medical Research Involving Human Subjects. JAMA. 27 de noviembre de 2013; 310(20):2191–4. https://doi.org/10.1001/jama.2013.281053 PMID: 24141714 \\n40. Kasapoglu Aksoy M, Birtane M, Taştekin N, Ekuklu G. The Effectiveness of Structured Group Education on Ankylosing Spondylitis Patients. J Clin Rheumatol. abril de 2017; 23(3):138–43. https://doi.org/10. 1097/RHU.0000000000000499 PMID: 28248799 \\n41. Fernandez-de-las-Penas C, Alonso-Blanco C, Alguacil-Diego IM, Miangolarra-Page JC. One-Year Follow-up of Two Exercise Interventions for the Management of Patients with Ankylosing Spondylitis: A Randomized Controlled Trial [Internet]. Vol. 85. 2006 [citado 27 de marzo de 2019]. 559 p. 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myofascial induction compared with its simulation on joint amplitude in people with axial spondylarthritis: Protocol of a randomized controlled clinical trial"}}},{"rowIdx":63,"cells":{"abstract":{"kind":"string","value":"PurposeIntensity-modulated radiotherapy (IMRT) and volumetric modulated arc therapy (VMAT), two advanced modes of high-precision radiotherapy (RT), have become standard of care in the treatment of head and neck cancer. The development in RT techniques has markedly increased the complexity of target volume definition and accurate treatment delivery. The aim of this study was to indirectly investigate the quality of current TV delineation and RT delivery by analyzing the patterns of treatment failure for head and neck cancer patients in our high-volume RT center.MethodsBetween 2004 and 2014, 385 patients with pharyngeal, laryngeal, and oral cavity tumors were curatively treated with primary RT (IMRT/VMAT). We retrospectively investigated locoregional recurrences (LRR), distant metastases (DM), and overall survival (OS).ResultsMedian follow-up was 6.4 years (IQR 4.7–8.3 years) during which time 122 patients (31.7%) developed LRR (22.1%) and DM (17.7%). The estimated 2- and 5-year locoregional control was 78.2% (95% CI 73.3, 82.3) and 74.2% (95% CI 69.0, 78.8). One patient developed a local recurrence outside the high-dose volume and five patients developed a regional recurrence outside the high-dose volume. Four patients (1.0%) suffered a recurrence in the electively irradiated neck and two patients had a recurrence outside the electively irradiated neck. No marginal failures were observed. The estimated 2- and 5-year DM-free survival rates were 83.3% (95% CI 78.9, 86.9) and 80.0% (95% CI 75.2, 84.0). The estimated 2- and 5-year OS rates were 73.6% (95% CI 68.9, 77.8) and 52. 6% (95% CI 47.3, 57.6). Median OS was 5.5 years (95% CI 4.5, 6.7).ConclusionTarget volume definition and treatment delivery were performed accurately, as only few recurrences occurred outside the high-dose regions and no marginal failures were observed. Research on dose intensification and identification of high-risk subvolumes might decrease the risk of locoregional relapses. The results of this study may serve as reference data for comparison with future studies, such as dose escalation or proton therapy trials."},"author":{"kind":"string","value":"Julie van der Veen|Annouschka Laenen|Heleen Bollen|Sandra Nuyts"},"content_list":{"kind":"list like","value":[{"bbox":"[290,240,847,297]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recurrence Patterns After IMRT/VMAT in Head and Neck Cancer ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[289,314,885,331]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Heleen Bollen<sup>1,2*</sup>, Julie van der Veen<sup>1,2†</sup>, Annouschka Laenen<sup>3</sup> and Sandra Nuyts<sup>1,2</sup> ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[289,342,889,381]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$^{1}$ Laboratory of Experimental Radiotherapy, Department of Oncology, KU Leuven, Leuven, Belgium, $^{2}$ Department of Radiation Oncology, Leuven Cancer Institute, University Hospitals Leuven, Leuven, Belgium, $^{3}$ Leuven Biostatistics and Statistical Bioinformatics Center, KU Leuven, Leuven, Belgium ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[289,401,924,525]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Purpose: Intensity-modulated radiotherapy (IMRT) and volumetric modulated arc therapy (VMAT), two advanced modes of high-precision radiotherapy (RT), have become standard of care in the treatment of head and neck cancer. The development in RT techniques has markedly increased the complexity of target volume definition and accurate treatment delivery. The aim of this study was to indirectly investigate the quality of current TV delineation and RT delivery by analyzing the patterns of treatment failure for head and neck cancer patients in our high-volume RT center. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[289,531,924,582]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Methods: Between 2004 and 2014, 385 patients with pharyngeal, laryngeal, and oral cavity tumors were curatively treated with primary RT (IMRT/VMAT). We retrospectively investigated locoregional recurrences (LRR), distant metastases (DM), and overall survival (OS). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[289,590,924,767]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Results: Median follow-up was 6.4 years (IQR 4.7-8.3 years) during which time 122 patients $(31.7\\%)$ developed LRR $(22.1\\%)$ and DM $(17.7\\%)$ . The estimated 2- and 5-year locoregional control was $78.2\\%$ $(95\\%$ CI 73.3, 82.3) and $74.2\\%$ $(95\\%$ CI 69.0, 78.8). One patient developed a local recurrence outside the high-dose volume and five patients developed a regional recurrence outside the high-dose volume. Four patients $(1.0\\%)$ suffered a recurrence in the electively irradiated neck and two patients had a recurrence outside the electively irradiated neck. No marginal failures were observed. The estimated 2- and 5-year DM-free survival rates were $83.3\\%$ $(95\\%$ CI 78.9, 86.9) and $80.0\\%$ $(95\\%$ CI 75.2, 84.0). The estimated 2- and 5-year OS rates were $73.6\\%$ $(95\\%$ CI 68.9, 77.8) and $52.6\\%$ $(95\\%$ CI 47.3, 57.6). Median OS was 5.5 years $(95\\%$ CI 4.5, 6.7). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[289,774,924,879]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conclusion: Target volume definition and treatment delivery were performed accurately, as only few recurrences occurred outside the high-dose regions and no marginal failures were observed. Research on dose intensification and identification of high-risk subvolumes might decrease the risk of locoregional relapses. The results of this study may serve as reference data for comparison with future studies, such as dose escalation or proton therapy trials. 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As surgical resection can be mutilating, radiation therapy (RT), with or without concurrent chemotherapy, has emerged as the treatment of choice in the management of local and locoregionally advanced HNC (2-4). Intensity-modulated radiotherapy (IMRT) and volumetric modulated arc therapy (VMAT) are two radiation techniques that enable steeper dose gradients, allowing better sparing of the surrounding structures compared with the older 3D techniques, thereby reducing toxicity (5) and improving quality of life (6, 7). The trade-off for these more conformal RT techniques is an increased reliance on precise TV definition and accurate treatment delivery. Tumor tissue that is not defined as target volume (TV) by the radiation oncologist will not receive the prescribed dose, and geographical misses, leading to locoregional recurrences, are a potential risk (8, 9). In addition, several studies have proven experience with more conformal RT techniques to be essential for optimal outcomes in HNC (10-12). IMRT was routinely implemented in University Hospitals Leuven for the definitive treatment of HNC in 2004. Since 2010, VMAT has become the standard of care. In preparation for the implementation of proton therapy in our RT center, we investigated the quality of our current TV delineation and RT delivery by retrospectively analyzing the incidence and location of local recurrence (LR) and regional recurrence (RR) compared with the TVs. Knowledge about treatment failure patterns is especially relevant when implementing more conformal RT techniques, such as proton therapy. In the treatment of HNC, efforts are continuously being made to optimize the therapeutic ratio to improve disease outcome while keeping toxicity to a minimum. By analyzing the patterns of failure in the HNC population, we hope to refine future strategies in TV delineation and dose escalation. Moreover, the results of this study may serve as reference data for comparison with future studies, such as dose escalation or proton therapy trials. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,664,349,682]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"MATERIALS AND METHODS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[73,693,238,709]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Patient Selection ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[73,709,485,916]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this retrospective analysis, patients treated with curative intent for a HNC with R(C)T between June 2004 and December 2014 were included, to allow a follow-up of at least 3 years. We included patients with primary pharyngeal, oral cavity, and laryngeal squamous cell carcinoma. We excluded patients previously treated with RT in the head and neck region, patients with metastatic disease, postoperative patients or patients who received induction chemotherapy, primary sinonasal or nasopharyngeal tumor patients, and patients who did not finish RT as planned. The medical files were reviewed for each patient. The study was approved by the ethical committee of University Hospitals Leuven/KU Leuven (S59803). All methods were performed in accordance with the relevant guidelines and regulations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,75,808,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Target Volume Delineation and Treatment Planning ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,110,922,492]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The gross tumor volume (GTV) was defined as the macroscopic tumor volume seen on planning CT and using information from clinical investigation and diagnostic and functional imaging. Patients with locally advanced disease underwent FDG-PET scan as staging exam, and PET scans were used \"side-by-side\" during the process of contouring with the diagnostic CT and/or MR scan. The clinical target volume of the primary tumor (CTVp) and adenopathies (CTVn) were created with a 3D $10\\mathrm{-mm}$ expansion around the GTV and cropped for anatomical boundaries, e.g., uninvolved bone and air. Neck regions at risk of harboring microscopic tumor cells were delineated using international guidelines (13-15) to create the elective CTV (CTVe) which received a lower dose than CTVp and CTVn. To ensure an adequate coverage of CTV, a planning target volume (PTV) was created by expanding CTV by $5\\mathrm{mm}$ . The clinical target volumes of the macroscopically affected tumor sites (CTVp and CTVn) were treated up to a normalized iso-effective dose in 2 Gy fractions (NID2 Gy) of $70\\mathrm{Gy}$ . CTVe was treated up to a NID2 Gy of $50\\mathrm{Gy}$ , except for 19 patients that were included in a dose deescalation trial and received a lower dose to CTVe up to a NID2 Gy of $40\\mathrm{Gy}$ (16). Concurrent chemotherapy (cisplatin $100\\mathrm{mg/m^2}$ , q3w) was offered to fit patients with advanced stage disease, according to hospital guidelines. Neck dissection post-RT was not routinely performed. Treatment plans were planned using IMRT/VMAT and tumors were classified according the American Joint Committee on Cancer seventh TNM edition (17). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,501,796,534]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recurrence Identification and Patient Evaluation ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,535,922,740]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recurrence and OS rates were measured starting at the start of RT until recurrence or death from any cause. CT or MRI images of the first recurrence (local/regional) were visually inspected and compared with the planning CT. Recurrences were defined as either local (LR), regional (RR), or distant metastases (DM) and further as 1) in CTVp or n, 2) marginal to CTVp or CTVn (overlap but also more than $50\\%$ of tumor load outside the original tumor site), 3) outside CTVp or CTVn, 4) outside CTVn but inside CTVe, and (5) outside CTVe. If the primary tumor or adenopathy was still visible on imaging 6 months after the start of treatment, this was classified as persistent disease (PD). Second primary (SP) HNCs were classified as such if the new tumor was more than $2\\mathrm{cm}$ from the index tumor, or if it was less than $2\\mathrm{cm}$ from the index tumor, but developed more than 3 years after RT. 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Locoregional ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,35,134,44]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bollen et al. 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The Cox proportional hazards model was used for analyzing the prognostic effect of patient or disease characteristics on oncological outcomes. Results are reported as hazard ratios with $95\\%$ confidence intervals. Univariate analysis was performed for several potential prognostic factors: age, sex, smoking status, stage, site, and tumor grade. Follow-up summary statistics were obtained using the Kaplan-Meier estimate of potential follow-up (18). Analyses were performed using SAS software (version 9.4 of the SAS System for Windows). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,77,606,92]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RESULTS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,103,732,118]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Patient Characteristics ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,121,922,238]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Patient characteristics of the 385 patients are shown in Table 1. Median age was 61 years old (range 34-89 years) and the majority were men (326 vs. 59 women). Seventeen patients had a multifocal tumor and two patients had an unknown primary. The primary tumor sites were the oropharynx $(46.2\\%)$ , hypopharynx $(23.9\\%)$ , supraglottis $(20.8\\%)$ , larynx $(10.6\\%)$ , and oral cavity $(3.9\\%)$ . Of the 178 oropharynx tumors, 39 were p16 positive, 42 were p16 negative, and 97 were of unknown ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[72,297,924,892]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0259/c471d26b977442bf2ee8b8114e98e9e2acd9e9d1fba7a28c2408b09a4f263228.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":"<table><tr><td>Patient characteristics</td><td></td><td>n = 385</td><td>%</td></tr><tr><td rowspan=\"2\">Gender</td><td>Male</td><td>326</td><td>84.7%</td></tr><tr><td>Female</td><td>59</td><td>15.3%</td></tr><tr><td rowspan=\"4\">Age (years), median (range)</td><td></td><td>61</td><td>(34-89)</td></tr><tr><td><60</td><td>160</td><td>41.6%</td></tr><tr><td>60-70</td><td>158</td><td>41.0%</td></tr><tr><td>>70</td><td>67</td><td>17.4%</td></tr><tr><td rowspan=\"8\">Subsite</td><td>Oropharynx</td><td>178</td><td>46.2%</td></tr><tr><td>Oral cavity</td><td>15</td><td>3.9%</td></tr><tr><td>Hypopharynx</td><td>92</td><td>23.9%</td></tr><tr><td>Larynx</td><td>41</td><td>10.6%</td></tr><tr><td>Supraglottis</td><td>80</td><td>20.8%</td></tr><tr><td>CUP</td><td>2</td><td>0.5%</td></tr><tr><td>Multifocal</td><td>17</td><td>4.4%</td></tr><tr><td>Unifocal</td><td>368</td><td>95.6%</td></tr><tr><td rowspan=\"4\">Grade</td><td>1</td><td>19</td><td>4.9%</td></tr><tr><td>2</td><td>106</td><td>27.5%</td></tr><tr><td>3</td><td>69</td><td>17.9%</td></tr><tr><td>Unknown</td><td>191</td><td>49.6%</td></tr><tr><td rowspan=\"5\">T stage</td><td>1</td><td>31</td><td>8.1%</td></tr><tr><td>2</td><td>120</td><td>31.2%</td></tr><tr><td>3</td><td>107</td><td>27.8%</td></tr><tr><td>4</td><td>125</td><td>32.5%</td></tr><tr><td>Unknown</td><td>2</td><td>0.5%</td></tr><tr><td rowspan=\"4\">N stage</td><td>0</td><td>116</td><td>30.1%</td></tr><tr><td>1</td><td>34</td><td>8.8%</td></tr><tr><td>2</td><td>222</td><td>57.7%</td></tr><tr><td>3</td><td>13</td><td>3.4%</td></tr><tr><td rowspan=\"4\">Stage</td><td>1</td><td>13</td><td>3.4%</td></tr><tr><td>2</td><td>51</td><td>13.2%</td></tr><tr><td>3</td><td>65</td><td>16.9%</td></tr><tr><td>4</td><td>256</td><td>66.5%</td></tr><tr><td rowspan=\"3\">P16 status in oropharyngeal tumors (n = 178)</td><td>Negative</td><td>42</td><td>23.6%</td></tr><tr><td>Positive</td><td>39</td><td>21.9%</td></tr><tr><td>Unknown</td><td>97</td><td>54.5%</td></tr><tr><td rowspan=\"3\">Smoking</td><td>Never</td><td>38</td><td>9.9%</td></tr><tr><td>Former</td><td>92</td><td>23.9%</td></tr><tr><td>Current</td><td>255</td><td>66.2%</td></tr><tr><td rowspan=\"3\">Ethyl</td><td>Never</td><td>154</td><td>40.0%</td></tr><tr><td>Former</td><td>85</td><td>22.1%</td></tr><tr><td>Current</td><td>146</td><td>37.9%</td></tr><tr><td rowspan=\"3\">Concomitant systemic therapy</td><td>None</td><td>132</td><td>34.3%</td></tr><tr><td>Chemotherapy</td><td>217</td><td>56.4%</td></tr><tr><td>EGFR inhibitor</td><td>36</td><td>9.4%</td></tr><tr><td rowspan=\"3\">RT dose, median (range)</td><td></td><td>72 Gy</td><td>(66-75 Gy)</td></tr><tr><td>66 Gy</td><td>16</td><td>4.2%</td></tr><tr><td>66.1-69.99 Gy</td><td>7</td><td>1.8%</td></tr></table>","table_caption":"[\"TABLE 1 | Patient characteristics. \"]","table_footnote":"[\"CUP, cancer of unknown primary; EGFR, epidermal growth factor. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[73,35,134,44]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bollen et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[727,35,924,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recurrence Patterns After IMRT/VMAT ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[73,951,294,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Oncology | www.frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[495,951,502,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[680,949,922,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"September 2021 | Volume 11 | Article 720052 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[73,77,485,193]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"status. Stage IV tumors were most common (66.5%), followed by stage III (16.9%), stage II (13.2%), and stage I (3.4%). More than half of patients were treated with concurrent chemotherapy (56.4%) and 36 were treated with an EGFR inhibitor (9.4%). Three hundred twenty-six patients were treated with accelerated RT (72 Gy in 6 weeks). Twenty-one patients were treated with adaptive RT as part of a trial and 19 patients received a lower dose to CTVe (40 Gy) in a dose de-escalation trial (16). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,207,149,220]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Survival ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[73,223,485,370]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The median follow-up period was 6.4 years (IQR 4.7-8.3 years). The estimated 2- and 5-year OS rates were $73.6\\%$ (95% CI 68.9, 77.8) and $52.56\\%$ (95% CI 47.3, 57.6). Median OS was 5.5 years (95% CI 4.5, 6.7). The type of recurrence had a significant impact on OS; 96 of 237 patients $(40.5\\%)$ with no recurrence died during follow-up vs. 44 of 60 patients $(73.3\\%, \\text{RR} 1.7, 95\\% \\text{CI} 1.4, 2.1)$ with LRR as first recurrence and 58 of 62 patients $(93.5\\%, \\text{RR} 2.3, 95\\% \\text{CI} 2.0, 2.7)$ with distant metastases. Regarding the development of a second primary in the head and neck, this had no significant impact on OS (10 of 26 patients died, $38.5\\%$ , RR 0.9, $95\\% \\text{CI} 0.6-1.6$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,385,273,399]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recurrence Patterns ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[73,401,485,474]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"One hundred twenty-two patients suffered a recurrence, of which the LR, RR, and DM distribution is shown in Figure 1 for first recurrence only. Among the patients with recurrence, the median time to failure was 9.6 months (range 3.3 months to 5.5 years). Fifty-four patients $(14.0\\%)$ had LR, 49 patients $(12.7\\%)$ had RR, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,77,922,179]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and 62 patients $(16.1\\%)$ developed DM as first site of recurrence. The estimated 2- and 5-year locoregional control was $78.2\\%$ (95% CI 73.3, 82.3) and $74.2\\%$ (95% CI 69.0, 78.8). On univariate analysis, a history of ethyl abuse, a higher tumor stage, and a higher tumor grade were significantly associated with more LRR (Table 2). The estimated 2- and 5-year DM-free survival rates were $83.3\\%$ (95% CI 78.9, 86.9) and $80.0\\%$ (95% CI 75.2, 84.0). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,192,640,207]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"T-Site Failure ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,207,922,414]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There were 53 LRs in the high-dose volume (CTVp), no marginal recurrences, and one isolated LR outside CTVp. The latter concerned a 46-year-old female patient with two synchronous tumors: a T2 retromolar trigone tumor and a T4 tumor in the vallecula (Figure 2A) with multiple lymph nodes (N2b). She was treated with accelerated RT to $72\\mathrm{Gy}$ concurrently with cisplatin. Ten months after the start of RT, there was a recurrence in the prelaryngeal space (Figure 2B) approximately $1\\mathrm{cm}$ caudal of the primary tumor in the vallecula, for which she underwent a total laryngectomy. Six months later, she developed a new recurrence next to the tracheostoma for which she had re-irradiation $(16\\times$ $3.125\\mathrm{Gy})$ . Half a year later, there was again local progression in combination with distant metastases. Palliative chemotherapy was started, but the patient died 6 months later. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,428,643,442]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"N-Site Failure ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,446,922,474]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Of the 49 patients with RRs, 44 had recurrences inside the high-dose volume (CTVn). Four patients developed recurrences in CTVe ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[248,517,749,843]","code_body":null,"code_caption":null,"image_caption":"[\"FIGURE 1 | Site of first recurrence. Number of patients with a recurrence in the different sites. Overlapping circles show combination possibilities. There were one isolated local recurrence outside CTVp and one isolated regional recurrence outside CTVe. The numbers in the shaded area represent persistent tumors after the end of treatment, included in the total number of recurrences, e.g., 10 of 16 RR were persistent after treatment. CTVe, elective clinical target volume; DM, distant metastases; HNC, head and neck cancer; LR, local recurrence; RR, regional recurrence; SP, second primary. \"]","image_footnote":"[]","img_path":"images/0776/ee0496359017ac9f822cb22c66fbef5625ea5b1bb6f21c892601aa84b9a3f825.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[73,35,134,44]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bollen et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[727,35,924,44]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recurrence Patterns After IMRT/VMAT ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[73,949,294,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Oncology | www.frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[495,951,502,960]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[680,949,922,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"September 2021 | Volume 11 | Article 720052 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[70,103,927,347]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0253/6e5c44270769b9857463f62f3ab4f8af5f85190fe4dc028e11264348d58fa6dc.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":"<table><tr><td>Variable</td><td>Test</td><td>Hazard ratio (95% CI)</td><td>P-value</td></tr><tr><td>Sex</td><td>Female vs. male</td><td>0.928 (0.514, 1.675)</td><td>0.8039</td></tr><tr><td>Age</td><td>+1 year</td><td>1.010 (0.987, 1.033)</td><td>0.3938</td></tr><tr><td>Smoking</td><td>Global test</td><td></td><td>0.3281</td></tr><tr><td></td><td>Current smoker vs. stopped >6 m</td><td>1.462 (0.843, 2.536)</td><td>0.1764</td></tr><tr><td></td><td>Current smoker vs. never</td><td>1.370 (0.655, 2.863)</td><td>0.4029</td></tr><tr><td></td><td>Stopped >6 m vs. never</td><td>0.937 (0.401, 2.189)</td><td>0.8805</td></tr><tr><td>Ethyl</td><td>Global test</td><td></td><td>0.0064</td></tr><tr><td></td><td>Stopped vs. yes</td><td>2.215 (1.290, 3.801)</td><td>0.0039</td></tr><tr><td></td><td>Stopped vs. none</td><td>1.997 (1.195, 3.340)</td><td>0.0083</td></tr><tr><td></td><td>Yes vs. none</td><td>0.902 (0.537, 1.513)</td><td>0.6956</td></tr><tr><td>Grade</td><td>+1 level</td><td>0.574 (0.369, 0.892)</td><td>0.0136</td></tr><tr><td>Highest TNM stagea</td><td>+1 level</td><td>1.935 (1.343, 2.788)</td><td>0.0004</td></tr><tr><td>Oral</td><td>Yes vs. no</td><td>1.888 (0.764, 4.664)</td><td>0.1683</td></tr><tr><td>Oropharynx</td><td>Yes vs. no</td><td>0.734 (0.475, 1.133)</td><td>0.1621</td></tr><tr><td>Glottis</td><td>Yes vs. no</td><td>0.939 (0.485, 1.817)</td><td>0.8513</td></tr><tr><td>Hypopharynx</td><td>Yes vs. no</td><td>1.435 (0.889, 2.316)</td><td>0.1396</td></tr><tr><td>Supraglottis</td><td>Yes vs. no</td><td>1.428 (0.885, 2.305)</td><td>0.1442</td></tr></table>","table_caption":"[\"TABLE 2 | Univariate analysis for locoregional recurrence. \"]","table_footnote":"[\"Continuous variables: $HR > (< )$ 1 means higher (lower) risk for increasing level. Categorical variables: pairwise tests only presented if significant global P-value. Binary variables/pairwise tests: $R > (< )$ 1 means higher (lower) risk for first category. \",\"CI, confidence interval; m, months; TNM, tumor, node, metastasis stage. \",\"${}^{a}$ In case patients had multiple tumors,the tumor with the highest TNM stage was used. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[70,409,487,526]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$(1.0\\%)$ , of which one also had a recurrence outside CTVe in ipsilateral level V 5 years after RT. Recurrence in level II and the retropharyngeal neck level (VIIa) were the most common (level II: one ipsilateral, three contralateral; level VIIa: two ipsilateral, one contralateral). There were two RRs in level Ib (one ipsilateral and one contralateral). One patient had an isolated RR outside the irradiated volume, in the ipsilateral retropharyngeal neck 4 months after RT (Table 3). There were no marginal recurrences. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,541,280,559]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Second Primary HNC ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,559,485,589]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Twenty-six patients developed SP in the head and neck region. In 8 patients, the SP developed less than $2\\mathrm{cm}$ from the index tumor more ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,409,924,468]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"than 3 years after treatment. In 3 patients, the SP was diagnosed less than 3 years after treatment of the index tumor but was more than $2\\mathrm{cm}$ from the index tumor, and in 15 patients, the SP occurred more than 3 years later and more than $2\\mathrm{cm}$ from the index tumor. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,500,640,517]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DISCUSSION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[507,528,926,589]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Thirty percent of HNC patients will develop a locoregional relapse, while therapy failure due to metastases is less common (19, 20). The prognosis for HNC patients after failure of first-line therapy is poor, with a median overall survival of less than 1 year (21). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[208,631,495,861]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0548/200e3878d473c5738f7bb9c15add4d171a5621413a0798352141a1b7267b7fa8.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[502,630,786,861]","code_body":null,"code_caption":null,"image_caption":"[\"FIGURE 2 | Axial CT scan images of the only patient with a local recurrence outside of the high-dose volume. (A) The primary tumor originating in the vallecula and (B) CT scan of the local recurrence in the prelaryngeal space, approximately $1\\\\mathrm{cm}$ caudal of the index tumor. \"]","image_footnote":"[]","img_path":"images/0719/19f1c325977ad9b14161140fd0c2537a2ce3e18f39c5802efd75b77574246cc0.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[73,35,134,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bollen et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[726,35,924,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recurrence Patterns After IMRT/VMAT ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[72,949,295,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Oncology | www.frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[492,949,504,960]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[680,949,924,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"September 2021 | Volume 11 | Article 720052 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[70,93,927,347]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0493/632f72b3d49510ac222101ecbfd9aa36333f017381cfbee0cd8aefc24e519f50.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":"<table><tr><td>No.</td><td>Sex, age, smoking</td><td>Subsite</td><td>Stage</td><td>Treatment</td><td>Failure type</td><td>Time to recurrence</td><td>Elective levels irradiated</td><td>Regional recurrence</td></tr><tr><td rowspan=\"4\">1</td><td rowspan=\"4\">Male, 46 y, former, multifocal</td><td>Piriform sinus (R)</td><td>Tis</td><td rowspan=\"4\">70 Gy (50 Gy) + cisplatin</td><td rowspan=\"4\">N+M</td><td rowspan=\"4\">5 years and 2 months</td><td rowspan=\"4\">Ipsi (R): Ib, II-IVa+b, Vc, VI, VIIa+b; Contra (L): II-IVa+b, Vc, VI, VIIa+b</td><td rowspan=\"4\">Ipsi (R): Ib, II, V, VIIa Contra (L): Ib, II, VIIa</td></tr><tr><td>Supraglottis (L)</td><td>T3N0</td></tr><tr><td>Esophagus (mid)</td><td>T2</td></tr><tr><td>Oropharynx (R)</td><td>Tis</td></tr><tr><td rowspan=\"3\">2</td><td rowspan=\"3\">Male, 63 y, current, multifocal</td><td>Floor of mouth (L)</td><td>T2</td><td rowspan=\"3\">72 Gy (46.4 Gy) + cisplatin</td><td rowspan=\"3\">T+N+M</td><td rowspan=\"3\">1 year and 6 months</td><td rowspan=\"3\">Ipsi (R): Ib, II-IVa, V, VIIb Contra (L): Ib, II-IVa</td><td rowspan=\"3\">Contra (L): II-III</td></tr><tr><td>T2N2b (R)</td><td>T1</td></tr><tr><td>Supraglottis 4. Glottis</td><td>T1</td></tr><tr><td>3</td><td>Male, 67 y, former</td><td>Piriform sinus</td><td>T3N2b</td><td>72 Gy (40 Gy) + cisplatin</td><td>N+M</td><td>4 months</td><td>Ipsi: Ib, II-IVa, V, Vla+b, VIIa+b Contra: II-IVa, Vla+b, VIIa</td><td>Contra: II</td></tr><tr><td>4</td><td>Male, 68 y, current</td><td>Piriform sinus</td><td>T4aN2b</td><td>72 Gy (40 Gy) + cisplatin</td><td>T+N</td><td>7 months</td><td>Ipsi: II-IVa, V, VIIa+b Contra: II-III</td><td>Ipsi: VIIa-b</td></tr><tr><td>5</td><td>Female, 75 y, former</td><td>Piriform sinus</td><td>T2N2b</td><td>72 Gy (40 Gy)</td><td>N</td><td>4 months</td><td>Ipsi: II-IVa, V Contra: II-III</td><td>Ipsi: VIIa</td></tr></table>","table_caption":"[\"TABLE 3 | Demographics of patients with regional recurrence outside CTVn. \"]","table_footnote":"[\"Four patients had a recurrence in CTVe, of which one (patient 1) also had a recurrence outside the irradiated volume (ipsilateral level V). One patient had an isolated regional recurrence outside the elective target volume (patient 5, ipsilateral retropharyngeal adenopathy). Dose in brackets shows dose to CTVe. TNM classification according to TNM7. Contra, contralateral; CTVe, elective clinical target volume; Ipsi, ipsilateral; L, left; N, nodal classification; R, right; T, tumor classification; y, years; M, metastasis. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[73,414,485,546]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In order to guide future attempts to improve the therapeutic ratio and outcomes for HNC patients, understanding of the patterns of treatment failure is essential. Particularly with the introduction of IMRT and VMAT, more conformal RT techniques allowing the prescribed radiation dose to be delivered precisely to the TV, concerns about an increased risk for marginal misses were raised. Indeed, the trade-off for these more conformal RT techniques is an increased reliance on precise TV definition and accurate treatment delivery. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,546,485,915]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The present study reports the patterns of recurrence after RT in 385 HNC patients treated between 2004 and 2014 with IMRT/VMAT at the University Hospitals Leuven. Thirty-one percent of patients suffered a LRR, which corresponds with previously reported recurrence rates in HNC patients treated with definitive radio (chemo)therapy (19, 20). History of ethyl abuse and a higher TNM stage were associated with more LRR. Only one patient developed a LR outside the CTVp. As for RR, five patients suffered a recurrence outside CTVn, of which two recurrences were located outside CTVe. There were no marginal recurrences, either local or regional. Demographics of patients with a RR outside the high-dose volume are summarized in Table 3. Of the five regional relapses seen in our patient population, all were originally LAHNSCC and underwent PET-CT scan. The PET scans were reviewed and no missed nodes were identified. In one patient, relapse occurred in ipsilateral level V, more than 5 years after RT. It is important to note that this patient had a multifocal tumor which makes elective level selection more complicated and comes with an increased RR risk. The other patient developed a RR 4 months after the start of RT in ipsilateral level VIIa. This patient suffered from a N2b (levels III and IV) hypopharynx tumor and level VIIa was not included in the CTVe. Looking at the planning CT retrospectively, it is possible that there was a lymph node initially, although it was not withheld on FDG-PET/CT. Several trials reported a higher incidence of LRR in the lower neck (22-24). In our study, no marginal recurrences were ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,413,922,649]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"observed, which provides reassurance about treatment quality and stresses the importance of guideline adherence for accurate neck level selection (13-15, 25). Nineteen patients were simultaneously included in a dose de-escalation trial, investigating the patterns of regional recurrences in the electively irradiated lymph node regions after dose de-escalation to $40\\mathrm{Gy}$ (EQD2 Gy) (16). The inclusion criteria of this study were previously untreated, histologically proven squamous cell carcinoma of the oral cavity, oropharynx, hypopharynx, or larynx, or cervical lymph node metastases of unknown primary cancer. All macroscopically affected tumor sites (both primary tumor and affected lymph nodes) were treated up to an EQD2 Gy of $70\\mathrm{Gy}$ . All 44 patients that suffered a recurrence inside CTVn were thus treated up to an EDQ2 of $70\\mathrm{Gy}$ . Of the four patients that developed a RR in CTVe, none were included in the de-escalation trial, and thus, all four patients received the standard elective dose, with an EQD2 up to $50\\mathrm{Gy}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,650,922,915]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The vast majority of relapses, both local and regional, occurred in-field. A number of trials aimed to analyze the patterns of treatment failures after (IM)RT in HNC patients. Gujral and Nutting reviewed the data from 5 prospective randomized controlled trials, 1 prospective phase II trial, and 10 retrospective comparative series (26). Two-year locoregional control rates for IMRT fluctuated between $59\\%$ and $98.7\\%$ . Only 1 of the 16 studies reported the rates on in-field and out-of-field failures and observed more relapses in the high-dose region (5). Our findings are consistent with existing literature, reporting locoregional failures to predominantly occur in high-dose volumes for both the older 3D (27) and more conformal RT techniques (28-43). Compared with previous studies, our analysis provides a larger patient cohort (22, 23, 27, 31, 32, 39, 43) and longer median follow-up time (22-24, 30-32, 35, 39). Leeman et al. reported the recurrence patterns of a large cohort of 1,000 patients and found neither marginal nor out-of-field failures (29). However, heterogeneity in all reported studies renders ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,35,134,44]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bollen et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[727,35,924,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recurrence Patterns After IMRT/VMAT ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[73,949,294,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Oncology | www.frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[495,951,502,960]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[680,949,922,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"September 2021 | Volume 11 | Article 720052 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[73,77,485,840]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"generalization difficult. Firstly, there are differences between patient and treatment cohorts: in terms of primary histology; anatomical sites and stages (8, 19, 28, 29, 33, 34, 36-40); different types of RT intent, i.e., primary curative or adjuvant (8, 28, 34, 38); and different types of intent and uses of chemotherapy (19, 36, 38, 39, 41). Secondly, a number of studies investigate tumor persistence as part of recurrences (19, 28) or do not specify the separation at all (29, 36). The determination of out-of-field failures, defined as failure that occurred outside the treatment field, is fairly straightforward in published reports. The definition of a marginal failure is, however, not as clear-cut (22, 30, 31, 39). In the current study, we defined marginal failure as a situation in which at least one-half of the volume of the recurrence appeared to be outside the original tumor site (CTVp or CTVn) (39). Using this definition, no marginal recurrences were found, providing reassurance that TV delineation, expansion for CTV and PTV, and treatment delivery were performed adequately. By all means, we must exercise caution generalizing our results, since other IMRT series do report the occurrence of marginal failures (8, 22, 24, 27, 31, 39, 43). Furthermore, most published data are coming from single-center cohorts, provided by large-volume centers with significant experience in the treatment of HNC. This should be taken into consideration, since variations in TV delineation and treatment quality are proven to affect LRR rates. Chen et al. evaluated the pattern of RR among 107 patients who presented for consideration of re-irradiation to a large tertiary center. They found $41\\%$ of recurrences to be a marginal miss, while $18\\%$ appeared to be a true miss (8). The higher incidence of true and marginal misses in this study, compared with previously mentioned reports, could be explained by the fact that patients received their initial treatment in several lower-volume RT departments with less experience with IMRT in the HNC population. Therefore, their results might paint a more realistic picture about the recurrence patterns of HNC patients. Indeed, several studies have found a worse OS among patients treated at low-volume RT centers, with incorrect TV delineation and radiotherapy planning as the main contributors to poor outcome (11, 12). Boero et al. showed that among HNC patients treated with IMRT, for every five additional patients treated per provider per year, the risk of mortality decreased by $21\\%$ (10). These findings were backed by the RTOG 0022 study, which noted higher failure rates among patients with major protocol violations in IMRT radiation plans (43). On top of that, several studies have reported a remarkable amount of heterogeneity with respect to TV delineation among RT centers, even when delineation guidelines are available (9, 44-46). The number of marginal or out-of-field recurrences was slightly lower than observed in previous studies, except for the large cohort of MSKCC, a center with a lot of expertise in the treatment of HNC (29). Our definition of marginal failure, which was narrower compared with several other studies (30-32, 38, 39), could also have contributed to the observed results. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,844,485,902]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Nevertheless, providing accurate guideline adherence, TV definition, and treatment delivery, all reported studies confirm the predominance of in-field recurrences after IMRT. True in-field recurrences likely represent more biologically resistant tumors, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,77,922,442]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"which could possibly be explained by, for example, the harboring of an increased proportion of cancer stem cells and/or hypoxic elements (47, 48). Several mechanisms involved in RT response have been described in HNSCC such as hypoxia, the presence of cancer stem cells (CSC), signaling pathways, DNA damage response (DDR), and cell death pathways. It is important to keep in mind that radioresistance cannot be explained by one single mechanism or protein, but rather by an interplay of different mechanisms. Considerable evidence has suggested that tumor hypoxia results in resistance to (C)RT and tumor recurrence (48, 49), setting the stage for dose intensification strategies. A recent review concluded that dose escalation could improve OS without increased toxicity, although follow-up periods were short in small cohorts, which could result in underreporting (50). The authors concluded that functional imaging modalities could help identify the true extent of the tumor and the region that could benefit most from dose escalation, without increased toxicity. In the meantime, results of five randomized controlled trials are awaited, investigating the benefit of RT dose escalation in HNC patients (NCT01212354, NCT03376386, NCT02352792, NCT02031250, NCT03865277). Another interesting track that deserves attention is dose escalation with proton therapy, as its unique characteristics allow better sparing of normal surrounding tissue, and therefore, dose escalation is less restricted by toxicity (NCT03513042). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,445,922,902]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The strength of the current study is that it concerns a large single-institution study, in which all patients were treated in a relatively uniform manner. This way, the confounder of variation in treatment quality is minimized, which allows for the analysis of the true patterns of recurrence. There was a long follow-up period (median 6.4 years) and a large patient cohort. Our study is limited by its retrospective nature and the heterogeneity of the patient cohort. P16 status of oropharyngeal carcinoma (OPC) was unknown for 97 patients. Differences in tumor stage, tumor subsite, and the use of concomitant systemic treatment could possibly affect the pattern of failure. OPC patients are overrepresented in our cohort, which follows the pattern of other large patient cohorts and is likely due to the increasing incidence of OPC. The current study is underpowered for a subgroup analysis. However, our results show a trend toward more LRR for hypopharyngeal, supraglottic, and oral cavity tumors (Table 2), which corresponds with the results of Leeman et al. (29). The subsites were not matched for varying stages of disease, which may affect differences in the observed outcomes. However, these variations may also more accurately reflect the clinical presentation that we deal with on a daily basis. All tumor stages [I-IV; classified according to the American Joint Committee on Cancer seventh TNM edition (17)] were included, which could affect the patterns of failure and increase the heterogeneity of the treatment. However, the primary aim of our study was to investigate the quality of current target volume delineation by analyzing the pattern of treatment failure. Since we do not adapt our CTV and PTV margins according to disease stage, we do not expect a significant impact of tumor stage on the recurrence pattern when assessing the accuracy of our GTV and CTV delineation. Another pitfall of the current study may be the ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,35,134,44]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bollen et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[727,35,924,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recurrence Patterns After IMRT/VMAT ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[73,949,294,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Oncology | www.frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[495,951,502,960]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[680,949,924,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"September 2021 | Volume 11 | Article 720052 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[73,77,484,134]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"cutoff period of 3 years or the cutoff distance of more than $2\\mathrm{cm}$ from the index tumor, to differentiate between a LR and SP. However, this definition was based on several previous publications (30, 51). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,170,210,186]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CONCLUSION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[73,197,485,345]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This large-institution study adds to the evidence of predominant treatment failure inside high-dose radiotherapy volumes, indicating that recurrences are mainly caused by tumor resistance. Our findings reinforce the need to focus on dose intensification and identification of high-risk subvolumes. No marginal recurrences were observed, providing reassurance about accurate TV delineation and the quality of treatment delivery. Caution must be exercised when generalizing these results, since experience with more conformal RT techniques seems a key prerequisite for favorable outcomes in the treatment of HNC patients. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,378,400,394]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DATA AVAILABILITY STATEMENT ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[73,406,484,436]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,480,210,496]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"REFERENCES ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,504,485,894]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. Fitzmaurice C, Allen C, Barber RM, Barregard L, Bhutta ZA, Brenner H, et al. Global, Regional, and National Cancer Incidence, Mortality, Years of Life Lost, Years Lived With Disability, and Disability-Adjusted Life-Years for 32 Cancer Groups, 1990 to 2015. JAMA Oncol (2017) 3:524-48. doi: 10.1001/jamaoncol.2016.5688 \",\"2. Pignon J-P, Maitre A, Maillard E, Bourhis J. Meta-Analysis of Chemotherapy in Head and Neck Cancer (MACH-NC): An Update on 93 Randomised Trials and 17,346 Patients. Radiother Oncol (2009) 92(1):4-14. doi: 10.1016/j.radonc.2009.04.014 \",\"3. Bourhis J, Overgaard J, Audry H, Ang KK, Saunders M, Bernier J, et al. Hyperfractionated or Accelerated Radiotherapy in Head and Neck Cancer: A Meta-Analysis. Lancet (2006) 368(9538):843-54. doi: 10.1016/S0140-6736(06)69121-6 \",\"4. Nuyts S, Dirix P, Clement PMJ, Vander Poorten V, Delaere P, Schoenaers J, et al. Impact of Adding Concomitant Chemotherapy to Hyperfractionated Accelerated Radiotherapy for Advanced Head-And-Neck Squamous Cell Carcinoma. Int J Radiat Oncol (2009) 73(4):1088-95. doi: 10.1016/j.ijrobp.2008.05.042 \",\"5. Nutting CM, Morden JP, Harrington KJ, Urbano TG, Bhide SA, Clark C, et al. Parotid-Sparing Intensity Modulated Versus Conventional Radiotherapy in Head and Neck Cancer (PARSPORT): A Phase 3 Multicenter Randomised Controlled Trial. Lancet Oncol (2011) 12(2):127-36. doi: 10.1016/S1470-2045(10)70290-4 \",\"6. Nutting CM, Morden JP, Beasley M, Bhide S, Cook A, De Winton E, et al. Results of a Multicenter Randomised Controlled Trial of Cochlear-Sparing Intensity-Modulated Radiotherapy Versus Conventional Radiotherapy in Patients With Parotid Cancer (COSTAR; CRUK/08/004). Eur J Cancer (2018) 103:249-58. doi: 10.1016/j.ejca.2018.08.006 \",\"7. Rathod S, Gupta T, Ghosh-Laskar S, Murthy V, Budrukkar A, Agarwal J. Quality-Of-Life (QOL) Outcomes in Patients With Head and Neck Squamous Cell Carcinoma (HNSCC) Treated With Intensity-Modulated Radiation Therapy (IMRT) Compared to Three-Dimensional Conformal \"]","page_idx":"7","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[512,77,715,92]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ETHICS STATEMENT ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,105,922,207]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The studies involving human participants were reviewed and approved by the Ethics Committee Research UZ/KU Leuven. Written informed consent for participation was not required for this study in accordance with the national legislation and the institutional requirements. Written informed consent was obtained from the individual(s) for the publication of any potentially identifiable images or data included in this article. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,242,773,256]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"AUTHOR CONTRIBUTIONS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,270,922,342]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conceptualization: JV and SN. Data curation: JV. Formal analysis: AL. Investigation: JV. Methodology: JV. Supervision: SN. Writing—original draft: JV and HB. Writing—review and editing: HB and SN. All authors contributed to the article and approved the submitted version. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,379,605,394]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"FUNDING ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[512,406,922,433]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"SN is appointed as Senior Clinical Investigator by FWO—Research Foundation Flanders. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,480,924,893]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"Radiotherapy (3D-CRT): Evidence From a Prospective Randomized Study. Oral Oncol (2013) 49(6):634-42. doi: 10.1016/j.oraloncology.2013.02.013 \",\"8. Chen AM, Chin R, Beron P, Yoshizaki T, Mikaeilian AG, Cao M. Inadequate Target Volume Delineation and Local-Regional Recurrence After Intensity-Modulated Radiotherapy for Human Papillomavirus-Positive Oropharynx Cancer. Radiother Oncol (2017) 123(3):412-8. doi: 10.1016/j.radonc.2017.04.015 \",\"9. Hong TS, Tome WA, Harari PM. Heterogeneity in Head and Neck IMRT Target Design and Clinical Practice. Radiother Oncol (2012) 103(1):92-8. doi: 10.1016/j.radonc.2012.02.010 \",\"10. Boero IJ, Paravati AJ, Xu B, Cohen EE, Mell LK, Le QT, et al. Importance of Radiation Oncologist Experience Among Patients With Head-And-Neck Cancer Treated With Intensity-Modulated Radiation Therapy. J Clin Oncol (2016) 34(7):684-90. doi: 10.1200/JCO.2015.63.9898 \",\"11. Peters LJ, O'Sullivan B, Giralt J, Fitzgerald TJ, Trotti A, Bernier J, et al. Critical Impact of Radiotherapy Protocol Compliance and Quality in the Treatment of Advanced Head and Neck Cancer: Results From TROG 02.02. J Clin Oncol (2010) 28(18):2996-3001. doi: 10.1200/JCO.2009.27.4498 \",\"12. Wuthrick EJ, Zhang Q, Machtay M, Rosenthal DI, Nguyen-Tan PF, Fortin A, et al. Institutional Clinical Trial Accrual Volume and Survival of Patients With Head and Neck Cancer. J Clin Oncol (2015) 33(2):156-64. doi: 10.1200/JCO.2014.56.5218 \",\"13. Gregoire V, Levendag P, Ang KK, Bernier J, Braaksma M, Budach V, et al. CT-Based Delineation of Lymph Node Levels and Related CTVs in the Node-Negative Neck: DAHANCA, EORTC, GORTEC, NCIC, RTOG Consensus Guidelines. Radiother Oncol (2003) 69(3):227-36. doi: 10.1016/j.radonc.2003.09.011 \",\"14. Gregoire V, Eisbruch A, Hamoir M, Levendag P. Proposal for the Delineation of the Nodal CTV in the Node-Positive and the Post-Operative Neck. Radiother Oncol (2006) 79(1):15-20. doi: 10.1016/j.radonc.2006.03.009 \",\"15. Gregoire V, Coche E, Cosnard G, Hamoir M, Reychler H. Selection and Delineation of Lymph Node Target Volumes in Head and Neck Conformal Radiotherapy. Proposal for Standardizing Terminology and Procedure Based on the Surgical Experience. Radiother Oncol (2000) 56(2):135-50. doi: 10.1016/s0167-8140(00)00202-4 \"]","page_idx":"7","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[73,35,134,44]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bollen et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[727,35,924,44]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recurrence Patterns After IMRT/VMAT ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[73,949,294,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Oncology | www.frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[495,951,502,960]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"8 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[680,949,922,960]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"September 2021 | Volume 11 | Article 720052 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[73,77,487,893]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"16. Nuyts S, Lambrecht M, Duprez F, Daisne JF, Van Gestel D, Van den Weyngaert D, et al. Reduction of the Dose to the Elective Neck in Head and Neck Squamous Cell Carcinoma, a Randomized Clinical Trial Using Intensity Modulated Radiotherapy (IMRT). Dosimetrical Analysis and Effect on Acute Toxicity. Radiother Oncol (2013) 109:323-9. doi: 10.1016/j.radonc.2013.06.044 \",\"17. Edge SB, Compton CC. The American Joint Committee on Cancer: The 7th Edition of the AJCC Cancer Staging Manual and the Future of TNM. Ann Surg Oncol (2010) 17(6):1471-4. doi: 10.1245/s10434-010-0985-4 \",\"18. Schemper M, Smith TL. A Note on Quantifying Follow-Up in Studies of Failure Time. Control Clin Trials (1996) 17(4):343-6. doi: 10.1016/0197-2456(96)00075-x \",\"19. Bayman E, Prestwich RJD, Speight R, Aspin L, Garratt L, Wilson S, et al. Patterns of Failure After Intensity-Modulated Radiotherapy in Head and Neck Squamous Cell Carcinoma Using Compartmental Clinical Target Volume Delineation. Clin Oncol (2014) 26(10):636-42. doi: 10.1016/j.clon.2014.05.001 \",\"20. Nuyts S, Dirix P, Clement PM, Vander Poorten V, Delaere P, Schoenaers J, et al. Impact of Adding Concomitant Chemotherapy to Hyperfractionated Accelerated Radiotherapy for Advanced Head-and-Neck Squamous Cell Carcinoma. Int J Radiat Oncol Biol Phys (2009) 73(4):1088-95. doi: 10.1016/j.ijrobp.2008.05.042 \",\"21. Vermorken JB, Mesia R, Rivera F, Remenar E, Kawecki A, Rottey S, et al. Platinum-Based Chemotherapy Plus Cetuximab in Head and Neck Cancer. N Engl J Med (2008) 359(11):1116-27. doi: 10.1056/NEJMoa0802656 \",\"22. Chao KS, Ozyigit G, Tran BN, Cengiz M, Dempsey JF, Low DA. Patterns of Failure in Patients Receiving Definitive and Postoperative IMRT for Head-and-Neck Cancer. Int J Radiat Oncol Biol Phys (2003) 55(2):312-21. doi: 10.1016/s0360-3016(02)03940-8 \",\"23. Daly ME, Le QT, Maxim PG, Loo BWJr, Kaplan MJ, Fischbein NJ, et al. Intensity-Modulated Radiotherapy in the Treatment of Oropharyngeal Cancer: Clinical Outcomes and Patterns of Failure. Int J Radiat Oncol Biol Phys (2010) 76(5):1339-46. doi: 10.1016/j.ijrobp.2009.04.006 \",\"24. Thorstad S, Hong A, Hope J, Lindsay PE, Haughey B, Deasey O, et al. Patterns of Failure in Patients Receiving Intensity Modulated Radiation Therapy (IMRT) for Head and Neck Cancer. Int J Radiat Oncol Biol Phys (2005) 63: S74. doi: 10.1016/j.ijrobp.2005.07.127 \",\"25. Biau J, Lapeyre M, Troussier I, Budach W, Giralt J, Grau C, et al. Selection of Lymph Node Target Volumes for Definitive Head and Neck Radiation Therapy: A 2019 Update. Radiother Oncol (2019) 134:1-9. doi: 10.1016/j.radonc.2019.01.018 \",\"26. Gujral DM, Nutting CM. Patterns of Failure, Treatment Outcomes and Late Toxicities of Head and Neck Cancer in the Current Era of IMRT. Oral Oncol (2018) 86:225-33. doi: 10.1016/j.oraloncology.2018.09.011 \",\"27. Oksuz DC, Prestwich RJ, Carey B, Wilson S, Senocak MS, Choudhury A, et al. Recurrence Patterns of Locally Advanced Head and Neck Squamous Cell Carcinoma After 3D Conformal (Chemo)-Radiotherapy. *Radiat Oncol* (2011) 6:54. doi: 10.1186/1748-717X-6-54 \",\"28. De Felice F, Thomas C, Barrington S, Pathmanathan A, Lei M, Urbano TG. Analysis of Loco-Regional Failures in Head and Neck Cancer After Radical Radiation Therapy. Oral Oncol (2015) 51(11):1051-5. doi: 10.1016/j.oraloncology.2015.08.004 \",\"29. Leeman JE, Li JG, Pei X, Venigalla P, Zumsteg ZS, Katsoulakis E, et al. Patterns of Treatment Failure and Postrecurrence Outcomes Among Patients With Locally Advanced Head and Neck Squamous Cell Carcinoma After Chemoradiotherapy Using Modern Radiation Techniques. JAMA Oncol (2017) 3(11):1487-94. doi: 10.1001/jamaoncol.2017.0973 \",\"30. Zukauskaite R, Hansen CR, Brink C, Johansen J, Asmussen JT, Grau C, et al. Analysis of CT-Verified Loco-Regional Recurrences After Definitive IMRT for HNSCC Using Site of Origin Estimation Methods. Acta Oncol (2017) 56 (11):1554-61. doi: 10.1080/0284186X.2017.1346384 \",\"31. Dawson LA, Anzai Y, Marsh L, Martel MK, Paulino A, Ship JA, et al. Patterns of Local-Regional Recurrence Following Parotid-Sparing Conformal and Segmental Intensity-Modulated Radiotherapy for Head and Neck Cancer. Int J Radiat Oncol Biol Phys (2000) 46(5):1117-26. doi: 10.1016/s0360-3016(99)00550-7 \",\"32. Shakam A, Scrimer R, Liu D, Mohamed M, Parliament M, Field GC, et al. Dose-Volume Analysis of Locoregional Recurrences in Head and Neck IMRT, \"]","page_idx":"8","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[510,75,924,893]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"as Determined by Deformable Registration: A Prospective Multi-Institutional Trial. Radiother Oncol (2011) 99(2):101-7. doi: 10.1016/j.radonc.2011.05.008 \",\"33. Caudell JJ, Meredith RF, Spencer SA, Keene KS, Dobelbower MC, Bonner JA. Margin on Gross Tumor Volume and Risk of Local Recurrence in Head-and-Neck Cancer. Int J Radiat Oncol Biol Phys (2010) 76(1):164-8. doi: 10.1016/j.ijrobp.2009.01.037 \",\"34. Ferreira BC, Marques RV, Khouri L, Santos T, Sa-Couto P, do Carmo Lopes M. Assessment and Topographic Characterization of Locoregional Recurrences in Head and Neck Tumors. Radiat Oncol (2015) 10:41. doi: 10.1186/s13014-015-0345-4 \",\"35. Due AK, Vogelius IR, Aznar MC, Bentzen SM, Berthelsen AK, Korreman SS, et al. Recurrences After Intensity Modulated Radiotherapy for Head and Neck Squamous Cell Carcinoma More Likely to Originate From Regions With High Baseline [18F]-FDG Uptake. Radiother Oncol (2014) 111(3):360-5. doi: 10.1016/j.radonc.2014.06.001 \",\"36. van den Bosch S, Dijkstra T, Verhoef LC, Zwijnenburg EM, Janssens GO, Kaanders JH. Patterns of Recurrence in Electively Irradiated Lymph Node Regions After Definitive Accelerated Intensity Modulated Radiation Therapy for Head and Neck Squamous Cell Carcinoma. Int J Radiat Oncol Biol Phys (2016) 94(4):766-74. doi: 10.1016/j.ijrobp.2015.12.002 \",\"37. Dandekar V, Morgan T, Turian J, Fidler MJ, Showel J, Nielsen T, et al. Patterns-Of-Failure After Helical Tomotherapy-Based Chemoradiotherapy for Head and Neck Cancer: Implications for CTV Margin, Elective Nodal Dose and Bilateral Parotid Sparing. Oral Oncol (2014) 50(5):520-6. doi: 10.1016/j.oraloncology.2014.02.009 \",\"38. Nevens D, Duprez F, Daisne JF, Schatteman J, van der Vorst A, De Neve W, et al. Recurrence Patterns After a Decreased Dose of 40Gy to the Elective Treated Neck in Head and Neck Cancer. Radiother Oncol (2017) 123(3):419-23. doi: 10.1016/j.radonc.2017.03.003 \",\"39. Schoenfeld GO, Amdur RJ, Morris CG, Li JG, Hinerman RW, Mendenhall WM. Patterns of Failure and Toxicity After Intensity-Modulated Radiotherapy for Head and Neck Cancer. Int J Radiat Oncol Biol Phys (2008) 71(2):377-85. doi: 10.1016/j.ijrobp.2007.10.010 \",\"40. Studer G, Luetolf UM, Glanzmann C. Locoregional Failure Analysis in Head-and-Neck Cancer Patients Treated With IMRT. Strahlenther Onkol (2007) 183(8):417-23. doi: 10.1007/s00066-007-1663-8 \",\"41. Garden AS, Dong L, Morrison WH, Stugis EM, Glisson BS, Frank SJ, et al. Patterns of Disease Recurrence Following Treatment of Oropharyngeal Cancer With Intensity Modulated Radiation Therapy. Int J Radiat Oncol Biol Phys (2013) 85(4):941-7. doi: 10.1016/j.ijrobp.2012.08.004 \",\"42. Kjems J, Gothelf AB, Hakansson K, Specht L, Kristensen CA, Friborg J. Elective Nodal Irradiation and Patterns of Failure in Head and Neck Cancer After Primary Radiation Therapy. Int J Radiat Oncol Biol Phys (2016) 94 (4):775-82. doi: 10.1016/j.ijrobp.2015.12.380 \",\"43. Eisbruch A, Harris J, Garden AS, Chao CK, Straube W, Harari PM, et al. Multi-Institutional Trial of Accelerated Hypofractionated Intensity-Modulated Radiation Therapy for Early-Stage Oropharyngeal Cancer (RTOG 00-22). Int J Radiat Oncol Biol Phys (2010) 76(5):1333-8. doi: 10.1016/j.ijrobp.2009.04.011 \",\"44. van der Veen J, Gulyban A, Nuyts S. Interobserver Variability in Delineation of Target Volumes in Head and Neck Cancer. Radiother Oncol (2019) 137:9-15. doi: 10.1016/j.radonc.2019.04.006 \",\"45. Pettit L, Hartley A, Bowden SJ, Mehanna H, Glaholm J, Cashmore J, et al. Variation in Volume Definition Between UK Head and Neck Oncologists Treating Oropharyngeal Carcinoma. Clin Oncol (2011) 23:654-5. doi: 10.1016/j.clon.2011.07.006 \",\"46. Hansen CR, Johansen J, Samsøe E, Andersen E, Petersen JBB, Jensen K, et al. Consequences of Introducing Geometric GTV to CTV Margin Expansion in DAHANCA Contouring Guidelines for Head and Neck Radiotherapy. Radiother Oncol (2018) 126:43-7. doi: 10.1016/j.radonc.2017.09.019 \",\"47. Hanns E, Job S, Coliat P, Wasylyk C, Ramolu L, Pencreach E, et al. Human Papillomavirus-Related Tumors of the Oropharynx Display a Lower Tumor Hypoxia Signature. Oral Oncol (2015) 51:848-56. doi: 10.1016/j.oraloncology.2015.06.003 \",\"48. Vlashi E, Chen AM, Boyrie S, Yu G, Nguyen A, Brower PA, et al. Radiation-Induced Dedifferentiation of Head and Neck Cancer Cells Into Cancer Stem Cells Depends on Human Papillomavirus Status. Int J Radiat Oncol Biol Phys (2016) 94(5):1198-206. doi: 10.1016/j.ijrobp.2016.01.005 \"]","page_idx":"8","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[73,35,134,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bollen et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[724,35,924,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recurrence Patterns After IMRT/VMAT ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[72,949,294,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Oncology | www.frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[494,951,502,960]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"9 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[680,949,922,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"September 2021 | Volume 11 | Article 720052 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[72,75,485,184]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"49. Hall EJ. The Oxygen Effects and Re-Oxygenation. In: Radiobiology for the Radiologist. Lippincott Williams & Wilkins (1994). p. 133-52. \",\"50. Atwell D, Elks J, Cahill K, Hearn N, Vignarajah D, Lagopoulos J, et al. A Review of Modern Radiation Therapy Dose Escalation in Locally Advanced Head and Neck Cancer. Clin Oncol (2020) 32(5):330-41. doi: 10.1016/j.clon.2019.12.004 \",\"51. Schwartz LH, Ozsahin M, Zhang GN, Touboul E, De Vataire F, Andolenko P, et al. Synchronous and Metachronous Head and Neck Carcinomas. Cancer (1994) 74(7):1933-8. doi: 10.1002/1097-0142(19941001)74:7<1933::aid-cncr 2820740718>3.0.co;2-x \"]","page_idx":"9","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[73,198,485,234]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conflict of Interest: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,75,922,137]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Publisher's Note: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,151,922,236]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Copyright © 2021 Bollen, van der Veen, Laenen and Nuyts. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,35,134,44]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bollen et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[727,35,924,46]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recurrence Patterns After IMRT/VMAT ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[73,951,294,961]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Oncology | www.frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[492,951,505,960]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[680,949,922,960]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"September 2021 | Volume 11 | Article 720052 ","text_format":null,"text_level":null,"type":"footer"}],"string":"[\n {\n \"bbox\": \"[290,240,847,297]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recurrence Patterns After IMRT/VMAT in Head and Neck Cancer \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[289,314,885,331]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Heleen Bollen<sup>1,2*</sup>, Julie van der Veen<sup>1,2†</sup>, Annouschka Laenen<sup>3</sup> and Sandra Nuyts<sup>1,2</sup> \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[289,342,889,381]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$^{1}$ Laboratory of Experimental Radiotherapy, Department of Oncology, KU Leuven, Leuven, Belgium, $^{2}$ Department of Radiation Oncology, Leuven Cancer Institute, University Hospitals Leuven, Leuven, Belgium, $^{3}$ Leuven Biostatistics and Statistical Bioinformatics Center, KU Leuven, Leuven, Belgium \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[289,401,924,525]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Purpose: Intensity-modulated radiotherapy (IMRT) and volumetric modulated arc therapy (VMAT), two advanced modes of high-precision radiotherapy (RT), have become standard of care in the treatment of head and neck cancer. The development in RT techniques has markedly increased the complexity of target volume definition and accurate treatment delivery. The aim of this study was to indirectly investigate the quality of current TV delineation and RT delivery by analyzing the patterns of treatment failure for head and neck cancer patients in our high-volume RT center. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[289,531,924,582]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Methods: Between 2004 and 2014, 385 patients with pharyngeal, laryngeal, and oral cavity tumors were curatively treated with primary RT (IMRT/VMAT). We retrospectively investigated locoregional recurrences (LRR), distant metastases (DM), and overall survival (OS). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[289,590,924,767]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Results: Median follow-up was 6.4 years (IQR 4.7-8.3 years) during which time 122 patients $(31.7\\\\%)$ developed LRR $(22.1\\\\%)$ and DM $(17.7\\\\%)$ . The estimated 2- and 5-year locoregional control was $78.2\\\\%$ $(95\\\\%$ CI 73.3, 82.3) and $74.2\\\\%$ $(95\\\\%$ CI 69.0, 78.8). One patient developed a local recurrence outside the high-dose volume and five patients developed a regional recurrence outside the high-dose volume. Four patients $(1.0\\\\%)$ suffered a recurrence in the electively irradiated neck and two patients had a recurrence outside the electively irradiated neck. No marginal failures were observed. The estimated 2- and 5-year DM-free survival rates were $83.3\\\\%$ $(95\\\\%$ CI 78.9, 86.9) and $80.0\\\\%$ $(95\\\\%$ CI 75.2, 84.0). The estimated 2- and 5-year OS rates were $73.6\\\\%$ $(95\\\\%$ CI 68.9, 77.8) and $52.6\\\\%$ $(95\\\\%$ CI 47.3, 57.6). Median OS was 5.5 years $(95\\\\%$ CI 4.5, 6.7). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[289,774,924,879]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conclusion: Target volume definition and treatment delivery were performed accurately, as only few recurrences occurred outside the high-dose regions and no marginal failures were observed. Research on dose intensification and identification of high-risk subvolumes might decrease the risk of locoregional relapses. The results of this study may serve as reference data for comparison with future studies, such as dose escalation or proton therapy trials. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[289,890,924,916]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords: radiotherapy, intensity-modulated radiotherapy, volumetric modulated arc therapy, recurrence, head and neck cancer, proton therapy, tumor resistance, delineation \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[142,462,267,476]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n 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\"text\": \"Specialty section: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[119,727,267,738]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This article was submitted to \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[161,740,267,752]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Radiation Oncology, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": 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\"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Received: 03 June 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[121,797,267,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Accepted: 30 August 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[99,810,267,820]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Published: 16 September 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[216,827,267,838]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Citation: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,840,267,852]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bollen H, van der Veen J, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,854,267,865]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Laenen A and Nuyts S (2021) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[99,866,267,876]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recurrence Patterns After IMRT/ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[99,879,267,889]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n 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Oncol. 11:720052. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[102,903,267,915]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"doi: 10.3389/fonc.2021.720052 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[72,30,105,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"# \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[107,42,203,57]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"frontiers \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[107,60,235,75]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"in Oncology \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[794,43,922,52]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ORIGINAL RESEARCH \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[764,53,922,64]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"published: 16 September 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[761,64,922,74]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"doi: 10.3389/fonc.2021.720052 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[873,92,922,129]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[72,949,295,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Oncology | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[492,951,502,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[680,949,924,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"September 2021 | Volume 11 | Article 720052 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[73,75,231,92]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"INTRODUCTION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,105,485,636]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Head and neck cancer (HNC) is the seventh most common cancer worldwide and is usually diagnosed in a locally advanced but curable stage (1). As surgical resection can be mutilating, radiation therapy (RT), with or without concurrent chemotherapy, has emerged as the treatment of choice in the management of local and locoregionally advanced HNC (2-4). Intensity-modulated radiotherapy (IMRT) and volumetric modulated arc therapy (VMAT) are two radiation techniques that enable steeper dose gradients, allowing better sparing of the surrounding structures compared with the older 3D techniques, thereby reducing toxicity (5) and improving quality of life (6, 7). The trade-off for these more conformal RT techniques is an increased reliance on precise TV definition and accurate treatment delivery. Tumor tissue that is not defined as target volume (TV) by the radiation oncologist will not receive the prescribed dose, and geographical misses, leading to locoregional recurrences, are a potential risk (8, 9). In addition, several studies have proven experience with more conformal RT techniques to be essential for optimal outcomes in HNC (10-12). IMRT was routinely implemented in University Hospitals Leuven for the definitive treatment of HNC in 2004. Since 2010, VMAT has become the standard of care. In preparation for the implementation of proton therapy in our RT center, we investigated the quality of our current TV delineation and RT delivery by retrospectively analyzing the incidence and location of local recurrence (LR) and regional recurrence (RR) compared with the TVs. Knowledge about treatment failure patterns is especially relevant when implementing more conformal RT techniques, such as proton therapy. In the treatment of HNC, efforts are continuously being made to optimize the therapeutic ratio to improve disease outcome while keeping toxicity to a minimum. By analyzing the patterns of failure in the HNC population, we hope to refine future strategies in TV delineation and dose escalation. Moreover, the results of this study may serve as reference data for comparison with future studies, such as dose escalation or proton therapy trials. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,664,349,682]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"MATERIALS AND METHODS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,693,238,709]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Patient Selection \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,709,485,916]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this retrospective analysis, patients treated with curative intent for a HNC with R(C)T between June 2004 and December 2014 were included, to allow a follow-up of at least 3 years. We included patients with primary pharyngeal, oral cavity, and laryngeal squamous cell carcinoma. We excluded patients previously treated with RT in the head and neck region, patients with metastatic disease, postoperative patients or patients who received induction chemotherapy, primary sinonasal or nasopharyngeal tumor patients, and patients who did not finish RT as planned. The medical files were reviewed for each patient. The study was approved by the ethical committee of University Hospitals Leuven/KU Leuven (S59803). All methods were performed in accordance with the relevant guidelines and regulations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,75,808,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Target Volume Delineation and Treatment Planning \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,110,922,492]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The gross tumor volume (GTV) was defined as the macroscopic tumor volume seen on planning CT and using information from clinical investigation and diagnostic and functional imaging. Patients with locally advanced disease underwent FDG-PET scan as staging exam, and PET scans were used \\\"side-by-side\\\" during the process of contouring with the diagnostic CT and/or MR scan. The clinical target volume of the primary tumor (CTVp) and adenopathies (CTVn) were created with a 3D $10\\\\mathrm{-mm}$ expansion around the GTV and cropped for anatomical boundaries, e.g., uninvolved bone and air. Neck regions at risk of harboring microscopic tumor cells were delineated using international guidelines (13-15) to create the elective CTV (CTVe) which received a lower dose than CTVp and CTVn. To ensure an adequate coverage of CTV, a planning target volume (PTV) was created by expanding CTV by $5\\\\mathrm{mm}$ . The clinical target volumes of the macroscopically affected tumor sites (CTVp and CTVn) were treated up to a normalized iso-effective dose in 2 Gy fractions (NID2 Gy) of $70\\\\mathrm{Gy}$ . CTVe was treated up to a NID2 Gy of $50\\\\mathrm{Gy}$ , except for 19 patients that were included in a dose deescalation trial and received a lower dose to CTVe up to a NID2 Gy of $40\\\\mathrm{Gy}$ (16). Concurrent chemotherapy (cisplatin $100\\\\mathrm{mg/m^2}$ , q3w) was offered to fit patients with advanced stage disease, according to hospital guidelines. Neck dissection post-RT was not routinely performed. Treatment plans were planned using IMRT/VMAT and tumors were classified according the American Joint Committee on Cancer seventh TNM edition (17). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,501,796,534]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recurrence Identification and Patient Evaluation \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,535,922,740]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recurrence and OS rates were measured starting at the start of RT until recurrence or death from any cause. CT or MRI images of the first recurrence (local/regional) were visually inspected and compared with the planning CT. Recurrences were defined as either local (LR), regional (RR), or distant metastases (DM) and further as 1) in CTVp or n, 2) marginal to CTVp or CTVn (overlap but also more than $50\\\\%$ of tumor load outside the original tumor site), 3) outside CTVp or CTVn, 4) outside CTVn but inside CTVe, and (5) outside CTVe. If the primary tumor or adenopathy was still visible on imaging 6 months after the start of treatment, this was classified as persistent disease (PD). Second primary (SP) HNCs were classified as such if the new tumor was more than $2\\\\mathrm{cm}$ from the index tumor, or if it was less than $2\\\\mathrm{cm}$ from the index tumor, but developed more than 3 years after RT. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,741,922,858]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Patients were seen 2 months after the end of therapy for a clinical evaluation which was repeated every 2 months for the first year. A CT or MRI scan was performed 4 months after the end of therapy to evaluate treatment response and once more during the first year of follow-up. Thereafter, imaging was only done in case of clinical suspicion of a recurrence. During the second, third, and fourth year, clinical follow-up was planned every 3, 4, and 6 months, respectively. Thereafter, a yearly review was planned. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,870,692,887]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Statistical Analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,888,922,917]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The patients were followed up from the date of start of RT to either date of death or the cutoff date April 2018. Locoregional \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,35,134,44]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bollen et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[727,35,924,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recurrence Patterns After IMRT/VMAT \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[73,949,294,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Oncology | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[495,951,502,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[680,949,924,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"September 2021 | Volume 11 | Article 720052 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[73,77,485,238]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"recurrence rates (LRR) and overall survival (OS) were estimated using the Kaplan-Meier method. The Cox proportional hazards model was used for analyzing the prognostic effect of patient or disease characteristics on oncological outcomes. Results are reported as hazard ratios with $95\\\\%$ confidence intervals. Univariate analysis was performed for several potential prognostic factors: age, sex, smoking status, stage, site, and tumor grade. Follow-up summary statistics were obtained using the Kaplan-Meier estimate of potential follow-up (18). Analyses were performed using SAS software (version 9.4 of the SAS System for Windows). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,77,606,92]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RESULTS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,103,732,118]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Patient Characteristics \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,121,922,238]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Patient characteristics of the 385 patients are shown in Table 1. Median age was 61 years old (range 34-89 years) and the majority were men (326 vs. 59 women). Seventeen patients had a multifocal tumor and two patients had an unknown primary. The primary tumor sites were the oropharynx $(46.2\\\\%)$ , hypopharynx $(23.9\\\\%)$ , supraglottis $(20.8\\\\%)$ , larynx $(10.6\\\\%)$ , and oral cavity $(3.9\\\\%)$ . Of the 178 oropharynx tumors, 39 were p16 positive, 42 were p16 negative, and 97 were of unknown \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[72,297,924,892]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0259/c471d26b977442bf2ee8b8114e98e9e2acd9e9d1fba7a28c2408b09a4f263228.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Patient characteristics</td><td></td><td>n = 385</td><td>%</td></tr><tr><td rowspan=\\\"2\\\">Gender</td><td>Male</td><td>326</td><td>84.7%</td></tr><tr><td>Female</td><td>59</td><td>15.3%</td></tr><tr><td rowspan=\\\"4\\\">Age (years), median (range)</td><td></td><td>61</td><td>(34-89)</td></tr><tr><td><60</td><td>160</td><td>41.6%</td></tr><tr><td>60-70</td><td>158</td><td>41.0%</td></tr><tr><td>>70</td><td>67</td><td>17.4%</td></tr><tr><td rowspan=\\\"8\\\">Subsite</td><td>Oropharynx</td><td>178</td><td>46.2%</td></tr><tr><td>Oral cavity</td><td>15</td><td>3.9%</td></tr><tr><td>Hypopharynx</td><td>92</td><td>23.9%</td></tr><tr><td>Larynx</td><td>41</td><td>10.6%</td></tr><tr><td>Supraglottis</td><td>80</td><td>20.8%</td></tr><tr><td>CUP</td><td>2</td><td>0.5%</td></tr><tr><td>Multifocal</td><td>17</td><td>4.4%</td></tr><tr><td>Unifocal</td><td>368</td><td>95.6%</td></tr><tr><td rowspan=\\\"4\\\">Grade</td><td>1</td><td>19</td><td>4.9%</td></tr><tr><td>2</td><td>106</td><td>27.5%</td></tr><tr><td>3</td><td>69</td><td>17.9%</td></tr><tr><td>Unknown</td><td>191</td><td>49.6%</td></tr><tr><td rowspan=\\\"5\\\">T stage</td><td>1</td><td>31</td><td>8.1%</td></tr><tr><td>2</td><td>120</td><td>31.2%</td></tr><tr><td>3</td><td>107</td><td>27.8%</td></tr><tr><td>4</td><td>125</td><td>32.5%</td></tr><tr><td>Unknown</td><td>2</td><td>0.5%</td></tr><tr><td rowspan=\\\"4\\\">N stage</td><td>0</td><td>116</td><td>30.1%</td></tr><tr><td>1</td><td>34</td><td>8.8%</td></tr><tr><td>2</td><td>222</td><td>57.7%</td></tr><tr><td>3</td><td>13</td><td>3.4%</td></tr><tr><td rowspan=\\\"4\\\">Stage</td><td>1</td><td>13</td><td>3.4%</td></tr><tr><td>2</td><td>51</td><td>13.2%</td></tr><tr><td>3</td><td>65</td><td>16.9%</td></tr><tr><td>4</td><td>256</td><td>66.5%</td></tr><tr><td rowspan=\\\"3\\\">P16 status in oropharyngeal tumors (n = 178)</td><td>Negative</td><td>42</td><td>23.6%</td></tr><tr><td>Positive</td><td>39</td><td>21.9%</td></tr><tr><td>Unknown</td><td>97</td><td>54.5%</td></tr><tr><td rowspan=\\\"3\\\">Smoking</td><td>Never</td><td>38</td><td>9.9%</td></tr><tr><td>Former</td><td>92</td><td>23.9%</td></tr><tr><td>Current</td><td>255</td><td>66.2%</td></tr><tr><td rowspan=\\\"3\\\">Ethyl</td><td>Never</td><td>154</td><td>40.0%</td></tr><tr><td>Former</td><td>85</td><td>22.1%</td></tr><tr><td>Current</td><td>146</td><td>37.9%</td></tr><tr><td rowspan=\\\"3\\\">Concomitant systemic therapy</td><td>None</td><td>132</td><td>34.3%</td></tr><tr><td>Chemotherapy</td><td>217</td><td>56.4%</td></tr><tr><td>EGFR inhibitor</td><td>36</td><td>9.4%</td></tr><tr><td rowspan=\\\"3\\\">RT dose, median (range)</td><td></td><td>72 Gy</td><td>(66-75 Gy)</td></tr><tr><td>66 Gy</td><td>16</td><td>4.2%</td></tr><tr><td>66.1-69.99 Gy</td><td>7</td><td>1.8%</td></tr></table>\",\n \"table_caption\": \"[\\\"TABLE 1 | Patient characteristics. \\\"]\",\n \"table_footnote\": \"[\\\"CUP, cancer of unknown primary; EGFR, epidermal growth factor. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[73,35,134,44]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bollen et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[727,35,924,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recurrence Patterns After IMRT/VMAT \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[73,951,294,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Oncology | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[495,951,502,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[680,949,922,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"September 2021 | Volume 11 | Article 720052 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[73,77,485,193]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"status. Stage IV tumors were most common (66.5%), followed by stage III (16.9%), stage II (13.2%), and stage I (3.4%). More than half of patients were treated with concurrent chemotherapy (56.4%) and 36 were treated with an EGFR inhibitor (9.4%). Three hundred twenty-six patients were treated with accelerated RT (72 Gy in 6 weeks). Twenty-one patients were treated with adaptive RT as part of a trial and 19 patients received a lower dose to CTVe (40 Gy) in a dose de-escalation trial (16). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,207,149,220]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Survival \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,223,485,370]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The median follow-up period was 6.4 years (IQR 4.7-8.3 years). The estimated 2- and 5-year OS rates were $73.6\\\\%$ (95% CI 68.9, 77.8) and $52.56\\\\%$ (95% CI 47.3, 57.6). Median OS was 5.5 years (95% CI 4.5, 6.7). The type of recurrence had a significant impact on OS; 96 of 237 patients $(40.5\\\\%)$ with no recurrence died during follow-up vs. 44 of 60 patients $(73.3\\\\%, \\\\text{RR} 1.7, 95\\\\% \\\\text{CI} 1.4, 2.1)$ with LRR as first recurrence and 58 of 62 patients $(93.5\\\\%, \\\\text{RR} 2.3, 95\\\\% \\\\text{CI} 2.0, 2.7)$ with distant metastases. Regarding the development of a second primary in the head and neck, this had no significant impact on OS (10 of 26 patients died, $38.5\\\\%$ , RR 0.9, $95\\\\% \\\\text{CI} 0.6-1.6$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,385,273,399]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recurrence Patterns \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,401,485,474]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"One hundred twenty-two patients suffered a recurrence, of which the LR, RR, and DM distribution is shown in Figure 1 for first recurrence only. Among the patients with recurrence, the median time to failure was 9.6 months (range 3.3 months to 5.5 years). Fifty-four patients $(14.0\\\\%)$ had LR, 49 patients $(12.7\\\\%)$ had RR, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,77,922,179]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and 62 patients $(16.1\\\\%)$ developed DM as first site of recurrence. The estimated 2- and 5-year locoregional control was $78.2\\\\%$ (95% CI 73.3, 82.3) and $74.2\\\\%$ (95% CI 69.0, 78.8). On univariate analysis, a history of ethyl abuse, a higher tumor stage, and a higher tumor grade were significantly associated with more LRR (Table 2). The estimated 2- and 5-year DM-free survival rates were $83.3\\\\%$ (95% CI 78.9, 86.9) and $80.0\\\\%$ (95% CI 75.2, 84.0). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,192,640,207]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"T-Site Failure \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,207,922,414]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There were 53 LRs in the high-dose volume (CTVp), no marginal recurrences, and one isolated LR outside CTVp. The latter concerned a 46-year-old female patient with two synchronous tumors: a T2 retromolar trigone tumor and a T4 tumor in the vallecula (Figure 2A) with multiple lymph nodes (N2b). She was treated with accelerated RT to $72\\\\mathrm{Gy}$ concurrently with cisplatin. Ten months after the start of RT, there was a recurrence in the prelaryngeal space (Figure 2B) approximately $1\\\\mathrm{cm}$ caudal of the primary tumor in the vallecula, for which she underwent a total laryngectomy. Six months later, she developed a new recurrence next to the tracheostoma for which she had re-irradiation $(16\\\\times$ $3.125\\\\mathrm{Gy})$ . Half a year later, there was again local progression in combination with distant metastases. Palliative chemotherapy was started, but the patient died 6 months later. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,428,643,442]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"N-Site Failure \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,446,922,474]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Of the 49 patients with RRs, 44 had recurrences inside the high-dose volume (CTVn). Four patients developed recurrences in CTVe \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[248,517,749,843]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIGURE 1 | Site of first recurrence. Number of patients with a recurrence in the different sites. Overlapping circles show combination possibilities. There were one isolated local recurrence outside CTVp and one isolated regional recurrence outside CTVe. The numbers in the shaded area represent persistent tumors after the end of treatment, included in the total number of recurrences, e.g., 10 of 16 RR were persistent after treatment. CTVe, elective clinical target volume; DM, distant metastases; HNC, head and neck cancer; LR, local recurrence; RR, regional recurrence; SP, second primary. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0776/ee0496359017ac9f822cb22c66fbef5625ea5b1bb6f21c892601aa84b9a3f825.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[73,35,134,44]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bollen et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[727,35,924,44]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recurrence Patterns After IMRT/VMAT \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[73,949,294,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Oncology | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[495,951,502,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[680,949,922,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"September 2021 | Volume 11 | Article 720052 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[70,103,927,347]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0253/6e5c44270769b9857463f62f3ab4f8af5f85190fe4dc028e11264348d58fa6dc.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Variable</td><td>Test</td><td>Hazard ratio (95% CI)</td><td>P-value</td></tr><tr><td>Sex</td><td>Female vs. male</td><td>0.928 (0.514, 1.675)</td><td>0.8039</td></tr><tr><td>Age</td><td>+1 year</td><td>1.010 (0.987, 1.033)</td><td>0.3938</td></tr><tr><td>Smoking</td><td>Global test</td><td></td><td>0.3281</td></tr><tr><td></td><td>Current smoker vs. stopped >6 m</td><td>1.462 (0.843, 2.536)</td><td>0.1764</td></tr><tr><td></td><td>Current smoker vs. never</td><td>1.370 (0.655, 2.863)</td><td>0.4029</td></tr><tr><td></td><td>Stopped >6 m vs. never</td><td>0.937 (0.401, 2.189)</td><td>0.8805</td></tr><tr><td>Ethyl</td><td>Global test</td><td></td><td>0.0064</td></tr><tr><td></td><td>Stopped vs. yes</td><td>2.215 (1.290, 3.801)</td><td>0.0039</td></tr><tr><td></td><td>Stopped vs. none</td><td>1.997 (1.195, 3.340)</td><td>0.0083</td></tr><tr><td></td><td>Yes vs. none</td><td>0.902 (0.537, 1.513)</td><td>0.6956</td></tr><tr><td>Grade</td><td>+1 level</td><td>0.574 (0.369, 0.892)</td><td>0.0136</td></tr><tr><td>Highest TNM stagea</td><td>+1 level</td><td>1.935 (1.343, 2.788)</td><td>0.0004</td></tr><tr><td>Oral</td><td>Yes vs. no</td><td>1.888 (0.764, 4.664)</td><td>0.1683</td></tr><tr><td>Oropharynx</td><td>Yes vs. no</td><td>0.734 (0.475, 1.133)</td><td>0.1621</td></tr><tr><td>Glottis</td><td>Yes vs. no</td><td>0.939 (0.485, 1.817)</td><td>0.8513</td></tr><tr><td>Hypopharynx</td><td>Yes vs. no</td><td>1.435 (0.889, 2.316)</td><td>0.1396</td></tr><tr><td>Supraglottis</td><td>Yes vs. no</td><td>1.428 (0.885, 2.305)</td><td>0.1442</td></tr></table>\",\n \"table_caption\": \"[\\\"TABLE 2 | Univariate analysis for locoregional recurrence. \\\"]\",\n \"table_footnote\": \"[\\\"Continuous variables: $HR > (< )$ 1 means higher (lower) risk for increasing level. Categorical variables: pairwise tests only presented if significant global P-value. Binary variables/pairwise tests: $R > (< )$ 1 means higher (lower) risk for first category. \\\",\\\"CI, confidence interval; m, months; TNM, tumor, node, metastasis stage. \\\",\\\"${}^{a}$ In case patients had multiple tumors,the tumor with the highest TNM stage was used. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[70,409,487,526]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$(1.0\\\\%)$ , of which one also had a recurrence outside CTVe in ipsilateral level V 5 years after RT. Recurrence in level II and the retropharyngeal neck level (VIIa) were the most common (level II: one ipsilateral, three contralateral; level VIIa: two ipsilateral, one contralateral). There were two RRs in level Ib (one ipsilateral and one contralateral). One patient had an isolated RR outside the irradiated volume, in the ipsilateral retropharyngeal neck 4 months after RT (Table 3). There were no marginal recurrences. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,541,280,559]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Second Primary HNC \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,559,485,589]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Twenty-six patients developed SP in the head and neck region. In 8 patients, the SP developed less than $2\\\\mathrm{cm}$ from the index tumor more \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,409,924,468]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"than 3 years after treatment. In 3 patients, the SP was diagnosed less than 3 years after treatment of the index tumor but was more than $2\\\\mathrm{cm}$ from the index tumor, and in 15 patients, the SP occurred more than 3 years later and more than $2\\\\mathrm{cm}$ from the index tumor. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,500,640,517]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DISCUSSION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,528,926,589]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Thirty percent of HNC patients will develop a locoregional relapse, while therapy failure due to metastases is less common (19, 20). The prognosis for HNC patients after failure of first-line therapy is poor, with a median overall survival of less than 1 year (21). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[208,631,495,861]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0548/200e3878d473c5738f7bb9c15add4d171a5621413a0798352141a1b7267b7fa8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[502,630,786,861]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIGURE 2 | Axial CT scan images of the only patient with a local recurrence outside of the high-dose volume. (A) The primary tumor originating in the vallecula and (B) CT scan of the local recurrence in the prelaryngeal space, approximately $1\\\\\\\\mathrm{cm}$ caudal of the index tumor. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0719/19f1c325977ad9b14161140fd0c2537a2ce3e18f39c5802efd75b77574246cc0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[73,35,134,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bollen et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[726,35,924,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recurrence Patterns After IMRT/VMAT \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[72,949,295,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Oncology | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[492,949,504,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[680,949,924,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"September 2021 | Volume 11 | Article 720052 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[70,93,927,347]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0493/632f72b3d49510ac222101ecbfd9aa36333f017381cfbee0cd8aefc24e519f50.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>No.</td><td>Sex, age, smoking</td><td>Subsite</td><td>Stage</td><td>Treatment</td><td>Failure type</td><td>Time to recurrence</td><td>Elective levels irradiated</td><td>Regional recurrence</td></tr><tr><td rowspan=\\\"4\\\">1</td><td rowspan=\\\"4\\\">Male, 46 y, former, multifocal</td><td>Piriform sinus (R)</td><td>Tis</td><td rowspan=\\\"4\\\">70 Gy (50 Gy) + cisplatin</td><td rowspan=\\\"4\\\">N+M</td><td rowspan=\\\"4\\\">5 years and 2 months</td><td rowspan=\\\"4\\\">Ipsi (R): Ib, II-IVa+b, Vc, VI, VIIa+b; Contra (L): II-IVa+b, Vc, VI, VIIa+b</td><td rowspan=\\\"4\\\">Ipsi (R): Ib, II, V, VIIa Contra (L): Ib, II, VIIa</td></tr><tr><td>Supraglottis (L)</td><td>T3N0</td></tr><tr><td>Esophagus (mid)</td><td>T2</td></tr><tr><td>Oropharynx (R)</td><td>Tis</td></tr><tr><td rowspan=\\\"3\\\">2</td><td rowspan=\\\"3\\\">Male, 63 y, current, multifocal</td><td>Floor of mouth (L)</td><td>T2</td><td rowspan=\\\"3\\\">72 Gy (46.4 Gy) + cisplatin</td><td rowspan=\\\"3\\\">T+N+M</td><td rowspan=\\\"3\\\">1 year and 6 months</td><td rowspan=\\\"3\\\">Ipsi (R): Ib, II-IVa, V, VIIb Contra (L): Ib, II-IVa</td><td rowspan=\\\"3\\\">Contra (L): II-III</td></tr><tr><td>T2N2b (R)</td><td>T1</td></tr><tr><td>Supraglottis 4. Glottis</td><td>T1</td></tr><tr><td>3</td><td>Male, 67 y, former</td><td>Piriform sinus</td><td>T3N2b</td><td>72 Gy (40 Gy) + cisplatin</td><td>N+M</td><td>4 months</td><td>Ipsi: Ib, II-IVa, V, Vla+b, VIIa+b Contra: II-IVa, Vla+b, VIIa</td><td>Contra: II</td></tr><tr><td>4</td><td>Male, 68 y, current</td><td>Piriform sinus</td><td>T4aN2b</td><td>72 Gy (40 Gy) + cisplatin</td><td>T+N</td><td>7 months</td><td>Ipsi: II-IVa, V, VIIa+b Contra: II-III</td><td>Ipsi: VIIa-b</td></tr><tr><td>5</td><td>Female, 75 y, former</td><td>Piriform sinus</td><td>T2N2b</td><td>72 Gy (40 Gy)</td><td>N</td><td>4 months</td><td>Ipsi: II-IVa, V Contra: II-III</td><td>Ipsi: VIIa</td></tr></table>\",\n \"table_caption\": \"[\\\"TABLE 3 | Demographics of patients with regional recurrence outside CTVn. \\\"]\",\n \"table_footnote\": \"[\\\"Four patients had a recurrence in CTVe, of which one (patient 1) also had a recurrence outside the irradiated volume (ipsilateral level V). One patient had an isolated regional recurrence outside the elective target volume (patient 5, ipsilateral retropharyngeal adenopathy). Dose in brackets shows dose to CTVe. TNM classification according to TNM7. Contra, contralateral; CTVe, elective clinical target volume; Ipsi, ipsilateral; L, left; N, nodal classification; R, right; T, tumor classification; y, years; M, metastasis. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[73,414,485,546]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In order to guide future attempts to improve the therapeutic ratio and outcomes for HNC patients, understanding of the patterns of treatment failure is essential. Particularly with the introduction of IMRT and VMAT, more conformal RT techniques allowing the prescribed radiation dose to be delivered precisely to the TV, concerns about an increased risk for marginal misses were raised. Indeed, the trade-off for these more conformal RT techniques is an increased reliance on precise TV definition and accurate treatment delivery. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,546,485,915]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The present study reports the patterns of recurrence after RT in 385 HNC patients treated between 2004 and 2014 with IMRT/VMAT at the University Hospitals Leuven. Thirty-one percent of patients suffered a LRR, which corresponds with previously reported recurrence rates in HNC patients treated with definitive radio (chemo)therapy (19, 20). History of ethyl abuse and a higher TNM stage were associated with more LRR. Only one patient developed a LR outside the CTVp. As for RR, five patients suffered a recurrence outside CTVn, of which two recurrences were located outside CTVe. There were no marginal recurrences, either local or regional. Demographics of patients with a RR outside the high-dose volume are summarized in Table 3. Of the five regional relapses seen in our patient population, all were originally LAHNSCC and underwent PET-CT scan. The PET scans were reviewed and no missed nodes were identified. In one patient, relapse occurred in ipsilateral level V, more than 5 years after RT. It is important to note that this patient had a multifocal tumor which makes elective level selection more complicated and comes with an increased RR risk. The other patient developed a RR 4 months after the start of RT in ipsilateral level VIIa. This patient suffered from a N2b (levels III and IV) hypopharynx tumor and level VIIa was not included in the CTVe. Looking at the planning CT retrospectively, it is possible that there was a lymph node initially, although it was not withheld on FDG-PET/CT. Several trials reported a higher incidence of LRR in the lower neck (22-24). In our study, no marginal recurrences were \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,413,922,649]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"observed, which provides reassurance about treatment quality and stresses the importance of guideline adherence for accurate neck level selection (13-15, 25). Nineteen patients were simultaneously included in a dose de-escalation trial, investigating the patterns of regional recurrences in the electively irradiated lymph node regions after dose de-escalation to $40\\\\mathrm{Gy}$ (EQD2 Gy) (16). The inclusion criteria of this study were previously untreated, histologically proven squamous cell carcinoma of the oral cavity, oropharynx, hypopharynx, or larynx, or cervical lymph node metastases of unknown primary cancer. All macroscopically affected tumor sites (both primary tumor and affected lymph nodes) were treated up to an EQD2 Gy of $70\\\\mathrm{Gy}$ . All 44 patients that suffered a recurrence inside CTVn were thus treated up to an EDQ2 of $70\\\\mathrm{Gy}$ . Of the four patients that developed a RR in CTVe, none were included in the de-escalation trial, and thus, all four patients received the standard elective dose, with an EQD2 up to $50\\\\mathrm{Gy}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,650,922,915]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The vast majority of relapses, both local and regional, occurred in-field. A number of trials aimed to analyze the patterns of treatment failures after (IM)RT in HNC patients. Gujral and Nutting reviewed the data from 5 prospective randomized controlled trials, 1 prospective phase II trial, and 10 retrospective comparative series (26). Two-year locoregional control rates for IMRT fluctuated between $59\\\\%$ and $98.7\\\\%$ . Only 1 of the 16 studies reported the rates on in-field and out-of-field failures and observed more relapses in the high-dose region (5). Our findings are consistent with existing literature, reporting locoregional failures to predominantly occur in high-dose volumes for both the older 3D (27) and more conformal RT techniques (28-43). Compared with previous studies, our analysis provides a larger patient cohort (22, 23, 27, 31, 32, 39, 43) and longer median follow-up time (22-24, 30-32, 35, 39). Leeman et al. reported the recurrence patterns of a large cohort of 1,000 patients and found neither marginal nor out-of-field failures (29). However, heterogeneity in all reported studies renders \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,35,134,44]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bollen et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[727,35,924,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recurrence Patterns After IMRT/VMAT \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[73,949,294,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Oncology | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[495,951,502,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[680,949,922,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"September 2021 | Volume 11 | Article 720052 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[73,77,485,840]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"generalization difficult. Firstly, there are differences between patient and treatment cohorts: in terms of primary histology; anatomical sites and stages (8, 19, 28, 29, 33, 34, 36-40); different types of RT intent, i.e., primary curative or adjuvant (8, 28, 34, 38); and different types of intent and uses of chemotherapy (19, 36, 38, 39, 41). Secondly, a number of studies investigate tumor persistence as part of recurrences (19, 28) or do not specify the separation at all (29, 36). The determination of out-of-field failures, defined as failure that occurred outside the treatment field, is fairly straightforward in published reports. The definition of a marginal failure is, however, not as clear-cut (22, 30, 31, 39). In the current study, we defined marginal failure as a situation in which at least one-half of the volume of the recurrence appeared to be outside the original tumor site (CTVp or CTVn) (39). Using this definition, no marginal recurrences were found, providing reassurance that TV delineation, expansion for CTV and PTV, and treatment delivery were performed adequately. By all means, we must exercise caution generalizing our results, since other IMRT series do report the occurrence of marginal failures (8, 22, 24, 27, 31, 39, 43). Furthermore, most published data are coming from single-center cohorts, provided by large-volume centers with significant experience in the treatment of HNC. This should be taken into consideration, since variations in TV delineation and treatment quality are proven to affect LRR rates. Chen et al. evaluated the pattern of RR among 107 patients who presented for consideration of re-irradiation to a large tertiary center. They found $41\\\\%$ of recurrences to be a marginal miss, while $18\\\\%$ appeared to be a true miss (8). The higher incidence of true and marginal misses in this study, compared with previously mentioned reports, could be explained by the fact that patients received their initial treatment in several lower-volume RT departments with less experience with IMRT in the HNC population. Therefore, their results might paint a more realistic picture about the recurrence patterns of HNC patients. Indeed, several studies have found a worse OS among patients treated at low-volume RT centers, with incorrect TV delineation and radiotherapy planning as the main contributors to poor outcome (11, 12). Boero et al. showed that among HNC patients treated with IMRT, for every five additional patients treated per provider per year, the risk of mortality decreased by $21\\\\%$ (10). These findings were backed by the RTOG 0022 study, which noted higher failure rates among patients with major protocol violations in IMRT radiation plans (43). On top of that, several studies have reported a remarkable amount of heterogeneity with respect to TV delineation among RT centers, even when delineation guidelines are available (9, 44-46). The number of marginal or out-of-field recurrences was slightly lower than observed in previous studies, except for the large cohort of MSKCC, a center with a lot of expertise in the treatment of HNC (29). Our definition of marginal failure, which was narrower compared with several other studies (30-32, 38, 39), could also have contributed to the observed results. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,844,485,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Nevertheless, providing accurate guideline adherence, TV definition, and treatment delivery, all reported studies confirm the predominance of in-field recurrences after IMRT. True in-field recurrences likely represent more biologically resistant tumors, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,77,922,442]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"which could possibly be explained by, for example, the harboring of an increased proportion of cancer stem cells and/or hypoxic elements (47, 48). Several mechanisms involved in RT response have been described in HNSCC such as hypoxia, the presence of cancer stem cells (CSC), signaling pathways, DNA damage response (DDR), and cell death pathways. It is important to keep in mind that radioresistance cannot be explained by one single mechanism or protein, but rather by an interplay of different mechanisms. Considerable evidence has suggested that tumor hypoxia results in resistance to (C)RT and tumor recurrence (48, 49), setting the stage for dose intensification strategies. A recent review concluded that dose escalation could improve OS without increased toxicity, although follow-up periods were short in small cohorts, which could result in underreporting (50). The authors concluded that functional imaging modalities could help identify the true extent of the tumor and the region that could benefit most from dose escalation, without increased toxicity. In the meantime, results of five randomized controlled trials are awaited, investigating the benefit of RT dose escalation in HNC patients (NCT01212354, NCT03376386, NCT02352792, NCT02031250, NCT03865277). Another interesting track that deserves attention is dose escalation with proton therapy, as its unique characteristics allow better sparing of normal surrounding tissue, and therefore, dose escalation is less restricted by toxicity (NCT03513042). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,445,922,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The strength of the current study is that it concerns a large single-institution study, in which all patients were treated in a relatively uniform manner. This way, the confounder of variation in treatment quality is minimized, which allows for the analysis of the true patterns of recurrence. There was a long follow-up period (median 6.4 years) and a large patient cohort. Our study is limited by its retrospective nature and the heterogeneity of the patient cohort. P16 status of oropharyngeal carcinoma (OPC) was unknown for 97 patients. Differences in tumor stage, tumor subsite, and the use of concomitant systemic treatment could possibly affect the pattern of failure. OPC patients are overrepresented in our cohort, which follows the pattern of other large patient cohorts and is likely due to the increasing incidence of OPC. The current study is underpowered for a subgroup analysis. However, our results show a trend toward more LRR for hypopharyngeal, supraglottic, and oral cavity tumors (Table 2), which corresponds with the results of Leeman et al. (29). The subsites were not matched for varying stages of disease, which may affect differences in the observed outcomes. However, these variations may also more accurately reflect the clinical presentation that we deal with on a daily basis. All tumor stages [I-IV; classified according to the American Joint Committee on Cancer seventh TNM edition (17)] were included, which could affect the patterns of failure and increase the heterogeneity of the treatment. However, the primary aim of our study was to investigate the quality of current target volume delineation by analyzing the pattern of treatment failure. Since we do not adapt our CTV and PTV margins according to disease stage, we do not expect a significant impact of tumor stage on the recurrence pattern when assessing the accuracy of our GTV and CTV delineation. Another pitfall of the current study may be the \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,35,134,44]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bollen et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[727,35,924,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recurrence Patterns After IMRT/VMAT \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[73,949,294,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Oncology | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[495,951,502,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[680,949,924,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"September 2021 | Volume 11 | Article 720052 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[73,77,484,134]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"cutoff period of 3 years or the cutoff distance of more than $2\\\\mathrm{cm}$ from the index tumor, to differentiate between a LR and SP. However, this definition was based on several previous publications (30, 51). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,170,210,186]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CONCLUSION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,197,485,345]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This large-institution study adds to the evidence of predominant treatment failure inside high-dose radiotherapy volumes, indicating that recurrences are mainly caused by tumor resistance. Our findings reinforce the need to focus on dose intensification and identification of high-risk subvolumes. No marginal recurrences were observed, providing reassurance about accurate TV delineation and the quality of treatment delivery. Caution must be exercised when generalizing these results, since experience with more conformal RT techniques seems a key prerequisite for favorable outcomes in the treatment of HNC patients. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,378,400,394]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DATA AVAILABILITY STATEMENT \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,406,484,436]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,480,210,496]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"REFERENCES \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,504,485,894]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. Fitzmaurice C, Allen C, Barber RM, Barregard L, Bhutta ZA, Brenner H, et al. Global, Regional, and National Cancer Incidence, Mortality, Years of Life Lost, Years Lived With Disability, and Disability-Adjusted Life-Years for 32 Cancer Groups, 1990 to 2015. JAMA Oncol (2017) 3:524-48. doi: 10.1001/jamaoncol.2016.5688 \\\",\\\"2. Pignon J-P, Maitre A, Maillard E, Bourhis J. Meta-Analysis of Chemotherapy in Head and Neck Cancer (MACH-NC): An Update on 93 Randomised Trials and 17,346 Patients. Radiother Oncol (2009) 92(1):4-14. doi: 10.1016/j.radonc.2009.04.014 \\\",\\\"3. Bourhis J, Overgaard J, Audry H, Ang KK, Saunders M, Bernier J, et al. Hyperfractionated or Accelerated Radiotherapy in Head and Neck Cancer: A Meta-Analysis. Lancet (2006) 368(9538):843-54. doi: 10.1016/S0140-6736(06)69121-6 \\\",\\\"4. Nuyts S, Dirix P, Clement PMJ, Vander Poorten V, Delaere P, Schoenaers J, et al. Impact of Adding Concomitant Chemotherapy to Hyperfractionated Accelerated Radiotherapy for Advanced Head-And-Neck Squamous Cell Carcinoma. Int J Radiat Oncol (2009) 73(4):1088-95. doi: 10.1016/j.ijrobp.2008.05.042 \\\",\\\"5. Nutting CM, Morden JP, Harrington KJ, Urbano TG, Bhide SA, Clark C, et al. Parotid-Sparing Intensity Modulated Versus Conventional Radiotherapy in Head and Neck Cancer (PARSPORT): A Phase 3 Multicenter Randomised Controlled Trial. Lancet Oncol (2011) 12(2):127-36. doi: 10.1016/S1470-2045(10)70290-4 \\\",\\\"6. Nutting CM, Morden JP, Beasley M, Bhide S, Cook A, De Winton E, et al. Results of a Multicenter Randomised Controlled Trial of Cochlear-Sparing Intensity-Modulated Radiotherapy Versus Conventional Radiotherapy in Patients With Parotid Cancer (COSTAR; CRUK/08/004). Eur J Cancer (2018) 103:249-58. doi: 10.1016/j.ejca.2018.08.006 \\\",\\\"7. Rathod S, Gupta T, Ghosh-Laskar S, Murthy V, Budrukkar A, Agarwal J. Quality-Of-Life (QOL) Outcomes in Patients With Head and Neck Squamous Cell Carcinoma (HNSCC) Treated With Intensity-Modulated Radiation Therapy (IMRT) Compared to Three-Dimensional Conformal \\\"]\",\n \"page_idx\": \"7\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[512,77,715,92]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ETHICS STATEMENT \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,105,922,207]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The studies involving human participants were reviewed and approved by the Ethics Committee Research UZ/KU Leuven. Written informed consent for participation was not required for this study in accordance with the national legislation and the institutional requirements. Written informed consent was obtained from the individual(s) for the publication of any potentially identifiable images or data included in this article. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,242,773,256]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"AUTHOR CONTRIBUTIONS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,270,922,342]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conceptualization: JV and SN. Data curation: JV. Formal analysis: AL. Investigation: JV. Methodology: JV. Supervision: SN. Writing—original draft: JV and HB. Writing—review and editing: HB and SN. All authors contributed to the article and approved the submitted version. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,379,605,394]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"FUNDING \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,406,922,433]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"SN is appointed as Senior Clinical Investigator by FWO—Research Foundation Flanders. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,480,924,893]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"Radiotherapy (3D-CRT): Evidence From a Prospective Randomized Study. Oral Oncol (2013) 49(6):634-42. doi: 10.1016/j.oraloncology.2013.02.013 \\\",\\\"8. Chen AM, Chin R, Beron P, Yoshizaki T, Mikaeilian AG, Cao M. Inadequate Target Volume Delineation and Local-Regional Recurrence After Intensity-Modulated Radiotherapy for Human Papillomavirus-Positive Oropharynx Cancer. Radiother Oncol (2017) 123(3):412-8. doi: 10.1016/j.radonc.2017.04.015 \\\",\\\"9. Hong TS, Tome WA, Harari PM. Heterogeneity in Head and Neck IMRT Target Design and Clinical Practice. Radiother Oncol (2012) 103(1):92-8. doi: 10.1016/j.radonc.2012.02.010 \\\",\\\"10. Boero IJ, Paravati AJ, Xu B, Cohen EE, Mell LK, Le QT, et al. Importance of Radiation Oncologist Experience Among Patients With Head-And-Neck Cancer Treated With Intensity-Modulated Radiation Therapy. J Clin Oncol (2016) 34(7):684-90. doi: 10.1200/JCO.2015.63.9898 \\\",\\\"11. Peters LJ, O'Sullivan B, Giralt J, Fitzgerald TJ, Trotti A, Bernier J, et al. Critical Impact of Radiotherapy Protocol Compliance and Quality in the Treatment of Advanced Head and Neck Cancer: Results From TROG 02.02. J Clin Oncol (2010) 28(18):2996-3001. doi: 10.1200/JCO.2009.27.4498 \\\",\\\"12. Wuthrick EJ, Zhang Q, Machtay M, Rosenthal DI, Nguyen-Tan PF, Fortin A, et al. Institutional Clinical Trial Accrual Volume and Survival of Patients With Head and Neck Cancer. J Clin Oncol (2015) 33(2):156-64. doi: 10.1200/JCO.2014.56.5218 \\\",\\\"13. Gregoire V, Levendag P, Ang KK, Bernier J, Braaksma M, Budach V, et al. CT-Based Delineation of Lymph Node Levels and Related CTVs in the Node-Negative Neck: DAHANCA, EORTC, GORTEC, NCIC, RTOG Consensus Guidelines. Radiother Oncol (2003) 69(3):227-36. doi: 10.1016/j.radonc.2003.09.011 \\\",\\\"14. Gregoire V, Eisbruch A, Hamoir M, Levendag P. Proposal for the Delineation of the Nodal CTV in the Node-Positive and the Post-Operative Neck. Radiother Oncol (2006) 79(1):15-20. doi: 10.1016/j.radonc.2006.03.009 \\\",\\\"15. Gregoire V, Coche E, Cosnard G, Hamoir M, Reychler H. Selection and Delineation of Lymph Node Target Volumes in Head and Neck Conformal Radiotherapy. Proposal for Standardizing Terminology and Procedure Based on the Surgical Experience. Radiother Oncol (2000) 56(2):135-50. doi: 10.1016/s0167-8140(00)00202-4 \\\"]\",\n \"page_idx\": \"7\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[73,35,134,44]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bollen et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[727,35,924,44]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recurrence Patterns After IMRT/VMAT \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[73,949,294,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Oncology | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[495,951,502,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"8 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[680,949,922,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"September 2021 | Volume 11 | Article 720052 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[73,77,487,893]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"16. Nuyts S, Lambrecht M, Duprez F, Daisne JF, Van Gestel D, Van den Weyngaert D, et al. Reduction of the Dose to the Elective Neck in Head and Neck Squamous Cell Carcinoma, a Randomized Clinical Trial Using Intensity Modulated Radiotherapy (IMRT). Dosimetrical Analysis and Effect on Acute Toxicity. Radiother Oncol (2013) 109:323-9. doi: 10.1016/j.radonc.2013.06.044 \\\",\\\"17. Edge SB, Compton CC. The American Joint Committee on Cancer: The 7th Edition of the AJCC Cancer Staging Manual and the Future of TNM. Ann Surg Oncol (2010) 17(6):1471-4. doi: 10.1245/s10434-010-0985-4 \\\",\\\"18. Schemper M, Smith TL. A Note on Quantifying Follow-Up in Studies of Failure Time. Control Clin Trials (1996) 17(4):343-6. doi: 10.1016/0197-2456(96)00075-x \\\",\\\"19. Bayman E, Prestwich RJD, Speight R, Aspin L, Garratt L, Wilson S, et al. Patterns of Failure After Intensity-Modulated Radiotherapy in Head and Neck Squamous Cell Carcinoma Using Compartmental Clinical Target Volume Delineation. Clin Oncol (2014) 26(10):636-42. doi: 10.1016/j.clon.2014.05.001 \\\",\\\"20. Nuyts S, Dirix P, Clement PM, Vander Poorten V, Delaere P, Schoenaers J, et al. Impact of Adding Concomitant Chemotherapy to Hyperfractionated Accelerated Radiotherapy for Advanced Head-and-Neck Squamous Cell Carcinoma. Int J Radiat Oncol Biol Phys (2009) 73(4):1088-95. doi: 10.1016/j.ijrobp.2008.05.042 \\\",\\\"21. Vermorken JB, Mesia R, Rivera F, Remenar E, Kawecki A, Rottey S, et al. Platinum-Based Chemotherapy Plus Cetuximab in Head and Neck Cancer. N Engl J Med (2008) 359(11):1116-27. doi: 10.1056/NEJMoa0802656 \\\",\\\"22. Chao KS, Ozyigit G, Tran BN, Cengiz M, Dempsey JF, Low DA. Patterns of Failure in Patients Receiving Definitive and Postoperative IMRT for Head-and-Neck Cancer. Int J Radiat Oncol Biol Phys (2003) 55(2):312-21. doi: 10.1016/s0360-3016(02)03940-8 \\\",\\\"23. Daly ME, Le QT, Maxim PG, Loo BWJr, Kaplan MJ, Fischbein NJ, et al. Intensity-Modulated Radiotherapy in the Treatment of Oropharyngeal Cancer: Clinical Outcomes and Patterns of Failure. Int J Radiat Oncol Biol Phys (2010) 76(5):1339-46. doi: 10.1016/j.ijrobp.2009.04.006 \\\",\\\"24. Thorstad S, Hong A, Hope J, Lindsay PE, Haughey B, Deasey O, et al. Patterns of Failure in Patients Receiving Intensity Modulated Radiation Therapy (IMRT) for Head and Neck Cancer. Int J Radiat Oncol Biol Phys (2005) 63: S74. doi: 10.1016/j.ijrobp.2005.07.127 \\\",\\\"25. Biau J, Lapeyre M, Troussier I, Budach W, Giralt J, Grau C, et al. Selection of Lymph Node Target Volumes for Definitive Head and Neck Radiation Therapy: A 2019 Update. Radiother Oncol (2019) 134:1-9. doi: 10.1016/j.radonc.2019.01.018 \\\",\\\"26. Gujral DM, Nutting CM. Patterns of Failure, Treatment Outcomes and Late Toxicities of Head and Neck Cancer in the Current Era of IMRT. Oral Oncol (2018) 86:225-33. doi: 10.1016/j.oraloncology.2018.09.011 \\\",\\\"27. Oksuz DC, Prestwich RJ, Carey B, Wilson S, Senocak MS, Choudhury A, et al. Recurrence Patterns of Locally Advanced Head and Neck Squamous Cell Carcinoma After 3D Conformal (Chemo)-Radiotherapy. *Radiat Oncol* (2011) 6:54. doi: 10.1186/1748-717X-6-54 \\\",\\\"28. De Felice F, Thomas C, Barrington S, Pathmanathan A, Lei M, Urbano TG. Analysis of Loco-Regional Failures in Head and Neck Cancer After Radical Radiation Therapy. Oral Oncol (2015) 51(11):1051-5. doi: 10.1016/j.oraloncology.2015.08.004 \\\",\\\"29. Leeman JE, Li JG, Pei X, Venigalla P, Zumsteg ZS, Katsoulakis E, et al. Patterns of Treatment Failure and Postrecurrence Outcomes Among Patients With Locally Advanced Head and Neck Squamous Cell Carcinoma After Chemoradiotherapy Using Modern Radiation Techniques. JAMA Oncol (2017) 3(11):1487-94. doi: 10.1001/jamaoncol.2017.0973 \\\",\\\"30. Zukauskaite R, Hansen CR, Brink C, Johansen J, Asmussen JT, Grau C, et al. Analysis of CT-Verified Loco-Regional Recurrences After Definitive IMRT for HNSCC Using Site of Origin Estimation Methods. Acta Oncol (2017) 56 (11):1554-61. doi: 10.1080/0284186X.2017.1346384 \\\",\\\"31. Dawson LA, Anzai Y, Marsh L, Martel MK, Paulino A, Ship JA, et al. Patterns of Local-Regional Recurrence Following Parotid-Sparing Conformal and Segmental Intensity-Modulated Radiotherapy for Head and Neck Cancer. Int J Radiat Oncol Biol Phys (2000) 46(5):1117-26. doi: 10.1016/s0360-3016(99)00550-7 \\\",\\\"32. Shakam A, Scrimer R, Liu D, Mohamed M, Parliament M, Field GC, et al. Dose-Volume Analysis of Locoregional Recurrences in Head and Neck IMRT, \\\"]\",\n \"page_idx\": \"8\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[510,75,924,893]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"as Determined by Deformable Registration: A Prospective Multi-Institutional Trial. Radiother Oncol (2011) 99(2):101-7. doi: 10.1016/j.radonc.2011.05.008 \\\",\\\"33. Caudell JJ, Meredith RF, Spencer SA, Keene KS, Dobelbower MC, Bonner JA. Margin on Gross Tumor Volume and Risk of Local Recurrence in Head-and-Neck Cancer. Int J Radiat Oncol Biol Phys (2010) 76(1):164-8. doi: 10.1016/j.ijrobp.2009.01.037 \\\",\\\"34. Ferreira BC, Marques RV, Khouri L, Santos T, Sa-Couto P, do Carmo Lopes M. Assessment and Topographic Characterization of Locoregional Recurrences in Head and Neck Tumors. Radiat Oncol (2015) 10:41. doi: 10.1186/s13014-015-0345-4 \\\",\\\"35. Due AK, Vogelius IR, Aznar MC, Bentzen SM, Berthelsen AK, Korreman SS, et al. Recurrences After Intensity Modulated Radiotherapy for Head and Neck Squamous Cell Carcinoma More Likely to Originate From Regions With High Baseline [18F]-FDG Uptake. Radiother Oncol (2014) 111(3):360-5. doi: 10.1016/j.radonc.2014.06.001 \\\",\\\"36. van den Bosch S, Dijkstra T, Verhoef LC, Zwijnenburg EM, Janssens GO, Kaanders JH. Patterns of Recurrence in Electively Irradiated Lymph Node Regions After Definitive Accelerated Intensity Modulated Radiation Therapy for Head and Neck Squamous Cell Carcinoma. Int J Radiat Oncol Biol Phys (2016) 94(4):766-74. doi: 10.1016/j.ijrobp.2015.12.002 \\\",\\\"37. Dandekar V, Morgan T, Turian J, Fidler MJ, Showel J, Nielsen T, et al. Patterns-Of-Failure After Helical Tomotherapy-Based Chemoradiotherapy for Head and Neck Cancer: Implications for CTV Margin, Elective Nodal Dose and Bilateral Parotid Sparing. Oral Oncol (2014) 50(5):520-6. doi: 10.1016/j.oraloncology.2014.02.009 \\\",\\\"38. Nevens D, Duprez F, Daisne JF, Schatteman J, van der Vorst A, De Neve W, et al. Recurrence Patterns After a Decreased Dose of 40Gy to the Elective Treated Neck in Head and Neck Cancer. Radiother Oncol (2017) 123(3):419-23. doi: 10.1016/j.radonc.2017.03.003 \\\",\\\"39. Schoenfeld GO, Amdur RJ, Morris CG, Li JG, Hinerman RW, Mendenhall WM. Patterns of Failure and Toxicity After Intensity-Modulated Radiotherapy for Head and Neck Cancer. Int J Radiat Oncol Biol Phys (2008) 71(2):377-85. doi: 10.1016/j.ijrobp.2007.10.010 \\\",\\\"40. Studer G, Luetolf UM, Glanzmann C. Locoregional Failure Analysis in Head-and-Neck Cancer Patients Treated With IMRT. Strahlenther Onkol (2007) 183(8):417-23. doi: 10.1007/s00066-007-1663-8 \\\",\\\"41. Garden AS, Dong L, Morrison WH, Stugis EM, Glisson BS, Frank SJ, et al. Patterns of Disease Recurrence Following Treatment of Oropharyngeal Cancer With Intensity Modulated Radiation Therapy. Int J Radiat Oncol Biol Phys (2013) 85(4):941-7. doi: 10.1016/j.ijrobp.2012.08.004 \\\",\\\"42. Kjems J, Gothelf AB, Hakansson K, Specht L, Kristensen CA, Friborg J. Elective Nodal Irradiation and Patterns of Failure in Head and Neck Cancer After Primary Radiation Therapy. Int J Radiat Oncol Biol Phys (2016) 94 (4):775-82. doi: 10.1016/j.ijrobp.2015.12.380 \\\",\\\"43. Eisbruch A, Harris J, Garden AS, Chao CK, Straube W, Harari PM, et al. Multi-Institutional Trial of Accelerated Hypofractionated Intensity-Modulated Radiation Therapy for Early-Stage Oropharyngeal Cancer (RTOG 00-22). Int J Radiat Oncol Biol Phys (2010) 76(5):1333-8. doi: 10.1016/j.ijrobp.2009.04.011 \\\",\\\"44. van der Veen J, Gulyban A, Nuyts S. Interobserver Variability in Delineation of Target Volumes in Head and Neck Cancer. Radiother Oncol (2019) 137:9-15. doi: 10.1016/j.radonc.2019.04.006 \\\",\\\"45. Pettit L, Hartley A, Bowden SJ, Mehanna H, Glaholm J, Cashmore J, et al. Variation in Volume Definition Between UK Head and Neck Oncologists Treating Oropharyngeal Carcinoma. Clin Oncol (2011) 23:654-5. doi: 10.1016/j.clon.2011.07.006 \\\",\\\"46. Hansen CR, Johansen J, Samsøe E, Andersen E, Petersen JBB, Jensen K, et al. Consequences of Introducing Geometric GTV to CTV Margin Expansion in DAHANCA Contouring Guidelines for Head and Neck Radiotherapy. Radiother Oncol (2018) 126:43-7. doi: 10.1016/j.radonc.2017.09.019 \\\",\\\"47. Hanns E, Job S, Coliat P, Wasylyk C, Ramolu L, Pencreach E, et al. Human Papillomavirus-Related Tumors of the Oropharynx Display a Lower Tumor Hypoxia Signature. Oral Oncol (2015) 51:848-56. doi: 10.1016/j.oraloncology.2015.06.003 \\\",\\\"48. Vlashi E, Chen AM, Boyrie S, Yu G, Nguyen A, Brower PA, et al. Radiation-Induced Dedifferentiation of Head and Neck Cancer Cells Into Cancer Stem Cells Depends on Human Papillomavirus Status. Int J Radiat Oncol Biol Phys (2016) 94(5):1198-206. doi: 10.1016/j.ijrobp.2016.01.005 \\\"]\",\n \"page_idx\": \"8\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[73,35,134,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bollen et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[724,35,924,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recurrence Patterns After IMRT/VMAT \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[72,949,294,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Oncology | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[494,951,502,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"9 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[680,949,922,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"September 2021 | Volume 11 | Article 720052 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[72,75,485,184]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"49. Hall EJ. The Oxygen Effects and Re-Oxygenation. In: Radiobiology for the Radiologist. Lippincott Williams & Wilkins (1994). p. 133-52. \\\",\\\"50. Atwell D, Elks J, Cahill K, Hearn N, Vignarajah D, Lagopoulos J, et al. A Review of Modern Radiation Therapy Dose Escalation in Locally Advanced Head and Neck Cancer. Clin Oncol (2020) 32(5):330-41. doi: 10.1016/j.clon.2019.12.004 \\\",\\\"51. Schwartz LH, Ozsahin M, Zhang GN, Touboul E, De Vataire F, Andolenko P, et al. Synchronous and Metachronous Head and Neck Carcinomas. Cancer (1994) 74(7):1933-8. doi: 10.1002/1097-0142(19941001)74:7<1933::aid-cncr 2820740718>3.0.co;2-x \\\"]\",\n \"page_idx\": \"9\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[73,198,485,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conflict of Interest: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,75,922,137]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Publisher's Note: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,151,922,236]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Copyright © 2021 Bollen, van der Veen, Laenen and Nuyts. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,35,134,44]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Bollen et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[727,35,924,46]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recurrence Patterns After IMRT/VMAT \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[73,951,294,961]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Oncology | www.frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[492,951,505,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[680,949,922,960]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"September 2021 | Volume 11 | Article 720052 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.3389/fonc.2021.720052"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Medicine and Health Sciences"},"sha256":{"kind":"string","value":"e3bd3967d7f8d7b67cfda2611d85b2f3fef1a34ea8e02db2d60cf7cbe65053f7"},"title":{"kind":"string","value":"Recurrence Patterns After IMRT/VMAT in Head and Neck Cancer"}}},{"rowIdx":64,"cells":{"abstract":{"kind":"string","value":""},"author":{"kind":"string","value":"Michael Perch|Isabelle Opitz|Mateja Ladan|Laura-Chiara Guglielmetti|Henrik Jessen Hansen|Didier Schneiter|Daniel Franzen|Walter Weder|Claudio Caviezel"},"content_list":{"kind":"list like","value":[{"bbox":"[88,83,910,108]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cite this article as: Caviezel C, Guglielmetti L-C, Ladan M, Hansen HJ, Perch M, Schneiter D et al. Lung volume reduction surgery as salvage procedure after previous use of endobronchial valves. Interact CardioVasc Thorac Surg 2020; doi:10.1093/icvts/ivaa261. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[94,134,903,182]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lung volume reduction surgery as salvage procedure after previous use of endobronchial valves ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[91,194,903,232]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Claudio Caviezel $^{\\text{a},\\ast}$ , Laura-Chiara Guglielmetti $^{\\text{a}}$ , Mateja Ladan $^{\\text{b}}$ , Henrik Jessen Hansen $^{\\text{b}}$ , Michael Perch $^{\\text{c}}$ , Didier Schneiter $^{\\text{a}}$ , Walter Weder $^{\\text{a}}$ , Isabelle Opitz $^{\\text{d},\\dagger}$ and Daniel Franzen $^{\\text{d},\\dagger}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,243,594,289]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"$^{a}$ Department of Thoracic Surgery, University Hospital Zurich, Zurich, Switzerland \",\"$^{b}$ Department of Cardiothoracic Surgery, Rigshospitalet, Copenhagen, Denmark \",\"$^{c}$ Department of Cardiology, Section for Lung Transplantation, Rigshospitalet, Copenhagen, Denmark \",\"$^{d}$ Department of Pulmonary Medicine, University Hospital Zurich, Zurich, Switzerland \"]","page_idx":"0","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[80,301,915,326]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"* Corresponding author. Department of Thoracic Surgery, University Hospital Zürich, 8006 Zürich, Switzerland. Tel: +41-43-253-88-77; e-mail: claudio.caviezel@usz.ch (C. Caviezel). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,335,589,348]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Received 3 September 2020; received in revised form 25 September 2020; accepted 4 October 2020 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[250,387,349,402]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Key question ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[145,411,455,455]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Is salvage lung volume reduction surgery (LVRS) reasonable after bronchoscopic lung volume reduction (BLVR) with valves? ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[246,489,352,507]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Key finding(s) ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[163,513,434,559]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"LVRS after BLVR with valves showed significant improvement of lung function with low morbidity and no mortality. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[222,594,377,611]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Take-home message ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[150,631,449,660]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"LVRS after previous BLVR with valves is reasonable at least as a salvage procedure. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[519,397,687,527]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/1008/5361327688d5fa732d385d808217d1c7aac86c9222ecc4d3a1beff148ac2a782.jpg","list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[521,527,687,656]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0158/5372251fc53544d317bd738bf6457c25a37d36b39250c2c4cf264cf14e872f2c.jpg","list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[689,398,859,527]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0560/e8745d8cb420f05dc12e67be0eff5dba6fffda236480985020cd266f55bdd58a.jpg","list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[689,527,857,656]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0368/edcae1feac39f18798ea59274168ca64fc8a1b1a62ed7e9d24f61775ded2aa45.jpg","list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[80,703,150,718]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[78,727,916,770]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"OBJECTIVES: Lung volume reduction (LVR) is an efficient and approved treatment for selected emphysema patients. There is some evidence that repeated LVR surgery (LVRS) might be beneficial, but there are no current data on LVRS after unsuccessful bronchoscopic LVR (BLVR) with endobronchial valves (EBVs). We hypothesize good outcome of LVRS after BLVR with valves. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,775,916,832]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"METHODS: In this study, we retrospectively investigated all patients who underwent LVRS between 2015 and 2019 at 2 centres after previous unsuccessful EBV treatment. They were further divided into subgroups with patients who never achieved the intended improvement after BLVR (primary failure) and patients whose benefit was fading over time due to the natural development of emphysema (secondary failure). Patients with severe air leak after BLVR and immediate concomitant LVRS and fistula closure thereafter were analysed separately. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,837,916,869]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RESULTS: A total of 38 patients were included. Of these, 19 patients had primary failure, 15 secondary failure and 4 were treated as an emergency due to severe air leak. At 3 months after LVRS, forced expiratory volume in 1 s had improved significantly by $12.5\\%$ $(P = 0.011)$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,878,470,892]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"These authors shared last authorship (contributed equally as senior authors). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,43,403,67]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Interactive CardioVascular and Thoracic Surgery (2020) 1-7 \ndoi:10.1093/ijvts/ivaa261 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[759,40,916,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ORIGINAL ARTICLE ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[78,917,820,931]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"© The Author(s) 2020. Published by Oxford University Press on behalf of the European Association for Cardio-Thoracic Surgery. All rights reserved. ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[955,106,973,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DIDARHOH ","text_format":null,"text_level":null,"type":"aside_text"},{"bbox":"[955,200,975,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Downloaded from https://academic.oup.com/icvts/advance-article/doi/10.1093/icvts/ivaa261/6024962 by guest on 11 December 2020 ","text_format":null,"text_level":null,"type":"aside_text"},{"bbox":"[78,68,915,97]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and there was no 90-day mortality. Considering subgroups, patients with primary failure after BLVR seem to profit more than those with secondary failure. Patients with severe air leak after BLVR did not profit from fistula closure with concomitant LVRS. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,102,916,132]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CONCLUSIONS: LVRS after previous BLVR with EBVs can provide significant clinical improvement with low morbidity, although results might not be as good as after primary LVRS. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,137,656,154]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords: LVRS $\\cdot$ BLVR $\\cdot$ Lung volume reduction surgery $\\cdot$ Endobronchial valves $\\cdot$ Emphysema ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[89,189,204,202]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ABBREVIATIONS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[91,217,383,230]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"BLVR Bronchoscopic lung volume reduction ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,231,282,243]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"EBVs Endobronchial valves ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,245,395,258]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"FEV1 Forced expiratory volume in first second ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,259,266,272]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IQR Interquartile range ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,273,289,286]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"LOS Length of hospital stay ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,287,292,300]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"LVR Lung volume reduction ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,301,341,315]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"LVRS Lung volume reduction surgery ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,316,303,327]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PFTs Pulmonary function fests ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,329,398,341]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Re-LVRS Repeated lung volume reduction surgery ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,343,253,355]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RV Residual volume ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,386,215,401]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"INTRODUCTION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[78,416,483,569]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Lung volume reduction (LVR) has been established as a beneficial and safe procedure in patients with pulmonary emphysema and hyperinflation. LVR surgery (LVRS) and bronchoscopic LVR (BLVR) with endobronchial valves (EBVs), coils or bronchial thermal vapour ablation are available [1-3]. All methods have been demonstrated to improve lung function, exercise capacity and quality of life in patients with both heterogeneous and homogeneous emphysema [4-7]. LVRS and BLVR with EBVs have been shown to prolong survival in highly selected patients [7, 8]. For the selection of patients and the optimal LVR procedure, it is recommendable to establish multidisciplinary conferences [9, 10]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,570,483,681]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"However, chronic obstructive pulmonary disease is a progressive disease and neither LVR procedure is able to stop the natural course with an annual lung function decline and recurring hyperinflation [11]. Therefore, initially successful LVR treatment effects may diminish over months or years [4, 12, 13]. Repeated LVRS (Re-LVRS) shows promising results in highly selected patients, who are slowly deteriorating after initially successful LVRS [14, 15]. Once a patient who already had LVR with EBVs is referred, LVRS might be offered. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,682,482,708]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To the best of our knowledge, there are so far no reports on LVRS after BLVR. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,708,482,751]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this retrospective study from 2 European centres, we aimed to investigate LVRS after previous BLVR with EBVs with focus on mortality, morbidity and outcome. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,775,295,789]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PATIENTS AND METHODS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,803,209,818]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ethical approval ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[78,832,482,860]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This study was performed under both institutional boards' ethical approval. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,860,483,931]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"All patients with previous BVLR with EBVs, who underwent LVRS at 1 of the 2 study centres (Department of Thoracic Surgery, University Hospital Zurich, Switzerland, and Department of Cardiothoracic Surgery, University Hospital Rigshospitalet, Copenhagen, Denmark) between December 2015 and December ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,182,915,279]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2019, were included. Previous BLVR had been performed at the institutions or elsewhere. Re-LVR were considered, when previous LVR was assessed to be unsuccessful (primary failure) according to the multidisciplinary emphysema board or initial benefit had faded over time (secondary failure). In- and exclusion criteria for LVRS were described previously [16, 17] and were similar in both study centres. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,279,915,336]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As many patients were referred from other centres, indications for EBV were not always according to our own policy and our usual emphysema board protocol. However, institutional guidelines from both study centres generally read as follows. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[527,337,704,350]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Primary scheduled for LVRs: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[513,356,807,397]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"- bilateral, upper-lobe predominant emphysema, \",\"- bulleous or paraseptal emphysema and \",\"concomitant nodule. \"]","page_idx":"1","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[527,413,774,427]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Primary scheduled for BLVR with valves: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,435,915,475]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"- heterogeneous emphysema markedly involving 1 whole lobe without collateral ventilation and \",\"- bilateral homogeneous emphysema without collateral ventilation. \"]","page_idx":"1","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[511,491,915,532]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Absent collateral ventilation must be proven by a interlobar fissure integrity score of $>85\\%$ (measured by StratX, Pulmonx, USA) and a negative Chartis measurement. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,534,915,575]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"All patients—for whom both LVRS and BLVR are possible—are fully offered and explained all alternatives. However, many patients already have their own firm preference. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,575,915,603]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"After LVRS, patients after previous BLVR were retrospectively allocated to 2 groups: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[513,610,916,749]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. Primary EBV non-responder (primary failure): no clinical improvement from beginning to 3 months after BLVR despite bronchoscopic revision attempt in case of absent target atelectasis (i.e. for valve replacement). \",\"2. Secondary EBV non-responder (secondary failure): loss of benefit after initial subjective and objective improvement following BLVR (with and without target lobar atelectasis). The time interval between BLVR and the time of 'secondary failure' was only driven by the patient's decision to announce subjective loss of the initial benefit. \"]","page_idx":"1","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[511,756,915,800]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Patients with high-flow fistula after BLVR who thereafter received surgery for combined fistula closure and LVRS (during the same hospitalization) were analysed separately. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,800,915,855]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Decisions to remove EBVs prior to LVRS depended on the presence of target atelectasis (absent target lobe atelectasis triggered indication to remove EBVs, whereas left in place in case of evidence of atelectasis). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,873,769,889]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Surgery and perioperative phase ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[511,902,915,931]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Preoperatively, all potential LVRS patients were functionally assessed with spirometry and bodyplethysmography. For cardiac ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[81,43,91,54]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[318,41,679,56]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Caviezel et al. / Interactive CardioVascular and Thoracic Surgery ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[955,200,975,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Downloaded from https://academic.oup.com/icvts/advance-article/doi/10.1093/icvts/ivaa261/6024962 by guest on 11 December 2020 ","text_format":null,"text_level":null,"type":"aside_text"},{"bbox":"[78,68,482,137]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"evaluation, a transthoracic echocardiogram was performed for screening for pulmonary hypertension. If systolic pulmonary arterial pressure exceeds $35\\mathrm{mmHg}$ , patients are further assessed with right heart catheterization. If mean pulmonary pressure exceeds $35\\mathrm{mmHg}$ , they are usually excluded from LVRS. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,139,482,237]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"LVRS target zones were selected using CT scans, perfusion scans, densitometries and intraoperative emphysema appearance. The area with most destruction in heterogeneous emphysema was approached in cases without persisting atelectasis and without functional valves in situ. In patients with persistent atelectasis and therefore valves left in situ, a target zone on the contralateral side was searched imperatively. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,237,483,322]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"LVRS was performed by video-assisted thoracoscopic surgery (VATS) or exceptionally by anterolateral thoracotomy in case of severe pleural adhesions. Pulmonary resection was performed with standard staplers and 1-2 chest tubes were placed applying suction with $2 - 5\\mathrm{cmH}_2\\mathrm{O}$ . All patients were extubated in the OR and transferred to intensive care or intermediate care units. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,340,349,356]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Follow-up and outcome measures ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[78,369,482,397]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Follow-up of pulmonary function fests (PFTs) was scheduled at 3 months postoperatively. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,398,483,455]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Primary outcome measures were improvement of forced expiratory volume in 1 s (FEV1) and residual volume (RV). Secondary end-points included length of hospital stay (LOS), chest tube time, postoperative complications and 90-day mortality. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,472,220,488]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Statistical analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[78,501,482,544]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Descriptive statistics were used to summarize patients' characteristics. Normality was assessed using the Shapiro-Wilk test. Normally distributed continuous variables were reported as mean and standard ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,68,915,168]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"deviation and compared using two-sample independent $t$ -tests. Non-normally distributed continuous variables were reported as median and range, and for comparisons between 2 groups, the Mann-Whitney $U$ -test was used. Paired continuous variables were compared using paired-samples $t$ -test. Categorical variables were summarized as frequencies (\\%) and compared using Pearson's chi-squared test or Fisher's exact test where applicable. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[513,192,588,207]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RESULTS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[511,222,916,451]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Thirty-eight patients were included (Fig. 1). Four patients had persistent pneumothorax with high-flow fistula even after the removal of EBVs. Nineteen patients were primary EBV non-responders despite previous bronchoscopic revision (group 1), and 15 patients had secondary failure (Group 2). Both groups were further divided into subgroups dependent on the presence of target atelectasis. Five patients were primary EBV non-responders despite the presence of a target atelectasis compared to 14 patients without atelectasis. In Group 2, 3 patients had loss of effect despite a persistent atelectasis after initially successful EBV treatment, and 12 patients with both loss of effect and atelectasis. EBVs were removed prior to LVRS in all patients without evidence of atelectasis. In those with persistent atelectasis $(n = 8)$ , EBVs were removed in 3 patients attributed to Group 1. In patients with secondary failure and persistent atelectasis $(n = 3)$ , EBVs were not removed and LVRS was performed contralateral. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,469,870,503]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"LVRS outcome in patients with primary or secondary failure after previous EBVs (n = 34) ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[511,515,915,544]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Nineteen patients were female (55.9%). Thirty patients had a VATS procedure (88.24%), while 4 patients had VATS converted ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,560,883,883]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 1: Flow chart of patients undergoing lung volume reduction surgery after previous bronchoscopic lung volume reduction using endobronchial valves $(n = 38)$ . BLVR: bronchoscopic lung volume reduction; LVRS: lung volume reduction surgery. \"]","image_footnote":"[]","img_path":"images/0861/23c2d781db243e7a7174b1bf301b7937401877d371a52d7c8ba74f12e1452695.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[318,41,679,56]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Cavigzel et al. / Interactive CardioVascular and Thoracic Surgery ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[906,43,916,54]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[955,106,973,159]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DIDAROHOL ","text_format":null,"text_level":null,"type":"aside_text"},{"bbox":"[955,200,973,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Downloaded from https://academic.oup.com/icvts/advance-article/doi/10.1093/icvts/ivaa261/6024962 by guest on 11 December 2020 ","text_format":null,"text_level":null,"type":"aside_text"},{"bbox":"[78,68,482,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"to a thoracotomy due to adhesions. The median LOS was 8 days [interquartile range (IQR) 6-13]. The median chest tube time was 5 days (IQR 3-11). Ten patients $(29.4\\%)$ had prolonged air leak longer than 7 days, and 1 patient had revision surgery for fistula closure $(2.9\\%)$ . One patient had a pneumothorax after chest tube removal and needed a new chest drain. No other relevant postoperative complications occurred. Ninety-day mortality after LVRS was zero. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,179,483,222]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Pre- and postoperative values of PFTs are displayed in Table 1. The median time interval between BLVR and LVRS was 10 months (IQR 6-14.25). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[83,354,480,553]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0775/a2d947df1b9cc13b70f54f5bfebe056c26bf2257f596b8484839ecb32bd548b7.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":"<table><tr><td>Mean (± SD)</td><td>Pre-LVRS</td><td>3 months post-LVRS</td><td>P-value</td></tr><tr><td>FEV1 (ml)</td><td>640 (± 210) \n(n = 32)</td><td>720 (± 141.1)</td><td>0.011</td></tr><tr><td>FEV1 (% predicted)</td><td>23 (± 3.6) \n(n = 34)</td><td>29.7 (± 6.4)</td><td>0.010</td></tr><tr><td>TLC (ml)</td><td>7661.4 (± 1853.3) \n(n = 26)</td><td>7599.6 (± 1782.8)</td><td>0.22</td></tr><tr><td>TLC (% predicted)</td><td>137.4 (± 23.8) \n(n = 27)</td><td>130 (± 18.0)</td><td>0.12</td></tr><tr><td>RV (ml)</td><td>6070 (± 1725.3) \n(n = 27)</td><td>5180 (± 720.2)</td><td>0.014</td></tr><tr><td>RV (% predicted)</td><td>272 (± 80.6) \n(n = 29)</td><td>0.70 (± 0.17)</td><td>0.002</td></tr><tr><td>DLCO (% predicted)</td><td>25 (± 5) \n(n = 34)</td><td>23.3 (± 5.8)</td><td>0.043</td></tr></table>","table_caption":"[\"Table 1: Pulmonary function tests pre- and post-lung volume reduction surgery in patients with primary and secondary endobronchial valve failure $(n = 34)$ \"]","table_footnote":"[\"DLCO: diffusion capacity; FEV1: forced expiratory volume in 1 s; RV: residual volume; SD: standard deviation; TLC: total lung capacity. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[511,68,890,118]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"LVRS outcome comparing patients operated for primary failure $(n = 19)$ versus secondary failure $(n = 15)$ ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[511,131,916,246]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There was no significant difference of LOS (9 vs 7 days), chest tube time (6 vs 4 days) and proportion of patients with prolonged air leak (43% vs 20%) between Groups 1 and 2. However, improvement of PFT values between pre- and post-LVRS was more pronounced in Group 1 (Table 2). The median time interval between BLVR and LVRS in patients with primary failure (9 months, IQR 5-13) did not differ significantly from patients with secondary failure (median 12 months, IQR 7-16, $P = 0.286$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,265,888,300]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"LVRS outcome in patients treated with EBVs but without achieving a lobar atelectasis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[511,311,916,441]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Concerning patients with absent atelectasis, there were no significant differences in LOS, chest tube time and proportion of prolonged air leak between the 14 patients with primary failure compared to the 12 patients in Group 2. Again, improvement of PFT values was distinctly better in the primary failure group (Table 3). The median time interval between BLVR and LVRS in patients with primary failure (9 months, IQR 4.75-13.5) did not differ significantly from patients with secondary failure (11.5 months, IQR 6.25-15.75, $P = 0.494$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,458,915,508]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"LVRS outcome in patients with (persistent) lobar atelectasis after EBVs (primary or secondary failure, n = 8) ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[511,521,916,608]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There were 8 patients with (persistent) atelectasis. Of these, EBVs were removed prior to LVRS in 3 patients, but in 5 patients, EBVs were left in situ and LVRS was performed at the contralateral side. In all 8 patients, there were no anaesthesiological or perioperative complications. The median LOS was 6 days (IQR 4-14 days), and a median chest tube time was 3 days (IQR 2- ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[83,665,915,892]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0109/ea0043db75e12f2fb6463b20ad5ea2fe51881c9823811c91b13f736f0ec062bf.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":"<table><tr><td rowspan=\"2\">Mean (± SD)</td><td colspan=\"3\">Primary failure (n = 19)</td><td colspan=\"3\">Secondary failure (n = 15)</td></tr><tr><td>Pre-LVRS</td><td>3 months post-LVRS</td><td>P-value</td><td>Pre-LVRS</td><td>3 months post-LVRS</td><td>P-value</td></tr><tr><td rowspan=\"2\">FEV1 (ml)</td><td>752.9 (± 303.9)</td><td>871.9 (± 306.8)</td><td>0.003</td><td>689.3 (± 220)</td><td>733.9 (± 182)</td><td>0.63</td></tr><tr><td>(n = 17)</td><td>(n = 16)</td><td></td><td>(n = 15)</td><td>(n = 13)</td><td></td></tr><tr><td rowspan=\"2\">FEV1 (% predicted)</td><td>26 (± 8.4)</td><td>31.3 (± 9.1)</td><td>0.002</td><td>26.9 (± 11.6)</td><td>28.9 (± 9.6)</td><td>0.77</td></tr><tr><td>(n = 19)</td><td>(n = 17)</td><td></td><td>(n = 15)</td><td>(n = 13)</td><td></td></tr><tr><td rowspan=\"2\">TLC (ml)</td><td>8464.6 (± 1917.6)</td><td>8334.3 (± 1879.6)</td><td>0.39</td><td>6859.2 (± 1443.1)</td><td>6742.5 (± 1248.9)</td><td>0.21</td></tr><tr><td>(n = 14)</td><td>(n = 14)</td><td></td><td>(n = 13)</td><td>(n = 12)</td><td></td></tr><tr><td rowspan=\"2\">TLC (% predicted)</td><td>143.8 (± 25.1)</td><td>136.7 (± 20.5)</td><td>0.39</td><td>131.0 (± 21.4)</td><td>122.3 (± 10.7)</td><td>0.21</td></tr><tr><td>(n = 13)</td><td>(n = 14)</td><td></td><td>(n = 13)</td><td>(n = 12)</td><td></td></tr><tr><td rowspan=\"2\">RV, ml</td><td>5144 (± 2437)</td><td>4244.9 (± 2365.1)</td><td>0.035</td><td>4812.3 (± 1263)</td><td>4413.3 (± 1274.9)</td><td>0.020</td></tr><tr><td>(n = 14)</td><td>(n = 14)</td><td></td><td>(n = 13)</td><td>(n = 12)</td><td></td></tr><tr><td rowspan=\"2\">RV (% predicted)</td><td>259.1 (± 57)</td><td>227.2 (± 54.5)</td><td>0.20</td><td>231.3 (± 43)</td><td>207.8 (± 42.8)</td><td>0.021</td></tr><tr><td>(n = 13)</td><td>(n = 14)</td><td></td><td>(n = 13)</td><td>(n = 12)</td><td></td></tr><tr><td rowspan=\"2\">RV/TLC</td><td>0.70 (± 0.04)</td><td>0.63 (± 0.06)</td><td>0.044</td><td>0.69 (± 0.06)</td><td>0.64 (± 0.07)</td><td>0.062</td></tr><tr><td>(n = 16)</td><td>(n = 15)</td><td></td><td>(n = 13)</td><td>(n = 12)</td><td></td></tr><tr><td rowspan=\"2\">DLCO (% predicted)</td><td>31.22 (±12.12)</td><td>36.2 (± 11.6)</td><td>0.15</td><td>32.76 (± 12.31)</td><td>32.4 (± 11.5)</td><td>0.22</td></tr><tr><td>(n = 19)</td><td>(n = 16)</td><td></td><td>(n = 15)</td><td>(n = 11)</td><td></td></tr></table>","table_caption":"[\"Table 2: Pulmonary function tests in patients with primary failure $\\\\left( {n = {19}}\\\\right)$ and in patients with secondary failure $\\\\left( {n = {15}}\\\\right)$ \"]","table_footnote":"[\"DLCO: diffusion capacity; FEV1: forced expiratory volume in 1 s; RV: residual volume; SD: standard deviation; TLC: total lung capacity. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[81,43,91,54]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[318,41,679,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Caviezel et al. / Interactive CardioVascular and Thoracic Surgery ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[955,200,975,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Downloaded from https://academic.oup.com/icvts/advance-article/doi/10.1093/icvts/ivaa261/6024962 by guest on 11 December 2020 ","text_format":null,"text_level":null,"type":"aside_text"},{"bbox":"[83,122,915,349]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0230/08933dfb0dd3a785c0c9d11d3dc428bbfdc677ca287b17d06692be07ea4505ae.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":"<table><tr><td rowspan=\"2\">Mean (± SD)</td><td colspan=\"3\">Primary failure, no atelectasis (n = 14)</td><td colspan=\"3\">Secondary failure, never or loss of atelectasis (n = 12)</td></tr><tr><td>Pre-LVRS</td><td>3 months post-LVRS</td><td>P-value</td><td>Pre-LVRS</td><td>3 months post-LVRS</td><td>P-value</td></tr><tr><td rowspan=\"2\">FEV1 (ml)</td><td>723.8 (± 291.6)</td><td>827.7 (± 274.6)</td><td>0.045</td><td>631.7 (± 180.2)</td><td>678 (± 149.7)</td><td>0.60</td></tr><tr><td>(n = 13)</td><td>(n = 13)</td><td></td><td>(n = 12)</td><td>(n = 10)</td><td></td></tr><tr><td rowspan=\"2\">FEV1 (%)</td><td>26.1 (± 9.2)</td><td>30.4 (± 9.7)</td><td>0.029</td><td>25.6 (± 11.1)</td><td>28 (± 9.3)</td><td>0.74</td></tr><tr><td>(n = 14)</td><td>(n = 13)</td><td></td><td>(n = 12)</td><td>(n = 10)</td><td></td></tr><tr><td rowspan=\"2\">TLC (ml)</td><td>8943.6 (± 1644.6)</td><td>8785.5 (± 1674.1)</td><td>0.42</td><td>6706.0 (± 1599.9)</td><td>6694.0 (± 1299.2)</td><td>0.37</td></tr><tr><td>(n = 12)</td><td>(n = 11)</td><td></td><td>(n = 10)</td><td>(n = 10)</td><td></td></tr><tr><td rowspan=\"2\">TLC (%)</td><td>149.3 (± 23.3)</td><td>143.1 (± 17.9)</td><td>0.44</td><td>134.1 (± 23.0)</td><td>123.3 (± 11.4)</td><td>0.28</td></tr><tr><td>(n = 11)</td><td>(n = 11)</td><td></td><td>(n = 10)</td><td>(n = 10)</td><td></td></tr><tr><td rowspan=\"2\">RV (ml)</td><td>5327.2 (± 2598.2)</td><td>4364.4 (± 2653)</td><td>0.030</td><td>4825 (± 1427.7)</td><td>4471 (± 1354.4)</td><td>0.084</td></tr><tr><td>(n = 12)</td><td>(n = 11)</td><td></td><td>(n = 10)</td><td>(n = 10)</td><td></td></tr><tr><td rowspan=\"2\">RV (%)</td><td>271.3 (± 53)</td><td>241.3 (± 50.7)</td><td>0.20</td><td>236.5 (± 48.1)</td><td>212.1 (± 45.5)</td><td>0.08</td></tr><tr><td>(n = 11)</td><td>(n = 11)</td><td></td><td>(n = 10)</td><td>(n = 10)</td><td></td></tr><tr><td rowspan=\"2\">RV/TLC</td><td>70.3 (± 4.2)</td><td>65.1 (± 6.5)</td><td>0.15</td><td>70.4 (± 6)</td><td>65.7 (± 7.8)</td><td>0.18</td></tr><tr><td>(n = 12)</td><td>(n = 11)</td><td></td><td>(n = 10)</td><td>(n = 10)</td><td></td></tr><tr><td rowspan=\"2\">DLCO</td><td>30.6 (± 11.2)</td><td>37 (± 12.4)</td><td>0.013</td><td>29.4 (± 10.2)</td><td>30.6 (± 11)</td><td>0.18</td></tr><tr><td>(n = 14)</td><td>(n = 13)</td><td></td><td>(n = 12)</td><td>(n = 9)</td><td></td></tr></table>","table_caption":"[\"Table 3: Pulmonary function tests in patients with primary failure and no atelectasis $(n = 14)$ and in patients with secondary failure and never or loss of atelectasis $(n = 12)$ \"]","table_footnote":"[\"DLCO: diffusion capacity; FEV1: forced expiratory volume in 1 s; RV: residual volume; SD: standard deviation; TLC: total lung capacity. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[83,468,480,679]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0566/5ec60364804a24534216f870ec77eab9e37f2a96df0e5e7780eb5117eddee625.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":"<table><tr><td>Mean (± SD)</td><td>Pre-LVRS</td><td>3 months post-LVRS</td><td>P-value</td></tr><tr><td rowspan=\"2\">FEV1 (ml)</td><td>878.6 (± 300.1)</td><td>990 (± 306)</td><td rowspan=\"2\">0.071</td></tr><tr><td>(n = 7)</td><td>(n = 6)</td></tr><tr><td rowspan=\"2\">FEV1 (%)</td><td>28 (± 9.9)</td><td>33.1 (± 8.8)</td><td rowspan=\"2\">0.12</td></tr><tr><td>(n = 8)</td><td>(n = 7)</td></tr><tr><td rowspan=\"2\">TLC (ml)</td><td>6567.5 (± 1119.5)</td><td>6330 (± 1282.5)</td><td rowspan=\"2\">0.24</td></tr><tr><td>(n = 5)</td><td>(n = 5)</td></tr><tr><td rowspan=\"2\">TLC (%)</td><td>117.6 (± 9.2)</td><td>114.7 (± 7.2)</td><td rowspan=\"2\">0.26</td></tr><tr><td>(n = 5)</td><td>(n = 5)</td></tr><tr><td rowspan=\"2\">RV (ml)</td><td>4480 (± 622.7)</td><td>3934 (± 850.7)</td><td rowspan=\"2\">0.013</td></tr><tr><td>(n = 5)</td><td>(n = 5)</td></tr><tr><td rowspan=\"2\">RV (%)</td><td>205.3 (± 16.2)</td><td>179.9 (± 28.8)</td><td rowspan=\"2\">0.007</td></tr><tr><td>(n = 5)</td><td>(n = 5)</td></tr><tr><td rowspan=\"2\">RV/TLC</td><td>67.6 (± 5)</td><td>58.4 (± 3)</td><td rowspan=\"2\">0.004</td></tr><tr><td>(n = 7)</td><td>(n = 6)</td></tr><tr><td rowspan=\"2\">DLCO</td><td>37.9 (± 15.3)</td><td>35.8 (± 9.8)</td><td rowspan=\"2\">0.54</td></tr><tr><td>(n = 8)</td><td>(n = 5)</td></tr></table>","table_caption":"[\"Table 4: Pulmonary function tests in patients with (persistent) atelectasis after bronchoscopic lung volume reduction (primary or secondary failure, $n = 8$ ) \"]","table_footnote":"[\"DLCO: diffusion capacity; FEV1: forced expiratory volume in 1 s; RV: residual volume; SD: standard deviation; TLC: total lung capacity. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[78,739,482,878]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"12 days). Three had a prolonged air leak $(37.5\\%)$ . At 3 months postoperatively, there was only a significant improvement of RV and RV/total lung capacity, but not in FEV1 (Table 4). In patients who had EBVs removed before LVRS $(n = 3)$ , the median LOS was 15 days (range 6-20) and the median chest tube time was 12 days (range 1-17). Two patients had a prolonged air leak. FEV1 improved from median $840\\mathrm{ml}$ (540-1370) to $1140\\mathrm{ml}$ (730-1550). Preoperative median RV was $3790\\mathrm{ml}$ (value only known for $n = 1$ patient), and postoperative median RV was $3795\\mathrm{ml}$ (range 2900-4690 ml). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,879,482,922]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In patients who did not have EBVs removed and received contralateral LVRS $(n = 5)$ , the median LOS was 5 days (range 2-11) and the median chest tube time was 3 days (range 2-10). One ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,394,915,450]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"patient had a prolonged air leak FEV1 improved from median 785 ml (range 640-1190) to 880 ml (range 780-1120). RV decreased from median 4475 ml (range 4210-5450) to 3830 (3360-4890). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,451,915,479]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The median time interval between BLVR and LVRS was 11 months (IQR 8.25-14). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,496,898,513]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Outcome of patients with high-flow fistula $(n = 4)$ ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,526,916,765]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Four patients with immediate pneumothorax after BLVR with EBVs developed a high-flow fistula. EBVs were removed during the same hospitalization (after a median of 11 days after BLVR, IQR 4.5-15.3), followed by VATS. All patients had a fistula at the non-target lobe, which was closed surgically. At the same time, LVRS was performed at the target lobe. The median postoperative LOS was 10.5 days (range 7-45). The median chest tube time was 19 days (range 2-43) with 1 patient having a postoperative chest tube time of 43 days and a second patient of 30 days. One 63-year-old female patient died because of suicide during inpatient rehabilitation 2 weeks after LVRS. One 58-year-old male patient reported no benefit at 3 months postoperatively. Complete pre- and postoperative PFTs were only available in 2 patients. The mean FEV1 increased from $550\\mathrm{ml}$ $(\\pm 210\\mathrm{ml})$ to $700\\mathrm{ml}$ $(\\pm 141\\mathrm{ml})$ and RV increased from $5470\\mathrm{ml}$ $(\\pm 720\\mathrm{ml})$ to $6070\\mathrm{ml}$ $(\\pm 1725\\mathrm{ml})$ . Diffusion capacity decreased from $25\\%$ $(\\pm 5\\%)$ predicted to $21\\%$ $(\\pm 5.8\\%)$ predicted. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,789,617,803]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DISCUSSION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,818,915,874]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This retrospective study aimed at investigating the postoperative outcome of patients who underwent LVRS after previous BLVR who despite initial success deteriorated or who were unsuccessful immediately after EBV treatment. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,874,916,931]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"LVRS in this sequence is not ideal and has to be considered as salvage therapy. Key issues of a successful LVRS include many aspects, but selecting the right volume and the right target area for resection in balance with physiologic parameters is of ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[318,41,679,56]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Cavigzel et al. / Interactive CardioVascular and Thoracic Surgery ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[906,43,916,54]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[955,106,973,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DIDARHOH ","text_format":null,"text_level":null,"type":"aside_text"},{"bbox":"[955,200,975,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Downloaded from https://academic.oup.com/icvts/advance-article/doi/10.1093/icvts/ivaa261/6024962 by guest on 11 December 2020 ","text_format":null,"text_level":null,"type":"aside_text"},{"bbox":"[83,69,482,234]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"paramount importance. After EBVs, a lobar atelectasis may be present and valves may be still in place causing disturbance of bronchial secret clearance. Both leads to compromises in LVRS treatment. Despite this, the presented data show that the procedure can be offered in some patients without 90-day mortality and a significant improvement in lung function. This information is important for interdisciplinary emphysema boards counselling patients. Patients must know that LVRS after EBV might still be possible but is not equally effective than performed at first choice. The concept of LVRS and EBV is distinctly different despite the common goal of reducing the hyperinflation of emphysematous lungs. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[83,235,482,456]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"EBVs have been shown to be an effective and safe LVR procedure in several randomized controlled trials [6, 18-20]. Improvement of FEV1 by $17 - 29.3\\%$ from baseline can be expected [5, 6, 18, 19, 21]. Considering FEV1, there is a responder rate ranging between $47\\%$ at 3 months and $59\\%$ at 6 months [6, 18]. There are several reasons for an unsuccessful EBV treatment, which can be classified as primary failure or as fading effect over time (secondary failure). While insufficient data are available on reasons for primary failure, there are several reported reasons for the loss of efficacy over time. According to 1-year follow-up data from the STELVIO trial, permanent removal of EBVs was required in $17\\%$ of the patients [22]. Reasons included recurrent pneumothorax, torsion of bronchus, pneumonia and granulation tissue. Besides other factors, the latter is maybe leading to paravalvular leakage and subsequent loss of atelectasis. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[83,458,482,596]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In both our study centres, patients who failed to develop an atelectasis or showed a loss of atelectasis after EBVs usually received revision bronchoscopy at 1-3 months after initial BLVR. Those who still had no profit despite EBV replacement were discussed at the multidisciplinary emphysema board. Performing LVRS in some of these, there was a significant improvement of PFT values at 3 months, which was more distinct in patients with primary failure after EBV treatment. Furthermore, there is a tendency that patients, undergoing unplanned LVRS due to highflow fistula after EBV treatment, might have a poor outcome. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[83,597,482,694]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Compared to primary LVRS with a reported improvement of FEV1 between $41\\%$ and $73\\%$ [23-26], efficiency of LVRS in patients with previous EBV therapy seems inferior. However, our study was not designed to answer this specific question, and comparison to historical LVRS cohorts is not without challenges. At least, postoperative morbidity and mortality seem comparable to primary LVRS. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[83,694,482,929]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Usually, target lobe atelectasis after EBVs is a good indicator for a favourable outcome with improvements of dyspnoea, quality of life (QoL), 6-minute walking distance (6-MWD) and PFT values [27]. However, in this study, we found 8 patients with unsuccessful EBV treatment in spite of an existing target lobe atelectasis. Reasons for these rare cases are not completely understood. Certainly, indications for EBVs (in particular hyperinflation) must be reviewed. The question arises, if EBVs should be removed or left in situ prior to LVRS in these cases. Interestingly, all patients with persistent atelectasis but loss of clinical benefit from EBVs improved after LVRS in FEV1 and RV. According to our experience, these patients can be safely treated with contralateral LVRS, leaving EBVs in situ. Conceptually, this corresponds to results from previous studies concerning Re-LVRS, and as such, the fading benefit after EBVs reflects the natural course of pulmonary emphysema [11, 14, 28]. The questions remain open, if valves should be removed prior to LVRS. A total lobar atelectasis ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,68,913,168]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"is a relevant intervention into the physiology of a patient with advanced emphysema. Lobectomy performed during LVRS should be reserved only to the rare cases of a total lobar destruction. In all our cases, better preserved lung parenchyma was left in place. If valves would be removed and the totally atelectatic lung would re-expand including the functioning lung parenchyma, a more balanced remodelling by LVRS would be possible [29]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,170,913,326]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"However, in patients with absent atelectasis, improvement was only significant in those with primary failure. This is surprising, since patients with secondary failure had experienced a temporary benefit after EBV treatment and thus, the principle of LVR was proven to succeed in these patients. Maybe in patients with secondary failure, the natural course of emphysema already was too advanced and prevented any LVR procedure to be beneficial again. According to our experience from these retrospective and heterogeneous data, EBV removal and subsequent LVRS are the promising options in patients with primary failure after previous EBV therapy. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,327,913,441]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Time interval between BLVR and LVRS did not differ significantly between patients with primary failure and those with secondary failure. Only in patients with primary failure, the decision for a further intervention usually was already made 3 months after BLVR. In patients with secondary failure, this decision was dependent on patient's subjective loss of benefit. Altogether, including planning time for LVRS and several other patient factors, this finding might be coincidental. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,441,913,600]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Pneumothorax occurs in $\\sim 15 - 30\\%$ of patients after BLVR with EBVs, which is usually treated with a chest tube drainage [5, 18, 30]. In $\\sim 70\\%$ of cases, these patients have a prolonged air leak continuing longer than 7 days [30]. In our study, there were 4 patients with persistent high-flow fistula despite previous EBV removal, as recommended [2]. In these patients, fistula closure by VATS followed by LVRS at the former EBV target lobe was performed. The outcome was poor. Although firm conclusions are not possible based on our small experience, combined fistula closure and LVRS might not be recommendable. Probably, a staged procedure is preferable. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,618,602,632]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Limitations ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,648,913,775]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There are some limitations to this study, essentially due to its retrospective nature and the relatively small and heterogeneous group of patients. Quality of EBV therapy was not assessed, and the definition of failure was not quantitative and, thus, only assessed by patient's statements. In addition, outcome of LVRS after EBV was only described by morbidity, mortality and PFTs, missing, i.e. 6-MWD or quality of life. Maximum benefit after LVRS is usually reached after 3-6 months, whereas our data only consist of 3-month follow-up. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,777,913,832]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The small patient's number in this study might be sufficient to point out the favourable outcome of LVRS after previous EBVs, while the conclusions drawn from the subgroup analyses might be interpreted with some caution. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,858,625,872]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CONCLUSION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,888,913,930]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"LVRS after previous BLVR with EBVs shows low morbidity and no mortality and significant clinical improvement can be expected, although results might not be as good as after primary LVRS. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[81,43,91,54]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[318,43,679,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Caviezel et al. / Interactive CardioVascular and Thoracic Surgery ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[957,200,975,796]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Downloaded from https://academic.oup.com/icvts/advance-article/doi/10.1093/icvts/ivaa261/6024962 by guest on 11 December 2020 ","text_format":null,"text_level":null,"type":"aside_text"},{"bbox":"[78,68,482,165]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conflict of interest: Daniel Franzen has received speaking honoraries and consultancy fees from Pulmonx SA. Walter Wedel has received honoraries from Medtronic for teaching and proctoring. Henrik Jessen Hansen participates in advisory boards for Medtronic, BD/Bard and Medela. Mateja Ladan has received a research grant from PulmonX SA. All other authors declare no conflict of interest. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,189,256,205]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Author contributions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[78,218,485,426]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Claudio Caviezel: Conceptualization; Data curation; Formal analysis; Investigation; Methodology; Project administration; Resources; Supervision; Validation; Visualization; Writing—original draft; Writing—review and editing. Laura-Chiara Guglielmetti: Data curation; Formal analysis; Investigation; Methodology; Validation; Writing—original draft; Writing—review and editing. Mateja Ladan: Data curation; Formal analysis; Investigation; Methodology; Validation; Writing—original draft; Writing—review and editing. Henrik Jessen Hansen: Data curation; Formal analysis; Methodology; Validation; Visualization; Writing—review and editing. Michael Perch: Formal analysis; Methodology; Validation; Writing—review and editing. Didier Schneiter: Data curation; Investigation; Resources; Writing—original draft; Writing—review and editing. Walter Weder: Investigation; Supervision; Validation; Writing—original draft; Writing—review and editing. Isabelle Opitz: Conceptualization; Data curation; Investigation; Methodology; Project administration; Resources; Supervision; Validation; Visualization; Writing—original draft; Writing—review and editing. Daniel Franzen: Conceptualization; Formal analysis; Investigation; Methodology; Project administration; Resources; Supervision; Validation; Visualization; Writing—original draft; Writing—review and editing. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,449,264,463]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Reviewer information ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[78,477,483,513]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Interactive CardioVascular and Thoracic Surgery thanks Clemens Aigner and the other, anonymous reviewer(s) for their contribution to the peer review process of this article. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[81,536,187,550]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"REFERENCES ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[86,564,483,896]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"[1] van Geffen WH, Slebos DJ, Herth FJ, Kemp SV, Weder W, Shah PL. Surgical and endoscopic interventions that reduce lung volume for emphysema: a systemic review and meta-analysis. Lancet Respir Med 2019; 7:313-24. \",\"[2] Slebos DJ, Shah PL, Herth FJ, Valipour A. Endobronchial valves for endoscopic lung volume reduction: best practice recommendations from expert panel on endoscopic lung volume reduction. Respiration 2017;93: 138-50. \",\"[3] Shah PL, Herth FJ, van Geffen WH, Deslee G, Slebos DJ. Lung volume reduction for emphysema. Lancet Respir Med 2017;5:147-56. \",\"[4] Bloch KE, Georgescu CL, Russi EW, Weder W. Gain and subsequent loss of lung function after lung volume reduction surgery in cases of severe emphysema with different morphologic patterns. J Thorac Cardiovasc Surg 2002;123:845-54. \",\"[5] Valipour A, Slebos DJ, Herth F, Darwiche K, Wagner M, Ficker JH et al. Endobronchial valve therapy in patients with homogeneous emphysema. Results from the IMPACT study. Am J Respir Crit Care Med 2016; 194:1073-82. \",\"[6] Davey C, Zoumot Z, Jordan S, McNulty WH, Carr DH, Hind MD et al. Bronchoscopic lung volume reduction with endobronchial valves for patients with heterogeneous emphysema and intact interlobar fissures (the BeLieVeR-HlFi study): a randomised controlled trial. Lancet 2015; 386:1066-73. \",\"[7] Criner GJ, Cordova F, Sternberg AL, Martinez FJ. The National Emphysema Treatment Trial (NETT) Part II: lessons learned about lung volume reduction surgery. Am J Respir Crit Care Med 2011;184:881-93. \",\"[8] Gompelmann D, Benjamin N, Bischoff E, Kontogianni K, Schuhmann M, Hoffmann H et al. Survival after endoscopic valve therapy in patients with severe emphysema. Respiration 2019;97:145-52. \"]","page_idx":"6","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[514,67,915,896]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"[9] Kouritas V, Milton R, Kefaloyannis E, Papagiannopoulos K, Brunelli A, Dimov D et al. The impact of a newly established multidisciplinary team on the interventional treatment of patients with emphysema. Clin Med Insights Circ Respir Pulm Med 2019;13:117954841985206. \",\"[10] Rathinam S, Oey I, Steiner M, Spyt T, Morgan MD, Waller DA. The role of the emphysema multidisciplinary team in a successful lung volume reduction surgery programmedagger. Eur J Cardiothorac Surg 2014;46: 1021-6. discussion 6. \",\"[11] Minai OA, Benditt J, Martinez FJ. Natural history of emphysema. Proc Am Thorac Soc 2008;5:468-74. \",\"[12] Hartman JE, Klooster K, Gortzak K, Hacken NHT, Slebos D-J. Long-term follow-up after bronchoscopic lung volume reduction treatment with coils in patients with severe emphysema. Respirology 2015;20:319-26. \",\"[13] Travaline JM, Gaughan JP, Furukawa S, Criner GJ. Effect of bilateral lung volume reduction surgery on FEV1 decline in severe emphysema. COPD 2005;2:203-8. \",\"[14] Kostron A, Horn-Tutic M, Franzen D, Kestenholz P, Schneiter D, Opitz I et al. Repeated lung volume reduction surgery is successful in selected patients. Eur J Cardiothorac Surg 2015;48:710-5. \",\"[15] Tacconi F, Pompeo E, Forcella D, Marino M, Varvaras D, Mineo TC. Lung volume reduction reoperations. Ann Thorac Surg 2008;85:1171-7. \",\"[16] Caviezel C, Aruldas C, Franzen D, Ulrich S, Inci I, Schneiter D et al. Lung volume reduction surgery in selected patients with emphysema and pulmonary hypertension. Eur J Cardiothorac Surg 2018;54:565-571. \",\"[17] Caviezel C, Schaffer N, Schneiter D, Franzen D, Inci I, Opitz I et al. Outcome after lung volume reduction surgery in patients with severely impaired diffusion capacity. Ann Thorac Surg 2018;105:379-85. \",\"[18] Klooster K, ten Hacken NH, Hartman JE, Kerstjens HA, van Rikxoort EM, Slebos DJ. Endobronchial valves for emphysema without interlobar collateral ventilation. N Engl J Med 2015;373:2325-35. \",\"[19] Kemp SV, Slebos DJ, Kirk A, Kornaszewska M, Carron K, Ek L et al. A multicenter randomized controlled trial of Zephyr Endobronchial Valve Treatment in Heterogeneous Emphysema (TRANSFORM). Am J Respir Crit Care Med 2017;196:1535-43. \",\"[20] Criner GJ, Sue R, Wright S, Dransfield M, Rivas-Perez H, Wiese T et al. A multicenter randomized controlled trial of Zephyr Endobronchial Valve Treatment in Heterogeneous Emphysema (LIBERATE). Am J Respir Crit Care Med 2018;198:1151-64. \",\"[21] Sciarba FC, Ernst A, Herth FJ, Strange C, Criner GJ, Marquette CH et al. A randomized study of endobronchial valves for advanced emphysema. N Engl J Med 2010;363:1233-44. \",\"[22] Klooster K, Hartman JE, Ten Hacken NH, Slebos DJ. One-year follow-up after endobronchial valve treatment in patients with emphysema without collateral ventilation treated in the STELVIO trial. Respiration 2017; 93:112-21. \",\"[23] Brenner M, McKenna RJ, Jr., Chen JC, Serna DL, Powell LL, Gelb AF et al. Relationship between amount of lung resected and outcome after lung volume reduction surgery. Ann Thorac Surg 2000;69:388-93. \",\"[24] Ciccone AM, Meyers BF, Guthrie TJ, Davis GE, Yusen RD, Lefrak SS et al. Long-term outcome of bilateral lung volume reduction in 250 consecutive patients with emphysema. J Thorac Cardiovasc Surg 2003;125: 513-25. \",\"[25] Tuti C, Lardinois D, Imfeld S, Korom S, Boehler A, Speich R et al. Lung-volume reduction surgery as an alternative or bridging procedure to lung transplantation. Ann Thorac Surg 2006;82:208-13. discussion 13. \",\"[26] Ginsburg ME, Thomashow BM, Bulman WA, Jellen PA, Whippo BA, Chiuzan C et al. The safety, efficacy, and durability of lung-volume reduction surgery: a 10-year experience. J Thorac Cardiovasc Surg 2016; 151:717-24 e1. \",\"[27] Herth FJ, Eberhardt R, Gompelmann D, Ficker JH, Wagner M, Ek L et al. Radiological and clinical outcomes of using Chartis to plan endobronchial valve treatment. Eur Respir J 2013;41:302-8. \",\"[28] Fiorelli A, Santini M. Bronchoscopic lung volume reduction: an alternative to repeated lung volume reduction surgery. Eur J Cardiothorac Surg 2016;50:584.1. \",\"[29] Caviezel CF, Schneiter D, Muehlmatter U, Frauenfelder T, Guglielmetti LC, Opitz I et al. Identification of target zones for lung volume reduction surgery using three dimensional computed tomography rendering. Eur Respir J Open Research 2020;6:00305-2020. \",\"[30] Gompelmann D, Herth FJ, Slebos DJ, Valipour A, Ernst A, Criner GJ et al. Pneumothorax following endobronchial valve therapy and its impact on clinical outcomes in severe emphysema. Respiration 2014;87:485-91. \"]","page_idx":"6","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[318,43,679,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"C. Caviezel et al. / Interactive CardioVascular and Thoracic Surgery ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[906,44,916,53]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[955,106,973,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DIDAH ","text_format":null,"text_level":null,"type":"aside_text"},{"bbox":"[955,200,975,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Downloaded from https://academic.oup.com/icvts/advance-article/doi/10.1093/icvts/ivaa261/6024962 by guest on 11 December 2020 ","text_format":null,"text_level":null,"type":"aside_text"}],"string":"[\n {\n \"bbox\": \"[88,83,910,108]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cite this article as: Caviezel C, Guglielmetti L-C, Ladan M, Hansen HJ, Perch M, Schneiter D et al. Lung volume reduction surgery as salvage procedure after previous use of endobronchial valves. Interact CardioVasc Thorac Surg 2020; doi:10.1093/icvts/ivaa261. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[94,134,903,182]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lung volume reduction surgery as salvage procedure after previous use of endobronchial valves \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,194,903,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Claudio Caviezel $^{\\\\text{a},\\\\ast}$ , Laura-Chiara Guglielmetti $^{\\\\text{a}}$ , Mateja Ladan $^{\\\\text{b}}$ , Henrik Jessen Hansen $^{\\\\text{b}}$ , Michael Perch $^{\\\\text{c}}$ , Didier Schneiter $^{\\\\text{a}}$ , Walter Weder $^{\\\\text{a}}$ , Isabelle Opitz $^{\\\\text{d},\\\\dagger}$ and Daniel Franzen $^{\\\\text{d},\\\\dagger}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,243,594,289]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"$^{a}$ Department of Thoracic Surgery, University Hospital Zurich, Zurich, Switzerland \\\",\\\"$^{b}$ Department of Cardiothoracic Surgery, Rigshospitalet, Copenhagen, Denmark \\\",\\\"$^{c}$ Department of Cardiology, Section for Lung Transplantation, Rigshospitalet, Copenhagen, Denmark \\\",\\\"$^{d}$ Department of Pulmonary Medicine, University Hospital Zurich, Zurich, Switzerland \\\"]\",\n \"page_idx\": \"0\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[80,301,915,326]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"* Corresponding author. Department of Thoracic Surgery, University Hospital Zürich, 8006 Zürich, Switzerland. Tel: +41-43-253-88-77; e-mail: claudio.caviezel@usz.ch (C. Caviezel). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,335,589,348]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Received 3 September 2020; received in revised form 25 September 2020; accepted 4 October 2020 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[250,387,349,402]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Key question \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[145,411,455,455]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Is salvage lung volume reduction surgery (LVRS) reasonable after bronchoscopic lung volume reduction (BLVR) with valves? \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[246,489,352,507]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Key finding(s) \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[163,513,434,559]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"LVRS after BLVR with valves showed significant improvement of lung function with low morbidity and no mortality. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[222,594,377,611]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Take-home message \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[150,631,449,660]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"LVRS after previous BLVR with valves is reasonable at least as a salvage procedure. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[519,397,687,527]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/1008/5361327688d5fa732d385d808217d1c7aac86c9222ecc4d3a1beff148ac2a782.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[521,527,687,656]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0158/5372251fc53544d317bd738bf6457c25a37d36b39250c2c4cf264cf14e872f2c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[689,398,859,527]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0560/e8745d8cb420f05dc12e67be0eff5dba6fffda236480985020cd266f55bdd58a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[689,527,857,656]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0368/edcae1feac39f18798ea59274168ca64fc8a1b1a62ed7e9d24f61775ded2aa45.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[80,703,150,718]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,727,916,770]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"OBJECTIVES: Lung volume reduction (LVR) is an efficient and approved treatment for selected emphysema patients. There is some evidence that repeated LVR surgery (LVRS) might be beneficial, but there are no current data on LVRS after unsuccessful bronchoscopic LVR (BLVR) with endobronchial valves (EBVs). We hypothesize good outcome of LVRS after BLVR with valves. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,775,916,832]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"METHODS: In this study, we retrospectively investigated all patients who underwent LVRS between 2015 and 2019 at 2 centres after previous unsuccessful EBV treatment. They were further divided into subgroups with patients who never achieved the intended improvement after BLVR (primary failure) and patients whose benefit was fading over time due to the natural development of emphysema (secondary failure). Patients with severe air leak after BLVR and immediate concomitant LVRS and fistula closure thereafter were analysed separately. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,837,916,869]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RESULTS: A total of 38 patients were included. Of these, 19 patients had primary failure, 15 secondary failure and 4 were treated as an emergency due to severe air leak. At 3 months after LVRS, forced expiratory volume in 1 s had improved significantly by $12.5\\\\%$ $(P = 0.011)$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,878,470,892]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"These authors shared last authorship (contributed equally as senior authors). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,43,403,67]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Interactive CardioVascular and Thoracic Surgery (2020) 1-7 \\ndoi:10.1093/ijvts/ivaa261 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[759,40,916,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ORIGINAL ARTICLE \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[78,917,820,931]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"© The Author(s) 2020. Published by Oxford University Press on behalf of the European Association for Cardio-Thoracic Surgery. All rights reserved. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[955,106,973,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DIDARHOH \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"aside_text\"\n },\n {\n \"bbox\": \"[955,200,975,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Downloaded from https://academic.oup.com/icvts/advance-article/doi/10.1093/icvts/ivaa261/6024962 by guest on 11 December 2020 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"aside_text\"\n },\n {\n \"bbox\": \"[78,68,915,97]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and there was no 90-day mortality. Considering subgroups, patients with primary failure after BLVR seem to profit more than those with secondary failure. Patients with severe air leak after BLVR did not profit from fistula closure with concomitant LVRS. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,102,916,132]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CONCLUSIONS: LVRS after previous BLVR with EBVs can provide significant clinical improvement with low morbidity, although results might not be as good as after primary LVRS. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,137,656,154]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords: LVRS $\\\\cdot$ BLVR $\\\\cdot$ Lung volume reduction surgery $\\\\cdot$ Endobronchial valves $\\\\cdot$ Emphysema \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[89,189,204,202]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ABBREVIATIONS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,217,383,230]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"BLVR Bronchoscopic lung volume reduction \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,231,282,243]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"EBVs Endobronchial valves \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,245,395,258]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"FEV1 Forced expiratory volume in first second \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,259,266,272]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IQR Interquartile range \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,273,289,286]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"LOS Length of hospital stay \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,287,292,300]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"LVR Lung volume reduction \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,301,341,315]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"LVRS Lung volume reduction surgery \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,316,303,327]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PFTs Pulmonary function fests \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,329,398,341]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Re-LVRS Repeated lung volume reduction surgery \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[91,343,253,355]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RV Residual volume \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,386,215,401]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"INTRODUCTION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,416,483,569]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Lung volume reduction (LVR) has been established as a beneficial and safe procedure in patients with pulmonary emphysema and hyperinflation. LVR surgery (LVRS) and bronchoscopic LVR (BLVR) with endobronchial valves (EBVs), coils or bronchial thermal vapour ablation are available [1-3]. All methods have been demonstrated to improve lung function, exercise capacity and quality of life in patients with both heterogeneous and homogeneous emphysema [4-7]. LVRS and BLVR with EBVs have been shown to prolong survival in highly selected patients [7, 8]. For the selection of patients and the optimal LVR procedure, it is recommendable to establish multidisciplinary conferences [9, 10]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,570,483,681]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"However, chronic obstructive pulmonary disease is a progressive disease and neither LVR procedure is able to stop the natural course with an annual lung function decline and recurring hyperinflation [11]. Therefore, initially successful LVR treatment effects may diminish over months or years [4, 12, 13]. Repeated LVRS (Re-LVRS) shows promising results in highly selected patients, who are slowly deteriorating after initially successful LVRS [14, 15]. Once a patient who already had LVR with EBVs is referred, LVRS might be offered. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,682,482,708]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To the best of our knowledge, there are so far no reports on LVRS after BLVR. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,708,482,751]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this retrospective study from 2 European centres, we aimed to investigate LVRS after previous BLVR with EBVs with focus on mortality, morbidity and outcome. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,775,295,789]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PATIENTS AND METHODS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,803,209,818]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ethical approval \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,832,482,860]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This study was performed under both institutional boards' ethical approval. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,860,483,931]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"All patients with previous BVLR with EBVs, who underwent LVRS at 1 of the 2 study centres (Department of Thoracic Surgery, University Hospital Zurich, Switzerland, and Department of Cardiothoracic Surgery, University Hospital Rigshospitalet, Copenhagen, Denmark) between December 2015 and December \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,182,915,279]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2019, were included. Previous BLVR had been performed at the institutions or elsewhere. Re-LVR were considered, when previous LVR was assessed to be unsuccessful (primary failure) according to the multidisciplinary emphysema board or initial benefit had faded over time (secondary failure). In- and exclusion criteria for LVRS were described previously [16, 17] and were similar in both study centres. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,279,915,336]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As many patients were referred from other centres, indications for EBV were not always according to our own policy and our usual emphysema board protocol. However, institutional guidelines from both study centres generally read as follows. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[527,337,704,350]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Primary scheduled for LVRs: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[513,356,807,397]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"- bilateral, upper-lobe predominant emphysema, \\\",\\\"- bulleous or paraseptal emphysema and \\\",\\\"concomitant nodule. \\\"]\",\n \"page_idx\": \"1\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[527,413,774,427]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Primary scheduled for BLVR with valves: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,435,915,475]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"- heterogeneous emphysema markedly involving 1 whole lobe without collateral ventilation and \\\",\\\"- bilateral homogeneous emphysema without collateral ventilation. \\\"]\",\n \"page_idx\": \"1\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[511,491,915,532]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Absent collateral ventilation must be proven by a interlobar fissure integrity score of $>85\\\\%$ (measured by StratX, Pulmonx, USA) and a negative Chartis measurement. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,534,915,575]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"All patients—for whom both LVRS and BLVR are possible—are fully offered and explained all alternatives. However, many patients already have their own firm preference. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,575,915,603]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"After LVRS, patients after previous BLVR were retrospectively allocated to 2 groups: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[513,610,916,749]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. Primary EBV non-responder (primary failure): no clinical improvement from beginning to 3 months after BLVR despite bronchoscopic revision attempt in case of absent target atelectasis (i.e. for valve replacement). \\\",\\\"2. Secondary EBV non-responder (secondary failure): loss of benefit after initial subjective and objective improvement following BLVR (with and without target lobar atelectasis). The time interval between BLVR and the time of 'secondary failure' was only driven by the patient's decision to announce subjective loss of the initial benefit. \\\"]\",\n \"page_idx\": \"1\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[511,756,915,800]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Patients with high-flow fistula after BLVR who thereafter received surgery for combined fistula closure and LVRS (during the same hospitalization) were analysed separately. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,800,915,855]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Decisions to remove EBVs prior to LVRS depended on the presence of target atelectasis (absent target lobe atelectasis triggered indication to remove EBVs, whereas left in place in case of evidence of atelectasis). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,873,769,889]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Surgery and perioperative phase \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,902,915,931]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Preoperatively, all potential LVRS patients were functionally assessed with spirometry and bodyplethysmography. For cardiac \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[81,43,91,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[318,41,679,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Caviezel et al. / Interactive CardioVascular and Thoracic Surgery \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[955,200,975,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Downloaded from https://academic.oup.com/icvts/advance-article/doi/10.1093/icvts/ivaa261/6024962 by guest on 11 December 2020 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"aside_text\"\n },\n {\n \"bbox\": \"[78,68,482,137]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"evaluation, a transthoracic echocardiogram was performed for screening for pulmonary hypertension. If systolic pulmonary arterial pressure exceeds $35\\\\mathrm{mmHg}$ , patients are further assessed with right heart catheterization. If mean pulmonary pressure exceeds $35\\\\mathrm{mmHg}$ , they are usually excluded from LVRS. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,139,482,237]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"LVRS target zones were selected using CT scans, perfusion scans, densitometries and intraoperative emphysema appearance. The area with most destruction in heterogeneous emphysema was approached in cases without persisting atelectasis and without functional valves in situ. In patients with persistent atelectasis and therefore valves left in situ, a target zone on the contralateral side was searched imperatively. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,237,483,322]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"LVRS was performed by video-assisted thoracoscopic surgery (VATS) or exceptionally by anterolateral thoracotomy in case of severe pleural adhesions. Pulmonary resection was performed with standard staplers and 1-2 chest tubes were placed applying suction with $2 - 5\\\\mathrm{cmH}_2\\\\mathrm{O}$ . All patients were extubated in the OR and transferred to intensive care or intermediate care units. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,340,349,356]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Follow-up and outcome measures \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,369,482,397]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Follow-up of pulmonary function fests (PFTs) was scheduled at 3 months postoperatively. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,398,483,455]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Primary outcome measures were improvement of forced expiratory volume in 1 s (FEV1) and residual volume (RV). Secondary end-points included length of hospital stay (LOS), chest tube time, postoperative complications and 90-day mortality. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,472,220,488]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Statistical analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,501,482,544]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Descriptive statistics were used to summarize patients' characteristics. Normality was assessed using the Shapiro-Wilk test. Normally distributed continuous variables were reported as mean and standard \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,68,915,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"deviation and compared using two-sample independent $t$ -tests. Non-normally distributed continuous variables were reported as median and range, and for comparisons between 2 groups, the Mann-Whitney $U$ -test was used. Paired continuous variables were compared using paired-samples $t$ -test. Categorical variables were summarized as frequencies (\\\\%) and compared using Pearson's chi-squared test or Fisher's exact test where applicable. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[513,192,588,207]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RESULTS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,222,916,451]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Thirty-eight patients were included (Fig. 1). Four patients had persistent pneumothorax with high-flow fistula even after the removal of EBVs. Nineteen patients were primary EBV non-responders despite previous bronchoscopic revision (group 1), and 15 patients had secondary failure (Group 2). Both groups were further divided into subgroups dependent on the presence of target atelectasis. Five patients were primary EBV non-responders despite the presence of a target atelectasis compared to 14 patients without atelectasis. In Group 2, 3 patients had loss of effect despite a persistent atelectasis after initially successful EBV treatment, and 12 patients with both loss of effect and atelectasis. EBVs were removed prior to LVRS in all patients without evidence of atelectasis. In those with persistent atelectasis $(n = 8)$ , EBVs were removed in 3 patients attributed to Group 1. In patients with secondary failure and persistent atelectasis $(n = 3)$ , EBVs were not removed and LVRS was performed contralateral. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,469,870,503]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"LVRS outcome in patients with primary or secondary failure after previous EBVs (n = 34) \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,515,915,544]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Nineteen patients were female (55.9%). Thirty patients had a VATS procedure (88.24%), while 4 patients had VATS converted \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,560,883,883]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 1: Flow chart of patients undergoing lung volume reduction surgery after previous bronchoscopic lung volume reduction using endobronchial valves $(n = 38)$ . BLVR: bronchoscopic lung volume reduction; LVRS: lung volume reduction surgery. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0861/23c2d781db243e7a7174b1bf301b7937401877d371a52d7c8ba74f12e1452695.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[318,41,679,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Cavigzel et al. / Interactive CardioVascular and Thoracic Surgery \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[906,43,916,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[955,106,973,159]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DIDAROHOL \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"aside_text\"\n },\n {\n \"bbox\": \"[955,200,973,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Downloaded from https://academic.oup.com/icvts/advance-article/doi/10.1093/icvts/ivaa261/6024962 by guest on 11 December 2020 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"aside_text\"\n },\n {\n \"bbox\": \"[78,68,482,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"to a thoracotomy due to adhesions. The median LOS was 8 days [interquartile range (IQR) 6-13]. The median chest tube time was 5 days (IQR 3-11). Ten patients $(29.4\\\\%)$ had prolonged air leak longer than 7 days, and 1 patient had revision surgery for fistula closure $(2.9\\\\%)$ . One patient had a pneumothorax after chest tube removal and needed a new chest drain. No other relevant postoperative complications occurred. Ninety-day mortality after LVRS was zero. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,179,483,222]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Pre- and postoperative values of PFTs are displayed in Table 1. The median time interval between BLVR and LVRS was 10 months (IQR 6-14.25). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,354,480,553]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0775/a2d947df1b9cc13b70f54f5bfebe056c26bf2257f596b8484839ecb32bd548b7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Mean (± SD)</td><td>Pre-LVRS</td><td>3 months post-LVRS</td><td>P-value</td></tr><tr><td>FEV1 (ml)</td><td>640 (± 210) \\n(n = 32)</td><td>720 (± 141.1)</td><td>0.011</td></tr><tr><td>FEV1 (% predicted)</td><td>23 (± 3.6) \\n(n = 34)</td><td>29.7 (± 6.4)</td><td>0.010</td></tr><tr><td>TLC (ml)</td><td>7661.4 (± 1853.3) \\n(n = 26)</td><td>7599.6 (± 1782.8)</td><td>0.22</td></tr><tr><td>TLC (% predicted)</td><td>137.4 (± 23.8) \\n(n = 27)</td><td>130 (± 18.0)</td><td>0.12</td></tr><tr><td>RV (ml)</td><td>6070 (± 1725.3) \\n(n = 27)</td><td>5180 (± 720.2)</td><td>0.014</td></tr><tr><td>RV (% predicted)</td><td>272 (± 80.6) \\n(n = 29)</td><td>0.70 (± 0.17)</td><td>0.002</td></tr><tr><td>DLCO (% predicted)</td><td>25 (± 5) \\n(n = 34)</td><td>23.3 (± 5.8)</td><td>0.043</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 1: Pulmonary function tests pre- and post-lung volume reduction surgery in patients with primary and secondary endobronchial valve failure $(n = 34)$ \\\"]\",\n \"table_footnote\": \"[\\\"DLCO: diffusion capacity; FEV1: forced expiratory volume in 1 s; RV: residual volume; SD: standard deviation; TLC: total lung capacity. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[511,68,890,118]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"LVRS outcome comparing patients operated for primary failure $(n = 19)$ versus secondary failure $(n = 15)$ \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,131,916,246]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There was no significant difference of LOS (9 vs 7 days), chest tube time (6 vs 4 days) and proportion of patients with prolonged air leak (43% vs 20%) between Groups 1 and 2. However, improvement of PFT values between pre- and post-LVRS was more pronounced in Group 1 (Table 2). The median time interval between BLVR and LVRS in patients with primary failure (9 months, IQR 5-13) did not differ significantly from patients with secondary failure (median 12 months, IQR 7-16, $P = 0.286$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,265,888,300]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"LVRS outcome in patients treated with EBVs but without achieving a lobar atelectasis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,311,916,441]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Concerning patients with absent atelectasis, there were no significant differences in LOS, chest tube time and proportion of prolonged air leak between the 14 patients with primary failure compared to the 12 patients in Group 2. Again, improvement of PFT values was distinctly better in the primary failure group (Table 3). The median time interval between BLVR and LVRS in patients with primary failure (9 months, IQR 4.75-13.5) did not differ significantly from patients with secondary failure (11.5 months, IQR 6.25-15.75, $P = 0.494$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,458,915,508]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"LVRS outcome in patients with (persistent) lobar atelectasis after EBVs (primary or secondary failure, n = 8) \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,521,916,608]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There were 8 patients with (persistent) atelectasis. Of these, EBVs were removed prior to LVRS in 3 patients, but in 5 patients, EBVs were left in situ and LVRS was performed at the contralateral side. In all 8 patients, there were no anaesthesiological or perioperative complications. The median LOS was 6 days (IQR 4-14 days), and a median chest tube time was 3 days (IQR 2- \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,665,915,892]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0109/ea0043db75e12f2fb6463b20ad5ea2fe51881c9823811c91b13f736f0ec062bf.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=\\\"2\\\">Mean (± SD)</td><td colspan=\\\"3\\\">Primary failure (n = 19)</td><td colspan=\\\"3\\\">Secondary failure (n = 15)</td></tr><tr><td>Pre-LVRS</td><td>3 months post-LVRS</td><td>P-value</td><td>Pre-LVRS</td><td>3 months post-LVRS</td><td>P-value</td></tr><tr><td rowspan=\\\"2\\\">FEV1 (ml)</td><td>752.9 (± 303.9)</td><td>871.9 (± 306.8)</td><td>0.003</td><td>689.3 (± 220)</td><td>733.9 (± 182)</td><td>0.63</td></tr><tr><td>(n = 17)</td><td>(n = 16)</td><td></td><td>(n = 15)</td><td>(n = 13)</td><td></td></tr><tr><td rowspan=\\\"2\\\">FEV1 (% predicted)</td><td>26 (± 8.4)</td><td>31.3 (± 9.1)</td><td>0.002</td><td>26.9 (± 11.6)</td><td>28.9 (± 9.6)</td><td>0.77</td></tr><tr><td>(n = 19)</td><td>(n = 17)</td><td></td><td>(n = 15)</td><td>(n = 13)</td><td></td></tr><tr><td rowspan=\\\"2\\\">TLC (ml)</td><td>8464.6 (± 1917.6)</td><td>8334.3 (± 1879.6)</td><td>0.39</td><td>6859.2 (± 1443.1)</td><td>6742.5 (± 1248.9)</td><td>0.21</td></tr><tr><td>(n = 14)</td><td>(n = 14)</td><td></td><td>(n = 13)</td><td>(n = 12)</td><td></td></tr><tr><td rowspan=\\\"2\\\">TLC (% predicted)</td><td>143.8 (± 25.1)</td><td>136.7 (± 20.5)</td><td>0.39</td><td>131.0 (± 21.4)</td><td>122.3 (± 10.7)</td><td>0.21</td></tr><tr><td>(n = 13)</td><td>(n = 14)</td><td></td><td>(n = 13)</td><td>(n = 12)</td><td></td></tr><tr><td rowspan=\\\"2\\\">RV, ml</td><td>5144 (± 2437)</td><td>4244.9 (± 2365.1)</td><td>0.035</td><td>4812.3 (± 1263)</td><td>4413.3 (± 1274.9)</td><td>0.020</td></tr><tr><td>(n = 14)</td><td>(n = 14)</td><td></td><td>(n = 13)</td><td>(n = 12)</td><td></td></tr><tr><td rowspan=\\\"2\\\">RV (% predicted)</td><td>259.1 (± 57)</td><td>227.2 (± 54.5)</td><td>0.20</td><td>231.3 (± 43)</td><td>207.8 (± 42.8)</td><td>0.021</td></tr><tr><td>(n = 13)</td><td>(n = 14)</td><td></td><td>(n = 13)</td><td>(n = 12)</td><td></td></tr><tr><td rowspan=\\\"2\\\">RV/TLC</td><td>0.70 (± 0.04)</td><td>0.63 (± 0.06)</td><td>0.044</td><td>0.69 (± 0.06)</td><td>0.64 (± 0.07)</td><td>0.062</td></tr><tr><td>(n = 16)</td><td>(n = 15)</td><td></td><td>(n = 13)</td><td>(n = 12)</td><td></td></tr><tr><td rowspan=\\\"2\\\">DLCO (% predicted)</td><td>31.22 (±12.12)</td><td>36.2 (± 11.6)</td><td>0.15</td><td>32.76 (± 12.31)</td><td>32.4 (± 11.5)</td><td>0.22</td></tr><tr><td>(n = 19)</td><td>(n = 16)</td><td></td><td>(n = 15)</td><td>(n = 11)</td><td></td></tr></table>\",\n \"table_caption\": \"[\\\"Table 2: Pulmonary function tests in patients with primary failure $\\\\\\\\left( {n = {19}}\\\\\\\\right)$ and in patients with secondary failure $\\\\\\\\left( {n = {15}}\\\\\\\\right)$ \\\"]\",\n \"table_footnote\": \"[\\\"DLCO: diffusion capacity; FEV1: forced expiratory volume in 1 s; RV: residual volume; SD: standard deviation; TLC: total lung capacity. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[81,43,91,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[318,41,679,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Caviezel et al. / Interactive CardioVascular and Thoracic Surgery \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[955,200,975,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Downloaded from https://academic.oup.com/icvts/advance-article/doi/10.1093/icvts/ivaa261/6024962 by guest on 11 December 2020 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"aside_text\"\n },\n {\n \"bbox\": \"[83,122,915,349]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0230/08933dfb0dd3a785c0c9d11d3dc428bbfdc677ca287b17d06692be07ea4505ae.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=\\\"2\\\">Mean (± SD)</td><td colspan=\\\"3\\\">Primary failure, no atelectasis (n = 14)</td><td colspan=\\\"3\\\">Secondary failure, never or loss of atelectasis (n = 12)</td></tr><tr><td>Pre-LVRS</td><td>3 months post-LVRS</td><td>P-value</td><td>Pre-LVRS</td><td>3 months post-LVRS</td><td>P-value</td></tr><tr><td rowspan=\\\"2\\\">FEV1 (ml)</td><td>723.8 (± 291.6)</td><td>827.7 (± 274.6)</td><td>0.045</td><td>631.7 (± 180.2)</td><td>678 (± 149.7)</td><td>0.60</td></tr><tr><td>(n = 13)</td><td>(n = 13)</td><td></td><td>(n = 12)</td><td>(n = 10)</td><td></td></tr><tr><td rowspan=\\\"2\\\">FEV1 (%)</td><td>26.1 (± 9.2)</td><td>30.4 (± 9.7)</td><td>0.029</td><td>25.6 (± 11.1)</td><td>28 (± 9.3)</td><td>0.74</td></tr><tr><td>(n = 14)</td><td>(n = 13)</td><td></td><td>(n = 12)</td><td>(n = 10)</td><td></td></tr><tr><td rowspan=\\\"2\\\">TLC (ml)</td><td>8943.6 (± 1644.6)</td><td>8785.5 (± 1674.1)</td><td>0.42</td><td>6706.0 (± 1599.9)</td><td>6694.0 (± 1299.2)</td><td>0.37</td></tr><tr><td>(n = 12)</td><td>(n = 11)</td><td></td><td>(n = 10)</td><td>(n = 10)</td><td></td></tr><tr><td rowspan=\\\"2\\\">TLC (%)</td><td>149.3 (± 23.3)</td><td>143.1 (± 17.9)</td><td>0.44</td><td>134.1 (± 23.0)</td><td>123.3 (± 11.4)</td><td>0.28</td></tr><tr><td>(n = 11)</td><td>(n = 11)</td><td></td><td>(n = 10)</td><td>(n = 10)</td><td></td></tr><tr><td rowspan=\\\"2\\\">RV (ml)</td><td>5327.2 (± 2598.2)</td><td>4364.4 (± 2653)</td><td>0.030</td><td>4825 (± 1427.7)</td><td>4471 (± 1354.4)</td><td>0.084</td></tr><tr><td>(n = 12)</td><td>(n = 11)</td><td></td><td>(n = 10)</td><td>(n = 10)</td><td></td></tr><tr><td rowspan=\\\"2\\\">RV (%)</td><td>271.3 (± 53)</td><td>241.3 (± 50.7)</td><td>0.20</td><td>236.5 (± 48.1)</td><td>212.1 (± 45.5)</td><td>0.08</td></tr><tr><td>(n = 11)</td><td>(n = 11)</td><td></td><td>(n = 10)</td><td>(n = 10)</td><td></td></tr><tr><td rowspan=\\\"2\\\">RV/TLC</td><td>70.3 (± 4.2)</td><td>65.1 (± 6.5)</td><td>0.15</td><td>70.4 (± 6)</td><td>65.7 (± 7.8)</td><td>0.18</td></tr><tr><td>(n = 12)</td><td>(n = 11)</td><td></td><td>(n = 10)</td><td>(n = 10)</td><td></td></tr><tr><td rowspan=\\\"2\\\">DLCO</td><td>30.6 (± 11.2)</td><td>37 (± 12.4)</td><td>0.013</td><td>29.4 (± 10.2)</td><td>30.6 (± 11)</td><td>0.18</td></tr><tr><td>(n = 14)</td><td>(n = 13)</td><td></td><td>(n = 12)</td><td>(n = 9)</td><td></td></tr></table>\",\n \"table_caption\": \"[\\\"Table 3: Pulmonary function tests in patients with primary failure and no atelectasis $(n = 14)$ and in patients with secondary failure and never or loss of atelectasis $(n = 12)$ \\\"]\",\n \"table_footnote\": \"[\\\"DLCO: diffusion capacity; FEV1: forced expiratory volume in 1 s; RV: residual volume; SD: standard deviation; TLC: total lung capacity. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[83,468,480,679]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0566/5ec60364804a24534216f870ec77eab9e37f2a96df0e5e7780eb5117eddee625.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Mean (± SD)</td><td>Pre-LVRS</td><td>3 months post-LVRS</td><td>P-value</td></tr><tr><td rowspan=\\\"2\\\">FEV1 (ml)</td><td>878.6 (± 300.1)</td><td>990 (± 306)</td><td rowspan=\\\"2\\\">0.071</td></tr><tr><td>(n = 7)</td><td>(n = 6)</td></tr><tr><td rowspan=\\\"2\\\">FEV1 (%)</td><td>28 (± 9.9)</td><td>33.1 (± 8.8)</td><td rowspan=\\\"2\\\">0.12</td></tr><tr><td>(n = 8)</td><td>(n = 7)</td></tr><tr><td rowspan=\\\"2\\\">TLC (ml)</td><td>6567.5 (± 1119.5)</td><td>6330 (± 1282.5)</td><td rowspan=\\\"2\\\">0.24</td></tr><tr><td>(n = 5)</td><td>(n = 5)</td></tr><tr><td rowspan=\\\"2\\\">TLC (%)</td><td>117.6 (± 9.2)</td><td>114.7 (± 7.2)</td><td rowspan=\\\"2\\\">0.26</td></tr><tr><td>(n = 5)</td><td>(n = 5)</td></tr><tr><td rowspan=\\\"2\\\">RV (ml)</td><td>4480 (± 622.7)</td><td>3934 (± 850.7)</td><td rowspan=\\\"2\\\">0.013</td></tr><tr><td>(n = 5)</td><td>(n = 5)</td></tr><tr><td rowspan=\\\"2\\\">RV (%)</td><td>205.3 (± 16.2)</td><td>179.9 (± 28.8)</td><td rowspan=\\\"2\\\">0.007</td></tr><tr><td>(n = 5)</td><td>(n = 5)</td></tr><tr><td rowspan=\\\"2\\\">RV/TLC</td><td>67.6 (± 5)</td><td>58.4 (± 3)</td><td rowspan=\\\"2\\\">0.004</td></tr><tr><td>(n = 7)</td><td>(n = 6)</td></tr><tr><td rowspan=\\\"2\\\">DLCO</td><td>37.9 (± 15.3)</td><td>35.8 (± 9.8)</td><td rowspan=\\\"2\\\">0.54</td></tr><tr><td>(n = 8)</td><td>(n = 5)</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 4: Pulmonary function tests in patients with (persistent) atelectasis after bronchoscopic lung volume reduction (primary or secondary failure, $n = 8$ ) \\\"]\",\n \"table_footnote\": \"[\\\"DLCO: diffusion capacity; FEV1: forced expiratory volume in 1 s; RV: residual volume; SD: standard deviation; TLC: total lung capacity. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[78,739,482,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"12 days). Three had a prolonged air leak $(37.5\\\\%)$ . At 3 months postoperatively, there was only a significant improvement of RV and RV/total lung capacity, but not in FEV1 (Table 4). In patients who had EBVs removed before LVRS $(n = 3)$ , the median LOS was 15 days (range 6-20) and the median chest tube time was 12 days (range 1-17). Two patients had a prolonged air leak. FEV1 improved from median $840\\\\mathrm{ml}$ (540-1370) to $1140\\\\mathrm{ml}$ (730-1550). Preoperative median RV was $3790\\\\mathrm{ml}$ (value only known for $n = 1$ patient), and postoperative median RV was $3795\\\\mathrm{ml}$ (range 2900-4690 ml). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,879,482,922]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In patients who did not have EBVs removed and received contralateral LVRS $(n = 5)$ , the median LOS was 5 days (range 2-11) and the median chest tube time was 3 days (range 2-10). One \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,394,915,450]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"patient had a prolonged air leak FEV1 improved from median 785 ml (range 640-1190) to 880 ml (range 780-1120). RV decreased from median 4475 ml (range 4210-5450) to 3830 (3360-4890). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,451,915,479]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The median time interval between BLVR and LVRS was 11 months (IQR 8.25-14). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,496,898,513]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Outcome of patients with high-flow fistula $(n = 4)$ \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,526,916,765]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Four patients with immediate pneumothorax after BLVR with EBVs developed a high-flow fistula. EBVs were removed during the same hospitalization (after a median of 11 days after BLVR, IQR 4.5-15.3), followed by VATS. All patients had a fistula at the non-target lobe, which was closed surgically. At the same time, LVRS was performed at the target lobe. The median postoperative LOS was 10.5 days (range 7-45). The median chest tube time was 19 days (range 2-43) with 1 patient having a postoperative chest tube time of 43 days and a second patient of 30 days. One 63-year-old female patient died because of suicide during inpatient rehabilitation 2 weeks after LVRS. One 58-year-old male patient reported no benefit at 3 months postoperatively. Complete pre- and postoperative PFTs were only available in 2 patients. The mean FEV1 increased from $550\\\\mathrm{ml}$ $(\\\\pm 210\\\\mathrm{ml})$ to $700\\\\mathrm{ml}$ $(\\\\pm 141\\\\mathrm{ml})$ and RV increased from $5470\\\\mathrm{ml}$ $(\\\\pm 720\\\\mathrm{ml})$ to $6070\\\\mathrm{ml}$ $(\\\\pm 1725\\\\mathrm{ml})$ . Diffusion capacity decreased from $25\\\\%$ $(\\\\pm 5\\\\%)$ predicted to $21\\\\%$ $(\\\\pm 5.8\\\\%)$ predicted. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[511,789,617,803]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DISCUSSION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,818,915,874]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This retrospective study aimed at investigating the postoperative outcome of patients who underwent LVRS after previous BLVR who despite initial success deteriorated or who were unsuccessful immediately after EBV treatment. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,874,916,931]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"LVRS in this sequence is not ideal and has to be considered as salvage therapy. Key issues of a successful LVRS include many aspects, but selecting the right volume and the right target area for resection in balance with physiologic parameters is of \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[318,41,679,56]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Cavigzel et al. / Interactive CardioVascular and Thoracic Surgery \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[906,43,916,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[955,106,973,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DIDARHOH \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"aside_text\"\n },\n {\n \"bbox\": \"[955,200,975,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Downloaded from https://academic.oup.com/icvts/advance-article/doi/10.1093/icvts/ivaa261/6024962 by guest on 11 December 2020 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"aside_text\"\n },\n {\n \"bbox\": \"[83,69,482,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"paramount importance. After EBVs, a lobar atelectasis may be present and valves may be still in place causing disturbance of bronchial secret clearance. Both leads to compromises in LVRS treatment. Despite this, the presented data show that the procedure can be offered in some patients without 90-day mortality and a significant improvement in lung function. This information is important for interdisciplinary emphysema boards counselling patients. Patients must know that LVRS after EBV might still be possible but is not equally effective than performed at first choice. The concept of LVRS and EBV is distinctly different despite the common goal of reducing the hyperinflation of emphysematous lungs. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,235,482,456]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"EBVs have been shown to be an effective and safe LVR procedure in several randomized controlled trials [6, 18-20]. Improvement of FEV1 by $17 - 29.3\\\\%$ from baseline can be expected [5, 6, 18, 19, 21]. Considering FEV1, there is a responder rate ranging between $47\\\\%$ at 3 months and $59\\\\%$ at 6 months [6, 18]. There are several reasons for an unsuccessful EBV treatment, which can be classified as primary failure or as fading effect over time (secondary failure). While insufficient data are available on reasons for primary failure, there are several reported reasons for the loss of efficacy over time. According to 1-year follow-up data from the STELVIO trial, permanent removal of EBVs was required in $17\\\\%$ of the patients [22]. Reasons included recurrent pneumothorax, torsion of bronchus, pneumonia and granulation tissue. Besides other factors, the latter is maybe leading to paravalvular leakage and subsequent loss of atelectasis. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,458,482,596]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In both our study centres, patients who failed to develop an atelectasis or showed a loss of atelectasis after EBVs usually received revision bronchoscopy at 1-3 months after initial BLVR. Those who still had no profit despite EBV replacement were discussed at the multidisciplinary emphysema board. Performing LVRS in some of these, there was a significant improvement of PFT values at 3 months, which was more distinct in patients with primary failure after EBV treatment. Furthermore, there is a tendency that patients, undergoing unplanned LVRS due to highflow fistula after EBV treatment, might have a poor outcome. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,597,482,694]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Compared to primary LVRS with a reported improvement of FEV1 between $41\\\\%$ and $73\\\\%$ [23-26], efficiency of LVRS in patients with previous EBV therapy seems inferior. However, our study was not designed to answer this specific question, and comparison to historical LVRS cohorts is not without challenges. At least, postoperative morbidity and mortality seem comparable to primary LVRS. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[83,694,482,929]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Usually, target lobe atelectasis after EBVs is a good indicator for a favourable outcome with improvements of dyspnoea, quality of life (QoL), 6-minute walking distance (6-MWD) and PFT values [27]. However, in this study, we found 8 patients with unsuccessful EBV treatment in spite of an existing target lobe atelectasis. Reasons for these rare cases are not completely understood. Certainly, indications for EBVs (in particular hyperinflation) must be reviewed. The question arises, if EBVs should be removed or left in situ prior to LVRS in these cases. Interestingly, all patients with persistent atelectasis but loss of clinical benefit from EBVs improved after LVRS in FEV1 and RV. According to our experience, these patients can be safely treated with contralateral LVRS, leaving EBVs in situ. Conceptually, this corresponds to results from previous studies concerning Re-LVRS, and as such, the fading benefit after EBVs reflects the natural course of pulmonary emphysema [11, 14, 28]. The questions remain open, if valves should be removed prior to LVRS. A total lobar atelectasis \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,68,913,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is a relevant intervention into the physiology of a patient with advanced emphysema. Lobectomy performed during LVRS should be reserved only to the rare cases of a total lobar destruction. In all our cases, better preserved lung parenchyma was left in place. If valves would be removed and the totally atelectatic lung would re-expand including the functioning lung parenchyma, a more balanced remodelling by LVRS would be possible [29]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,170,913,326]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"However, in patients with absent atelectasis, improvement was only significant in those with primary failure. This is surprising, since patients with secondary failure had experienced a temporary benefit after EBV treatment and thus, the principle of LVR was proven to succeed in these patients. Maybe in patients with secondary failure, the natural course of emphysema already was too advanced and prevented any LVR procedure to be beneficial again. According to our experience from these retrospective and heterogeneous data, EBV removal and subsequent LVRS are the promising options in patients with primary failure after previous EBV therapy. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,327,913,441]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Time interval between BLVR and LVRS did not differ significantly between patients with primary failure and those with secondary failure. Only in patients with primary failure, the decision for a further intervention usually was already made 3 months after BLVR. In patients with secondary failure, this decision was dependent on patient's subjective loss of benefit. Altogether, including planning time for LVRS and several other patient factors, this finding might be coincidental. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,441,913,600]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Pneumothorax occurs in $\\\\sim 15 - 30\\\\%$ of patients after BLVR with EBVs, which is usually treated with a chest tube drainage [5, 18, 30]. In $\\\\sim 70\\\\%$ of cases, these patients have a prolonged air leak continuing longer than 7 days [30]. In our study, there were 4 patients with persistent high-flow fistula despite previous EBV removal, as recommended [2]. In these patients, fistula closure by VATS followed by LVRS at the former EBV target lobe was performed. The outcome was poor. Although firm conclusions are not possible based on our small experience, combined fistula closure and LVRS might not be recommendable. Probably, a staged procedure is preferable. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,618,602,632]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Limitations \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,648,913,775]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There are some limitations to this study, essentially due to its retrospective nature and the relatively small and heterogeneous group of patients. Quality of EBV therapy was not assessed, and the definition of failure was not quantitative and, thus, only assessed by patient's statements. In addition, outcome of LVRS after EBV was only described by morbidity, mortality and PFTs, missing, i.e. 6-MWD or quality of life. Maximum benefit after LVRS is usually reached after 3-6 months, whereas our data only consist of 3-month follow-up. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,777,913,832]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The small patient's number in this study might be sufficient to point out the favourable outcome of LVRS after previous EBVs, while the conclusions drawn from the subgroup analyses might be interpreted with some caution. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,858,625,872]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CONCLUSION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,888,913,930]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"LVRS after previous BLVR with EBVs shows low morbidity and no mortality and significant clinical improvement can be expected, although results might not be as good as after primary LVRS. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[81,43,91,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[318,43,679,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Caviezel et al. / Interactive CardioVascular and Thoracic Surgery \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[957,200,975,796]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Downloaded from https://academic.oup.com/icvts/advance-article/doi/10.1093/icvts/ivaa261/6024962 by guest on 11 December 2020 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"aside_text\"\n },\n {\n \"bbox\": \"[78,68,482,165]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conflict of interest: Daniel Franzen has received speaking honoraries and consultancy fees from Pulmonx SA. Walter Wedel has received honoraries from Medtronic for teaching and proctoring. Henrik Jessen Hansen participates in advisory boards for Medtronic, BD/Bard and Medela. Mateja Ladan has received a research grant from PulmonX SA. All other authors declare no conflict of interest. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,189,256,205]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Author contributions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,218,485,426]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Claudio Caviezel: Conceptualization; Data curation; Formal analysis; Investigation; Methodology; Project administration; Resources; Supervision; Validation; Visualization; Writing—original draft; Writing—review and editing. Laura-Chiara Guglielmetti: Data curation; Formal analysis; Investigation; Methodology; Validation; Writing—original draft; Writing—review and editing. Mateja Ladan: Data curation; Formal analysis; Investigation; Methodology; Validation; Writing—original draft; Writing—review and editing. Henrik Jessen Hansen: Data curation; Formal analysis; Methodology; Validation; Visualization; Writing—review and editing. Michael Perch: Formal analysis; Methodology; Validation; Writing—review and editing. Didier Schneiter: Data curation; Investigation; Resources; Writing—original draft; Writing—review and editing. Walter Weder: Investigation; Supervision; Validation; Writing—original draft; Writing—review and editing. Isabelle Opitz: Conceptualization; Data curation; Investigation; Methodology; Project administration; Resources; Supervision; Validation; Visualization; Writing—original draft; Writing—review and editing. Daniel Franzen: Conceptualization; Formal analysis; Investigation; Methodology; Project administration; Resources; Supervision; Validation; Visualization; Writing—original draft; Writing—review and editing. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,449,264,463]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Reviewer information \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,477,483,513]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Interactive CardioVascular and Thoracic Surgery thanks Clemens Aigner and the other, anonymous reviewer(s) for their contribution to the peer review process of this article. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[81,536,187,550]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"REFERENCES \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[86,564,483,896]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"[1] van Geffen WH, Slebos DJ, Herth FJ, Kemp SV, Weder W, Shah PL. Surgical and endoscopic interventions that reduce lung volume for emphysema: a systemic review and meta-analysis. Lancet Respir Med 2019; 7:313-24. \\\",\\\"[2] Slebos DJ, Shah PL, Herth FJ, Valipour A. Endobronchial valves for endoscopic lung volume reduction: best practice recommendations from expert panel on endoscopic lung volume reduction. Respiration 2017;93: 138-50. \\\",\\\"[3] Shah PL, Herth FJ, van Geffen WH, Deslee G, Slebos DJ. Lung volume reduction for emphysema. Lancet Respir Med 2017;5:147-56. \\\",\\\"[4] Bloch KE, Georgescu CL, Russi EW, Weder W. Gain and subsequent loss of lung function after lung volume reduction surgery in cases of severe emphysema with different morphologic patterns. J Thorac Cardiovasc Surg 2002;123:845-54. \\\",\\\"[5] Valipour A, Slebos DJ, Herth F, Darwiche K, Wagner M, Ficker JH et al. Endobronchial valve therapy in patients with homogeneous emphysema. Results from the IMPACT study. Am J Respir Crit Care Med 2016; 194:1073-82. \\\",\\\"[6] Davey C, Zoumot Z, Jordan S, McNulty WH, Carr DH, Hind MD et al. Bronchoscopic lung volume reduction with endobronchial valves for patients with heterogeneous emphysema and intact interlobar fissures (the BeLieVeR-HlFi study): a randomised controlled trial. Lancet 2015; 386:1066-73. \\\",\\\"[7] Criner GJ, Cordova F, Sternberg AL, Martinez FJ. The National Emphysema Treatment Trial (NETT) Part II: lessons learned about lung volume reduction surgery. Am J Respir Crit Care Med 2011;184:881-93. \\\",\\\"[8] Gompelmann D, Benjamin N, Bischoff E, Kontogianni K, Schuhmann M, Hoffmann H et al. Survival after endoscopic valve therapy in patients with severe emphysema. Respiration 2019;97:145-52. \\\"]\",\n \"page_idx\": \"6\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[514,67,915,896]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"[9] Kouritas V, Milton R, Kefaloyannis E, Papagiannopoulos K, Brunelli A, Dimov D et al. The impact of a newly established multidisciplinary team on the interventional treatment of patients with emphysema. Clin Med Insights Circ Respir Pulm Med 2019;13:117954841985206. \\\",\\\"[10] Rathinam S, Oey I, Steiner M, Spyt T, Morgan MD, Waller DA. The role of the emphysema multidisciplinary team in a successful lung volume reduction surgery programmedagger. Eur J Cardiothorac Surg 2014;46: 1021-6. discussion 6. \\\",\\\"[11] Minai OA, Benditt J, Martinez FJ. Natural history of emphysema. Proc Am Thorac Soc 2008;5:468-74. \\\",\\\"[12] Hartman JE, Klooster K, Gortzak K, Hacken NHT, Slebos D-J. Long-term follow-up after bronchoscopic lung volume reduction treatment with coils in patients with severe emphysema. Respirology 2015;20:319-26. \\\",\\\"[13] Travaline JM, Gaughan JP, Furukawa S, Criner GJ. Effect of bilateral lung volume reduction surgery on FEV1 decline in severe emphysema. COPD 2005;2:203-8. \\\",\\\"[14] Kostron A, Horn-Tutic M, Franzen D, Kestenholz P, Schneiter D, Opitz I et al. Repeated lung volume reduction surgery is successful in selected patients. Eur J Cardiothorac Surg 2015;48:710-5. \\\",\\\"[15] Tacconi F, Pompeo E, Forcella D, Marino M, Varvaras D, Mineo TC. Lung volume reduction reoperations. Ann Thorac Surg 2008;85:1171-7. \\\",\\\"[16] Caviezel C, Aruldas C, Franzen D, Ulrich S, Inci I, Schneiter D et al. Lung volume reduction surgery in selected patients with emphysema and pulmonary hypertension. Eur J Cardiothorac Surg 2018;54:565-571. \\\",\\\"[17] Caviezel C, Schaffer N, Schneiter D, Franzen D, Inci I, Opitz I et al. Outcome after lung volume reduction surgery in patients with severely impaired diffusion capacity. Ann Thorac Surg 2018;105:379-85. \\\",\\\"[18] Klooster K, ten Hacken NH, Hartman JE, Kerstjens HA, van Rikxoort EM, Slebos DJ. Endobronchial valves for emphysema without interlobar collateral ventilation. N Engl J Med 2015;373:2325-35. \\\",\\\"[19] Kemp SV, Slebos DJ, Kirk A, Kornaszewska M, Carron K, Ek L et al. A multicenter randomized controlled trial of Zephyr Endobronchial Valve Treatment in Heterogeneous Emphysema (TRANSFORM). Am J Respir Crit Care Med 2017;196:1535-43. \\\",\\\"[20] Criner GJ, Sue R, Wright S, Dransfield M, Rivas-Perez H, Wiese T et al. A multicenter randomized controlled trial of Zephyr Endobronchial Valve Treatment in Heterogeneous Emphysema (LIBERATE). Am J Respir Crit Care Med 2018;198:1151-64. \\\",\\\"[21] Sciarba FC, Ernst A, Herth FJ, Strange C, Criner GJ, Marquette CH et al. A randomized study of endobronchial valves for advanced emphysema. N Engl J Med 2010;363:1233-44. \\\",\\\"[22] Klooster K, Hartman JE, Ten Hacken NH, Slebos DJ. One-year follow-up after endobronchial valve treatment in patients with emphysema without collateral ventilation treated in the STELVIO trial. Respiration 2017; 93:112-21. \\\",\\\"[23] Brenner M, McKenna RJ, Jr., Chen JC, Serna DL, Powell LL, Gelb AF et al. Relationship between amount of lung resected and outcome after lung volume reduction surgery. Ann Thorac Surg 2000;69:388-93. \\\",\\\"[24] Ciccone AM, Meyers BF, Guthrie TJ, Davis GE, Yusen RD, Lefrak SS et al. Long-term outcome of bilateral lung volume reduction in 250 consecutive patients with emphysema. J Thorac Cardiovasc Surg 2003;125: 513-25. \\\",\\\"[25] Tuti C, Lardinois D, Imfeld S, Korom S, Boehler A, Speich R et al. Lung-volume reduction surgery as an alternative or bridging procedure to lung transplantation. Ann Thorac Surg 2006;82:208-13. discussion 13. \\\",\\\"[26] Ginsburg ME, Thomashow BM, Bulman WA, Jellen PA, Whippo BA, Chiuzan C et al. The safety, efficacy, and durability of lung-volume reduction surgery: a 10-year experience. J Thorac Cardiovasc Surg 2016; 151:717-24 e1. \\\",\\\"[27] Herth FJ, Eberhardt R, Gompelmann D, Ficker JH, Wagner M, Ek L et al. Radiological and clinical outcomes of using Chartis to plan endobronchial valve treatment. Eur Respir J 2013;41:302-8. \\\",\\\"[28] Fiorelli A, Santini M. Bronchoscopic lung volume reduction: an alternative to repeated lung volume reduction surgery. Eur J Cardiothorac Surg 2016;50:584.1. \\\",\\\"[29] Caviezel CF, Schneiter D, Muehlmatter U, Frauenfelder T, Guglielmetti LC, Opitz I et al. Identification of target zones for lung volume reduction surgery using three dimensional computed tomography rendering. Eur Respir J Open Research 2020;6:00305-2020. \\\",\\\"[30] Gompelmann D, Herth FJ, Slebos DJ, Valipour A, Ernst A, Criner GJ et al. Pneumothorax following endobronchial valve therapy and its impact on clinical outcomes in severe emphysema. Respiration 2014;87:485-91. \\\"]\",\n \"page_idx\": \"6\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[318,43,679,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"C. Caviezel et al. / Interactive CardioVascular and Thoracic Surgery \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[906,44,916,53]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[955,106,973,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DIDAH \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"aside_text\"\n },\n {\n \"bbox\": \"[955,200,975,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Downloaded from https://academic.oup.com/icvts/advance-article/doi/10.1093/icvts/ivaa261/6024962 by guest on 11 December 2020 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"aside_text\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.1093/icvts/ivaa261"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Medicine and Health Sciences"},"sha256":{"kind":"string","value":"ea39a50f89f95e431fa3d47ed579a5eb5bae40f3b65e8677730eef4880592c6c"},"title":{"kind":"string","value":"Lung volume reduction surgery as salvage procedure after previous use of endobronchial valves"}}},{"rowIdx":65,"cells":{"abstract":{"kind":"string","value":""},"author":{"kind":"string","value":"Darwis|P Didiansari|Y Lumoindong|E B Demmallino|M A Ramlan"},"content_list":{"kind":"list like","value":[{"bbox":"[60,131,243,144]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PAPER $\\cdot$ OPEN ACCESS ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[60,157,640,204]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Effectiveness of the Bansos Rastra distribution program in Kajang District, Bulukumba Regency ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[60,219,568,232]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To cite this article: Darwis et al 2021 IOP Conf. Ser.: Earth Environ. Sci. 807 032065 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[60,274,391,286]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"View the article online for updates and enhancements. 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Engage. Champion. Empower. Accelerate. 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Prosperous Rice Social Assistance (BansosRastra) is a program organized by the Indonesian government that aims to increase and open up access for poor families to food, especially rice. The effectiveness of BansosRastra distribution can be assessed through the achievement of the 6R targets (six right), namely the right target recipient, the right amount, the right price, the right time, the right administration and the right quality. The large number of poor households in Kajang District, Bulukumba Regency requires an evaluation of the effectiveness of the BansosRastra program so that the program runs optimally and avoids conflicts related to aid distribution. This study aims to analyze the effectiveness of the distribution of the BansosRastra program in Kajang District, Bulukumba Regency, South Sulawesi. This research uses quantitative methods. The results of the study concluded that the implementation of the BansosRastra Program in Kajang District, Bulukumba Regency was classified as ineffective. The level of effectiveness of the implementation of the BansosRastraProgram is low, at only $42.37\\%$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,623,242,637]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1. Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[112,638,880,697]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Food is a basic human need that is essential and must be fulfilled. Even food is the right of every human being. So important that it can be said that food is a milestone in life in a region, as well as in a country. Meeting people's needs is the responsibility and obligation of the government. Especially in Indonesia, which has a large area and a large population, it is very prone to food crises [1]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,699,880,772]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Currently, Indonesia is still facing the problem of poverty and food insecurity that must be tackled together by the government and society because the number of Indonesians who can afford to buy rice at normal market prices is still very limited. This is because there are still many people with low incomes so that it affects their purchasing power for food needs. Therefore, the government needs to make a policy to regulate and control national food security. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,772,880,878]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Prior to 1998, rice policies in Indonesia included a ceiling price, which was the limit for the highest consumer price limit, so that the government carried out Pure Market Operations (PMO) to reduce the price of rice that was charged by the public. This subsidy policy in the price of rice is given to all levels of society, both poor and capable (general subsidies). Since July 1998 with the onset of the economic crisis, the rice subsidy policy has been specifically given to certain groups of people (targeted subsidies) through Special Market Operations (SMO). With the change in subsidy policy from general targeted to targeted subsidy, subsidies are only given to the poor [2]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,879,880,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"After several years of running this program, in 2002 the name of the program was changed to Raskin (Rice for Poor Families) as an effort to shaIDRen the meaning of social welfare programs to ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,64,226,77]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IC-FSSAT 2021 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[774,64,880,79]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Publishing ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[112,80,880,96]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Conf. Series: Earth and Environmental Science 807 (2021) 032065 doi:10.1088/1755-1315/807/3/032065 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[112,913,198,934]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CC BY ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[198,913,880,936]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[112,938,371,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Published under licence by IOP Publishing Ltd ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[497,940,507,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[115,133,880,223]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"increase and open access for poor families to food, especially rice. This name was also followed by a change in the mechanism for distributing aid and targeting beneficiaries. Raskin allocation for each district and city is determined based on the calculation of the poor population by the Central Bureau of Statistics (CBS). In each district and city, the local government again determines the allocation for each distribution point. At the lowest level, the village government determines which households are entitled to receive Raskin [3]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,223,880,329]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In 2015, the Raskin program changed its name to Rastra (Beras Sejahtera). Rastra is the same program as Raskin, but what sets it apart is its thought patterns. After two years of the Raskin program, from 2017 to 2018, the government gradually began to change the Rastra subsidy program scheme from a subsidy pattern to a food social assistance pattern in order to increase the effectiveness and accuracy of program targeting, in accordance with the mandate of the Indonesian President at a Limited Cabinet Meeting regarding the Poverty and Economic Inequality Reduction Program on March 16 2016, so that social assistance and subsidies are distributed non-cash [4]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,330,880,508]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Food social assistance can be distributed in the form of in-kind (rice) or in non-cash form. Food social assistance in the form of in kind (rice) or known as Prosperous Rice Social Assistance (Bansos Rastra) is a social assistance program in the form of medium quality rice from the government to BRF as much as $10\\mathrm{kg}$ and is distributed every month without any redemption price / fee. Food assistance in the form of non-cash or referred to as Non-Cash Food Assistance (NCFA), which is a social food assistance program, in which BRF receives social assistance of IDR 110.000 which is transferred each month by the government through an electronic money mechanism which is used only to purchase food in the form of rice, sugar, flour, cooking oil, etc. in foodstuff traders or called e-warong in cooperation with distribution banks, according to the quantity and quality and at the desired time. NCFA has a broader objective, namely to provide more balanced nutrition, provide choice, control and access to financial services for the poor and vulnerable, encourage people's retail businesses, and streamline their budgets [4]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,508,880,645]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"One of the districts that has the highest number of poor families in South Sulawesi Province is Kajang District, Bulukumba Regency, which has 3,885 poor families who are entitled to receive the Bansos Rastra Program. With the largest number of poor households, the effectiveness of the implementation of the Bansos Rastra Program in Kajang District, Bulukumba Regency must be evaluated so that this program runs optimally and avoids conflicts or problems related to the distribution of Bansos Rastra so that indicators of success can be achieved. Because if the effectiveness of the Bansos Rastra Program is low, it will affect the survival of poor families and their access to food, especially rice. Based on this, this study aims to analyze the effectiveness of social assistance program distribution in Kajang District, Bulukumba Regency. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,659,285,671]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2. Research methods ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[112,673,880,762]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This research was conducted in Kajang District, Bulukumba Regency, South Sulawesi Province in March 2019. The population in this study were people who received assistance from the BansosRastra Program from the local government as many as 3.885 families with a total sample size of 131 peoples. The sample was determined using Proportionate Stratified Random Sampling because Kajang District has 17 villages and two sub-districts where each village in Kajang District has different numbers of poor families. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,763,880,822]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The effectiveness of the distribution of the BansosRastra Program will be analyzed using quantitative descriptive methods. The quantitative method used is descriptive statistics with the aim of describing the social symptoms related to the effectiveness of the BansosRastra Program in Kajang District. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,853,880,884]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Knowing the level of effectiveness of the Bansos Rastra Program at each BRF, it is calculated using the following formula [5]: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,64,226,77]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IC-FSSAT 2021 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[774,64,880,79]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Publishing ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[112,80,880,93]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Conf. Series: Earth and Environmental Science 807 (2021) 032065 doi:10.1088/1755-1315/807/3/032065 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[497,939,507,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[305,133,690,160]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\text {Program effectiveness} = \\frac {\\text {Realization} x 100 \\%}{\\text {Target}} (1)\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[137,166,235,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"information: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[137,180,603,195]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Realization = The amount of accuracy that BRF achieved; ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[137,195,747,211]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Target = The sum of all the accuracy that the BRF should have achieved. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,225,880,255]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The success rate category of distribution effectiveness according to the Ministry of Home Affairs Research and Development, namely [6]: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[137,256,640,316]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1) Value effectiveness coefficient $< 40\\\\%$ : Very ineffective; \",\"2) Value effectiveness coefficient $40 - 60\\\\%$ : Ineffective; \",\"3) Value effectiveness coefficient $61 - 80\\\\%$ : Effective enough; \",\"4) Value effectiveness coefficient $>80\\\\%$ : Very effective. \"]","page_idx":"3","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[112,330,319,344]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3. Results and discussion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[112,344,880,391]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The effectiveness of the distribution of BansosRastra can be assessed through the achievement of the 6R (six right) targets, namely the right target recipient, the right amount, the right price, the right time, the right administration and the right quality [7]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,404,270,419]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.1. Right on target ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[112,419,880,479]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Bansos Rastra program is aimed at poor or food insecure (low income) families who cannot afford to buy rice at normal market prices due to their low income levels, and the increasingly expensive prices of basic necessities[8]. Poor families who are entitled to receive the Bansos Rastra Program are then called BRF (Benefit Recipients Families). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,480,882,734]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In determining whether a family is categorized as poor or not, CBS uses 14 feasible and operational indicators to determine whether the family is eligible and entitled to receive assistance from the Bansos Rastra Program. The indicators are 1) the floor area of the building where he lives is less than $8\\mathrm{m}^2$ per person; 2) the floor of the building where he lives is made of cheap soil / bamboo / wood; 3) the walls of the buildings he lives in are made of bamboo / thatch / low quality wood or walls without plaster; 4) does not have a defecation facility / another household uses one latrine; 5) household lighting sources do not use electricity; 6) drinking water comes from unprotected wells / springs / rivers / rainwater; 7) fuel for daily cooking is firewood / charcoal / kerosene; 8) only consume meat / milk / chicken once a week; 9) only buy one new set of clothes a year; 10) can only eat once / twice a day; 11) unable to pay for medical expenses at the puskesmas / polyclinic; 12) income or income of the household head below IDR 600.000 per month; 13) The last education of the head of the household is not going to school / not completing elementary school (ES) / only ES; 14) do not have savings or items that can be easily sold with a minimum value of IDR 500.000. A family is categorized as poor if the family meets a minimum of nine poverty indicators. Therefore, the Bansos Rastra Program is said to be right on target if the sample families have a minimum of 9 (nine) out of 14 (fourteen) poverty indicators [9]. The results of data analysis regarding the accuracy of targeting the beneficiary families of the Bansos Rastra program can be seen in table 1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[200,784,798,848]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0657/0daf76952b20116b29d0bd9b2b49414798706637c1adb72625f4576e3f61f486.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":"<table><tr><td>No.</td><td>Information</td><td>Number of BRF</td><td>Percentage</td></tr><tr><td>1.</td><td>Right on target (≥ 9 indicators)</td><td>79</td><td>60 %</td></tr><tr><td>2.</td><td>Not on target (< 9 indicators)</td><td>52</td><td>40 %</td></tr><tr><td colspan=\"2\">Total</td><td>131</td><td>100 %</td></tr></table>","table_caption":"[\"Table 1. The accuracy of targeting beneficiary families of the Bansos Rastra program in Kajang District, Bulukumba Regency, 2018. \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[112,848,882,895]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Based on the data in table 1, it shows that not all BRF who received Bansos Rastra were included in the right target category, namely the fulfillment of at least nine poverty indicators. The results of data analysis show that the BRF which is included in the right target category, namely the fulfillment ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,64,226,77]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IC-FSSAT 2021 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[774,64,880,79]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Publishing ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[112,80,880,93]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Conf. Series: Earth and Environmental Science 807 (2021) 032065 doi:10.1088/1755-1315/807/3/032065 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[497,939,507,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[110,133,882,239]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"of a minimum of nine poverty indicators is as many as 79 BRF (60%) of the sampled families, while 52 BRF (40%) is included in the category not on target because it has less than nine poverty indicators. This happens because there are two distribution systems for Bansos Rastra in villages or sub-districts in Kajang District, namely implementing an even distribution system for all families in the village without referring to the BRF list established by the Ministry of Social Affairs and distributing Bansos Rastra assistance to families registered in the list BRF which has been established by the Ministry of Social Affairs. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,253,257,268]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.2. Right amount ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[110,268,880,328]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The allocation of social assistance in the form of rice provided to BRF is $10\\mathrm{kg}$ per BRF per month or equivalent to $120\\mathrm{kg}$ per BRF in 2018[4]. The results of data analysis regarding the accuracy of the amount of social assistance in the form of rice received by beneficiary families of the Bansos Rastra program in Kajang District can be seen in table 2. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[200,384,800,467]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0447/90cba5a378e358a9c1a93bebcfdbda2a231124a86b852e3557dbab755b41faea.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":"<table><tr><td>No.</td><td>Information</td><td>Number of BRF</td><td>Percentage</td></tr><tr><td>1.</td><td>Right amount (≥ 10 kg per BRF)</td><td>47</td><td>36 %</td></tr><tr><td>2.</td><td>Not right amount (<10 kg per BRF)</td><td>84</td><td>64 %</td></tr><tr><td colspan=\"2\">Total</td><td>131</td><td>100 %</td></tr></table>","table_caption":"[\"Table 2. The accuracy of the amount of social assistance in the form of rice received by beneficiary families of the Bansos Rastra Program in Kajang District, Bulukumba Regency, 2018. \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[110,472,880,563]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Table 2 shows that the families who received Bansos Rastra were included in the right category, namely receiving social assistance in the form of rice from the government at $10\\mathrm{kg}$ per BRF every month or the equivalent of $120\\mathrm{kg}$ per BRF in 2018 was 47 BRF $(36\\%)$ , while the family 84 BRF $(64\\%)$ received assistance of less than $10\\mathrm{kg}$ per month or less than $120\\mathrm{kg}$ in 2018. This shows that the implementation of the Bansos Rastra distribution program in Kajang District is in the inaccurate category. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,577,238,590]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.3. Right price ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[110,592,880,697]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Implementation of the Bansos Rastra Program which is distributed to BRF without being charged a price or redemption fee[4]. The exact price provisions are also used as an indicator of price accuracy in this study, namely the Bansos Rastra program is said to be right in price if BRF does not pay or does not pay fees in redeeming Bansos Rastra, and the Bansos Rastra Program is said to be not the right price if BRF in redeeming Bansos Rastra is charged a fee. The results of data analysis regarding the accuracy of prices charged to beneficiary families of the Bansos Rastra program in Kajang District can be seen in table 3. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[200,753,798,820]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0485/78768d35cf71e21032e853b30a3f2f0268ee50bede0370993d2ecd3d9773e501.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":"<table><tr><td>No.</td><td>Information</td><td>Number of BRF</td><td>Percentage</td></tr><tr><td>1.</td><td>Right price</td><td>116</td><td>89 %</td></tr><tr><td>2.</td><td>Not right price</td><td>15</td><td>11 %</td></tr><tr><td></td><td>Total</td><td>131</td><td>100 %</td></tr></table>","table_caption":"[\"Table 3. The accuracy of the Bansos Rastra price charged to beneficiary families of the Bansos Rastra program in Kajang District, Bulukumba Regency, 2018. \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[110,833,880,894]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Table 3 shows that there are BRFs who received Bansos Rastra including the right price category, namely not paying in redeeming Bansos Rastra as much as 116 BRF (89%), while the other 15 BRF were categorized as incorrect prices. This happened because the implementers in several villages in Kajang District applied a voluntary fee at the time of redemption of Bansos Rastra. The costs referred ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,64,226,77]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IC-FSSAT 2021 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[774,64,880,79]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Publishing ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[112,80,880,93]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Conf. Series: Earth and Environmental Science 807 (2021) 032065 doi:10.1088/1755-1315/807/3/032065 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[497,939,507,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[110,134,880,164]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"to are money for transporting rice, food and drinking fees for field officers, paid as much as they can to the distribution operator. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[110,164,880,254]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"When viewed based on the prevailing laws and regulations and guidelines for the distribution of the Bansos Rastra Program, this violates because it is not in accordance with what has been determined. Everything related to the issue of costs in the distribution of the Bansos Rastra Program, has been borne by the Bulukumba Regency APBD, the South Sulawesi Provincial APBD, and the National Budget at the Ministry of Social Affairs, so the cost at the time of redemption of Bansos Rastra to BRF is completely free in this case. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,268,233,282]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.4. Right time ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[110,282,880,388]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The implementation of the Bansos Rastra Program is said to be right time if the implementation time for the distribution of the Bansos Rastra Program to BRF is in accordance with the distribution plan. In 2018, the distribution time for Bansos Rastra is planned for 12 times or held every month. The time and procedure for distribution is adjusted to the situation and conditions as well as the needs of the area[4]. The indicator of timeliness in this study is if BRF receives Bansos Rastra every month or 12 times a year. The results of data analysis regarding the timeliness of distributing Bansos Rastra to beneficiary families of the Bansos Rastra program in Kajang District can be seen in table 4. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[200,445,796,511]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0122/41495b1ec713b6bff360d201e0766f7335acd3878f042cbdafc09ed4668af20f.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":"<table><tr><td>No.</td><td>Information</td><td>Number of BRF</td><td>Percentage</td></tr><tr><td>1.</td><td>Right time</td><td>11</td><td>8 %</td></tr><tr><td>2.</td><td>Not right time</td><td>120</td><td>92 %</td></tr><tr><td></td><td>Total</td><td>131</td><td>100 %</td></tr></table>","table_caption":"[\"Table 4. The timeliness of the distribution of the Bansos Rastra program to beneficiary families of the Bansos Rastra program in Kajang District, Bulukumba Regency, 2018. \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[110,517,880,652]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Table 4 shows that the majority of BRF who received Bansos Rastra in Kajang District were categorized as not on time, namely that the distribution of Bansos Rastra Program to BRF was almost never carried out every month. This happened because at the beginning of 2018 the distribution of the Bansos Rastra Program in Bulukumba District experienced obstacles at the Ministry of Social Affairs so that it had an impact on disbursement in Kajang District. In addition, the Bansos Rastra transportation which is a third party from Perum BULOG only provides one transportation for each district. So that Bulukumba Regency only has one transportation that will transport Bansos Rastra from the BULOG warehouse to all distribution points in each sub-district in Bulukumba Regency in turn. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,665,253,682]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.5. Right quality ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[110,682,880,757]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The right quality is defined as the fulfillment of the rice quality requirements according to the policy for the procurement of grain / rice and the distribution of rice by the government, namely the rice distributed is of medium quality with the rules that are not broken, odorless, colorless and not lice.[4]. The results of data analysis regarding the accuracy of Bansos Rastra quality for beneficiary families of the Bansos Rastra program in Kajang District can be seen in table 5. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,64,226,77]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IC-FSSAT 2021 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[774,64,880,79]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Publishing ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[112,80,880,93]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Conf. Series: Earth and Environmental Science 807 (2021) 032065 doi:10.1088/1755-1315/807/3/032065 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[497,939,507,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[200,183,796,250]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0804/8dbca933f040381727d198cc29b9e87cde629d2a94b82895083bd5d540bdd3b8.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":"<table><tr><td>No.</td><td>Information</td><td>Number of BRF</td><td>Percentage</td></tr><tr><td>1.</td><td>Right quality</td><td>56</td><td>43 %</td></tr><tr><td>2.</td><td>Not right quality</td><td>75</td><td>57 %</td></tr><tr><td></td><td>Total</td><td>131</td><td>100 %</td></tr></table>","table_caption":"[\"Table 5. The Accuracy of the quality of the Bansos Rastra program to beneficiary families of the Bansos Rastra program in Kajang District, Bulukumba Regency, 2018. \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[112,255,880,347]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Table 5 shows that the implementation of the Bansos Rastra Distribution Program in Kajang District is included in the category of inappropriate quality. There are 75 BRF or $57\\%$ getting rice with poor quality, namely broken, colored, smelly and lice. Based on the recognition from Bulog Bulukumba, the cause of damage to Bansos Rastra's rice was due to the long time span of storing rice in the warehouse which resulted in the rice being damaged due to fungus. Perum Bulog did not conduct a thorough rice quality check because the amount of assistance was quite large. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,353,310,366]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.6. Right administration ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[110,367,880,458]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Administrative accuracy is defined as the correct and complete fulfillment of administrative requirements[4]. The administrative right indicator in this study is that every time BRF redeems Bansos Rastra, BRF shows the BRF Card or other card that has been determined by the village government as evidence of being registered in the list of beneficiaries as recipients of Bansos Rastra. The results of data analysis regarding the administrative accuracy of recipient families to redeem Bansos Rastra in Kajang District can be seen in table 6. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[200,500,796,565]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0158/c4d75efd0ba68be13636d3296a6301dbc1177fc9ac383255de435e960463f5a2.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":"<table><tr><td>No.</td><td>Information</td><td>Number of BRF</td><td>Percentage</td></tr><tr><td>1.</td><td>Right administration</td><td>24</td><td>18 %</td></tr><tr><td>2.</td><td>Not right administration</td><td>107</td><td>82 %</td></tr><tr><td></td><td>Total</td><td>131</td><td>100 %</td></tr></table>","table_caption":"[\"Table 6. Administrative accuracy of beneficiary families to redeem Bansos Rastra in Kajang District, Bulukumba Regency, 2018. \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[112,571,880,722]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Table 6 shows that the administrative process in the implementation of the distribution of the Bansos Rastra program in Kajang District is included in the administration inappropriate category. There are $82\\%$ or 107 BRF who redeem Bansos Rastra in the wrong category by showing someone else's BRF card to redeem Bansos Rastra or even not bringing or showing the Bansos Rastra recipient card to pick up rice at the distribution point. This happened because there were several areas in Kajang District when distributing Bansos Rastra, they only signed the BRF as proof of receiving assistance, without showing the BRF card that had been assigned by the village government. Regions that do not carry out administration assume that the administrative process has been carried out when the Bansos Rastra program was first distributed, namely the process of family cards or KTPs to show that the BRF is registered on the Beneficiary List. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,736,583,751]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.7. Analysis effectiveness level of the Bansos Rastra program ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[112,751,880,812]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Referring to the results of the 6R target analysis achieved by BRF Bansos Rastra, an overview of the frequency distribution regarding the effectiveness of the Bansos Rastra Program in Kajang District is obtained. The analysis of the proportion of the effectiveness level of the implementation of the Bansos Rastra Program in Kajang District, namely: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,64,226,77]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IC-FSSAT 2021 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[774,64,880,79]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Publishing ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[112,80,880,93]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Conf. Series: Earth and Environmental Science 807 (2021) 032065 doi:10.1088/1755-1315/807/3/032065 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[497,940,507,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[307,133,689,158]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\text {Program effectiveness} = \\frac {\\text {Realization} x 100 \\%}{\\text {Target}} (2)\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[396,159,600,173]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\text {P r o g r a m E f f e c t i v e n e s s}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[115,173,815,244]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\begin{array}{l} = \\frac {(\\sum t a r g e t + \\sum a m o u n t + \\sum p r i c e + \\sum t i m e + \\sum q u a l i t y + \\sum a d m i n s t r a t i o n) x 100 \\%}{T a r g e t} \\\\ = \\frac {3 3 3 \\times 1 0 0 \\%}{7 8 6} \\\\ = 42.37 \\% \\\\ \\end{array}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[112,252,211,265]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"information: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,267,537,282]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Realization = Total number of BRFs achieving accuracy; ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,282,692,298]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Target $=$ The total number of BRFs that reached their supposed precision. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[221,298,410,311]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(131 families x 6 = 786). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,319,880,425]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It can be concluded that the implementation of the Bansos Rastra Program in Kajang District, Bulukumba Regency is classified as ineffective. It can be seen in the percentage results of the effectiveness level of the implementation of the Bansos Rastra program only $42.37\\%$ . According to the Ministry of Home Affairs Research and Development, the success rate category of distribution effectiveness, the percentage is in the range of $40 - 60\\%$ is in the ineffective category [6]. This can be seen clearly in the timeliness indicator, where the majority of BRF Bansos Rastra do not receive assistance every month or do not comply with the predetermined plans. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,439,230,451]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4. Conclusion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[112,453,880,514]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The implementation of the BansosRastra program in Kajang District, Bulukumba Regency is classified as ineffective. The effectiveness level of the implementation of the BansosRastra Program is only $42.37\\%$ , this is because the 6R has not been achieved in its implementation, namely the right target, the right amount, the right price, the right time, the right quality and the right administration. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,527,205,541]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[115,543,880,843]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"[1] Harmoko P 2010 Efektivitas Program Raskin Di Kecamatan Banjarsari Kota Surakarta Tahun 2009 \",\"[2] Sudarmono 2006 Politik Beras dan Ketahanan Bangsa (Jakarta) \",\"[3] Kementerian Koordinator Bidang Kesegahteraan Rakyat Republik Indonesia 2014 Pedoman Umum Raskin 2014 \",\"[4] Kementerian Koordinator Bidang Kesjahteraan Rakyat Republik Indonesia 2018 Pedoman Umum Bantuan Sosial Beras Sejahtera 2018 \",\"[5] Dewi I A C, Budhi M K S and Marhaeni A A I. 2012 Efektivitas program Jaminan Kesehatan Bali Mandara (JKBM) di Kecamatan Gianyar Kabupaten Gianyar E-Jurnal Ekon. dan Bisnis Univ. Udayana1 \",\"[6] Arnaya P Y and Utama M S 2012 Efektivitas kegiatan simpan pinjam khusus bagi kaum perempuan rumah tangga miskin di Kecamatan Denpasar Timur E-Jurnal Ekon. Pembang. Univ. Udayana1 44409 \",\"[7] Hastuti 2012 Tinjauan Efektivitas Pelaksanaan Raskin dalam Mencapai Enam Tepat (SMERU Research Institute) \",\"[8] Winarni W 2013 Study Implementasi program raskin Di Desa Kebumen Kecamatan Sukorejo Kabupaten Kendal Serat Acitya1 27 \",\"[9] Isdijoso W, Suryahadi A and Akhmadi A 2016 Penetapan kriteria dan variabel pendataan penduduk miskin yang komprehensif dalam rangka perlindungan penduduk miskin di Kabupaten/Kota Kertas Kerja SMERU,(September) 1-16 \"]","page_idx":"7","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[112,64,226,77]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IC-FSSAT 2021 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[774,65,880,79]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Publishing ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[112,80,880,93]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Conf. Series: Earth and Environmental Science 807 (2021) 032065 doi:10.1088/1755-1315/807/3/032065 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[497,939,507,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7 ","text_format":null,"text_level":null,"type":"page_number"}],"string":"[\n {\n \"bbox\": \"[60,131,243,144]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PAPER $\\\\cdot$ OPEN ACCESS \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,157,640,204]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Effectiveness of the Bansos Rastra distribution program in Kajang District, Bulukumba Regency \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,219,568,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To cite this article: Darwis et al 2021 IOP Conf. Ser.: Earth Environ. Sci. 807 032065 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,274,391,286]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"View the article online for updates and enhancements. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[673,131,838,149]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"You may also like \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[699,157,932,271]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"- Farmer performance in beef cattle production Intibona Village, Indonesia A A Amrawaty, Hastang and Sumarni \\\",\\\"- The strategy of strengthening business of salt-boiled fish scad (Decapterus russelli) in Bulukumba Regency \\\\nA M Ichwan, S S Adhawati and Jusni \\\",\\\"- Analysis of corn agribusiness revenue in Bulukumba Regency \\\\nH Hatima, A R Siregar and Jusni \\\"]\",\n \"page_idx\": \"0\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[84,779,156,815]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0770/03a2256e2fd4ebe7240225c0d8edf1f0306497d3a7c7d23f6e1ce01349439f33.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[157,785,410,802]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Electrochemical Society \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[159,801,410,809]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Advancing solid state & electrochemical science & technology \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[159,815,410,834]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"242nd ECS Meeting \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[159,839,440,855]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Oct 9 - 13, 2022 • Atlanta, GA, US \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[157,863,531,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract submission deadline: April 8, 2022 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[157,889,583,904]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Connect. Engage. Champion. Empower. Accelerate. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[159,913,410,927]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"MOVE SCIENCE FORWARD \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[468,802,529,846]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0426/d57dfed65bef5536486aa594f73d1f5f3bb2243569267dd0e29fd1b83dfc4521.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[534,815,721,831]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Submit your abstract \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[692,771,937,855]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0505/70aeef5afa3f5119e65dde8d4abd5e915ed0df62119402ad29168a87955a43e7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[690,856,936,938]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0339/d877d25a6e2d7164fc6990dd131c87677aa9fd9bfc97103622855145217577b8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[60,42,586,58]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Conference Series: Earth and Environmental Science \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[115,228,835,275]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Effectiveness of the Bansos Rastra distribution program in Kajang District, Bulukumba Regency \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[233,307,815,324]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Darwis, P Didiansari, Y Lumoindong, E B Demmallino and M A Ramlan \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[233,329,811,360]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Agribusiness Study Program, Faculty of Agriculture, Hasanuddin University, Makassar, 90245, Indonesia \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[233,388,470,405]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"E-mail: darwisali@yahoo.co.id \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[233,418,882,596]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract. Prosperous Rice Social Assistance (BansosRastra) is a program organized by the Indonesian government that aims to increase and open up access for poor families to food, especially rice. The effectiveness of BansosRastra distribution can be assessed through the achievement of the 6R targets (six right), namely the right target recipient, the right amount, the right price, the right time, the right administration and the right quality. The large number of poor households in Kajang District, Bulukumba Regency requires an evaluation of the effectiveness of the BansosRastra program so that the program runs optimally and avoids conflicts related to aid distribution. This study aims to analyze the effectiveness of the distribution of the BansosRastra program in Kajang District, Bulukumba Regency, South Sulawesi. This research uses quantitative methods. The results of the study concluded that the implementation of the BansosRastra Program in Kajang District, Bulukumba Regency was classified as ineffective. The level of effectiveness of the implementation of the BansosRastraProgram is low, at only $42.37\\\\%$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,623,242,637]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1. Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,638,880,697]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Food is a basic human need that is essential and must be fulfilled. Even food is the right of every human being. So important that it can be said that food is a milestone in life in a region, as well as in a country. Meeting people's needs is the responsibility and obligation of the government. Especially in Indonesia, which has a large area and a large population, it is very prone to food crises [1]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,699,880,772]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Currently, Indonesia is still facing the problem of poverty and food insecurity that must be tackled together by the government and society because the number of Indonesians who can afford to buy rice at normal market prices is still very limited. This is because there are still many people with low incomes so that it affects their purchasing power for food needs. Therefore, the government needs to make a policy to regulate and control national food security. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,772,880,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Prior to 1998, rice policies in Indonesia included a ceiling price, which was the limit for the highest consumer price limit, so that the government carried out Pure Market Operations (PMO) to reduce the price of rice that was charged by the public. This subsidy policy in the price of rice is given to all levels of society, both poor and capable (general subsidies). Since July 1998 with the onset of the economic crisis, the rice subsidy policy has been specifically given to certain groups of people (targeted subsidies) through Special Market Operations (SMO). With the change in subsidy policy from general targeted to targeted subsidy, subsidies are only given to the poor [2]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,879,880,909]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"After several years of running this program, in 2002 the name of the program was changed to Raskin (Rice for Poor Families) as an effort to shaIDRen the meaning of social welfare programs to \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,64,226,77]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IC-FSSAT 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[774,64,880,79]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Publishing \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[112,80,880,96]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Conf. Series: Earth and Environmental Science 807 (2021) 032065 doi:10.1088/1755-1315/807/3/032065 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[112,913,198,934]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CC BY \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[198,913,880,936]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[112,938,371,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Published under licence by IOP Publishing Ltd \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[497,940,507,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[115,133,880,223]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"increase and open access for poor families to food, especially rice. This name was also followed by a change in the mechanism for distributing aid and targeting beneficiaries. Raskin allocation for each district and city is determined based on the calculation of the poor population by the Central Bureau of Statistics (CBS). In each district and city, the local government again determines the allocation for each distribution point. At the lowest level, the village government determines which households are entitled to receive Raskin [3]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,223,880,329]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In 2015, the Raskin program changed its name to Rastra (Beras Sejahtera). Rastra is the same program as Raskin, but what sets it apart is its thought patterns. After two years of the Raskin program, from 2017 to 2018, the government gradually began to change the Rastra subsidy program scheme from a subsidy pattern to a food social assistance pattern in order to increase the effectiveness and accuracy of program targeting, in accordance with the mandate of the Indonesian President at a Limited Cabinet Meeting regarding the Poverty and Economic Inequality Reduction Program on March 16 2016, so that social assistance and subsidies are distributed non-cash [4]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,330,880,508]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Food social assistance can be distributed in the form of in-kind (rice) or in non-cash form. Food social assistance in the form of in kind (rice) or known as Prosperous Rice Social Assistance (Bansos Rastra) is a social assistance program in the form of medium quality rice from the government to BRF as much as $10\\\\mathrm{kg}$ and is distributed every month without any redemption price / fee. Food assistance in the form of non-cash or referred to as Non-Cash Food Assistance (NCFA), which is a social food assistance program, in which BRF receives social assistance of IDR 110.000 which is transferred each month by the government through an electronic money mechanism which is used only to purchase food in the form of rice, sugar, flour, cooking oil, etc. in foodstuff traders or called e-warong in cooperation with distribution banks, according to the quantity and quality and at the desired time. NCFA has a broader objective, namely to provide more balanced nutrition, provide choice, control and access to financial services for the poor and vulnerable, encourage people's retail businesses, and streamline their budgets [4]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,508,880,645]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"One of the districts that has the highest number of poor families in South Sulawesi Province is Kajang District, Bulukumba Regency, which has 3,885 poor families who are entitled to receive the Bansos Rastra Program. With the largest number of poor households, the effectiveness of the implementation of the Bansos Rastra Program in Kajang District, Bulukumba Regency must be evaluated so that this program runs optimally and avoids conflicts or problems related to the distribution of Bansos Rastra so that indicators of success can be achieved. Because if the effectiveness of the Bansos Rastra Program is low, it will affect the survival of poor families and their access to food, especially rice. Based on this, this study aims to analyze the effectiveness of social assistance program distribution in Kajang District, Bulukumba Regency. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,659,285,671]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2. Research methods \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,673,880,762]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This research was conducted in Kajang District, Bulukumba Regency, South Sulawesi Province in March 2019. The population in this study were people who received assistance from the BansosRastra Program from the local government as many as 3.885 families with a total sample size of 131 peoples. The sample was determined using Proportionate Stratified Random Sampling because Kajang District has 17 villages and two sub-districts where each village in Kajang District has different numbers of poor families. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,763,880,822]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The effectiveness of the distribution of the BansosRastra Program will be analyzed using quantitative descriptive methods. The quantitative method used is descriptive statistics with the aim of describing the social symptoms related to the effectiveness of the BansosRastra Program in Kajang District. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,853,880,884]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Knowing the level of effectiveness of the Bansos Rastra Program at each BRF, it is calculated using the following formula [5]: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,64,226,77]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IC-FSSAT 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[774,64,880,79]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Publishing \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[112,80,880,93]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Conf. Series: Earth and Environmental Science 807 (2021) 032065 doi:10.1088/1755-1315/807/3/032065 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[497,939,507,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[305,133,690,160]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\text {Program effectiveness} = \\\\frac {\\\\text {Realization} x 100 \\\\%}{\\\\text {Target}} (1)\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[137,166,235,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"information: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[137,180,603,195]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Realization = The amount of accuracy that BRF achieved; \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[137,195,747,211]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Target = The sum of all the accuracy that the BRF should have achieved. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,225,880,255]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The success rate category of distribution effectiveness according to the Ministry of Home Affairs Research and Development, namely [6]: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[137,256,640,316]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1) Value effectiveness coefficient $< 40\\\\\\\\%$ : Very ineffective; \\\",\\\"2) Value effectiveness coefficient $40 - 60\\\\\\\\%$ : Ineffective; \\\",\\\"3) Value effectiveness coefficient $61 - 80\\\\\\\\%$ : Effective enough; \\\",\\\"4) Value effectiveness coefficient $>80\\\\\\\\%$ : Very effective. \\\"]\",\n \"page_idx\": \"3\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[112,330,319,344]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3. Results and discussion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,344,880,391]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The effectiveness of the distribution of BansosRastra can be assessed through the achievement of the 6R (six right) targets, namely the right target recipient, the right amount, the right price, the right time, the right administration and the right quality [7]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,404,270,419]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.1. Right on target \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,419,880,479]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Bansos Rastra program is aimed at poor or food insecure (low income) families who cannot afford to buy rice at normal market prices due to their low income levels, and the increasingly expensive prices of basic necessities[8]. Poor families who are entitled to receive the Bansos Rastra Program are then called BRF (Benefit Recipients Families). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,480,882,734]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In determining whether a family is categorized as poor or not, CBS uses 14 feasible and operational indicators to determine whether the family is eligible and entitled to receive assistance from the Bansos Rastra Program. The indicators are 1) the floor area of the building where he lives is less than $8\\\\mathrm{m}^2$ per person; 2) the floor of the building where he lives is made of cheap soil / bamboo / wood; 3) the walls of the buildings he lives in are made of bamboo / thatch / low quality wood or walls without plaster; 4) does not have a defecation facility / another household uses one latrine; 5) household lighting sources do not use electricity; 6) drinking water comes from unprotected wells / springs / rivers / rainwater; 7) fuel for daily cooking is firewood / charcoal / kerosene; 8) only consume meat / milk / chicken once a week; 9) only buy one new set of clothes a year; 10) can only eat once / twice a day; 11) unable to pay for medical expenses at the puskesmas / polyclinic; 12) income or income of the household head below IDR 600.000 per month; 13) The last education of the head of the household is not going to school / not completing elementary school (ES) / only ES; 14) do not have savings or items that can be easily sold with a minimum value of IDR 500.000. A family is categorized as poor if the family meets a minimum of nine poverty indicators. Therefore, the Bansos Rastra Program is said to be right on target if the sample families have a minimum of 9 (nine) out of 14 (fourteen) poverty indicators [9]. The results of data analysis regarding the accuracy of targeting the beneficiary families of the Bansos Rastra program can be seen in table 1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[200,784,798,848]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0657/0daf76952b20116b29d0bd9b2b49414798706637c1adb72625f4576e3f61f486.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>No.</td><td>Information</td><td>Number of BRF</td><td>Percentage</td></tr><tr><td>1.</td><td>Right on target (≥ 9 indicators)</td><td>79</td><td>60 %</td></tr><tr><td>2.</td><td>Not on target (< 9 indicators)</td><td>52</td><td>40 %</td></tr><tr><td colspan=\\\"2\\\">Total</td><td>131</td><td>100 %</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 1. The accuracy of targeting beneficiary families of the Bansos Rastra program in Kajang District, Bulukumba Regency, 2018. \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[112,848,882,895]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Based on the data in table 1, it shows that not all BRF who received Bansos Rastra were included in the right target category, namely the fulfillment of at least nine poverty indicators. The results of data analysis show that the BRF which is included in the right target category, namely the fulfillment \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,64,226,77]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IC-FSSAT 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[774,64,880,79]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Publishing \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[112,80,880,93]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Conf. Series: Earth and Environmental Science 807 (2021) 032065 doi:10.1088/1755-1315/807/3/032065 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[497,939,507,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[110,133,882,239]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"of a minimum of nine poverty indicators is as many as 79 BRF (60%) of the sampled families, while 52 BRF (40%) is included in the category not on target because it has less than nine poverty indicators. This happens because there are two distribution systems for Bansos Rastra in villages or sub-districts in Kajang District, namely implementing an even distribution system for all families in the village without referring to the BRF list established by the Ministry of Social Affairs and distributing Bansos Rastra assistance to families registered in the list BRF which has been established by the Ministry of Social Affairs. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,253,257,268]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.2. Right amount \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[110,268,880,328]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The allocation of social assistance in the form of rice provided to BRF is $10\\\\mathrm{kg}$ per BRF per month or equivalent to $120\\\\mathrm{kg}$ per BRF in 2018[4]. The results of data analysis regarding the accuracy of the amount of social assistance in the form of rice received by beneficiary families of the Bansos Rastra program in Kajang District can be seen in table 2. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[200,384,800,467]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0447/90cba5a378e358a9c1a93bebcfdbda2a231124a86b852e3557dbab755b41faea.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>No.</td><td>Information</td><td>Number of BRF</td><td>Percentage</td></tr><tr><td>1.</td><td>Right amount (≥ 10 kg per BRF)</td><td>47</td><td>36 %</td></tr><tr><td>2.</td><td>Not right amount (<10 kg per BRF)</td><td>84</td><td>64 %</td></tr><tr><td colspan=\\\"2\\\">Total</td><td>131</td><td>100 %</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 2. The accuracy of the amount of social assistance in the form of rice received by beneficiary families of the Bansos Rastra Program in Kajang District, Bulukumba Regency, 2018. \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[110,472,880,563]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Table 2 shows that the families who received Bansos Rastra were included in the right category, namely receiving social assistance in the form of rice from the government at $10\\\\mathrm{kg}$ per BRF every month or the equivalent of $120\\\\mathrm{kg}$ per BRF in 2018 was 47 BRF $(36\\\\%)$ , while the family 84 BRF $(64\\\\%)$ received assistance of less than $10\\\\mathrm{kg}$ per month or less than $120\\\\mathrm{kg}$ in 2018. This shows that the implementation of the Bansos Rastra distribution program in Kajang District is in the inaccurate category. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,577,238,590]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.3. Right price \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[110,592,880,697]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Implementation of the Bansos Rastra Program which is distributed to BRF without being charged a price or redemption fee[4]. The exact price provisions are also used as an indicator of price accuracy in this study, namely the Bansos Rastra program is said to be right in price if BRF does not pay or does not pay fees in redeeming Bansos Rastra, and the Bansos Rastra Program is said to be not the right price if BRF in redeeming Bansos Rastra is charged a fee. The results of data analysis regarding the accuracy of prices charged to beneficiary families of the Bansos Rastra program in Kajang District can be seen in table 3. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[200,753,798,820]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0485/78768d35cf71e21032e853b30a3f2f0268ee50bede0370993d2ecd3d9773e501.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>No.</td><td>Information</td><td>Number of BRF</td><td>Percentage</td></tr><tr><td>1.</td><td>Right price</td><td>116</td><td>89 %</td></tr><tr><td>2.</td><td>Not right price</td><td>15</td><td>11 %</td></tr><tr><td></td><td>Total</td><td>131</td><td>100 %</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 3. The accuracy of the Bansos Rastra price charged to beneficiary families of the Bansos Rastra program in Kajang District, Bulukumba Regency, 2018. \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[110,833,880,894]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Table 3 shows that there are BRFs who received Bansos Rastra including the right price category, namely not paying in redeeming Bansos Rastra as much as 116 BRF (89%), while the other 15 BRF were categorized as incorrect prices. This happened because the implementers in several villages in Kajang District applied a voluntary fee at the time of redemption of Bansos Rastra. The costs referred \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,64,226,77]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IC-FSSAT 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[774,64,880,79]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Publishing \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[112,80,880,93]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Conf. Series: Earth and Environmental Science 807 (2021) 032065 doi:10.1088/1755-1315/807/3/032065 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[497,939,507,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[110,134,880,164]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"to are money for transporting rice, food and drinking fees for field officers, paid as much as they can to the distribution operator. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[110,164,880,254]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"When viewed based on the prevailing laws and regulations and guidelines for the distribution of the Bansos Rastra Program, this violates because it is not in accordance with what has been determined. Everything related to the issue of costs in the distribution of the Bansos Rastra Program, has been borne by the Bulukumba Regency APBD, the South Sulawesi Provincial APBD, and the National Budget at the Ministry of Social Affairs, so the cost at the time of redemption of Bansos Rastra to BRF is completely free in this case. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,268,233,282]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.4. Right time \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[110,282,880,388]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The implementation of the Bansos Rastra Program is said to be right time if the implementation time for the distribution of the Bansos Rastra Program to BRF is in accordance with the distribution plan. In 2018, the distribution time for Bansos Rastra is planned for 12 times or held every month. The time and procedure for distribution is adjusted to the situation and conditions as well as the needs of the area[4]. The indicator of timeliness in this study is if BRF receives Bansos Rastra every month or 12 times a year. The results of data analysis regarding the timeliness of distributing Bansos Rastra to beneficiary families of the Bansos Rastra program in Kajang District can be seen in table 4. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[200,445,796,511]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0122/41495b1ec713b6bff360d201e0766f7335acd3878f042cbdafc09ed4668af20f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>No.</td><td>Information</td><td>Number of BRF</td><td>Percentage</td></tr><tr><td>1.</td><td>Right time</td><td>11</td><td>8 %</td></tr><tr><td>2.</td><td>Not right time</td><td>120</td><td>92 %</td></tr><tr><td></td><td>Total</td><td>131</td><td>100 %</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 4. The timeliness of the distribution of the Bansos Rastra program to beneficiary families of the Bansos Rastra program in Kajang District, Bulukumba Regency, 2018. \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[110,517,880,652]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Table 4 shows that the majority of BRF who received Bansos Rastra in Kajang District were categorized as not on time, namely that the distribution of Bansos Rastra Program to BRF was almost never carried out every month. This happened because at the beginning of 2018 the distribution of the Bansos Rastra Program in Bulukumba District experienced obstacles at the Ministry of Social Affairs so that it had an impact on disbursement in Kajang District. In addition, the Bansos Rastra transportation which is a third party from Perum BULOG only provides one transportation for each district. So that Bulukumba Regency only has one transportation that will transport Bansos Rastra from the BULOG warehouse to all distribution points in each sub-district in Bulukumba Regency in turn. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,665,253,682]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.5. Right quality \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[110,682,880,757]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The right quality is defined as the fulfillment of the rice quality requirements according to the policy for the procurement of grain / rice and the distribution of rice by the government, namely the rice distributed is of medium quality with the rules that are not broken, odorless, colorless and not lice.[4]. The results of data analysis regarding the accuracy of Bansos Rastra quality for beneficiary families of the Bansos Rastra program in Kajang District can be seen in table 5. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,64,226,77]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IC-FSSAT 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[774,64,880,79]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Publishing \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[112,80,880,93]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Conf. Series: Earth and Environmental Science 807 (2021) 032065 doi:10.1088/1755-1315/807/3/032065 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[497,939,507,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[200,183,796,250]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0804/8dbca933f040381727d198cc29b9e87cde629d2a94b82895083bd5d540bdd3b8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>No.</td><td>Information</td><td>Number of BRF</td><td>Percentage</td></tr><tr><td>1.</td><td>Right quality</td><td>56</td><td>43 %</td></tr><tr><td>2.</td><td>Not right quality</td><td>75</td><td>57 %</td></tr><tr><td></td><td>Total</td><td>131</td><td>100 %</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 5. The Accuracy of the quality of the Bansos Rastra program to beneficiary families of the Bansos Rastra program in Kajang District, Bulukumba Regency, 2018. \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[112,255,880,347]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Table 5 shows that the implementation of the Bansos Rastra Distribution Program in Kajang District is included in the category of inappropriate quality. There are 75 BRF or $57\\\\%$ getting rice with poor quality, namely broken, colored, smelly and lice. Based on the recognition from Bulog Bulukumba, the cause of damage to Bansos Rastra's rice was due to the long time span of storing rice in the warehouse which resulted in the rice being damaged due to fungus. Perum Bulog did not conduct a thorough rice quality check because the amount of assistance was quite large. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,353,310,366]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.6. Right administration \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[110,367,880,458]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Administrative accuracy is defined as the correct and complete fulfillment of administrative requirements[4]. The administrative right indicator in this study is that every time BRF redeems Bansos Rastra, BRF shows the BRF Card or other card that has been determined by the village government as evidence of being registered in the list of beneficiaries as recipients of Bansos Rastra. The results of data analysis regarding the administrative accuracy of recipient families to redeem Bansos Rastra in Kajang District can be seen in table 6. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[200,500,796,565]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0158/c4d75efd0ba68be13636d3296a6301dbc1177fc9ac383255de435e960463f5a2.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>No.</td><td>Information</td><td>Number of BRF</td><td>Percentage</td></tr><tr><td>1.</td><td>Right administration</td><td>24</td><td>18 %</td></tr><tr><td>2.</td><td>Not right administration</td><td>107</td><td>82 %</td></tr><tr><td></td><td>Total</td><td>131</td><td>100 %</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 6. Administrative accuracy of beneficiary families to redeem Bansos Rastra in Kajang District, Bulukumba Regency, 2018. \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[112,571,880,722]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Table 6 shows that the administrative process in the implementation of the distribution of the Bansos Rastra program in Kajang District is included in the administration inappropriate category. There are $82\\\\%$ or 107 BRF who redeem Bansos Rastra in the wrong category by showing someone else's BRF card to redeem Bansos Rastra or even not bringing or showing the Bansos Rastra recipient card to pick up rice at the distribution point. This happened because there were several areas in Kajang District when distributing Bansos Rastra, they only signed the BRF as proof of receiving assistance, without showing the BRF card that had been assigned by the village government. Regions that do not carry out administration assume that the administrative process has been carried out when the Bansos Rastra program was first distributed, namely the process of family cards or KTPs to show that the BRF is registered on the Beneficiary List. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,736,583,751]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3.7. Analysis effectiveness level of the Bansos Rastra program \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,751,880,812]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Referring to the results of the 6R target analysis achieved by BRF Bansos Rastra, an overview of the frequency distribution regarding the effectiveness of the Bansos Rastra Program in Kajang District is obtained. The analysis of the proportion of the effectiveness level of the implementation of the Bansos Rastra Program in Kajang District, namely: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,64,226,77]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IC-FSSAT 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[774,64,880,79]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Publishing \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[112,80,880,93]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Conf. Series: Earth and Environmental Science 807 (2021) 032065 doi:10.1088/1755-1315/807/3/032065 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[497,940,507,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"6 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[307,133,689,158]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\text {Program effectiveness} = \\\\frac {\\\\text {Realization} x 100 \\\\%}{\\\\text {Target}} (2)\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[396,159,600,173]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\text {P r o g r a m E f f e c t i v e n e s s}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[115,173,815,244]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\begin{array}{l} = \\\\frac {(\\\\sum t a r g e t + \\\\sum a m o u n t + \\\\sum p r i c e + \\\\sum t i m e + \\\\sum q u a l i t y + \\\\sum a d m i n s t r a t i o n) x 100 \\\\%}{T a r g e t} \\\\\\\\ = \\\\frac {3 3 3 \\\\times 1 0 0 \\\\%}{7 8 6} \\\\\\\\ = 42.37 \\\\% \\\\\\\\ \\\\end{array}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[112,252,211,265]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"information: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,267,537,282]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Realization = Total number of BRFs achieving accuracy; \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,282,692,298]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Target $=$ The total number of BRFs that reached their supposed precision. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[221,298,410,311]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(131 families x 6 = 786). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,319,880,425]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It can be concluded that the implementation of the Bansos Rastra Program in Kajang District, Bulukumba Regency is classified as ineffective. It can be seen in the percentage results of the effectiveness level of the implementation of the Bansos Rastra program only $42.37\\\\%$ . According to the Ministry of Home Affairs Research and Development, the success rate category of distribution effectiveness, the percentage is in the range of $40 - 60\\\\%$ is in the ineffective category [6]. This can be seen clearly in the timeliness indicator, where the majority of BRF Bansos Rastra do not receive assistance every month or do not comply with the predetermined plans. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,439,230,451]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4. Conclusion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,453,880,514]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The implementation of the BansosRastra program in Kajang District, Bulukumba Regency is classified as ineffective. The effectiveness level of the implementation of the BansosRastra Program is only $42.37\\\\%$ , this is because the 6R has not been achieved in its implementation, namely the right target, the right amount, the right price, the right time, the right quality and the right administration. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,527,205,541]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,543,880,843]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"[1] Harmoko P 2010 Efektivitas Program Raskin Di Kecamatan Banjarsari Kota Surakarta Tahun 2009 \\\",\\\"[2] Sudarmono 2006 Politik Beras dan Ketahanan Bangsa (Jakarta) \\\",\\\"[3] Kementerian Koordinator Bidang Kesegahteraan Rakyat Republik Indonesia 2014 Pedoman Umum Raskin 2014 \\\",\\\"[4] Kementerian Koordinator Bidang Kesjahteraan Rakyat Republik Indonesia 2018 Pedoman Umum Bantuan Sosial Beras Sejahtera 2018 \\\",\\\"[5] Dewi I A C, Budhi M K S and Marhaeni A A I. 2012 Efektivitas program Jaminan Kesehatan Bali Mandara (JKBM) di Kecamatan Gianyar Kabupaten Gianyar E-Jurnal Ekon. dan Bisnis Univ. Udayana1 \\\",\\\"[6] Arnaya P Y and Utama M S 2012 Efektivitas kegiatan simpan pinjam khusus bagi kaum perempuan rumah tangga miskin di Kecamatan Denpasar Timur E-Jurnal Ekon. Pembang. Univ. Udayana1 44409 \\\",\\\"[7] Hastuti 2012 Tinjauan Efektivitas Pelaksanaan Raskin dalam Mencapai Enam Tepat (SMERU Research Institute) \\\",\\\"[8] Winarni W 2013 Study Implementasi program raskin Di Desa Kebumen Kecamatan Sukorejo Kabupaten Kendal Serat Acitya1 27 \\\",\\\"[9] Isdijoso W, Suryahadi A and Akhmadi A 2016 Penetapan kriteria dan variabel pendataan penduduk miskin yang komprehensif dalam rangka perlindungan penduduk miskin di Kabupaten/Kota Kertas Kerja SMERU,(September) 1-16 \\\"]\",\n \"page_idx\": \"7\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[112,64,226,77]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IC-FSSAT 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[774,65,880,79]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Publishing \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[112,80,880,93]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Conf. Series: Earth and Environmental Science 807 (2021) 032065 doi:10.1088/1755-1315/807/3/032065 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[497,939,507,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"7 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.1088/1755-1315/807/3/032065"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Life Sciences"},"sha256":{"kind":"string","value":"7199d745246bd85537f4ee97a97b447361acc48d727f71291b4740eea7099edf"},"title":{"kind":"string","value":"Effectiveness of the Bansos Rastra distribution program in Kajang District, Bulukumba Regency"}}},{"rowIdx":66,"cells":{"abstract":{"kind":"string","value":"Restoration of the locomotor function is one of the most essential problems of medicine. This problem becomes especially essential, when we deal with those children, who can't walk from their birth. Term “abilitation” is used just in relation to such children, what means not retraining children the right stereotype of walking, but appearance of this skill and its further adequate development. Nowadays, more and more specialists, who occupy themselves with abilitation of children of the early age, realize necessity of the instrumental estimation of the locomotor function. This phenomenon is connected with the fact, that in contrast with the score system of estimation, instrumental estimation of walking is maximally concrete and exact, and complex investigation of the biomechanical and innervative structure of walking gives complete idea of the locomotor disturbances of the child. Authors of this article investigated biomechanical and innervative structure of walking of 4 healthy children at the age of 2-2.5 years. On the basis of these investigations, it has been revealed, that the following peculiarities of walking are characteristic of children at the age of 2: remarkable diminution of the mean walking velocity, increase of cadence and shortening of the step length, essential transformation of the temporal structure of walking (growth of duration of the stance and double-support phase and decrease of duration of the swing phase), flexion position of the lower extremities, displacement of all extreme values of the angular displacements to the right along the temporal axis, considerable prolongation of the maximums of activity, absence of the second maximum of activity in a number of muscles, severe disturbances in the resonance properties of the lower extremities. All these peculiarities point out to the immaturity of the process of locomotion in children at the age of 2. The received data are necessary first of all as the standard ones, because walking parameters of children with the locomotor disorders of the early age must be compared with the corresponding parameters of the healthy children of the same age. © 2021. Batysheva T.T., Pismennaya E.V., Petrushanskaya K.A., Pisareva M.V., Kovina M.V. All Rights Reserved."},"author":{"kind":"string","value":"Batysheva, T.T.|Pismennaya, E.V.|Petrushanskaya, K.A.|Pisareva, M.V.|Kovina, M.V."},"content_list":{"kind":"list like","value":[{"bbox":"[196,192,848,247]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PECULIARITIES OF THE BIOMECHANICAL AND INNERVATIVE STRUCTURE OF WALKING OF HEALTHY CHILDREN OF THE EARLY AGE ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[265,275,779,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"T.T. Batysheva<sup>1</sup>, E.V. Pismennaya<sup>2</sup>, K.A. Petrushanskaya<sup>3</sup>, M.V. Pisareva<sup>4</sup>, M.V. Kovina<sup>4</sup> ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[137,319,905,405]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"$^{1}$ Scientific-practical Centre of Children Psychoneurology, 74 Michurinsky Avenue, Moscow, 119602, Russian Federation, e-mail: info@npcdp.ru \",\"$^{2}$ ExoAtlet LLC, 15 1-st Truzhenikov Side Street, Moscow, 119121, Russian Federation, e-mail: info@exoatlet.ru \",\"<sup>3</sup>M.F. Vladimirsky Moscow Regional Research Clinical Institute, 61/2-1 Shchepkin Street, Moscow, 129110, Russian Federation, e-mail: moniki@monikiweb.ru \",\"<sup>4</sup> Regional Children Clinical Hospital, 27 Tutaevskoe Highway, Yaroslavl, 150042, Russian Federation, e-mail: adm@odkb76.ru \"]","page_idx":"0","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[231,434,907,764]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract. Restoration of the locomotor function is one of the most essential problems of medicine. This problem becomes especially essential, when we deal with those children, who can't walk from their birth. Term \"abilitation\" is used just in relation to such children, what means not retraining children the right stereotype of walking, but appearance of this skill and its further adequate development. Nowadays, more and more specialists, who occupy themselves with abilitation of children of the early age, realize necessity of the instrumental estimation of the locomotor function. This phenomenon is connected with the fact, that in contrast with the score system of estimation, instrumental estimation of walking is maximally concrete and exact, and complex investigation of the biomechanical and innervative structure of walking gives complete idea of the locomotor disturbances of the child. Authors of this article investigated biomechanical and innervative structure of walking of 4 healthy children at the age of 2-2.5 years. On the basis of these investigations, it has been revealed, that the following peculiarities of walking are characteristic of children at the age of 2: remarkable diminution of the mean walking velocity, increase of cadence and shortening of the step length, essential transformation of the temporal structure of walking (growth of duration of the stance and double-support phase and decrease of duration of the swing phase), flexion position of the lower extremities, displacement of all extreme values of the angular displacements to the right along the temporal axis, considerable prolongation of the maximums of activity, absence of the second maximum of activity in a number of muscles, severe disturbances in the resonance properties of the lower extremities. All these peculiarities point out to the immaturity of the process of locomotion in children at the age of 2. The received data are necessary first of all as the standard ones, because walking parameters of children with the locomotor disorders of the early age must be compared with the corresponding parameters of the healthy children of the same age. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[231,790,907,819]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Key words: abilitation, children of the early age, biomechanical and innervative structure of walking, EMG-pattern of muscles. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[240,41,806,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[147,148,460,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DOI: 10.15593/RJBiomech/2021.4.08 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[643,98,739,165]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"# ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[744,101,887,149]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Russian Journal of Biomechanics ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[746,151,843,161]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"www.biomech.ru ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[136,868,873,964]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\odot$ Batysheva T.T., Pismennaya E.V., Petrushanskaya K.A., Pisareva M.V., Kovina M.V., 2021 \nTatyana T. Batysheva, Ph.D., Director of Scientific-practical Centre of Children Psychoneurology, Moscow \nElena V. Pismennaya, Ph.D., Senior Researcher of ExoAtlet LLC, Moscow \nKira A. Petrushanskaya, Ph.D., Senior Researcher of M.F. Vladimirsky Moscow Regional Research Clinical Institute, Moscow \nMarina V. Pisareva, Chief Physician of Regional Children Clinical Hospital, Yaroslavl \nMarina V. Kovina, Head of Department of Speech Pathology and Neurorehabilitation, Yaroslavl ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[411,99,537,116]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"INTRODUCTION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[92,123,858,187]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"At present, there exist a small number of works, devoted to the biomechanical and innervative structure of walking of children of the early age [2, 4-6, 8, 10-12]. A.S. Vitenson and L.N. Samsonova recorded electrical activity of the shank and femur muscles during walking of children at the age from 1.5 to 4 years [11]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,189,858,254]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It has been established, that the following changes occur with increase of child's age: growth of walking velocity, decrease of cadence of walking and rise of the step length, transformation of the temporal structure of walking, growth of amplitude of angular displacements at the joints of the lower extremities. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,255,858,369]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"But the main attention in their work was paid to change of electrical activity of muscles during the locomotor cycle. It has been established, that with increase of child's age concentration of the muscle efforts takes place during the locomotor cycle; maxima of activity are diminished in duration and become more exact. Such transformation is observed by the age of 4. For all this activity of muscles-extensors decreases more essentially than activity of muscles-flexors, and their EMG-pattern is more similar to that distribution of activity, which is peculiar to adults. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,370,858,483]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The same tendency of concentration of activity in the narrow time intervals was noted in monograph of D.P. Bukreeva and coauthors, who investigated electrical activity of a number of muscles in children at the age from 3 to 15 [2]. Nevertheless, these authors did not receive the averaged graphs of distribution of electrical activity of muscles during the locomotor cycle. But just distribution of electrical activity of muscles during the locomotor cycle, i.e. EMG-pattern of muscles is of the standard importance and permits to realize the comparative analysis of work of muscles in norm and in children with pathology of the locomotor system. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,483,858,549]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"That's why, the purpose of this work is to present our first results of investigations of the biomechanical and innervative structure of walking of healthy children at the age of 2 and to compare parameters of their walking with the corresponding parameters of walking of healthy adults and healthy children at the age of 7. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[292,571,657,587]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"MATERIAL AND METHOD OF INVESTIGATIONS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[92,594,858,659]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Biomechanical complex \"Liker\" was applied for investigation of the biomechanical and innervative structure of walking. Investigations were carried out in the department of pathology of speech and neurorehabilitation of State Institution of Public Health \"Yaroslavl Children Clinical Hospital\". ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,659,858,791]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 orthopedically healthy children (children of workers of the hospital) at the age from 2 to 2.5 were under consideration. All 4 children were thoroughly examined by orthopedist. The following parameters of walking were investigated: main parameters (walking velocity, cadence, double step length, duration of the locomotor cycle), temporal parameters of walking (duration of the support, swing and double support phases, rhythm coefficient), kinematic parameters of walking (angular displacements at the main joints of the lower extremities - hip and knee) and also electrical activity of four symmetrical muscles of the lower extremities (m. tibialis anterior, m. gastrocnemius medialis, m. rectus femoris and m. biceps femoris). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[92,791,858,840]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Children walked along the horizontal pass of length $l = 10$ m. During investigation children walked 8-9 passes. All data of walking of children of the early age were compared with data of walking of healthy adults and healthy children at the age of 7 [7-10]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[436,860,512,878]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RESULTS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[92,885,858,917]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our investigations showed, that remarkable increase of the walking cadence up to 141 steps/min is noted in children at the age of 2 in comparison with healthy adults and children at ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[181,41,766,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"T.T. Batysheva, E.V. Pismennaya, K.A. Petrushanskaya, M.V. Pisareva, M.V. Kovina ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[90,952,117,963]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"376 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[297,951,855,963]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[136,99,907,151]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"the age of 7. It typical that healthy two-year-old children very easily change walking to running inside one pass. At the same time, double step length is very inconsiderable – $0.5\\mathrm{m}$ . That's why walking velocity in children at the age of 2–2.5 is rather low – $0.59\\mathrm{m/s}$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[139,200,905,319]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0773/4a72b5ea04a85f14595d098fb5ed38db9905ffe7d0d7f170fd6eeef8cfc8cd4b.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":"<table><tr><td>Parameters</td><td>Adults</td><td>Children at the age of 7</td><td>Children at the age of 2-2,5</td></tr><tr><td>Duration of the cycle, s</td><td>1,21 ± 0.03</td><td>0.99 ± 0.02</td><td>0.85 ± 0.02</td></tr><tr><td>Double step length, m</td><td>1.42 ± 0.02</td><td>1.04 ± 0.03</td><td>0.50 ± 0.03</td></tr><tr><td>Walking velocity, m/s</td><td>1.17 ± 0.02</td><td>1.05 ± 0.02</td><td>0.59 ± 0.04</td></tr><tr><td>Walking cadence, steps/min</td><td>99 ± 1</td><td>121 ± 2</td><td>141 ± 3</td></tr></table>","table_caption":"[\"Table 1 \",\"Main parameters of walking of healthy adults and healthy children at the age of 7 and 2-2.5 \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[136,334,905,382]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Slight increase of duration of the stance phase up to $64.7\\%$ and diminution of duration of the swing phase up to $35.3\\%$ is observed in healthy children at the age of 2 in comparison with the healthy adults and children at the age of 7. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[136,382,905,432]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It typical that duration of the double-support phase grows in healthy two-year-old children, what points to weakening of the static stability. The latter fact, is probably, connected with walking on the slightly flexed legs. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[139,483,915,684]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0762/9432f6494241cbe4e0a8555f58207549bdd70f42fbe066fcbccbd0a310c54555.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":"<table><tr><td rowspan=\"2\">Parameters</td><td colspan=\"2\">Healthy adults</td><td colspan=\"2\">Children at the age of 7</td><td colspan=\"2\">Children at the age of 2–2.5</td></tr><tr><td>Left</td><td>Right</td><td>Left</td><td>Right</td><td>Left</td><td>Right</td></tr><tr><td>Duration of the stance phase</td><td>62.7 ± 0.7</td><td>62.4 ± 0.9</td><td>62.0 ± 0.9</td><td>60.6 ± 0.8</td><td>64.7 ± 0.8</td><td>64.1 ± 1.0</td></tr><tr><td>Duration of the swing phase</td><td>37.3 ± 0.6</td><td>37.6 ± 0.4</td><td>38.0 ± 0.5</td><td>39.4 ± 0.7</td><td>35.3 ± 0.6</td><td>35.9 ± 0.8</td></tr><tr><td>Duration of the double-support phase</td><td>12.7 ± 0.4</td><td>12.4 ± 0.6</td><td>10.2 ± 0.7</td><td>12.4 ± 0.7</td><td>15.1 ± 0.6</td><td>13.7 ± 0.6</td></tr><tr><td>Rhythm coefficient</td><td colspan=\"2\">0.99</td><td colspan=\"2\">0.96</td><td colspan=\"2\">0.98</td></tr></table>","table_caption":"[\"Table 2 \",\"Temporal parameters of walking of healthy adults and healthy children at the age of 7 and 2-2.5 \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[136,699,905,747]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Investigations showed that essential change of the kinematic parameters of walking is observed in two-year-old children. Kinematic parameters of healthy adults and healthy children at the age of 7 and 2 are presented in Fig 1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[136,749,905,847]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The curve of the angular displacements at HJ during walking in healthy adults has three extreme values: two of them correspond to flexion at the joint and one - to extension. Each cycle begins from flexion, which lasts all time of the heel-support, then extension at HJ takes place, which achieves the maximal value in the middle of the fore-foot support, then again flexion at HJ in the swing phase occurs, which is changed by the small extension at the end of the locomotor cycle [4-12]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[136,846,905,929]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"During walking of adult healthy examinees, the curve of the knee angle (KA) consists of two half-waves – with small and large amplitude. The first half-wave characterizes flexion at KJ in the stance phase, which holds mostly shock-absorption significance (reduction of loading, transferring on the supporting leg), the second half-wave represents flexion at KJ in the swing phase. Practically complete extension follows each flexion [4-12]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[161,41,880,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Peculiarities of the biomechanical and innervative structure of walking of healthy children of the early age ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[137,950,700,964]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[884,951,912,963]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"377 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[87,99,860,198]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The following changes in the curves of angular displacements are noted in children at the age of 7: firstly, all extreme values are displaced to the left along the temporal axis, secondly, change of a number of amplitude values is observed, namely, increase of angle of extension at KJ both in the stance and in the swing phase, what supposes walking on slightly-flexed legs, thirdly, spatial asymmetry is noted, i.e. difference in amplitude of the right and left lower extremities [10]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[87,198,860,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As may be seen from fig. 1, the form of HA in two-year-old children does not practically differ from the form of adults. But in children at the age of 2 amplitude of movements is a little bit less than in adults $(34^{\\circ})$ , for all this value of flexion is greater than in adult norm both in the stance and in the swing phase $-29^{\\circ}$ , while extension at HJ is weakly expressed $(-5^{\\circ})$ , that's why the whole curve is lifted above the level of zero line, what is connected with walking on the slightly flexed legs. The horizontal plateau in the beginning of the stance phase, coinciding in time with double-support phase, is absent in children at the age of 2. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[90,313,860,479]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"At the same time more remarkable changes are seen in the curve at a KA in children at the age of 2. First of all, amplitude of movements at KJ decreases up to $55^{\\circ}$ . Secondly, flexion at KJ is very long, it ends only at $23\\%$ of the cycle. Value of knee flexion in the stance phase exceeds the corresponding value in adults $-35^{\\circ}$ . For all this, extension in the stance phase is also very long, and, besides, it is realized incompletely, value of extension comprises $26^{\\circ}$ . Decrease of amplitude of movements at KJ takes place also because of diminution of the main flexion at KJ up to $65^{\\circ}$ , extreme value of the main flexion is timed to $t = 80\\%$ of the locomotor cycle, i.e. is displaced to the right along the temporal axis. So, increase of flexion at KJ in the stance phase, diminution of amplitude of movements and displacement of all extreme values to right along the temporal axis is typical of the knee angle of healthy children at the age of 2. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[89,491,868,706]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 1. Kinematic parameters of walking of healthy adult examinees, healthy children at the age of 7 and healthy children at the age of 2. KA - knee angle, HA - hip angle. Podograms are below the graphs. Along the abscissa axis - duration of the locomotor cycle in $\\\\%$ , along the ordinate axis - angular displacements at the joints of the lower extremities in degrees \"]","image_footnote":"[]","img_path":"images/0113/b229895a92bca442e90416f56457f6209fcb326c98d16577f640ad095b5c4a8d.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[87,809,858,858]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"But distribution of electrical activity of muscles during the locomotor cycle is of the greatest interest. EMG-pattern of 4 symmetrical muscles of the lower extremities in adult examinees and children at the age of 7 and 2 is demonstrated in Fig. 2. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[87,858,860,925]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"M. tibialis anterior (Fig. 2, a). While walking of healthy adult examinees electrical activity of m. tibialis anterior during the locomotor cycle has two maxima: the first - high-begins at $t = 95\\%$ of the subsequent cycle, gives the greatest wave in the limits of the first $10\\%$ of the following cycle and ends by $15\\%$ of the cycle. The functional purpose of the first ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[181,41,766,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"T.T. Batysheva, E.V. Pismennaya, K.A. Petrushanskaya, M.V. Pisareva, M.V. Kovina ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[89,951,119,963]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"378 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[295,950,858,964]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[134,99,904,165]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"maximum - lifting of the foot before touching the support, its gradual lowering on the support surface and rotation of the shank forward at the fixed foot. The second, less expressed maximum, begins at $t = 55\\%$ and ends at $t = 80\\%$ of the cycle. This maximum promotes the dorsal flexion at an AJ in the first half of the swing phase [7-11]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[134,166,904,231]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"EMG-pattern of m. tibialis anterior in children at the age of 7 is similar with the EMG-pattern of the adult healthy examinees, but some distinctions are noticed. At the age of 7 intermediate interpeaked activity increases. Besides, asymmetry of activity is observed in the right and left lower extremities [4, 5, 9]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[134,231,905,362]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"During walking of children at the age of 2 considerable growth of the second maximum of activity of m. tibialis anterior may be seen. Maxima of activity of children at the age of 2 are more prolonged in contrast with adults and seven-year-old children. So, the first maximum begins at the level of $t = 86\\%$ of the subsequent cycle and ends at the level of $23 - 24\\%$ of the following cycle. The second maximum, promoting the dorsal flexion at an AJ in the swing phase, is also more long $-53\\% < t < 86\\%$ . So remarkable growth of duration of the period of activity of this muscle up to $71\\%$ of the cycle is typical of children at the age of 2 in contrast with adults and children at the age of 7. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[134,363,905,461]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"M. gastrocnemius medialis (Fig. 2, b). During walking of healthy adult examinees electrical activity of m. gastrocnemius medialis has maximum between $t = 20\\%$ and $t = 60\\%$ of the cycle with the peak, timed to $t = 45\\%$ of the cycle. Function of this activity consists in regulation of movement of the shank forward (in the eccentric regime of work) until $50\\%$ of the cycle and in realization of flexion at KJ and plantar flexion at an AJ at the end of the stance phase [7-11]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[134,461,905,542]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"During walking of children at the age of 7 EMG-pattern of m. gastrocnemius is characterized by more early appearance of maximum of activity, greater level of the minimal activity, more high level of excitation in the beginning of the stance phase and during the swing phase. Relationship among the minimal and maximal activity of m. gastrocnemius in healthy adults comprises 8:1, in children at the age of $7 - 4:1$ [4, 5, 9]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[134,542,905,673]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"During walking of children at the age of 2 maximum of activity of m. gastrocnemius medialis is prolonged for the most part of the stance phase, what is connected with walking on slightly flexed legs (Fig. 2, b). As may be seen from Fig. 1, all kinematic curves in healthy children at the age of 2 are displaced above zero line, which determines standing with the straightened legs. Such position corresponds with the flexion position of the lower extremities, which is remained during the locomotor cycle. However, one more wave of activity may appear in two-year-old children - from $t = 61\\%$ till $t = 86\\%$ of the locomotor cycle. It is believed, that this wave of activity is intended for flexion at KJ. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[134,674,905,804]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"M. rectus femoris (Fig. 2, c). During walking of healthy adult examines electrical activity of m. rectus femoris has two maxima. The first maximum arises at the end of the subsequent cycle at the level of $t = 85\\%$ and ends at $t = 20\\%$ of the following cycle. The second small maximum is timed to a period $55\\% < t < 75\\%$ . Activity of this muscle at the end of the swing phase $(85\\% < t < 100\\%)$ promotes extension at KJ. During heel-support $(0\\% < t < 15\\%)$ and later m. rectus femoris works in the eccentric regime. Activity of muscle in this period is intended for counteraction of flexion at KJ. Activity of the second maximum is intended for hip flexion at the end of the stance phase [7-11]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[134,804,904,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Two maxima are also observed in electrical activity of m. rectus femoris during walking of seven-year-old children. The first maximum is very prolonged, it arises at the end of the subsequent cycle at the level of $90\\%$ and ends only by $60\\%$ of the following cycle. The second maximum, intended for hip flexion, is inconsiderable, both by value and duration. Evident asymmetry of maxima is revealed [4, 5, 9]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[136,887,904,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"During walking of children at the age of 2 EMG-pattern of m. rectus femoris acquires the two-peaked form, the first maximum has boundaries between $t = 95\\%$ from the subsequent ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[161,41,880,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Peculiarities of the biomechanical and innervative structure of walking of healthy children of the early age ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[137,948,700,963]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[884,951,912,963]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"379 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[87,99,860,181]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"cycle and $t = 30\\%$ of the following cycle, the second maximum lasts from $t = 30\\%$ till $t = 50\\%$ of the cycle. EMG-pattern of this muscle in two-year-old children resembles in its form the curve of the vertical component $R_{z}$ of the ground reaction force. Essential growth of the level of the intermediate interpeaked activity is observed. The second maximum of activity, promoting flexion at HJ, is absent in children at the age of 2. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[87,181,860,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"M. biceps femoris (Fig. 2, d). During walking of healthy examinees electrical activity of m. biceps femoris has maximum, which lasts from $t = 85\\%$ of the subsequent cycle till $t = 40\\%$ of the following cycle. Peak of maximum falls on $t = 90 - 92\\%$ of the subsequent cycle. The functional purpose of this wave of activity consists in slowing down extension of the shank at the end of the swing phase and creation of the small angle of flexion in the beginning of the step and extension at HJ in the first two thirds of the stance phase. The additional wave of activity may appear in the boundary of the stance and swing phases ( $45\\% < t < 65\\%$ ), which is intended, probably for flexion at KJ [7-11]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[89,313,858,346]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Increase of activity of this muscle is noticed in children at the age of 7. Nevertheless, slight asymmetry of EMG-pattern in the right and left lower extremities may be traced [4, 5, 9]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[87,347,860,464]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Two maxima of activity may be also revealed in children at the age of 2. But the level of the minimal interpeaked activity is much higher in comparison with seven-year-old children and adult examinees. The first maximum begins in the subsequent cycle at the level of $t = 90\\%$ , its peak is timed to $t = 20\\%$ of the following cycle and ends only by $t = 50\\%$ . The second maximum of activity is much higher in comparison with the first one, this maximum is timed to the period from $50\\% < t < 80\\%$ of the locomotor cycle. It is believed, that this maximum is intended for deceleration of quick extension at KJ in the swing phase. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[94,474,850,834]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 2. Electrical activity of muscles of the lower extremities in adult healthy examinees and healthy children at the age of 7 and 2: $a - \\\\mathrm{m}$ tibialis anterior; $b - \\\\mathrm{m}$ gastrocnemius medialis; $c - \\\\mathrm{m}$ rectus femoris; $d - \\\\mathrm{m}$ biceps femoris. Podograms are below the graphs. Along the abscissa axis – duration of the locomotor cycle in %, along the ordinate axis – electrical activity of muscles in % to the maximal value during the locomotor cycle \"]","image_footnote":"[]","img_path":"images/0732/0f5fa937f3cf98fafa714851f21116b9aabdd7916e7430b818bd7f6aa635285e.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[181,41,766,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"T.T. Batysheva, E.V. Pismennaya, K.A. Petrushanskaya, M.V. Pisareva, M.V. Kovina ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[90,951,117,963]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"380 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[295,950,858,964]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[136,99,904,130]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The received data point out, that walking of children at the age of 2-2.5 is characterized by the following peculiarities: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[137,133,905,344]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1) remarkable diminution of the mean walking velocity exclusively at the expense of shortening of the step length and increase of cadence of walking; \",\"2) considerable disturbance of the resonance phenomena as a consequence of discrepancy of high cadence of walking and low step length; \",\"3) change of the temporal structure of the step: growth of duration of the stance and double-support phase and diminution of duration of the swing phase; \",\"4) presence of the flexion position of the lower extremities, what leads to decrease of amplitude of angular displacements and their lifting above the level of zero line; \",\"5) displacement of all extreme values of angular displacements to the right along the temporal axis, in spite of higher cadence; \",\"6) considerable increase of duration of maxima of activity of all muscles during the locomotor cycle, essential growth of the minimal interpeaked activity; \",\"7) absence of the second maximum of activity in some muscles (m. rectus femoris). \"]","page_idx":"6","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[396,368,648,382]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DISCUSSION OF THE RESULTS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[136,391,904,439]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In spite of actuality of all received data, we consider expedient to examine only some of them. As may be seen from the received results, signs of immaturity of walking may be noticed in healthy children of the early age. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[136,441,905,505]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Flexion position of the lower extremities is one of such signs. This position is expressed in appearance of the initial angle of flexion at all joints (displacement of curves of angular displacements above the level of zero line). This fact leads to remarkable diminution of the step length and consequently, to deceleration of walking. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[136,506,905,602]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Shortening of the step length is accompanied by decrease of amplitude of angular displacements at the joints of the lower extremities. Change of the temporal structure of walking in children at the age of 2 is another consequence of diminution of walking velocity, and namely increase of duration of the stance and double-support phase and decrease of duration of the swing phase. Displacement of all extreme values to the right along the temporal axis takes place as a consequence of change of the temporal structure of walking [7-10]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[136,605,905,718]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Discrepancy of change of cadence and double step length is another important indication of immaturity of walking in children of the early age. As may be seen from the received data, very high cadence of walking (141 step/min) is accompanied by low double step length $(0.5\\mathrm{m})$ . This phenomenon leads to disorder of the resonance phenomena, i.e. to discrepancy of frequency of the forced oscillations with frequency of own oscillations of the lower extremities. In healthy adults and children at the age of 7 growth of walking velocity is accompanied by increase of both cadence and the step length. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[136,719,905,832]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As investigations of A.S. Vitenson and L.N. Samsonova demonstrated, there is not strict correlation between cadence and step length in children at the age of 2 [11]. It is believed, that association of mechanisms, controlling these both variables, is formed by conditioned reflex way in the process of ontogenesis. As a result, greater step length begins to correspond to higher cadence, what causes increase of the walking velocity [7-10]. Disorder of the resonance phenomena leads to increase of the energetic cost of walking in children of the early age. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[136,834,905,931]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Question of a change of electrical activity of muscles in children of the early age is of the special interest. As may be seen from the received data, expressed prolongation of maxima of activity of all muscles is noticed in children at the age of 2. Walking on the slightly flexed legs furthers formation of such stereotype. For all this it is typical, that activity of muscles-extensors (m. gastrocnemius and m. rectus femoris) acquires the form of the wide plateau, which involves two zones of the stance phase - zone of the maximal activity (zone $M$ ) and zone ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[161,41,880,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Peculiarities of the biomechanical and innervative structure of walking of healthy children of the early age ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[137,950,700,963]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[884,951,910,963]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"381 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[87,99,858,181]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"of the temperate activity (zone $T$ ). Walking on the slightly flexed legs is the reason of prolongation of activity. Really, all kinematic curves of HA and KA are displaced above zero line, which reflects standing with the straightened legs. Such position of the curves of the angular displacements corresponds to the flexion position of the lower extremities, which is retained during the locomotor cycle. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[87,181,855,231]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Consequently, EMG-pattern is rather immature in children of the early age, because high interpeakd activity, increase of duration of activity and diminution of its amplitude is observed. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[87,231,857,281]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Change of the pattern of activity of m. gastrocnemius is of the special interest. A number of authors consider, that there are two EMG-patterns of this muscle in children of the early age [4, 5, 12]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[87,281,858,379]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The first EMG- pattern is immature, because it differs essentially from pattern of activity of the adult healthy examinees. It consists in remarkable increase of the maximum of activity of m. gastrocnemius from the middle of the swing phase and in the beginning of the stance phase. Activity of this muscle in the middle and at the end of the swing phase leads to decrease in the required minimum distance of the foot from the support surface. Activity in the stance phase disturbs the process of absorption and transformation of energy of m. triceps surae. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[87,379,858,428]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The second, mature pattern of activity of m. gastrocnemius consists in concentration of activity in the single-support phase, in the period from 10 to $45\\%$ of the locomotor cycle. Such pattern resembles in its form EMG-pattern of adult healthy man. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[87,429,858,508]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From the authors' point of view, immature EMG-pattern of m. gastrocnemius in children of the early age is connected with late maturation of its phase activity. They consider, that deceleration of conduction because of incomplete myelination and greater distance of the sensory component of the reflex ark are the important factors of maturation delay of gastrocnemius function [5, 12]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[87,510,858,608]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As investigations of A.S. Vitenson and A.S. Vitenson and coauthors demonstrated, reconstruction of the internal structure of movements is not finished by the age of 7 [9, 10]. This fact may be explained in the following way: the static-geometrical parameters of the man's body and the force characteristic of the trunk muscles are essentially changed with age, and that's why constant correction of the motional act is required, especially in conditions of transformation of the main indices of walking - cadence and step length. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[87,608,858,707]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is necessary to note, that EMG-pattern of muscles, corresponding to adult healthy examinees, is formed only by the age of 13 in healthy children. From our point of view, the received results of investigations of the biomechanical and innervative structure of walking of children of the early age are quite necessary for revealing the locomotor disturbances in children of early age, for choice of the adequate methods of abilitation and retracing this process with an account of age-specific peculiarities of children. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[415,728,532,743]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"REFERENCES ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[90,750,858,916]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. Andriacchi T.P., Ogle J.A., Galante J.O. Walking speed as a basis for normal and abnormal gait measurement. Journal of Biomechanics, 1977, vol. 10, no. 4, pp. 261-268. \",\"2. Bukreeva D.P., Kosilov S.A., Tambiae A.P. Vozrastnye osobennosti tsiklicheskikh dvizheniy detey i podrostkov [Age-specific peculiarities of the cyclic movements of children and teenagers]. Moscow: Pedagogika, 1975, 159 p. (in Russian). \",\"3. Filatov V.I. Klinicheskaya biomekhanika [Clinical biomechanics]. Ed. by Prof. V.I. Filatov. Leningrad-Moscow, 1980, 199 p. (in Russian). \",\"4. Skvortsov D.V. Diagnostika dvigatelnoy patologii instrumentalnyi metodami. Analiz pokhodki. Stabilometiya [Diagnostics of the locomotor pathology by means of the instrumental methods. Gait analysis. Stabilometry]. Moscow:MBN, 2007, 638 p. (in Russian). \",\"5. Sutherland D.H., Olshen R., Cooper B.A., Woo S.L.Y. The development of mature gait. The Journal of Biomechanics and Joint Surgery, 1980, vol. 62-A, no 3, pp. 336-353. \"]","page_idx":"7","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[181,41,766,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"T.T. Batysheva, E.V. Pismennaya, K.A. Petrushanskaya, M.V. Pisareva, M.V. Kovina ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[90,951,117,963]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"382 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[295,950,858,964]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[139,99,905,332]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"6. Todd F.N., Lamoreux L.W., Skinner S.R., Johanson M.E., St. Helen R., Moran S.A., Ashley K.A. Variation in the gait of normal children //Journal of Biomechanics and Joint Surgery. -1989. - Vol. 71A. - P. 196-204. \",\"7. Vitenson A.S. Zakonomernosti normalnoi i patologicheskoy khodby cheloveka [Peculiarities of normal and pathological human walking]. Moscow: Zerkalo-M, 1998, 272 p. (in Russian). \",\"8. Vitenson A.S. Issledovanie biomechanicheskikh i neyrophiziologicheskikh zakonomernostey normalnoy i patologicheskoy khodby cheloveka [Investigation of the biomechanical and neurophysiological peculiarities of normal and pathological human walking]. Ph. D. Thesis. Moscow, 1982 (in Russian). \",\"9. Vitenson A.S., Petrushanskaya K.A. Ot estestvennogo k iskusstvennomy upravleniyu lokomotsiey [From natural to artificial control of locomotion]. Moscow: MBN, 2003, 448 p. (in Russian). \",\"10. Vitenson A.S., Petrushanskaya K.A., Spivak B.G., Matveeva I.A., Gritsenko G.P., Sutchenkov I.A. Peculiarities of biomechanical structure of walking of healthy children of different age groups //Russian Journal of biomechanics, 2013, vol. 17, no. 1, pp. 69-83. \",\"11. Vitenson A.S., Samsonova L.N. Biomehanicheskaya i innervatsionnaya struktura khodby detey rannego vozrasta [Biomechanical and innervative structure of walking of children of the early age]. Protezirovanie i protezostroenie, 1973, no. 31, pp. 39-46 (in Russian). \",\"12. Winter D. The biomechanics and motor control of human gait. Waterloo: University of Waterloo Press, 1990, 145 p. \"]","page_idx":"8","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[139,350,304,363]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Received 8 August 2021 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[161,41,880,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Peculiarities of the biomechanical and innervative structure of walking of healthy children of the early age ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[139,950,700,963]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[884,952,912,963]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"383 ","text_format":null,"text_level":null,"type":"page_number"}],"string":"[\n {\n \"bbox\": \"[196,192,848,247]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PECULIARITIES OF THE BIOMECHANICAL AND INNERVATIVE STRUCTURE OF WALKING OF HEALTHY CHILDREN OF THE EARLY AGE \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[265,275,779,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"T.T. Batysheva<sup>1</sup>, E.V. Pismennaya<sup>2</sup>, K.A. Petrushanskaya<sup>3</sup>, M.V. Pisareva<sup>4</sup>, M.V. Kovina<sup>4</sup> \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[137,319,905,405]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"$^{1}$ Scientific-practical Centre of Children Psychoneurology, 74 Michurinsky Avenue, Moscow, 119602, Russian Federation, e-mail: info@npcdp.ru \\\",\\\"$^{2}$ ExoAtlet LLC, 15 1-st Truzhenikov Side Street, Moscow, 119121, Russian Federation, e-mail: info@exoatlet.ru \\\",\\\"<sup>3</sup>M.F. Vladimirsky Moscow Regional Research Clinical Institute, 61/2-1 Shchepkin Street, Moscow, 129110, Russian Federation, e-mail: moniki@monikiweb.ru \\\",\\\"<sup>4</sup> Regional Children Clinical Hospital, 27 Tutaevskoe Highway, Yaroslavl, 150042, Russian Federation, e-mail: adm@odkb76.ru \\\"]\",\n \"page_idx\": \"0\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[231,434,907,764]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract. Restoration of the locomotor function is one of the most essential problems of medicine. This problem becomes especially essential, when we deal with those children, who can't walk from their birth. Term \\\"abilitation\\\" is used just in relation to such children, what means not retraining children the right stereotype of walking, but appearance of this skill and its further adequate development. Nowadays, more and more specialists, who occupy themselves with abilitation of children of the early age, realize necessity of the instrumental estimation of the locomotor function. This phenomenon is connected with the fact, that in contrast with the score system of estimation, instrumental estimation of walking is maximally concrete and exact, and complex investigation of the biomechanical and innervative structure of walking gives complete idea of the locomotor disturbances of the child. Authors of this article investigated biomechanical and innervative structure of walking of 4 healthy children at the age of 2-2.5 years. On the basis of these investigations, it has been revealed, that the following peculiarities of walking are characteristic of children at the age of 2: remarkable diminution of the mean walking velocity, increase of cadence and shortening of the step length, essential transformation of the temporal structure of walking (growth of duration of the stance and double-support phase and decrease of duration of the swing phase), flexion position of the lower extremities, displacement of all extreme values of the angular displacements to the right along the temporal axis, considerable prolongation of the maximums of activity, absence of the second maximum of activity in a number of muscles, severe disturbances in the resonance properties of the lower extremities. All these peculiarities point out to the immaturity of the process of locomotion in children at the age of 2. The received data are necessary first of all as the standard ones, because walking parameters of children with the locomotor disorders of the early age must be compared with the corresponding parameters of the healthy children of the same age. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[231,790,907,819]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Key words: abilitation, children of the early age, biomechanical and innervative structure of walking, EMG-pattern of muscles. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[240,41,806,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[147,148,460,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DOI: 10.15593/RJBiomech/2021.4.08 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[643,98,739,165]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"# \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[744,101,887,149]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Russian Journal of Biomechanics \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[746,151,843,161]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"www.biomech.ru \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[136,868,873,964]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\odot$ Batysheva T.T., Pismennaya E.V., Petrushanskaya K.A., Pisareva M.V., Kovina M.V., 2021 \\nTatyana T. Batysheva, Ph.D., Director of Scientific-practical Centre of Children Psychoneurology, Moscow \\nElena V. Pismennaya, Ph.D., Senior Researcher of ExoAtlet LLC, Moscow \\nKira A. Petrushanskaya, Ph.D., Senior Researcher of M.F. Vladimirsky Moscow Regional Research Clinical Institute, Moscow \\nMarina V. Pisareva, Chief Physician of Regional Children Clinical Hospital, Yaroslavl \\nMarina V. Kovina, Head of Department of Speech Pathology and Neurorehabilitation, Yaroslavl \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[411,99,537,116]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"INTRODUCTION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,123,858,187]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"At present, there exist a small number of works, devoted to the biomechanical and innervative structure of walking of children of the early age [2, 4-6, 8, 10-12]. A.S. Vitenson and L.N. Samsonova recorded electrical activity of the shank and femur muscles during walking of children at the age from 1.5 to 4 years [11]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,189,858,254]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It has been established, that the following changes occur with increase of child's age: growth of walking velocity, decrease of cadence of walking and rise of the step length, transformation of the temporal structure of walking, growth of amplitude of angular displacements at the joints of the lower extremities. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,255,858,369]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"But the main attention in their work was paid to change of electrical activity of muscles during the locomotor cycle. It has been established, that with increase of child's age concentration of the muscle efforts takes place during the locomotor cycle; maxima of activity are diminished in duration and become more exact. Such transformation is observed by the age of 4. For all this activity of muscles-extensors decreases more essentially than activity of muscles-flexors, and their EMG-pattern is more similar to that distribution of activity, which is peculiar to adults. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,370,858,483]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The same tendency of concentration of activity in the narrow time intervals was noted in monograph of D.P. Bukreeva and coauthors, who investigated electrical activity of a number of muscles in children at the age from 3 to 15 [2]. Nevertheless, these authors did not receive the averaged graphs of distribution of electrical activity of muscles during the locomotor cycle. But just distribution of electrical activity of muscles during the locomotor cycle, i.e. EMG-pattern of muscles is of the standard importance and permits to realize the comparative analysis of work of muscles in norm and in children with pathology of the locomotor system. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,483,858,549]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"That's why, the purpose of this work is to present our first results of investigations of the biomechanical and innervative structure of walking of healthy children at the age of 2 and to compare parameters of their walking with the corresponding parameters of walking of healthy adults and healthy children at the age of 7. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[292,571,657,587]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"MATERIAL AND METHOD OF INVESTIGATIONS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,594,858,659]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Biomechanical complex \\\"Liker\\\" was applied for investigation of the biomechanical and innervative structure of walking. Investigations were carried out in the department of pathology of speech and neurorehabilitation of State Institution of Public Health \\\"Yaroslavl Children Clinical Hospital\\\". \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,659,858,791]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 orthopedically healthy children (children of workers of the hospital) at the age from 2 to 2.5 were under consideration. All 4 children were thoroughly examined by orthopedist. The following parameters of walking were investigated: main parameters (walking velocity, cadence, double step length, duration of the locomotor cycle), temporal parameters of walking (duration of the support, swing and double support phases, rhythm coefficient), kinematic parameters of walking (angular displacements at the main joints of the lower extremities - hip and knee) and also electrical activity of four symmetrical muscles of the lower extremities (m. tibialis anterior, m. gastrocnemius medialis, m. rectus femoris and m. biceps femoris). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,791,858,840]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Children walked along the horizontal pass of length $l = 10$ m. During investigation children walked 8-9 passes. All data of walking of children of the early age were compared with data of walking of healthy adults and healthy children at the age of 7 [7-10]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[436,860,512,878]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RESULTS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[92,885,858,917]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our investigations showed, that remarkable increase of the walking cadence up to 141 steps/min is noted in children at the age of 2 in comparison with healthy adults and children at \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[181,41,766,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"T.T. Batysheva, E.V. Pismennaya, K.A. Petrushanskaya, M.V. Pisareva, M.V. Kovina \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[90,952,117,963]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"376 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[297,951,855,963]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[136,99,907,151]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"the age of 7. It typical that healthy two-year-old children very easily change walking to running inside one pass. At the same time, double step length is very inconsiderable – $0.5\\\\mathrm{m}$ . That's why walking velocity in children at the age of 2–2.5 is rather low – $0.59\\\\mathrm{m/s}$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[139,200,905,319]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0773/4a72b5ea04a85f14595d098fb5ed38db9905ffe7d0d7f170fd6eeef8cfc8cd4b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Parameters</td><td>Adults</td><td>Children at the age of 7</td><td>Children at the age of 2-2,5</td></tr><tr><td>Duration of the cycle, s</td><td>1,21 ± 0.03</td><td>0.99 ± 0.02</td><td>0.85 ± 0.02</td></tr><tr><td>Double step length, m</td><td>1.42 ± 0.02</td><td>1.04 ± 0.03</td><td>0.50 ± 0.03</td></tr><tr><td>Walking velocity, m/s</td><td>1.17 ± 0.02</td><td>1.05 ± 0.02</td><td>0.59 ± 0.04</td></tr><tr><td>Walking cadence, steps/min</td><td>99 ± 1</td><td>121 ± 2</td><td>141 ± 3</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 1 \\\",\\\"Main parameters of walking of healthy adults and healthy children at the age of 7 and 2-2.5 \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[136,334,905,382]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Slight increase of duration of the stance phase up to $64.7\\\\%$ and diminution of duration of the swing phase up to $35.3\\\\%$ is observed in healthy children at the age of 2 in comparison with the healthy adults and children at the age of 7. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[136,382,905,432]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It typical that duration of the double-support phase grows in healthy two-year-old children, what points to weakening of the static stability. The latter fact, is probably, connected with walking on the slightly flexed legs. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[139,483,915,684]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0762/9432f6494241cbe4e0a8555f58207549bdd70f42fbe066fcbccbd0a310c54555.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=\\\"2\\\">Parameters</td><td colspan=\\\"2\\\">Healthy adults</td><td colspan=\\\"2\\\">Children at the age of 7</td><td colspan=\\\"2\\\">Children at the age of 2–2.5</td></tr><tr><td>Left</td><td>Right</td><td>Left</td><td>Right</td><td>Left</td><td>Right</td></tr><tr><td>Duration of the stance phase</td><td>62.7 ± 0.7</td><td>62.4 ± 0.9</td><td>62.0 ± 0.9</td><td>60.6 ± 0.8</td><td>64.7 ± 0.8</td><td>64.1 ± 1.0</td></tr><tr><td>Duration of the swing phase</td><td>37.3 ± 0.6</td><td>37.6 ± 0.4</td><td>38.0 ± 0.5</td><td>39.4 ± 0.7</td><td>35.3 ± 0.6</td><td>35.9 ± 0.8</td></tr><tr><td>Duration of the double-support phase</td><td>12.7 ± 0.4</td><td>12.4 ± 0.6</td><td>10.2 ± 0.7</td><td>12.4 ± 0.7</td><td>15.1 ± 0.6</td><td>13.7 ± 0.6</td></tr><tr><td>Rhythm coefficient</td><td colspan=\\\"2\\\">0.99</td><td colspan=\\\"2\\\">0.96</td><td colspan=\\\"2\\\">0.98</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 2 \\\",\\\"Temporal parameters of walking of healthy adults and healthy children at the age of 7 and 2-2.5 \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[136,699,905,747]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Investigations showed that essential change of the kinematic parameters of walking is observed in two-year-old children. Kinematic parameters of healthy adults and healthy children at the age of 7 and 2 are presented in Fig 1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[136,749,905,847]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The curve of the angular displacements at HJ during walking in healthy adults has three extreme values: two of them correspond to flexion at the joint and one - to extension. Each cycle begins from flexion, which lasts all time of the heel-support, then extension at HJ takes place, which achieves the maximal value in the middle of the fore-foot support, then again flexion at HJ in the swing phase occurs, which is changed by the small extension at the end of the locomotor cycle [4-12]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[136,846,905,929]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"During walking of adult healthy examinees, the curve of the knee angle (KA) consists of two half-waves – with small and large amplitude. The first half-wave characterizes flexion at KJ in the stance phase, which holds mostly shock-absorption significance (reduction of loading, transferring on the supporting leg), the second half-wave represents flexion at KJ in the swing phase. Practically complete extension follows each flexion [4-12]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[161,41,880,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Peculiarities of the biomechanical and innervative structure of walking of healthy children of the early age \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[137,950,700,964]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[884,951,912,963]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"377 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[87,99,860,198]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The following changes in the curves of angular displacements are noted in children at the age of 7: firstly, all extreme values are displaced to the left along the temporal axis, secondly, change of a number of amplitude values is observed, namely, increase of angle of extension at KJ both in the stance and in the swing phase, what supposes walking on slightly-flexed legs, thirdly, spatial asymmetry is noted, i.e. difference in amplitude of the right and left lower extremities [10]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[87,198,860,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As may be seen from fig. 1, the form of HA in two-year-old children does not practically differ from the form of adults. But in children at the age of 2 amplitude of movements is a little bit less than in adults $(34^{\\\\circ})$ , for all this value of flexion is greater than in adult norm both in the stance and in the swing phase $-29^{\\\\circ}$ , while extension at HJ is weakly expressed $(-5^{\\\\circ})$ , that's why the whole curve is lifted above the level of zero line, what is connected with walking on the slightly flexed legs. The horizontal plateau in the beginning of the stance phase, coinciding in time with double-support phase, is absent in children at the age of 2. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[90,313,860,479]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"At the same time more remarkable changes are seen in the curve at a KA in children at the age of 2. First of all, amplitude of movements at KJ decreases up to $55^{\\\\circ}$ . Secondly, flexion at KJ is very long, it ends only at $23\\\\%$ of the cycle. Value of knee flexion in the stance phase exceeds the corresponding value in adults $-35^{\\\\circ}$ . For all this, extension in the stance phase is also very long, and, besides, it is realized incompletely, value of extension comprises $26^{\\\\circ}$ . Decrease of amplitude of movements at KJ takes place also because of diminution of the main flexion at KJ up to $65^{\\\\circ}$ , extreme value of the main flexion is timed to $t = 80\\\\%$ of the locomotor cycle, i.e. is displaced to the right along the temporal axis. So, increase of flexion at KJ in the stance phase, diminution of amplitude of movements and displacement of all extreme values to right along the temporal axis is typical of the knee angle of healthy children at the age of 2. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[89,491,868,706]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 1. Kinematic parameters of walking of healthy adult examinees, healthy children at the age of 7 and healthy children at the age of 2. KA - knee angle, HA - hip angle. Podograms are below the graphs. Along the abscissa axis - duration of the locomotor cycle in $\\\\\\\\%$ , along the ordinate axis - angular displacements at the joints of the lower extremities in degrees \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0113/b229895a92bca442e90416f56457f6209fcb326c98d16577f640ad095b5c4a8d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[87,809,858,858]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"But distribution of electrical activity of muscles during the locomotor cycle is of the greatest interest. EMG-pattern of 4 symmetrical muscles of the lower extremities in adult examinees and children at the age of 7 and 2 is demonstrated in Fig. 2. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[87,858,860,925]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"M. tibialis anterior (Fig. 2, a). While walking of healthy adult examinees electrical activity of m. tibialis anterior during the locomotor cycle has two maxima: the first - high-begins at $t = 95\\\\%$ of the subsequent cycle, gives the greatest wave in the limits of the first $10\\\\%$ of the following cycle and ends by $15\\\\%$ of the cycle. The functional purpose of the first \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[181,41,766,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"T.T. Batysheva, E.V. Pismennaya, K.A. Petrushanskaya, M.V. Pisareva, M.V. Kovina \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[89,951,119,963]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"378 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[295,950,858,964]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[134,99,904,165]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"maximum - lifting of the foot before touching the support, its gradual lowering on the support surface and rotation of the shank forward at the fixed foot. The second, less expressed maximum, begins at $t = 55\\\\%$ and ends at $t = 80\\\\%$ of the cycle. This maximum promotes the dorsal flexion at an AJ in the first half of the swing phase [7-11]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,166,904,231]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"EMG-pattern of m. tibialis anterior in children at the age of 7 is similar with the EMG-pattern of the adult healthy examinees, but some distinctions are noticed. At the age of 7 intermediate interpeaked activity increases. Besides, asymmetry of activity is observed in the right and left lower extremities [4, 5, 9]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,231,905,362]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"During walking of children at the age of 2 considerable growth of the second maximum of activity of m. tibialis anterior may be seen. Maxima of activity of children at the age of 2 are more prolonged in contrast with adults and seven-year-old children. So, the first maximum begins at the level of $t = 86\\\\%$ of the subsequent cycle and ends at the level of $23 - 24\\\\%$ of the following cycle. The second maximum, promoting the dorsal flexion at an AJ in the swing phase, is also more long $-53\\\\% < t < 86\\\\%$ . So remarkable growth of duration of the period of activity of this muscle up to $71\\\\%$ of the cycle is typical of children at the age of 2 in contrast with adults and children at the age of 7. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,363,905,461]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"M. gastrocnemius medialis (Fig. 2, b). During walking of healthy adult examinees electrical activity of m. gastrocnemius medialis has maximum between $t = 20\\\\%$ and $t = 60\\\\%$ of the cycle with the peak, timed to $t = 45\\\\%$ of the cycle. Function of this activity consists in regulation of movement of the shank forward (in the eccentric regime of work) until $50\\\\%$ of the cycle and in realization of flexion at KJ and plantar flexion at an AJ at the end of the stance phase [7-11]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,461,905,542]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"During walking of children at the age of 7 EMG-pattern of m. gastrocnemius is characterized by more early appearance of maximum of activity, greater level of the minimal activity, more high level of excitation in the beginning of the stance phase and during the swing phase. Relationship among the minimal and maximal activity of m. gastrocnemius in healthy adults comprises 8:1, in children at the age of $7 - 4:1$ [4, 5, 9]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,542,905,673]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"During walking of children at the age of 2 maximum of activity of m. gastrocnemius medialis is prolonged for the most part of the stance phase, what is connected with walking on slightly flexed legs (Fig. 2, b). As may be seen from Fig. 1, all kinematic curves in healthy children at the age of 2 are displaced above zero line, which determines standing with the straightened legs. Such position corresponds with the flexion position of the lower extremities, which is remained during the locomotor cycle. However, one more wave of activity may appear in two-year-old children - from $t = 61\\\\%$ till $t = 86\\\\%$ of the locomotor cycle. It is believed, that this wave of activity is intended for flexion at KJ. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,674,905,804]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"M. rectus femoris (Fig. 2, c). During walking of healthy adult examines electrical activity of m. rectus femoris has two maxima. The first maximum arises at the end of the subsequent cycle at the level of $t = 85\\\\%$ and ends at $t = 20\\\\%$ of the following cycle. The second small maximum is timed to a period $55\\\\% < t < 75\\\\%$ . Activity of this muscle at the end of the swing phase $(85\\\\% < t < 100\\\\%)$ promotes extension at KJ. During heel-support $(0\\\\% < t < 15\\\\%)$ and later m. rectus femoris works in the eccentric regime. Activity of muscle in this period is intended for counteraction of flexion at KJ. Activity of the second maximum is intended for hip flexion at the end of the stance phase [7-11]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[134,804,904,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Two maxima are also observed in electrical activity of m. rectus femoris during walking of seven-year-old children. The first maximum is very prolonged, it arises at the end of the subsequent cycle at the level of $90\\\\%$ and ends only by $60\\\\%$ of the following cycle. The second maximum, intended for hip flexion, is inconsiderable, both by value and duration. Evident asymmetry of maxima is revealed [4, 5, 9]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[136,887,904,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"During walking of children at the age of 2 EMG-pattern of m. rectus femoris acquires the two-peaked form, the first maximum has boundaries between $t = 95\\\\%$ from the subsequent \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[161,41,880,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Peculiarities of the biomechanical and innervative structure of walking of healthy children of the early age \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[137,948,700,963]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[884,951,912,963]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"379 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[87,99,860,181]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"cycle and $t = 30\\\\%$ of the following cycle, the second maximum lasts from $t = 30\\\\%$ till $t = 50\\\\%$ of the cycle. EMG-pattern of this muscle in two-year-old children resembles in its form the curve of the vertical component $R_{z}$ of the ground reaction force. Essential growth of the level of the intermediate interpeaked activity is observed. The second maximum of activity, promoting flexion at HJ, is absent in children at the age of 2. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[87,181,860,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"M. biceps femoris (Fig. 2, d). During walking of healthy examinees electrical activity of m. biceps femoris has maximum, which lasts from $t = 85\\\\%$ of the subsequent cycle till $t = 40\\\\%$ of the following cycle. Peak of maximum falls on $t = 90 - 92\\\\%$ of the subsequent cycle. The functional purpose of this wave of activity consists in slowing down extension of the shank at the end of the swing phase and creation of the small angle of flexion in the beginning of the step and extension at HJ in the first two thirds of the stance phase. The additional wave of activity may appear in the boundary of the stance and swing phases ( $45\\\\% < t < 65\\\\%$ ), which is intended, probably for flexion at KJ [7-11]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[89,313,858,346]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Increase of activity of this muscle is noticed in children at the age of 7. Nevertheless, slight asymmetry of EMG-pattern in the right and left lower extremities may be traced [4, 5, 9]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[87,347,860,464]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Two maxima of activity may be also revealed in children at the age of 2. But the level of the minimal interpeaked activity is much higher in comparison with seven-year-old children and adult examinees. The first maximum begins in the subsequent cycle at the level of $t = 90\\\\%$ , its peak is timed to $t = 20\\\\%$ of the following cycle and ends only by $t = 50\\\\%$ . The second maximum of activity is much higher in comparison with the first one, this maximum is timed to the period from $50\\\\% < t < 80\\\\%$ of the locomotor cycle. It is believed, that this maximum is intended for deceleration of quick extension at KJ in the swing phase. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[94,474,850,834]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 2. Electrical activity of muscles of the lower extremities in adult healthy examinees and healthy children at the age of 7 and 2: $a - \\\\\\\\mathrm{m}$ tibialis anterior; $b - \\\\\\\\mathrm{m}$ gastrocnemius medialis; $c - \\\\\\\\mathrm{m}$ rectus femoris; $d - \\\\\\\\mathrm{m}$ biceps femoris. Podograms are below the graphs. Along the abscissa axis – duration of the locomotor cycle in %, along the ordinate axis – electrical activity of muscles in % to the maximal value during the locomotor cycle \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0732/0f5fa937f3cf98fafa714851f21116b9aabdd7916e7430b818bd7f6aa635285e.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[181,41,766,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"T.T. Batysheva, E.V. Pismennaya, K.A. Petrushanskaya, M.V. Pisareva, M.V. Kovina \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[90,951,117,963]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"380 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[295,950,858,964]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[136,99,904,130]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The received data point out, that walking of children at the age of 2-2.5 is characterized by the following peculiarities: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[137,133,905,344]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1) remarkable diminution of the mean walking velocity exclusively at the expense of shortening of the step length and increase of cadence of walking; \\\",\\\"2) considerable disturbance of the resonance phenomena as a consequence of discrepancy of high cadence of walking and low step length; \\\",\\\"3) change of the temporal structure of the step: growth of duration of the stance and double-support phase and diminution of duration of the swing phase; \\\",\\\"4) presence of the flexion position of the lower extremities, what leads to decrease of amplitude of angular displacements and their lifting above the level of zero line; \\\",\\\"5) displacement of all extreme values of angular displacements to the right along the temporal axis, in spite of higher cadence; \\\",\\\"6) considerable increase of duration of maxima of activity of all muscles during the locomotor cycle, essential growth of the minimal interpeaked activity; \\\",\\\"7) absence of the second maximum of activity in some muscles (m. rectus femoris). \\\"]\",\n \"page_idx\": \"6\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[396,368,648,382]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DISCUSSION OF THE RESULTS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[136,391,904,439]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In spite of actuality of all received data, we consider expedient to examine only some of them. As may be seen from the received results, signs of immaturity of walking may be noticed in healthy children of the early age. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[136,441,905,505]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Flexion position of the lower extremities is one of such signs. This position is expressed in appearance of the initial angle of flexion at all joints (displacement of curves of angular displacements above the level of zero line). This fact leads to remarkable diminution of the step length and consequently, to deceleration of walking. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[136,506,905,602]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Shortening of the step length is accompanied by decrease of amplitude of angular displacements at the joints of the lower extremities. Change of the temporal structure of walking in children at the age of 2 is another consequence of diminution of walking velocity, and namely increase of duration of the stance and double-support phase and decrease of duration of the swing phase. Displacement of all extreme values to the right along the temporal axis takes place as a consequence of change of the temporal structure of walking [7-10]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[136,605,905,718]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Discrepancy of change of cadence and double step length is another important indication of immaturity of walking in children of the early age. As may be seen from the received data, very high cadence of walking (141 step/min) is accompanied by low double step length $(0.5\\\\mathrm{m})$ . This phenomenon leads to disorder of the resonance phenomena, i.e. to discrepancy of frequency of the forced oscillations with frequency of own oscillations of the lower extremities. In healthy adults and children at the age of 7 growth of walking velocity is accompanied by increase of both cadence and the step length. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[136,719,905,832]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As investigations of A.S. Vitenson and L.N. Samsonova demonstrated, there is not strict correlation between cadence and step length in children at the age of 2 [11]. It is believed, that association of mechanisms, controlling these both variables, is formed by conditioned reflex way in the process of ontogenesis. As a result, greater step length begins to correspond to higher cadence, what causes increase of the walking velocity [7-10]. Disorder of the resonance phenomena leads to increase of the energetic cost of walking in children of the early age. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[136,834,905,931]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Question of a change of electrical activity of muscles in children of the early age is of the special interest. As may be seen from the received data, expressed prolongation of maxima of activity of all muscles is noticed in children at the age of 2. Walking on the slightly flexed legs furthers formation of such stereotype. For all this it is typical, that activity of muscles-extensors (m. gastrocnemius and m. rectus femoris) acquires the form of the wide plateau, which involves two zones of the stance phase - zone of the maximal activity (zone $M$ ) and zone \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[161,41,880,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Peculiarities of the biomechanical and innervative structure of walking of healthy children of the early age \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[137,950,700,963]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[884,951,910,963]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"381 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[87,99,858,181]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"of the temperate activity (zone $T$ ). Walking on the slightly flexed legs is the reason of prolongation of activity. Really, all kinematic curves of HA and KA are displaced above zero line, which reflects standing with the straightened legs. Such position of the curves of the angular displacements corresponds to the flexion position of the lower extremities, which is retained during the locomotor cycle. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[87,181,855,231]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Consequently, EMG-pattern is rather immature in children of the early age, because high interpeakd activity, increase of duration of activity and diminution of its amplitude is observed. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[87,231,857,281]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Change of the pattern of activity of m. gastrocnemius is of the special interest. A number of authors consider, that there are two EMG-patterns of this muscle in children of the early age [4, 5, 12]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[87,281,858,379]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The first EMG- pattern is immature, because it differs essentially from pattern of activity of the adult healthy examinees. It consists in remarkable increase of the maximum of activity of m. gastrocnemius from the middle of the swing phase and in the beginning of the stance phase. Activity of this muscle in the middle and at the end of the swing phase leads to decrease in the required minimum distance of the foot from the support surface. Activity in the stance phase disturbs the process of absorption and transformation of energy of m. triceps surae. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[87,379,858,428]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The second, mature pattern of activity of m. gastrocnemius consists in concentration of activity in the single-support phase, in the period from 10 to $45\\\\%$ of the locomotor cycle. Such pattern resembles in its form EMG-pattern of adult healthy man. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[87,429,858,508]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From the authors' point of view, immature EMG-pattern of m. gastrocnemius in children of the early age is connected with late maturation of its phase activity. They consider, that deceleration of conduction because of incomplete myelination and greater distance of the sensory component of the reflex ark are the important factors of maturation delay of gastrocnemius function [5, 12]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[87,510,858,608]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As investigations of A.S. Vitenson and A.S. Vitenson and coauthors demonstrated, reconstruction of the internal structure of movements is not finished by the age of 7 [9, 10]. This fact may be explained in the following way: the static-geometrical parameters of the man's body and the force characteristic of the trunk muscles are essentially changed with age, and that's why constant correction of the motional act is required, especially in conditions of transformation of the main indices of walking - cadence and step length. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[87,608,858,707]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is necessary to note, that EMG-pattern of muscles, corresponding to adult healthy examinees, is formed only by the age of 13 in healthy children. From our point of view, the received results of investigations of the biomechanical and innervative structure of walking of children of the early age are quite necessary for revealing the locomotor disturbances in children of early age, for choice of the adequate methods of abilitation and retracing this process with an account of age-specific peculiarities of children. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[415,728,532,743]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"REFERENCES \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[90,750,858,916]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. Andriacchi T.P., Ogle J.A., Galante J.O. Walking speed as a basis for normal and abnormal gait measurement. Journal of Biomechanics, 1977, vol. 10, no. 4, pp. 261-268. \\\",\\\"2. Bukreeva D.P., Kosilov S.A., Tambiae A.P. Vozrastnye osobennosti tsiklicheskikh dvizheniy detey i podrostkov [Age-specific peculiarities of the cyclic movements of children and teenagers]. Moscow: Pedagogika, 1975, 159 p. (in Russian). \\\",\\\"3. Filatov V.I. Klinicheskaya biomekhanika [Clinical biomechanics]. Ed. by Prof. V.I. Filatov. Leningrad-Moscow, 1980, 199 p. (in Russian). \\\",\\\"4. Skvortsov D.V. Diagnostika dvigatelnoy patologii instrumentalnyi metodami. Analiz pokhodki. Stabilometiya [Diagnostics of the locomotor pathology by means of the instrumental methods. Gait analysis. Stabilometry]. Moscow:MBN, 2007, 638 p. (in Russian). \\\",\\\"5. Sutherland D.H., Olshen R., Cooper B.A., Woo S.L.Y. The development of mature gait. The Journal of Biomechanics and Joint Surgery, 1980, vol. 62-A, no 3, pp. 336-353. \\\"]\",\n \"page_idx\": \"7\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[181,41,766,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"T.T. Batysheva, E.V. Pismennaya, K.A. Petrushanskaya, M.V. Pisareva, M.V. Kovina \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[90,951,117,963]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"382 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[295,950,858,964]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[139,99,905,332]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"6. Todd F.N., Lamoreux L.W., Skinner S.R., Johanson M.E., St. Helen R., Moran S.A., Ashley K.A. Variation in the gait of normal children //Journal of Biomechanics and Joint Surgery. -1989. - Vol. 71A. - P. 196-204. \\\",\\\"7. Vitenson A.S. Zakonomernosti normalnoi i patologicheskoy khodby cheloveka [Peculiarities of normal and pathological human walking]. Moscow: Zerkalo-M, 1998, 272 p. (in Russian). \\\",\\\"8. Vitenson A.S. Issledovanie biomechanicheskikh i neyrophiziologicheskikh zakonomernostey normalnoy i patologicheskoy khodby cheloveka [Investigation of the biomechanical and neurophysiological peculiarities of normal and pathological human walking]. Ph. D. Thesis. Moscow, 1982 (in Russian). \\\",\\\"9. Vitenson A.S., Petrushanskaya K.A. Ot estestvennogo k iskusstvennomy upravleniyu lokomotsiey [From natural to artificial control of locomotion]. Moscow: MBN, 2003, 448 p. (in Russian). \\\",\\\"10. Vitenson A.S., Petrushanskaya K.A., Spivak B.G., Matveeva I.A., Gritsenko G.P., Sutchenkov I.A. Peculiarities of biomechanical structure of walking of healthy children of different age groups //Russian Journal of biomechanics, 2013, vol. 17, no. 1, pp. 69-83. \\\",\\\"11. Vitenson A.S., Samsonova L.N. Biomehanicheskaya i innervatsionnaya struktura khodby detey rannego vozrasta [Biomechanical and innervative structure of walking of children of the early age]. Protezirovanie i protezostroenie, 1973, no. 31, pp. 39-46 (in Russian). \\\",\\\"12. Winter D. The biomechanics and motor control of human gait. Waterloo: University of Waterloo Press, 1990, 145 p. \\\"]\",\n \"page_idx\": \"8\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[139,350,304,363]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Received 8 August 2021 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[161,41,880,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Peculiarities of the biomechanical and innervative structure of walking of healthy children of the early age \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[139,950,700,963]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ISSN 1812-5123. Russian Journal of Biomechanics. 2021. Vol. 21, No. 4: 375-383 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[884,952,912,963]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"383 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.15593/rjbiomech/2021.4.08"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Medicine and Health Sciences"},"sha256":{"kind":"string","value":"4a0f94976542a2c1c001039e99cd3be6deb7651e96e5e0e8e660670c6360c6d5"},"title":{"kind":"string","value":"Peculiarities of the biomechanical and innervative structure of walking of healthy children of the early age"}}},{"rowIdx":67,"cells":{"abstract":{"kind":"string","value":""},"author":{"kind":"string","value":"Geneviève Morneau-Vaillancourt|Olakunle Oginni|Elham Assary|Georgina Krebs|Ellen J. Thompson|Elisavet Palaiologou|Celestine Lockhart|Louise Arseneault|Thalia C. Eley"},"content_list":{"kind":"list like","value":[{"bbox":"[90,91,909,171]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A cross-lagged twin study of emotional symptoms, social isolation and peer victimisation from early adolescence to emerging adulthood ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[124,192,877,237]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Genevieve Morneau-Vaillancourt,1 $\\textcircled{1}$ Olakunle Oginni,1,2 $\\textcircled{1}$ Elham Assary, D Georgina Krebs,1,3 $\\textcircled{1}$ Ellen J. Thompson,1,4 $\\textcircled{1}$ Elisavet Palaiologou, Celestine Lockhart,1 $\\textcircled{1}$ Louise Arseneault,1 $\\textcircled{1}$ and Thalia C. Eley1,5 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,240,926,311]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1Social,Geneticand Developmental PsychiatryCentre,Instituteof Psychiatry,Psychology and Neuroscience,King's College London,London,UK;²DepartmentofMental Health,Obafemi AwolowoUniversity,Ie-Ife,Nigeria;Research Departmentof Clinical,Educationaland HealthPsychology,UniversityCollege London,London,UK; 4Departmentof Twin Research and Genetic EpidemiologyScholof Life Courseand Population Sciences, King's Coege London, London,UK; 5National Institute for Health Research (NIHR) Biomedical Research Centre, South London and Maudsley Hospital, London, UK ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,338,924,611]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Background: Emotional symptoms, such as anxiety and depresive symptoms, are common during adolescence, often persist over time,and can precede the emergence of severe anxiety and depresive disorders. Studies suggest that a vicious cycle of reciprocal infuences between emotional symptoms and interpersonal difculties may explain why some adolescents sufer frompersisting emotional symptoms.However,theroleofdiferent typesofinterpersonal difficulties,suchassocial isolationandpervictimisation,inthesereciproalasociationsisstillunclear.Inaddition, the lack of longitudinal twin studies conducted on emotional symptoms during adolescence means that the genetic and environmental contributions to these relationships during adolescence remain unknown.Methods: Participants $( N = 1 5 , 8 6 9 )$ from the Twins Early Development Study completed self-reports of emotional symptoms,social isolation and peer victimisationat12,16 and 21 years old.A phenotypic cross-lagged model examined reciprocal asociations between variables overtime,and agenetic extensionofthis model examined the aetiologyofthe relationships between variables at eachtimepoint. Results: First,emotional symptoms were reciprocally andindependently associated with both social isolation and peer victimisation over time,indicating that diferent forms of interpersonal dificulties uniquely contributed to emotional symptoms during adolescence and vice versa. Second,early peer victimisation predicted later emotional symptoms via social isolation in mid-adolescence, indicating that social isolation may constitute an intermediate pathway through which peer victimisation predicts longer-term emotional symptoms. Finaly,individual diferences inemotionalsymptoms were mostlyaccountedforbynon-shared environmentalfctors at each timepoint,and both gene-environment and individual-specific environmental mechanisms were involved in therelationships between emotional symptoms and interpersonal diffculties.Conclusions: Our study highlights the necesity to intervene early inadolescence to prevent the escalation of emotional symptoms over time and to consider socialisolation and peer victimisation as importantrisk factors for the long-term persistence of emotional symptoms. Keywords: Anxiety: depression; peer relationships; twins; longitudinal studies. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,649,183,662]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[75,664,480,890]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Emotional symptoms,also referred to as internalising problems, comprise a broad range of anxiety and depressive symptoms， such as worry，fear and sadness (Caspi，Houts，Fisher，Danese，& Moffitt,2023). Adolescence is a critical period for the development of emotional symptoms,with approximately $10 \\%$ to $2 5 \\%$ ofadolescentssuffering from severe clinically relevant anxiety and depressive symptoms (Lu, 2019； Racine et al.， 2021; Tiirikainen，Haravuori，Ranta,Kaltiala-Heino，& Martunen, 2019). Emotional symptoms are associated with detrimental long-term outcomes, including academic disruption and serious mental and physical health problems(Cohen， Edmondson，& Kronish, 2015;Johnson,Dupuis, Piche,Clayborne, & Colman， 2018； López-López et al.， 2020). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,643,924,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Unfortunately,emotional symptoms often remain untreated and show continuity into adulthood, where they can escalate into full-syndrome and recurrent anxietyanddepressivedisorders(Johnson et al., 2018; Kwong et al., 2019; Naicker, Galambos, Zeng，Senthilselvan，& Colman，2013；Wolitzky-Taylor et al., 2O14). It is therefore crucial to identify the mechanisms by which emotional symptoms develop and persist during adolescence,as this will help identify young individuals at risk and inform early intervention strategies. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,799,924,926]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"One pathway contributing to the continuity of emotional symptoms may be negative interpersonal experiences,such as peer victimisation.Peer victimisation affects $1 0 \\% { - } 3 5 \\%$ of adolescents and is both a precursor and an outcome of emotional symptoms(Cook， Williams， Guerra， Kim，& Sadek, 2010; Modecki, Minchin, Harbaugh, Guerra, & Runions，2014； Moore et al.， 2017； Reijntjes, Kamphuis,Prinzie,& Telch,2O1O).Several studies haveexaminedthebidirectional associations between emotional problems and peer victimisation by using cross-lagged models. Although many sup-port a transactional model where emotional symptoms and peer victimisation mutually reinforce each other over time (e.g. Forrest, Gibson, Halligan, & St Clair,2020; Smith,Hébert,Brendgen,& Blais,2022; Zhu et al.，2022)，others align with a symptomdriven model,where emotional symptoms precede and perhaps evoke peer victimisation (e.g., Barzeva, Richards，Meeus，& Oldehinkel，202O；Lee& Vaillancourt, 2019; Pabian & Vandebosch, 2016). There is also evidence supporting an interpersonal risk model,where peer victimisation predicts later emotional symptoms (Williford et al.,2012). These mixed findings could be explained by the fact that most studies relied on narrow definitions of anxiety and depressive symptoms and evaluated them separately. Yet，adopting a broader definition of emotional symptoms,especially within a longitudinal context, may more accurately depict the experience of adolescents affected by these problems. Anxiety and depressive symptoms often co-occur, are characterised by sequential comorbidity (i.e., where one predicts the emergence of the other), share genetic risks and can be classified under a broader dimension of internalising problems (Caspi et al.，2023； Jacobson & Newman， 2017； Melton, Croarkin, Strawn, & McClintock, 2016; Middeldorp, Cath, van Dyck,& Boomsma, 2005). The few cross-lagged studies that have assessed emotional symptoms broadly (i.e.，both anxiety and depressive symptoms) consistently show that emotional symptoms are reciprocally associated with peer victimisation， thereby supporting the transactional model (e.g.,Boyes,Bowes, Cluver,Ward, & Badcock, 2014; Smith et al., 2022). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,941,926,978]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ 2023TheAuthors.JoualofChildPsychologyndPsychatrypublishdbyohnWiley&SonsLdonehalffAsociationforChildandAdolescent Mental Health. ThisisanoenacsartidertesfteatieComsAtrbutoesechpeitsusistrtiodeprodctio medium, provided the original work is properly cited. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[73,47,551,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Journal of Child Psychology and Psychiatry 64:11 (2023), pp 1569-1582 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[75,921,415,932]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conflict of interest statement: No conflicts declared. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[764,0,1000,43]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates The Association ACAMH for Childand Adolescent Mental Health ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[761,47,924,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"doi:10.1111/jcpp.13847 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[75,8,322,43]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The Journal of Child Psychology and Psychiatry ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[75,78,480,607]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[75,606,480,932]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In addition,other types of interpersonal experiences during adolescence,such as social isolation, may be involved in these reciprocal associations. Emotionally vulnerable adolescents often withdraw socially because of low mood， loss of interest,or worrying about social situations (Barzeva et al., 2020; Barzeva, Meeus,& Oldehinkel, 2019; Rubin, Coplan, & Bowker,2009). They are also more likely to be rejected or excluded by their peers，who can be reluctant to initiate interactions with adolescents who exhibit negative emotions (Barzeva et al., 2020; Bowker，Nelson，Markovic，& Luster，2013；Oh et al., 20o8). Consequently, socially isolated adolescents are often viewed as vulnerable targets for peer victimisation (Acquah,Topalli,Wilson,Junttila,& Niemi, 2016; Sentse, Prinzie,& Salmivalli, 2017). In turn, their desire not to be exposed to peer victimisation mightlead them to avoid social situations (Hutzell & Payne, 2O12),or peers who are unwilling to affiliate with victims might reject them (Salmivalli, Lagerspetz,Bjorkqvist, Osterman,& Kaukiainen，1996). This cascade of negative social experiences could ultimately aggravate emotional symptoms, as lacking positive peer relationships and opportunities to practise socialising often contributes to emotional symptoms(Barzevaetal., 2020; Matthews et al.,2O16). As such, social isolation may be an intermediate pathway through which emotional symptoms lead to peer victimisation and vice versa. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,80,924,164]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,168,924,577]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Longitudinal twin studies have shed light on the mechanisms underlying these pathways by showing that genetic and environmental factors independently contribute to the associations between emotional symptoms，social isolation and peer victimisation (e.g.，Matthews et al., 2016; Schaefer et al., 2018). Social isolation and peer victimisation do not occur randomly and are both heritable $( 3 0 \\%$ to $7 0 \\%$ ， suggesting that genetic factors contribute to individual differences in these social difficulties (Fisher et al.，2015；Johansson et al.，2022；Matthews et al.,2016; Veldkamp et al., 2019). Evidence suggests a genetic overlap between emotional symptoms and negative peer experiences,meaning that a genetic disposition for emotional symptoms enhances the risk of being exposed to negative social experiences via gene-environment correlations (Brendgen， Zheng, Vitaro,Dionne,& Boivin,2022). Furthermore,twin studies show that negative peer interactions increase emotional symptoms via non-shared environmental pathways,providing evidence that these associations are unconfounded by pre-existing genetic dispositions or by features of the familial environment shared by both twins (Arseneault et al.， 20o8； Schaefer et al.，2018；Silberg et al.，2016)． Therefore，the mechanisms by which emotional symptoms are related to interpersonal diffculties involve a mix of gene-environmentcorrelationsandindividualspecific environmental influences. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,579,922,889]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Despite these findings,questions remain unanswered.First,most cross-lagged studies on emotional symptoms solely focused on peer victimisation without considering other types of interpersonal difficulties and were conducted over short periods (e.g., 1 or 2 years).As such, it remains unclear whether social isolation acts as an intermediate pathway in the reciprocal associations between emotional symptoms and peer victimisation,and whether these associations persist over an extended period and across life transitions, such as from adolescence and into adulthood. Second, the extent to which genetic and environmental influences contribute to individual differences in emotional symptoms,social isolation and peer victimisation from adolescence to emerging adulthood,as well as to their relationships over this period, remain unknown.Documenting these genetic and environmental influences within a longitudinal perspective could enhance our understanding of the mechanisms underlying the persistence of emotional symptoms and its relationships with interpersonal difficulties over time. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,891,922,932]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The present study aimed to address these gaps by testing (a) the reciprocal phenotypic associations between emotional symptoms,social isolation and peer victimisation from 12 to 21 years old,and (b) the genetic and environmental influences and overlap of emotional symptoms, social isolation and peer victimisation at 12,16 and 21 years old. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,945,922,969]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ 2023 TheAuthors.JoumalofChild Psychologyand Psychiatry publishedbyJohn Wiley&Sons LtdonbehalfofAssociation for Child and Adolescent Mental Health. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[603,49,924,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J Child Psychol Psychiatr 2023; 64(11): 1569-82 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[77,48,398,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1570 Genevieve Morneau-Vaillancourt et al. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[75,80,480,135]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[75,166,152,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Methods ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[75,180,166,193]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Participants ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[75,202,480,508]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Participants were from the Twins Early Development Study (TEDS),a longitudinal cohort of twins in England and Wales (Lockhart et al.,2023).TEDS recruited more than 13,000 families from a national register of all twins born in England and Wales between 1994 and 1996;and the twins have been followed up into early adulthood.TEDS has assessed the twins' behavioural， cognitive and emotional development from infancy to early adulthood.The present study uses twins' reports of emotional symptoms，social isolation and peer victimisation at 12,16 and 21 years old (Table 1 provides age estimates for each variable).A total of 15,869 participants $5 4 \\%$ females) for whom data were available for at least one measure were included in the phenotypic model.Among these participants,only those for whom zygosity was available were included in the genetic model $\\scriptstyle { [ N = \\ 1 5 , 6 9 5 }$ ，including 2,727 complete monozygotic [MZ],4,837 complete dizygotic [DZ],138 incomplete MZ and 429 incomplete DZ pairs; Table S1 provides a description of twin pairs for each variable).When twins were about 18 months old, zygosity was determined using a parent-reported questionnaire of physical similarity (Price et al.，2Ooo).When unclear, zygosity was determined using DNA testing.Participants who were born in extreme perinatal conditions $N = 2 9 6$ twin pairs or 592 individuals),had severe medical conditions $( N = 9 5$ individuals),or for whom zygosity and/or gender were unknown $( N = 1 0 5$ twin pairs or 210 individuals) were excluded. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[75,530,247,544]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ethical considerations ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[75,552,480,575]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ethical approval for TEDS at the 12-，16- and 21-year assessment waves was granted by the King's College London ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,82,924,115]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Research Ethics Committee (reference:05-Q0706-228 and PNM 0910 104)．TEDS collected informed consent from parents or participants prior to each wave of data collection. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,139,591,151]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Measures ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,159,924,279]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Emotional symptoms and social isolation. Twins answered questions from the Strengths and Difficulties Questionnaire (SQD；Goodman，1997) to assess emotional symptoms and social isolation.Participants evaluated how things had been for them over the last three or six months ona three-point scale ( $0 =$ \\`not true'; $1 =$ quite true'; $2 =$ very true'). We computed emotional symptoms and social isolation scores by deriving the average of at least three items and multiplying this by the total number of items for each subscale to minimise the impact of missing data. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,280,924,516]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The emotional symptoms subscale includes five items assessing the extent to which participants experience anxiety and depressive symptoms (e.g.，I worry a lot',I am often unhappy，downhearted,or tearful').Emotional symptoms scores ranged from O to 1O.Internal consistency was assessed using ordinal alpha coefficients (Gadermann，Guhn，& Zumbo，2012) and was satisfactory at 12 $( \\mathsf { \\alpha } ) = \\mathsf { \\alpha } . 7 9 )$ ，16 $( \\alpha = . 7 8 )$ and 21 years old $( \\alpha = . 8 6 )$ . Both emotional symptoms and peer problems SDQ subscales show good convergent validity as well as measurement invariance over time (Goodman& Scott, 1999; Speyer,Auyeng,& Murray,2022;Vugteveen,de Bildt,Theunissen,Reijneveld,& Timmerman,2021). They both correlate with similar measures from other widely used scales,such as the Child Behaviour Checklist (CBCL) and the Youth Self-Report (YSR；Achenbach，1991).The SDQ emotional symptoms subscale is strongly associated with anxiety and depression from the CBCL and YSR,and the SDQ peer problems subscale is strongly correlated with social problems from the CBCL and YSR (Goodman & Scott,1999; Vugteveen et al., 2021). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,517,924,574]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We assessed social isolation using four out of the five items of the SDQ peer problems subscale,removing the item ‘Others pick on me or bully me’to avoid overlap with the peer victimisation measure.The other four items assessed the extent to which participants spend time on their own or are socially accepted by peers (e.g.，I am usually on my own.I generally play alone or keep to myself). Scores ranged from O to 8.Internal consistency was satisfactory at 12 $( \\mathsf { x } = . 7 0 )$ ，16 $( \\mathsf { \\alpha } ) = \\mathsf { \\beta } . 6 7 )$ and 21 years old $( \\mathsf { \\Omega } \\mathsf { \\Omega } ) = \\mathsf { \\Omega } . 6 8 )$ ，as indicated by ordinal alpha coefficients.We also conducted sensitivity analyses using the full five-item peer problems subscale to see whether removing one item would affect the reliability of the scale. Results suggest that it did not,as they were highly similar to results obtained using the four-item social isolation measure (see Table S2, Figure S1 and S2). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[72,609,926,877]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0023/3f1da5d7ce6868a59a0079569f76274586fcd4d66ce1d713ee0bd009e9dde784.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":"<table><tr><td></td><td></td><td colspan=\"2\"> MZ Pairs</td><td colspan=\"2\">DZ Pairs</td><td colspan=\"6\"></td></tr><tr><td></td><td>N</td><td>Complete</td><td>Incomplete</td><td></td><td>Complete Incomplete</td><td>Age (SD)</td><td>Mean (SD)</td><td>Range</td><td>Skew</td><td></td><td>Kurtosis</td></tr><tr><td>12 years old</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Emotional</td><td>11,681</td><td>2,092</td><td>33</td><td>3,678</td><td>59</td><td>11.28 (.70)</td><td>2.19 (2.06)</td><td></td><td>0-10</td><td>1.02</td><td>3.71</td></tr><tr><td>symptoms Social isolation</td><td>11,675</td><td>2,090</td><td>35</td><td>3,674</td><td>63</td><td>11.28 (.70)</td><td></td><td>1.15 (1.34)</td><td>0-8</td><td>1.51</td><td>5.73</td></tr><tr><td>Peer</td><td>11,729</td><td>2,104</td><td>23</td><td>3,710</td><td>28</td><td>11.28 (.70)</td><td></td><td>3.13 (2.88)</td><td>0-10</td><td>.71</td><td>2.45</td></tr><tr><td>victimisation 16 years old</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Emotional</td><td>10,196</td><td>1,811</td><td>23</td><td>3,209</td><td>66</td><td>16.32 (.68)</td><td>2.75 (2.24)</td><td></td><td>0-10</td><td>.75</td><td>2.99</td></tr><tr><td>symptoms Social isolation</td><td>10,181</td><td>1,804</td><td>30</td><td>3,202</td><td>72</td><td>16.32 (.68)</td><td></td><td>1.47 (1.38)</td><td>0-8</td><td>1.16</td><td>4.44</td></tr><tr><td>Peer</td><td>5,236</td><td>893</td><td>178</td><td>1,416</td><td>416</td><td>16.48 (.27)</td><td></td><td>2.93 (2.72)</td><td>0-10</td><td>.75</td><td>2.70</td></tr><tr><td>victimisation 21 years old</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Emotional</td><td>9,501</td><td>1,271</td><td>1,275</td><td>2,113</td><td>2,454</td><td>22.26 (.91)</td><td>3.61 (2.70)</td><td></td><td>0-10</td><td>.50</td><td>2.29</td></tr><tr><td>symptoms Social isolation</td><td>9,501</td><td>1,270</td><td>1,276</td><td>2,112</td><td>2,453</td><td>22.26 (.91)</td><td></td><td>2.08 (1.68)</td><td>0-8</td><td>.76</td><td>3.23</td></tr><tr><td>Peer victimisation</td><td>8,382</td><td>1,155</td><td>1,343</td><td>1,883</td><td>2,556</td><td>22.84 (.88)</td><td></td><td>1.78 (2.21)</td><td>0-10</td><td>1.26</td><td>4.00</td></tr></table>","table_caption":"[\"Table 1Descriptive statistics for emotional symptoms,socialisolationand peer victimisationat12,16and 21 years old \"]","table_footnote":"[\"Emotionalsymptoms refer tosymptoms ofanxietyand depresion.Peer victimisationat16 years old was measured inaseparate web survey that was completed by fewerparticipants forfinancialreasons.Values areforrawscores (Table S4 shows descriptive statistics for transformed scores). DZ,dizygotic twin; MZ,monozygotic twin; $N ,$ number of participants; $S D$ ，standard deviation; Skew, skewness. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[73,49,235,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"doi:10.1111/jcpp.13847 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[70,945,926,969]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ 2023TheAuthors.JoumalofChildPsychologyand Psychiatry publishedbyJohn Wiley&Sons Ltdonbehalfof Association for Child and Adolescent Mental Health. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[468,48,922,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cross-lagged twin study emotional symptoms adolescence 1571 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[75,80,480,198]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[75,214,480,475]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Peer victimisation. We used the self-reported Multidimensional Peer Victimisation Scale (Mynard & Joseph,2000) to assess peer victimisation at 12,16 and 21 years old. Using a three-point scale $\\scriptstyle ( 0 = { \\mathrm { \\overleftarrow { n o t } } }$ at all'; $1 = \\cdot _ { \\mathrm { o n c e } } ,$ ； $2 = \\cdot _ { \\mathrm { m o r e } }$ than once'，participants answered five items about the extent to which they had experienced acts of physical,verbal and social peer victimisation from other students (at 12 and 16 years) or someone excluding their partner or family (at 21 years),in the past year (e.g.，‘How often has someone refused to talk to you?'). The scale focuses specifically on victimisation by peers at 12 and 16 years,whereas the definition of victimisation is broader at 21 years,encompassing victimisation by individuals from wider social networks (e.g.,work colleagues).Despite these differences,the validity and reliability of the Multidimensional Peer Victimisation Scale has been shown to be comparable across developmental periods, including emerging adulthood (Joseph & Stockton，2018)．We computed peer victimisation scores by multiplying the average across at least three items by the total number of items,and scores ranged from O to 1O.Ordinal alpha coefficients indicated satisfactory internal consistency at 12( $\\langle \\alpha = . 8 5 \\rangle$ ，16 $\\left( \\alpha = . 8 8 \\right)$ and 21 years old $( \\mathsf { \\pmb { x } } = . 8 7 )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[75,499,220,513]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Statistical analyses ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[75,521,480,806]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Phenotypic analyses. We first transformed all variables using a square root $+ 1$ transformation to correct for positive skewness.We conducted descriptive statistics and correlations using RStudio (version 2022.2.2.485；RStudio Team，2022). The analysis plan was pre-registered (https://osf.io/sjb5a/? view_only $r =$ lce2374f6e15467ebd2cf626538fbe19). To examine sex differences,we conducted independent samples t-tests for all variables (see Table S3).We found that females showed higher emotional symptoms than males and that males were more victimised by their peers than females at 12,16 and 21 years old.We also found that males were more socially isolated than femalesat12 years old.To see whether these sex differences were reflected in the patterns of associations between variables,we ran a multi-group phenotypic crosslagged model (see Figure S3A; Figure 2B).We found that a cross-lagged modelinwhich parameterswere freely estimated for females and males showed a better fit than a model in which parameters were constrained to equality across sexes. However,these differences were very small in terms of effect sizes and confidence intervals for most estimates overlapped across femalesand males.Therefore,we decided not to test a multi-group genetic cross-lagged model and to report analyses that controlled for age and sex,as is standard practice for twin model fitting (McGue & Bouchard, 1984). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[75,808,480,925]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To examine the bidirectional phenotypic associations between emotional symptoms, social isolation and peer victimisation at all three timepoints,we specified a crosslagged panel model using the lavaan package (Rosseel, 2012) to obtain structural equation modelling ft indices (using fullinformation maximum likelihood [FIMLl). The cross-lagged model estimated the reciprocal associations between all variables from 12 to 16 years and also from 16 to 21 years. It accounted for the stability across timepoints and preexisting relationships between variables, thereby assessing the prediction of incremental changes in variables over time (Adachi & Willoughby, 2015).We then examined the extent to which social isolation at 16 years mediated the bidirectional associationsbetweenemotionalsymptomsandpeer victimisation by estimating indirect effects and their $9 5 \\%$ confidence intervals (CI) using bootstrapping with 1,000 iterations. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,80,924,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,164,922,246]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Asa sensitivity check，and because twin models are estimated by nesting participants within twin pairs,we ran the model in lavaan using one twin per pair who was randomly chosen $( N = 7 , 9 2 6 )$ and then using twin pairs in OpenMx $( N = 8 , 2 6 2$ twin pairs;Boker et al.,2011).Estimates were similar across these models,so we report the results of the model using all participants $( N = 1 5 , 8 6 9 )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,262,924,439]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Genetic analyses. We then examined the genetic and environmental mechanismsunderlying the associations between variables using twin modelling. The twin method compares DZ twins,who share on average $50 \\%$ of their segregating genes,to MZ twins,who share $100 \\%$ of their genes.Because twins grow up in the same familial environment,contributions from shared environmental factors are expected to increase the similarity between twins.Conversely, non-shared environmental factors unique to each twin are expected to increase differences between twins.The twin method relies on these assumptions to decompose phenotypic variance and covariance into genetic (A),shared (C) and nonshared (E) environmental factors (Rijsdijk & Sham,2002). Non-shared environmental factors (E) also include measurement error. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,441,922,759]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We first ran cross-twin correlations (i.e.,between twins of the same family) and univariate twin models to estimate the extent to whichA,CandE factors contributed to eachvariable individually.We estimated cross-twin cross-trait correlations between variables (e.g.,between trait1 for twin 1 and trait 2 for twin 2) by running a constrained saturated model,where means and variances were constrained to equality across zygosity and birth order while covariances were constrained to be symmetrical for MZ and DZ twin pairs.We then specified a twin extension of the phenotypic cross-lagged model, initially described by Burt，McGue，Krueger，and Iacono (2005). Variances (at 12 years) and residual variances (at 16 and 21 years) were decomposed into ACE components.Genetic and environmental correlations were estimated at each timepoint by examining the extent to which genetic and environmental factors account for the covariance between variables using cross-twin cross-trait correlations.In addition,gene-environment correlations were examined by focusing on the extent to which genetic factors associated with emotional symptoms overlapped with the other two‘environmental’ outcomes,that is social isolation and peer victimisation. Such genetic overlap would indicate that a genetic vulnerability for emotional symptoms increases the risk of experiencing interpersonal difficulties,thereby providing evidence for gene-environment correlations. The genetic cross-lagged model was fitted using FIML and parameter $9 5 \\%$ confidence intervals were estimated in OpenMx. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,787,579,800]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Results ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,802,924,857]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Table 1 presents descriptive statistics of raw scores (Table S4 presents descriptive statistics for transformed variables),and Table 2 presents phenotypic correlations between variables. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[519,876,764,889]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Phenotypic cross-lagged model ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,898,924,925]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We examined the bidirectional associations between emotional symptoms, social isolation and peer victimisation from 12 to 21 years old using a phenotypic cross-lagged model (Figure 1). Fit indices indicated a good fit of the model: Root Mean Square Error of Approximation $( \\mathrm { R M S E A } ) = . 0 4$ , Comparative Fit Index $\\left( \\mathrm { C F I } \\right) = \\ . 9 8$ , Tucker-Lewis Index $( \\mathrm { T L I } ) = . 9 1$ Cross-lagged paths were statistically significant for the most part $( \\beta = . 0 3$ to .12),except for the crosslagged path from social isolation at 16 years to peer victimisation at 21 years: $\\beta = . 0 1$ $1 9 5 \\%$ CI: -.02, .03). These cross-lagged associations were observed while accounting for the moderate stability in all variables from 12 to 16 years $( \\beta = . 2 6$ to .33) and from 16 to 21 years $( \\beta = . 3 0$ to .35)．They also accounted for the pre-existing correlations between variables,which were weak to moderate within each of the timepoints $( r = . 0 3$ to .35). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[68,945,924,969]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ 2023TheAuthors.JoumalofChildPsychologyand Psychiatry publishedbyJohn Wiley&Sons Ltdonbehalfof Association for Child and Adolescent Mental Health. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[603,49,924,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J Child Psychol Psychiatr 2023;64(11): 1569-82 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[77,48,396,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1572 Genevieve Morneau-Vaillancourt et al. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[517,80,924,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[164,93,357,928]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0452/1d7d06556e26495ad493aa2d2158d0944bbcaf38f5055e109953f8dd1002800b.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":"<table><tr><td></td><td>1.</td><td></td><td>2.</td><td>3.</td><td>4.</td><td>5.</td><td>6.</td><td>7.</td><td>8.</td></tr><tr><td>1. Emotional symptoms 12 years old</td><td>1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>2. Social isolation 12 years old</td><td>.35 (.33, .37)</td><td>1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>3.Peer victimisation 12 years old</td><td>.32 (.31, .34)</td><td></td><td>.27 (.25,28)</td><td>1</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>4.Emotional symptoms 16 years old 5.Social isolation 16 years old</td><td>.34 (.32, .36)</td><td></td><td>.15 (.13, .17)</td><td>.14 (.12, .17)</td><td>1</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>.20 (.17, .22)</td><td></td><td>.30 (.28, .32)</td><td>.16 (.14, .18)</td><td>.34 (.32, .36)</td><td>1</td><td></td><td></td><td></td></tr><tr><td>6.Peer victimisation 16 years old</td><td>.18 (.15, .21)</td><td></td><td>.16 (.13, .19)</td><td>.31 (.29, .34)</td><td>.23 (.20,.25)</td><td>.25 (.22,.28)</td><td>1</td><td>1</td><td></td></tr><tr><td>7. Emotional symptoms 21 years old 8.Social isolation 21 years old</td><td>.25 (.23,.28)</td><td></td><td>.12 (.10, .14) .21 (.19,.23)</td><td>.14 (.12, .16)</td><td>.39 (.37, .41)</td><td>.21 (.19, .24)</td><td>.18 (.15,.22)</td><td></td><td>1</td></tr><tr><td></td><td>.17 (.15, .19)</td><td></td><td></td><td>.14 (.11, .16)</td><td>.20 (.18,.22)</td><td>.35 (.33, .37)</td><td>.20 (.17, .23)</td><td>.38 (.37, .40)</td><td></td></tr><tr><td>9.Peer victimisation 21 years old</td><td>.09 (.06, .11)</td><td></td><td>.10 (.07, .12)</td><td>.18 (.15, .20)</td><td>.13 (.11, .16)</td><td>.11 (.08, .13)</td><td>.33 (.30, .36)</td><td>.21 (.19,.23)</td><td>.12 (.10,.1</td></tr></table>","table_caption":"[\"\"]","table_footnote":"[\"\"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[517,307,924,577]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Overall， thephenotypic cross-lagged model showed that emotional symptoms at 12 and 16 years were independently and reciprocally associated with both social isolation and peer victimisation at 16 and 21 years,respectively.We then examined whether some of the cross-lagged paths were stronger than others by comparing models in which the selected parameters were constrained to equality with the unconstrained model in Figure 1 (Table S5 presents these analyses).We found that emotional symptoms equally predicted social isolation and peer victimisation and that conversely,both forms of interpersonal difficulties equally predicted emotional symptoms.In addition,both forms of interpersonal difficulties at 16 years predicted more subsequent emotional symptoms than at 12 years,indicating that social isolation and peer victimisation may carry higher risk for emotional symptoms in mid-adolescence, compared to early adolescence. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,579,924,733]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Notably, the model showed that peer victimisation at 12 years indirectly predicted emotional symptoms at 21 years via social isolation at 16 years: $\\beta = . 0 0 4$ $( 9 5 \\% \\mathrm { ~ C I } = . 0 0 3$ ，.007). This indirect effect explained $3 \\%$ of the correlation between peer victimisation at 12 years and emotional symptoms at 21 years $\\left( . 0 0 4 / . 1 4 1 = . 0 2 8 \\right)$ ． However，we did not observe the reverse sequence of associations,as emotional symptoms at 12 years did not predict peer victimisation at 21 years via social isolation: $\\beta = . 0 0 1$ $1 9 5 \\%$ $\\mathrm { C I } = - . 0 0 2$ , .003). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[519,755,737,769]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Genetic cross-lagged model ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,777,924,932]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Before testing the full genetic cross-lagged model, we inspected the cross-twin correlations and univariate twin models. Cross-twin correlations suggested significant A (i.e.，additive genetic) and $\\mathbf { E }$ (nonshared environmental contributions for most variables，as DZ correlations were about half of MZ correlations (i.e.，indicating A influences) and MZ correlations were less than 1 (i.e.，indicating E influences).Univariate twin models confirmed that most variables were accounted for by A and E influences (Table 3, see Table S6 for fit comparisons of univariate twin models).We then specified the genetic cross-lagged model informed by cross-twin cross-trait correlations (Table S7; Burt et al., 2005). We first tested an ACE model and then dropped C parameters that were non-significant or close to 0, one at a time. We compared nested models using likelihood ratio tests. The final model included AE factors for most variables,except for victimisation at 12 years old,which had a significant C component. The genetic model had a better ft than the phenotypic cross-lagged model， suggestingthat adding A and $\\mathbf { E }$ components improved model fit (likelihood ratio test, $p < . 0 0 1$ ; Akaike's Information Criterion $[ \\mathrm { A I C } ] = 2 3 7 , 4 6 3 . 6 0$ and 234,391.90 for the phenotypic and genetic models，respectively [lower AIC values suggesting better fit]). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,49,235,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"doi:10.1111/jcpp.13847 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[67,945,922,969]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ 2023TheAuthors.JoumalofChildPsychologyand Psychiatry publishedbyJohn Wiley&Sons Ltdon behalf of Association for Child and Adolescent Mental Health. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[468,48,924,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cross-lagged twin study emotional symptoms adolescence 1573 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[82,80,914,400]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 1 Phenotypic cross-lagged panel model. $N = 1 5 , 8 6 9$ .Emotional symptoms refer to symptoms of anxiety and depression.Values on single-headedarrowsarestandardisedpartialregressoncoeficients.Thesecoeficientscanbesquaredtocalculatetheproportionof varianceexplained.Valuesondouble-headedarrows betweenvariableswithintesametimepointarecorelationcoeficients;the correlationsatags16and2areresiduals,dexingtherelationshipetwenvariablesattatagethatisnotexplainedbytiralier association.Solidlines indicate statisticallysignificant parameters.The doted lines indicate non-significant parameters \"]","image_footnote":"[]","img_path":"images/0277/3ecc8481be5516b1f95e7ddb6225d5017b5530912474f0f284fb12e1b524a6a6.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[72,542,929,718]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0747/f53da1ff011dc2305a2f33053a48074be945dd095cd4481284092320e31862a9.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":"<table><tr><td></td><td>rMZ</td><td>rDZ</td><td>A</td><td>C</td><td>E</td></tr><tr><td>12 years old</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Emotional symptoms</td><td>.40 (.36, .44)</td><td>.20 (.17,.23)</td><td>.40 (.37, .43)</td><td></td><td>.60 (.57, .63)</td></tr><tr><td>Social isolation</td><td>.37 (.33, .41)</td><td>.21 (.18,.24)</td><td>.39 (.36, .42)</td><td></td><td>.61 (.58, .64)</td></tr><tr><td>Peer victimisation</td><td>.57 (.54, .60)</td><td>.33 (.30, .36)</td><td>.45 (.37, .52)</td><td>.11 (.05, .18)</td><td>.44 (.41, .47)</td></tr><tr><td>16 years old</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Emotional symptoms</td><td>.42 (.38, .46)</td><td>.16 (.13, .20)</td><td>.40 (.36, .43)</td><td></td><td>.60 (.57, .64)</td></tr><tr><td>Social isolation</td><td>.41 (.37, .45)</td><td>.20 (.17,.23)</td><td>.41 (.38, .44)</td><td></td><td>.59 (.56, .62)</td></tr><tr><td>Peer victimisation</td><td>.46 (.41, .51)</td><td>.20 (.15,.25)</td><td>.45 (.41, .50)</td><td></td><td>.55 (.50, .59)</td></tr><tr><td>21 years old</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Emotional symptoms</td><td>.34 (.30, .39)</td><td>.19 (.15, .23)</td><td>.35 (.31, .39)</td><td></td><td>.65 (.61, .69)</td></tr><tr><td>Social isolation</td><td>.39 (.35,.43)</td><td>.19 (.16,.23)</td><td>.39 (.35,.42)</td><td></td><td>.61 (.58, .65)</td></tr><tr><td>Peer victimisation</td><td>.29 (.24,.34)</td><td>.14 (.10, .18)</td><td>.29 (.25,.34)</td><td></td><td>.71 (.66,.75)</td></tr></table>","table_caption":"[\"Table3Cross-twincorrlationsandunivarateetimatesforemotioalsymptom,socialisolationandpeervctimisatinat12,16 and 21 years old \"]","table_footnote":"[\"Emotional symptoms refer to symptoms of anxiety and depression. $9 5 \\\\%$ confidence intervals are in parentheses.A,additive genetic factors;C,sardeviromtalactors;Enoaredeioetalctrs;DZ,o-twnorelatiosforgotitwins;, cross-twin correlations for monozygotic twins. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[75,783,480,925]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,783,924,868]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,869,924,925]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The genetic model, presented in Figure 2, shows that A and $\\mathrm { E }$ estimates for emotional symptoms and social isolation at 12 years were similar,with E accounting for a larger proportion of variance $( 6 1 \\% )$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[68,945,924,969]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ 2023TheAuthors.JoumalofChildPsychologyand Psychiatry publishedbyJohn Wiley&Sons Ltdonbehalfof Association for Child and Adolescent Mental Health. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[603,49,924,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J Child Psychol Psychiatr 2023; 64(11): 1569-82 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[75,48,396,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1574 Genevieve Morneau-Vaillancourt et al. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[84,80,912,409]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 2 Genetic cross-lagged panel model. $N = 1 5 , 6 9 5 ,$ including 2,727 complete MZ,4,837 complete DZ, 138 incomplete MZ and 429 incomplete DZpairs.Emotional symptomsrefertosymptomsofanxietyanddepresion.Aadditivegeneticfactors;Csharedenvironmentalfactors;Eenvironmentalfactors.Valuesonsingle-eadedarrowsarestandardsedpathestimates;valuesondubleeaded arrowsarecorrelationcoeficients.Solidlines indicatestatisticallysignificantparameters.Thedottedlines indicatenonsignificant parameters.Proportionofvariance explainedbyAandEcomponentscanbeestimatedbysquaring theirrespectivepath estimates \"]","image_footnote":"[]","img_path":"images/0109/f35af659f3a3934cfd4bd00f8e38e34cab91c6fafe7855061e1b520bf720fe09.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[75,517,480,898]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"than A $( 3 9 \\% )$ . Findings were somewhat different for peer victimisation at 12 years,with A $( 5 0 \\% )$ and $\\mathtt { E }$ $( 4 4 \\% )$ explaining comparable proportions of variance (C,or non-shared environmental factors,explained $6 \\%$ of the variance in peer victimisation at 12 years old).At 16 and 21 years,most of the variance was accounted for bynewA $20 \\%$ to $3 2 \\%$ and $\\mathrm { E }$ （ $56 \\%$ to $67 \\%$ ） components， indicating time-specific genetic and non-shared environmental contributions. The remaining part of the variance in outcome variables was accounted for by previous A and E influences transmitted via stability ( $2 \\%$ to $7 \\%$ and cross-lagged paths ( $0 1 \\%$ to $1 \\%$ . Total transmitted contributions accounted for 9 to $1 3 \\%$ of the variance in outcomes at 16 and 21 years old,indicating that outcome variables were only partially explained by variables at previous timepoints. These contributions from previous timepoints,along with their genetic and environmental components,are presented in Figure 3 and Table 4. Overall, findings from the genetic cross-lagged model indicate that, for the most part, non-shared environmental influences accounted for the largest proportion of variability in emotional symptoms,social isolation and peer victimisation at all timepoints and that most of the variance in outcome variables at 16 and 21 years was explained by new aetiological influences emerging over time. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,901,480,928]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Genetic and environmental correlations were mostlysignificant， indicatingthatemotional symptoms，social isolation and peer victimisation shared both genetic and environmental risks at each timepoint. Genetic correlations were moderate at 12 years and were weaker at later timepoints ( $_ { r G }$ at 12 years $= . 4 6$ to .60; rG at 16 years $=$ .33 to .48; rG at 21 years $= . 0 0$ to .45). These findings suggest that gene-environment correlations partially account for the relationships between emotional symptoms and both social isolation and peer victimisation. Non-shared environmental correlations were weak but comparable at each timepoint(rEat 12 years $= . 1 2$ to .19；rE at 16 years $\\ c = . 0 7$ to .20; rE at 21 years $= . 0 6$ and .27)，indicating that individual-specific environmental pathways modestly accounted for the associations between interpersonal difficulties and emotional symptoms. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,517,924,744]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,772,608,784]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Discussion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,788,924,927]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The present study is the first to use a genetically sensitive design to document the reciprocal associations between emotional symptoms (i.e.，anxiety and depressive symptoms) and two different forms of interpersonal difficulties， social isolation and peer victimisation,over an extended period during adolescence and early adulthood. First,we found that emotional symptoms predicted increases in social isolation and peer victimisation at later timepoints, and that in turn， social isolation andpeer victimisation independently contributed to emotional symptoms over time. We also found a significant longitudinal mediation where social isolation in mid-adolescence partly explained the long-term link between peer victimisation at 12 years and emotional problems at 21 years. Second, twin analyses showed that, for the most part,non-shared environmental influences mostly explained individual differences in all variables during adolescence and early adulthood and that both gene-environment andnon-sharedenvironmentalcorrelations accounted for their associations at each timepoint. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[67,945,922,969]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ 2023TheAuthors.JoumalofChildPsychologyand Psychiatry publishedbyJohn Wiley&Sons Ltdon behalf of Association for Child and Adolescent Mental Health. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[73,49,235,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"doi:10.1111/jcpp.13847 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[468,48,924,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cross-lagged twin study emotional symptoms adolescence 1575 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[84,79,915,431]","code_body":null,"code_caption":null,"image_caption":"[\"Figure3Proportionofvarianceexplainedinoutcomesfromthegeneticcros-laggedmodelpresentedinFigure2.Emotionalsymptoms refertoanxietyanddepressonsymptoms.Eachbarrepresents thevariance inoutcomevariablesexplainedbygeneticandnon-shared environmentalcontributionsfrompreviousvariables(iecontributionstransmitedovertime).Forexample,thetopbarhowsthe geneticandnon-sharedenvironmentalcontributionstoemotionalsymptomsat16yearsoldfromvariablesat12yearsold.Theleft-hand sideofthebarshowsthegeneticcontributions,andtheright-andsidshowsthenon-sharedenvironmentalcontributions.achcolour representsthepredictingvariablefromwhichthesecontributionsoriginate.Thepinksectionontheeft-andsideofthebarshows,for example,that approximately $4 \\\\%$ of the variance in emotional symptoms at 16 years old is explained by genetic contributions unique to emotionalsymptomsat12yearsold(theexactestimateisprovidedinTable4).Bycomparison,thepinksectionontheright-andside shows that approximately $6 \\\\%$ of the variance in emotional symptoms at 16 years old is explained by non-shared environmental factors contributions unique to emotional symptoms at 12 years old. \"]","image_footnote":"[]","img_path":"images/0762/bef12d863c15a22de167423508befdc26157341acfcc211c05d7b302476f3cf1.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[75,595,480,764]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[75,765,480,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our findings align with prior studies supporting a transactional model of reciprocal influences between emotional symptoms and negative peer interactions (e.g.,Smith et al., 2022),but also extend them by including another aspect of interpersonal difficulties. Prior research has mainly focused on the links between emotional symptoms and peer victimisation without considering that adolescents experiencing anxiety and depressive symptoms may also experience other,maybe less obvious,types of interpersonal difficulties such as social isolation. Our study shows that social isolation carries as much risk as peer victimisation when it comes to exacerbating emotional symptoms during adolescence.As peer relationships become more salient during adolescence, young individuals who show persistent difficulties with social relationships may be viewed as deviating from the peer group norms (Brown & Larson, 2009; Oh et al., 2008;Rubin, Bukowski, & Parker, 2006). This may be the case for emotionally vulnerable adolescents,who often struggle with anxiety in social situations (Nilsen, Karevold, Roysamb,Gustavson,& Mathiesen， 2013；Pickard，Rijsdijk， Happé，& Mandy, 2017). Consequently, these adolescents may miss out on opportunities to create meaningful relationships and gain confidence in their social abilities,which may ultimately exacerbate emotional problems over time (Rubin et al., 2009). Our results also suggest that social isolation and peer victimisation have a greater impact on long-term emotional symptoms if experienced at 16 years,compared to earlier. Experiencing interpersonal difficulties in midadolescence may affect young people's capacity to navigate the transition to early adulthood,a period marking important career and educational choices. As a result,it may be harderfor young individuals who lack the social skills and confidence necessary to establish a supportive social network to maintain emotional well-being during this transition (Nikitin & Freund, 2008). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,595,924,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[68,945,922,969]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ 2023 TheAuthors.JoumalofChildPsychologyand Psychiatry publishedbyJohn Wiley&Sons Ltdonbehalfof Association for Child and Adolescent Mental Health. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[601,49,924,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J Child Psychol Psychiatr 2023; 64(11): 1569-82 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[77,48,396,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1576 Genevieve Morneau-Vaillancourt et al. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[73,80,480,122]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,80,924,122]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,184,927,907]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0016/0f692444fe2bfb24d1faa5af2b3b55e60c8a707d7d7d94abd6ade40244fa66a1.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":"<table><tr><td></td><td>A</td><td>E</td><td>Total phenotypic variance explained</td></tr><tr><td>Contributions to emotional symptoms at age 16</td><td></td><td></td><td></td></tr><tr><td>Unique effects of</td><td></td><td></td><td></td></tr><tr><td>Stability from emotional symptoms age 12</td><td>.04 (11%)</td><td>.06 (9%)</td><td>.10</td></tr><tr><td>Cross-lagged from social isolation age 12</td><td>.0001 (.04%)</td><td>.0002 (.03%)</td><td>.0003</td></tr><tr><td>Cross-lagged from peer victimisation age 12</td><td>.001 (.1%)</td><td>.0003 (.1%)</td><td>.001</td></tr><tr><td>Common effects of pre-existing correlation with</td><td>.003 (1%)</td><td>.001 (.2%)</td><td>.004</td></tr><tr><td>Social isolation age 12</td><td></td><td></td><td></td></tr><tr><td>Peer victimisation age 12</td><td>.005 (1%)</td><td>.001 (.2%)</td><td>.006</td></tr><tr><td>Residual factors</td><td>.30 (86%)</td><td>.59 (90%)</td><td>.89</td></tr><tr><td>Total AE components</td><td>.35 (100%)</td><td>.65 (100%)</td><td>1.00</td></tr><tr><td>Contributions to social isolation at age 16</td><td></td><td></td><td></td></tr><tr><td>Unique effects of</td><td>.02 (6%)</td><td></td><td></td></tr><tr><td>Stability from social isolation age 12</td><td></td><td>.03 (5%)</td><td>.06</td></tr><tr><td>Cross-lagged from emotional symptoms age 12</td><td>.003 (1%)</td><td>.005 (1%)</td><td>.01</td></tr><tr><td>Cross-lagged from peer victimisation age 12</td><td>.002 (1%)</td><td>.001 (.2%)</td><td>.003</td></tr><tr><td>Common effects of pre-existing correlation with</td><td>.01 (3%)</td><td></td><td></td></tr><tr><td>Emotional symptoms age 12</td><td>.01 (2%)</td><td>.005 (1%)</td><td>.01</td></tr><tr><td>Peer victimisation age 12</td><td></td><td>.002 (.3%)</td><td>.01</td></tr><tr><td>Residual factors</td><td>.31 (88%)</td><td>.60 (93%)</td><td>.91</td></tr><tr><td>Total AE components</td><td>.35 (100%)</td><td>.65 (100%)</td><td>1.00</td></tr><tr><td>Contributions to peer victimisation at age 16</td><td></td><td></td><td></td></tr><tr><td>Unique effects of</td><td>.04 (9%)</td><td>.03 (5%)</td><td></td></tr><tr><td>Stability from peer victimisation age 12</td><td>.002 (1%)</td><td>.003 (1%)</td><td>.07</td></tr><tr><td>Cross-lagged from emotional symptoms age 12</td><td>.002 (1%)</td><td>.003 (1%)</td><td>.005</td></tr><tr><td>Cross-lagged from social isolation age 12</td><td></td><td></td><td>.005</td></tr><tr><td>Common effects of pre-existing correlation with Emotional symptoms age 12</td><td>.01 (2%)</td><td>.003 (.4%)</td><td>.01</td></tr><tr><td></td><td>.01 (2%)</td><td>.002 (.4%)</td><td></td></tr><tr><td>Social isolation age 12</td><td>.33 (85%)</td><td></td><td>.01</td></tr><tr><td>Residual factors</td><td></td><td>.56 (93%)</td><td>.89</td></tr><tr><td>Total AE components</td><td>.39 (100%)</td><td>.61 (100%)</td><td>.99a</td></tr><tr><td>Contributions to emotional symptoms at age 21 Unique effects of</td><td></td><td></td><td></td></tr><tr><td>Stability from emotional symptoms age 16</td><td>.0351 (14%)</td><td>.0680 (9%)</td><td>.1031</td></tr><tr><td>Cross-lagged from social isolation age 16</td><td>.0013 (1%)</td><td>.0025 (.4%)</td><td>.0038</td></tr><tr><td>Cross-lagged from peer victimisation age 16</td><td>.0034 (1%)</td><td>.0059 (1%)</td><td>.0093</td></tr><tr><td>Common effects of pre-existing correlation with</td><td></td><td></td><td></td></tr><tr><td>Social isolation age 16</td><td>.0065 (3%)</td><td>.0053 (1%)</td><td>.0117</td></tr><tr><td>Peer victimisation age 16</td><td>.0073 (3%)</td><td>.0032 (.4%)</td><td>.0105</td></tr><tr><td>Residual factors</td><td>.1980 (79%)</td><td>.6376 (88%)</td><td>.8356</td></tr><tr><td>Total AE components (excluding those at 12 years)</td><td>.2516 (100%)</td><td>.7225 (100%)</td><td>.9741</td></tr><tr><td>Contributions to social isolation at age 21</td><td></td><td></td><td></td></tr><tr><td>Unique effects of</td><td></td><td></td><td></td></tr><tr><td>Stability from social isolation age 16</td><td>.02 (8%)</td><td>.05 (7%)</td><td>.07</td></tr><tr><td>Cross-lagged from emotional symptoms age 16</td><td>.002 (1%)</td><td>.003 (1%)</td><td>.01</td></tr><tr><td>Cross-lagged from peer victimisation age 16</td><td>.01 (2%)</td><td>.01 (1%)</td><td>.01</td></tr><tr><td>Common effects of pre-existing correlation with</td><td></td><td></td><td></td></tr><tr><td>Emotional symptoms age 16</td><td>.01 (2%)</td><td>.01 (1%)</td><td>.01</td></tr><tr><td>Peer victimisation age 16</td><td>.01 (3%)</td><td>.003 (.4%)</td><td>.01</td></tr><tr><td>Residual factors</td><td>.24 (84%)</td><td>.63 (91%)</td><td>.87</td></tr><tr><td>Total AE components (excluding those at 12 years)</td><td>.29 (100%)</td><td>.69 (100%)</td><td>.98</td></tr><tr><td>Contributions to peer victimisation at age 21 Unique effects of</td><td></td><td></td><td></td></tr><tr><td>Stability from peer victimisation age 16</td><td>.03 (14%)</td><td>.06 (8%)</td><td>.09</td></tr><tr><td>Cross-lagged from emotional symptoms age 16</td><td>.001 (.4%)</td><td>.002 (.3%)</td><td>.003</td></tr><tr><td>Cross-lagged from social isolation age 16</td><td>.00003 (.01%)</td><td>.0001 (.01%)</td><td>.0001</td></tr><tr><td>Common effects of pre-existing correlation with</td><td></td><td></td><td></td></tr><tr><td>Emotional symptoms age 16</td><td>.004 (2%)</td><td>.002 (.2%)</td><td>.01</td></tr></table>","table_caption":"[\"Table4 Proportions of genetic and environmental varianceaccounted for in emotional symptoms,social isolation and peer victimisation at 16 and 21 years old \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[850,909,924,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(continues) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,49,235,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"doi:10.1111/jcpp.13847 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[468,48,924,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cross-lagged twin study emotional symptoms adolescence 1577 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[72,944,926,969]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ 2023TheAuthors.JoumalofChildPsychologyand Psychiatry publishedbyJohn Wiley&Sons LtdonbehalfofAssociation for Child and Adolescent Mental Health. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[72,97,926,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0424/1ec62a984c86c48e8aa89e255b91ffab471d3f236747bc4206b8f09acd5d51b9.jpg","list_items":null,"page_idx":"9","sub_type":null,"table_body":"<table><tr><td></td><td>A</td><td>E</td><td>Total phenotypic variance explained</td></tr><tr><td>Social isolation age 16</td><td>.001 (.3%)</td><td>.0003 (.04%)</td><td>.001</td></tr><tr><td>Residual factors</td><td>.21 (83%)</td><td>.67 (91%)</td><td>.88</td></tr><tr><td>Total AE components (excluding those at 12 years)</td><td>.25 (100%)</td><td>.74 (100%)</td><td>.98</td></tr></table>","table_caption":"[\"Table 4 (continued) \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[72,183,926,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Unique A (i.e.,additive genetic) and $\\mathbf { E }$ (non-shared environmental) effects transmitted via stability and cross-lagged paths were calculatedbysquaring theproductoftephenotypicpathbytheAorEpathonthe previous corresponding variable.Forexample, theAeffects transmitedviastabilityofemotionalsymptomsfrom12 to16yearsold werecalculatedbasedonestimatesshownin Figure 2: $( . 3 2 ^ { * } . 6 2 ) ^ { 2 } = . 0 4$ . Common A and $\\mathbf { E }$ effects of pre-existing correlations were computed by multiplying by two the product of all paths connecting two variables via prior A or $\\mathbf { E }$ influences.For example,the common A effects on emotional symptoms at 16 years duetothe pre-existingcorrelation between emotionalsymptomsand socialisolationwere basedonestimatesshown in Figure 2: $2 ^ { * } ( . 3 2 ^ { * } . 6 2 ^ { * } . 6 0 ^ { * } . 6 2 ^ { * } . 0 2 ) = . 0 0 3$ \nTo avoid overcrowding the table,contributions tooutcomes atage21years only included efects fromAandEfactorsatage 16, which explains why the total phenotypic variance at age 21 years is smaller than 1. \nIn parentheses are the percentages of contributions to total A and $\\mathbf { E }$ influences of each variable. \naProportions do not add up to one because peer victimisation at 12 years old had a C component. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[75,332,480,819]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In addition, our results provide some evidence that social isolation in late adolescence may be an intermediate pathway for the relationship between peer victimisation and emotional symptoms.Peer victimisation in early adolescence predicted later emotional symptoms in emerging adulthood via social isolation， suggesting that adolescents who experience peer victimisation during the first years of secondary school may develop a tendency to withdraw from social interactions (Barzeva et al., 2020; Oh et al.，2Oo8)． In turn，being socially isolated reduces opportunities for socialising and establishing new relationships during the transition to emerging adulthood, thereby leading to more distress and emotional symptoms over time (Oh et al.，2008； Rubin et al.， 2006,2009).However, social isolation did not significantly explain the link between emotional symptoms in early adolescence and peer victimisation in emerging adulthood,as social isolation in mid-adolescence did not significantly predict later peer victimisation. This could be explained by the increased freedom to decide whether and with whom to interact socially in young adulthood (e.g.，at work or university； Nikitin & Freund, 20o8). Young adults may be in a better position to avoid social situations conducive to peer victimisation. Peer victimisation may also be less common once adolescents finish secondary school and are no longer exposed to their peers every day. Therefore, although social isolation in midadolescence was associated with later increases in emotional symptoms, our findings suggest that being socially isolated at 16 years may carry less risk for peer victimisation in emerging adulthood. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[75,820,480,932]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We further found that non-shared environmental influences accounted for most of individual differences in emotional symptoms,social isolation and peer victimisation and that the magnitude of these influences was consistent across timepoints.Because the cross-lagged model accounted for the stability and prior relationships between variables,non-shared environmental contributions at 16 and 21 years old can be interpreted as explaining incremental change in variables over time. As such，this finding is consistent with longitudinal twin studies showing that change in emotional and social problems is primarily accounted for by environmental influences (Brendgen et al., 2022; Matthews et al., 2022; Nivard et al., 2015). Some of these non-shared environmental contributions could include risk factors with a broader impact on both emotional and social domains such as negative relationships with parents and peers (Kaufman,Kretschmer,Huitsing,& Veenstra, 2019). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,337,924,492]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,493,924,932]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our findings further suggest that both gene-environment non-shared environmental correlations partially accounted for the associationsbetween emotional symptoms, social isolation and peer victi-misation at each timepoint. This finding is in line with studies showing genetic and environmental overlap between emotional symptoms and negative peer experiences but also extends them by providing evidence of these etiological processes at different periods during adolescence and early adulthood (Brendgen et al., 2013, 2022; Matthews et al., 2016; Schaefer et al.,2018； Silberg et al.，2016; Smith et al.,2021). Whereas gene-environment correlations suggest that the risk of experiencing interpersonal difficulties may be traced back to adolescents'genetic predisposition to emotional symptoms,non-shared environmental correlations point to the possibility that the relationships between emotional symptoms and interpersonal difficulties are partly due to direct phenotypic pathways (i.e., unconfounded by genetic and familial vulnerabilities). Interestingly, the geneenvironment correlations were larger at 12 years than at later timepoints,while the magnitude of nonshared environmental correlations was consistent across timepoints.This could be explained by the fact that stabilityin behavioural traits,whichis accounted for in the cross-lagged model，is substantially accounted for by genetic factors (Nivard et al., 2015). Nonetheless, this means that new etiological processes emerge over time, perhaps suggesting that the genetic and non-shared environmental factors responsible for the relationships between emotional symptoms and interpersonal difficulties are not the same at12,16 and 21 years. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[68,945,922,969]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ 2023 TheAuthors.JoumalofChildPsychologyand Psychiatry publishedbyJohn Wiley&Sons Ltdonbehalfof Association for Child and Adolescent Mental Health. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[603,49,924,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"J Child Psychol Psychiatr 2023; 64(11): 1569-82 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[77,48,396,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1578 Genevieve Morneau-Vaillancourt et al. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[75,80,480,122]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"","text_format":null,"text_level":null,"type":"text"},{"bbox":"[75,121,480,664]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our results must be interpreted while considering some limitations. First, the cross-lagged model does not disentangle between- from within-person effects and cannot be used to answer causal inference hypotheses (Hamaker, Kuiper,& Grasman， 2015). Our findings could either reflect between-person processes，where adolescents who manifest more emotional symptoms than their peers are also more victimised, or within-person processes，where adolescents'fluctuations in emotional symptoms lead to more peer victimisation than what they usually experience. We could have examined these within-person associations more specifically by testing a random-intercept cross-lagged model (Hamaker et al., 2015). However, because we aimed to under-stand why some adolescents were at higher risk for emotional symptoms than others and examine the risk factors associated with the persistence of emo-tional symptoms, the cross-lagged panel model was more appropriate. The cross-lagged panel model was also suitable to twin decomposition，as shown in previous studies using the same twin model(e.g., Burt et al., 20o5),and examining the genetic and environ-mental associations between variables was a central aim of our study. Further methodological work would have been required to apply the twin method to a random-intercept cross-lagged model,as this has never been done before. Such methodological work was beyond the scope of the present study. Second, some of the variances in self-reported measures may be due to self-perception biases and may be respon-sible for inflating within-person correlations between variables (i.e.,common method variance) or increas-ing non-shared environmental variance in twin analyses. Third, assessments were separated by a large time interval (i.e.，4-5 years) and this may have affected our capacity to identify the specific developmental periods at which risks were most important. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,692,178,706]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conclusions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[75,708,480,862]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In summary,our study shows that the mechanisms underlying emotional symptoms during adolescence involve reciprocal associations with social isolation and peer victimisation and that both genetic and environmental influences play a role in these mechanisms. If replicated,our findings highlight the necessity to intervene early in adolescence and to pay attention to the different types of interpersonal difficulties that emotionally vulnerable adolescents may experience,including social isolation and peer victimisation,to prevent their exacerbation over time. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[519,86,724,102]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Supporting information ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,103,924,141]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Additional supporting information may be found online in the Supporting Information section at the end of the article: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[515,156,924,413]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Table S1. Number of complete and incomplete twin pairs for each variable. \nTable S2.Sensitivity correlations between emotional symptoms,peer problems and peer victimisation at 12, 16 and 21 years old. \nTable S3. Independent sample t-tests to examine sex differences. \nTable S4. Descriptive statistics for transformed variables at 12,16 and 21 years old. \nTable S5. Comparison of phenotypic cross-lagged models. \nTable S6. Fit comparisons for univariate twin models. Table S7. Cross-twin cross-trait correlations, \nFigure S1.Phenotypic cross-lagged model using the full peer problems scale (sensitivity analysis). \nFigure S2. Genetic cross-lagged model using the full peer problems scale (sensitivity analysis). \nFigure S3.Multi-group phenotypic cross-lagged model (sex differences). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,441,690,455]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acknowledgements ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,455,924,755]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The authors gratefully acknowledge the ongoing contribution of the participants in the Twins Early Development Study (TEDS) and their families. TEDSis supported by a program grant from the UK Medical Research Council (MR/V012878/1 and previously MR/ M021475/1)，with additional support from the US National Institutes of Health (AG046938). This study presents independent research part-funded by the National Institute for Health Research (NIHR) Biomedical Research Centre at South London and Maudsley NHS Foundation Trust and King's College London. The views expressed are those of the authors and not necessarily those of the NHS,the NIHR，or the Department of Health.G.M.-V. is supported by postdoctoral fellowships from the Social Sciences and Humanities Research Council of Canada (756-2021- 0516) and Fonds de recherche du Québec Société et Culture (2022-B3Z-297753). For the purposes of open access,the author has applied a Creative Commons Attribution (CC BY) licence to any Accepted Author Manuscript version arising from this submission. The authors have declared that they have no competing or potential conflicts of interest. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,783,658,797]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Correspondence ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,799,924,862]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Thalia C. Eley，Social，Genetic and Developmental Psychiatry Centre, Institute of Psychiatry，Psychology and Neuroscience，King's College London，Memory Lane,London， SE5 8AF，UK; Email: thalia.eley@kcl. ac.uk ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,945,926,969]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\circledcirc$ 2023 TheAuthors.JoumalofChild Psychologyand Psychiatry publishedbyJohn Wiley&Sons LtdonbehalfofAssociation for Child and Adolescent Mental Health. ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[73,49,235,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"doi:10.1111/jcpp.13847 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[468,48,924,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cross-lagged twin study emotional symptoms adolescence 1579 ","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[84,129,179,145]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Key points ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[82,151,919,307]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Emotional symptoms,including anxiety and depressive symptoms,and peer victimisation are reciprocally associated during adolescence,and social isolation may explain these bidirectional relationships. \nBoth genetic and environmental influences account for individual differences in emotional symptoms, social isolation and peer victimisation,as well as for their associations. \nFindings showed that emotional symptoms were reciprocally and independently associated with social isolation and peer victimisation over time and that social isolation in mid-adolescence partly explained th link between early peer victimisation and later emotional problems. \nDifferent gene-environment and non-shared environmental correlations explained the associations between emotional symptoms and interpersonal difficulties at each timepoint. \nThis study underscores the necesity to pay attention to social isolation and peer victimisation during adolescence to prevent the exacerbation of emotional symptoms over time. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,356,169,369]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[72,369,482,928]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Achenbach，T.M. (1991)．Manual for the Child Behavior Checklist/4-18and 1991 profile. Burlington,VT: University of Vermont, Department of Psychiatry. \nAcquah, E.O., Topalli, P.Z., Wilson, M.L., Junttila,N.,& Niemi, P.M.(2016).Adolescent loneliness and social anxiety as predictors of bullying victimisation. International Journal of Adolescence and Youth,21,320-331. \nAdachi,P.,& Willoughby,T. (2015). Interpreting effect sizes when controlling for stability effects in longitudinal autoregressive models: Implications for psychological science. European Journal of Developmental Psychology，12，116- 128. \nArseneault, L., Milne, B.J., Taylor, A., Adams, F., Delgado, K, Caspi,A.，& Moffitt, T.E. (2008).Being bullied as an environmentally mediated contributing factor to children's internalizing problems. Archives of Pediatrics & Adolescent Medicine,162,145-150. \nBarzeva, S.A., Meeus,W.H.J.,& Oldehinkel, A.J. (2019). Social withdrawal in adolescence and early adulthood: Measure-ment issues, normative development, and distinct trajectories. Journal of Abnormal Child Psychology, 47, 865-879. \nBarzeva,S.A.,Richards,J.S.,Meeus,W.H.J.,& Oldehinkel, A.J.(202O). The social withdrawal and social anxiety feedback loop and the role of peer victimization and acceptance in the pathways. Development and Psychopathology,32, 1402-1417. \nBoker,S., Neale,M., Maes,H.,Wilde,M., Spiegel, M.,Brick,T., ... Fox，J. (2011).OpenMx: An open source extended structural equation modeling framework. Psychometrika, 76, 306-317. \nBowker, J.C., Nelson,L.J., Markovic,A.,& Luster, S. (2013). Social withdrawal during adolescence and emerging adulthood.In R.J. Coplan & J.C. Bowker (Eds.), The handbook of solitude: Psychological perspectives on social isolation, social withdrawal, and being alone (pp. 167-183). New York: Wiley-Blackwell. \nBoyes,M.E.,Bowes,L., Cluver,L.D.,Ward, C.L.,& Badcock, N.A. (2014). Bullying victimisation, internalising symptoms, and conduct problems in south African children and adolescents: A longitudinal investigation. Journal of Abnormal Child Psychology, 42, 1313-1324. \nBrendgen, M., Vitaro,F., Barker, E.D., Girard,A., Dionne, G., Tremblay，R.E.，& Boivin，M.(2013)．Do other people's plights matter? A genetically informed twin study of the role of social context in the link between peer victimization and children's aggression and _depression symptoms. Developmental Psychology, 49, 327-340. \nBrendgen,M., Zheng,Y., Vitaro,F.,Dionne,G.,& Boivin,M. (2022). Gene-environment interplay linking peer victimiza-tion with adolescents' trajectories of depressive symptoms. Journal of the American Academy of Child & Adolescent Psychiatry,62,261-271. \nBrown，B.B.，& Larson，J. (2009)．Peer relationships in adolescence.In R.M.Lerner & L. Steinberg (Eds.),Handbook of adolescent psychology. John Wiley & Sons, Inc. \nBurt, S.A., McGue, M., Krueger,R.F.,& Iacono,W.G. (2005). How are parent-child conflict and childhood externalizing symptoms related over time? Results from a genetically informative cross-lagged study. Development and Psychopathology,17,145-165. \nCaspi,A.，Houts，R.，Fisher，H.,Danese,A.，& Moffitt，T. (2023). The general factor of psychopathology (p): Choosing among competing models and interpreting p. Clinical Psychological Science. \nCohen, B.E., Edmondson, D.,& Kronish, I.M. (2015). State of the art review: Depression, stress,anxiety, and cardiovascular disease.American Journal of Hypertension, 28,1295- 1302. \nCook, C.R., Williams, K.R., Guerra, N.G., Kim, T.E.,& Sadek, S. (201O).Predictors of bullying and victimization in childhood and adolescence: A meta-analytic investigation. School Psychology Quarterly,25,65-83. \nFisher，H.L.，Caspi，A.，Moffitt，T.E.，Wertz,J.，Gray，R., Newbury, J.,... Arseneault, L. (20i5). Measuring adoles-cents'exposure to victimization: The Environmental Risk (E-risk) Longitudinal Twin Study. Development and Psychopathology,27,1399-1416. \nForrest,C.L.,Gibson,J.L.,Halligan,S.L.,& St Clair，M.C. (2020). A cross-lagged analysis of emotion regulation, peer problems,and emotional problems in children with and without early language difficulties: Evidence from the Millennium Cohort Study. Journal of Speech, Language, and Hearing Research, 63,1227-1239. \nGadermann,A.M.,Guhn，M.,& Zumbo,B.D. (2012).Estimating ordinal reliability for Likert-type and ordinal item response data: A conceptual，empirical，and practical guide. Practical Assessment， Research & Evaluation， 17, 1-13. \nGoodman, R. (1997). The Strengths and Difficulties Questionnaire: A research note. Journal of Child Psychology and Psychiatry, 38, 581-586. \nGoodman, R., & Scott, S. (1999). Comparing the Strengths and Difficulties Questionnaire and the Child Behavior Checklist: Is_small beautiful? Jourmal of Abnormal Child Psychology, 27, 17-24. \nHamaker, E.L., Kuiper,R.M.,& Grasman, R.P.P.P. (2015).A critique of the cross-lagged panel model. Psychological Methods,20,102-116. \nHutzell,K.L.,& Payne,A.A. (2012). The impact of bullying victimization on school avoidance.Youth Violence and Juvenile Justice,10, 370-385. \nJacobson, N.C., & Newman, M.G. (2017). Anxiety and depression as bidirectional risk factors for one another:A metaanalysis of longitudinal studies.Psychological Bulletin, 143, 1155-1200. \nJohansson，A.，Huhtamaki，A.，Sainio，M.，Kaljonen，A., Boivin, M.,& Salmivalli, C. (2022). Heritability of bullying and victimization in children and adolescents: Moderation by the KiVa antibullying program. Journal of Clinical Child & Adolescent Psychology, 51, 505-514. \nJohnson, D., Dupuis, G., Piche,J., Clayborne, Z.,& Colman, I. (2018).Adult mental health outcomes of adolescent depres-sion: A systematic review. Depression and Anxiety, 35, 700- 716. \nJoseph, S.,& Stockton, H. (2018). The multidimensional peer victimization scale: A systematic review. Aggression and Violent Behavior,42(September-October 2018),96-114. \nKaufman, T.M.L., Kretschmer, T., Huitsing, G.,& Veenstra, R. (2019). Caught in a vicious cycle? Explaining bidirectional spillover between parent-child relationships and peer victimization. Development and Psychopathology, 32, 11-20. \nKwong,A.S.F.,López-López,J.A.,Hammerton,G.,Manley,D., Timpson, N.J., Leckie,G.,& Pearson, R.M. (2019). Genetic and environmental risk factors associated with trajectories of depression symptoms from adolescence to young adulthood. JAMA Network Open,2,1-14. \nLee,K.S.,& Vaillancourt, T. (2019).A four-year prospective study of bullying,anxiety,and disordered eating behavior across early adolescence. Child Psychiatry and Human Development, 50, 815-825. \nLockhart,C.,Ahmadzadeh,Y.,Breen,G.,Bright,J.,Bristow, S.,Boyd,A.,...Eley,T.C. (2023). Twins Early Development Study (TEDS): A genetically sensitive investigation of mental health outcomes in the mid-twenties. JCPP Advances, e12154. https://doi.0rg/10.1002/jcv2.12154 \nL6pez-López, J.A.， Kwong, A.S.F.， Washbrook, E.， Pearson, R.M.，Tilling,K.，Fazel,M.S.，... Hammerton，G. (2020). Trajectories of depressive symptoms and adult educational and employment outcomes.BJPsych Open,6,e6. \nLu,W. (2019). Adolescent depression: National trends,risk factors，and healthcare disparities. American Journal of Health Behavior, 43,181-194. \nMatthews,T.,Danese,A.,Wertz,J., Odgers,C.L.,Ambler,A., Mofitt，T.E.，& Arseneault,L. (2016). Social isolation, loneliness and depression in young adulthood: A behavioural genetic_analysis. Social Psychiatry and Psychiatric Epidemiology,51,339-348. \nMatthews,T.,Qualter,P.,Bryan,B.T.,Caspi,A.,Danese,A., Mofitt, T.E.，... Arseneault, L. (2022). The developmental course of loneliness in adolescence: Implications for mental health, educational attainment, and psychosocial functioning.Development and Psychopathology,35,537-546. \nMcGue,M.,& Bouchard, T.J.(1984).Adjustment of twin data for the effects of age and sex. Behavior Genetics,14,325-343. \nMelton,T.H.,Croarkin,P.E., Strawn,J.R.,& McClintock,S.M. (2016). Comorbid anxiety and depressive symptoms in children and adolescents: A systematic review and analysis. Journal of Psychiatric Practice, 22, 84-89. \nMiddeldorp, C.M., Cath,D.C., van Dyck, R.,& Boomsma, D.I. (2005). The co-morbidity of anxiety and depression in the perspective of genetic epidemiology.A review of twin and family studies.Psychological Medicine,35,611-624. \nModecki，K.，Minchin，J.，Harbaugh，A.，Guerra，N.，& Runions,K. (2014). Bullying prevalence across contexts: A meta-analysis measuring_cyber and traditional bullying. Journal of Adolescent Health, 55, 602-611. \nMoore,S.，Norman,R.，Suetani, S.， Thomas,H.， Sly, P.，& Scott,J. (2017). Consequences of bullying victimization in childhood and adolescence:A systematic review and metaanalysis. World Journal of Psychiatry, 7, 60-76. \nMynard, H., & Joseph, S. (2000). Development of the multidimensional peer-victimization scale. Aggressive Behavior, 26,169-178. \nNaicker,K.，Galambos,N.L., Zeng,Y.，Senthilselvan，A.，& Colman, I. (2013). Social,demographic,and health outcomes in the 10 years following adolescent depression. Journal of Adolescent Health, 52, 533-538. \nNikitin,J.,& Freund, A.M. (2oo8). The role of social approach and avoidance motives for subjective well-being and the successful transition to adulthood.Applied Psychology,57 (s1), 90-111. \nNilsen，W.，Karevold，E.，Roysamb，E.，Gustavson，K.，& Mathiesen, K.S. (2013). Social skills and depressive symp-toms across adolescence: Social support as a mediator in girls versus boys. Journal of Adolescence, 36, 11-20. \nNivard, M.G., Dolan, C.V., Kendler,K.S., Kan,K.J., Willemsen, G.，van Beijsterveldt,C.E.M.，... Boomsma，D.I. (2015). Stability in symptoms of anxiety and depression as a function of genotype and environment: A longitudinal twin study from ages 3 to 63 years. Psychological Medicine, 45, 1039-1049. \nOh,W.,Rubin,K.H.,Bowker,J.C.,Booth-LaForce,C.,Rose-Krasnor,L.,& Laursen,B. (2008). Trajectories of social withdrawal from middle childhood to early adolescence. Journal of Abnormal Child Psychology, 36, 553-566. \nPabian, S.,& Vandebosch,H. (2016). An investigation of shortterm longitudinal associations between social anxiety and victimization and perpetration of traditional bullying and cyberbullying. Journal of Youth and Adolescence, 45, 328- 339. \nPickard,H.,Rijsdijk,F.,Happé,F.,& Mandy,W.(2017).Are social and communication difficulties a risk factor for the development of social anxiety? Journal of the American Academy of Child & Adolescent Psychiatry, 56, 344-351.e3. \nPrice,T.S., Freeman,B.,Craig,I.,Petrill, S.A.,Ebersole,L.,& Plomin, R. (200o).Infant zygosity_can be assigned by parental report questionnaire data. Twin Research, 3, 129- 133. \nRacine,N., McArthur,B.A., Cooke,J.E., Eirich,R., Zhu,J.,& Madigan, S. (2021). Global prevalence of depressive and anxiety symptoms in children and adolescents_ during COVID-19:A meta-analysis.JAMA Pediatrics,175,1142- 1150. \nReijntjes,A.,Kamphuis,J.H.,Prinzie,P.,& Telch,M.J. (2010). Peer victimization and internalizing problems in children: A meta-analysis of longitudinal studies. Child Abuse& Neglect, 34,244-252. \nRijsdijk,F.V.，& Sham, P.C. (2002). Analytic approaches to twin data using structural equation models.Briefings in Bioinformatics,3,119-133. \nRosseel，Y. (2012). Lavaan: An R package for structural equation modeling. Journal of Statistical Software,48,1-36. \nRStudio Team. (2022). RStudio: Integrated development environment for R. Boston,MA: RStudio, PBC. Available from: http://www.rstudio.com/ \nRubin，K.H.，Bukowski，W.M.，& Parker,J.G.(2006).Peer interactions,relationships,and groups.In N.Eisenberg,W. Damon, & R.M. Lerner (Eds.), Handbook of child psychology: Social, emotional, and personality development (pp. 571- 645). John Wiley & Sons,Inc. \nRubin，K.H.，Coplan，R.J.，& Bowker，J.C. (2009).Social withdrawal in childhood. Annual Review of Psychology, 60, 141-171. \nSalmivalli,C.,Lagerspetz,K.,Bjorkqvist,K., Osterman,K.,& Kaukiainen, A. (1996). Bullying as a group process: Participant roles and their relations to social status within the group. Aggressive Behavior, 22, 1-15. \nSchaefer,J.D.,Moffitt,T.E.,Arseneault,L.,Danese,A.,Fisher, H.L.,Houts,R.,...Caspi,A. (2018).Adolescent victimization and early-adult psychopathology:Approaching causal inference using a longitudinal twin study to rule out noncausal explanations.Clinical Psychological Science,6,352-371. \nSentse，M.,Prinzie,P.,& Salmivalli,C. (2017). Testing the direction of longitudinal paths between victimization,peer rejection,and different types of internalizing problems in adolescence.Journal of Abnormal Child Psychology，45, 1013-1023. \nSilberg,J.L.,Copeland,W.,Linker,J.,Moore,A.A.,Roberson-Nay，R.，& York，T.P.(2016)．Psychiatric outcomes of bullying victimization: A study of discordant monozygotic twins.Psychological Medicine,46,1875-1883. \nSmith,K.,Brendgen,M.,Hébert,M.,Vitaro,F.,Dionne,G.,& Boivin,M. (2021). Links between peer victimization, dating violence victimization and depression in adolescence:A genetically-informed study.Journal of Clinical Child and Adolescent Psychology,1-12,1-12. \nSmith,K.,Hébert,M.,Brendgen,M.,& Blais,M. (2022).The mediating role of internalizing problems between peer victimization and dating violence victimization:A test of the stress generation hypothesis.Journal of Interpersonal Violence,37,NP11271-NP11295. \nSpeyer,L.G.,Auyeng,B.,& Murray,A.L. (2022).Longitudinal invariance of the Strengths and Difficulties Questionnaire across ages 4 to 16 in the ALSPAC sample.Assessment. https://doi.0rg/10.1177/10731911221128948 \nTiirikainen,K.,Haravuori,H.,Ranta,K.,Kaltiala-Heino,R.,& Marttunen,M. (2019).Psychometric properties of the 7-item Generalized Anxiety Disorder scale (GAD-7) in a large representative sample of Finnish adolescents.Psychiatry Research,272,30-35. \nVeldkamp，S.A.M.，Boomsma，D.I.，de Zeeuw，E.L.，van Beijsterveldt, C.E.M.,Bartels,M.,Dolan,C.V.,& van Bergen, E.(2019).Genetic and environmental influences on diferent forms of bullying perpetration,bullying victimization,and their co-occurrence.Behavior Genetics,49,432-443. \nVugteveen,J.,de Bildt,A.,Theunissen,M.,Reijneveld, S.A.,& Timmerman,M. (2021).Validity aspects of the Strengths and Difficulties Questionnaire (SDQ) adolescent self-report and parent-report versions among Dutch adolescents. Assessment, 29, 601-616. \nWilliford,A.,Boulton,A.,Noland,B.,Little,T.D.,Kärnä,A.,& Salmivalli，C. (2012). Effects of the KiVa anti-bullying program on adolescents' depression,anxiety,and perception of peers. Journal of abnormal child psychology,40,289-300. https://doi.0rg/10.1007/s10802-011-9551-1 \nWolitzky-Taylor,K.,Dour,H., Zinbarg,R.,Mineka,S.,Vrshek-Schallhorn，S.，Epstein，A.，...Craske，M.G． (2014). Experiencing core symptoms of anxiety and unipolar mood disorders in late adolescence predicts disorder onset in early adulthood.Depression and Anxiety,31,207-213. \nZhu,X.,Griffths,H.,Eisner,M.,Hepp,U.,Ribeaud,D.,& Murray，A.L. (2022)． Developmental relations between bullying victimization and suicidal ideation in middle adolescence and emerging adulthood: Do internalizing problems and substance use mediate their links? Journal of Youth and Adolescence,51,1745-1759. 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","text_format":null,"text_level":null,"type":"discarded"},{"bbox":"[517,425,794,438]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"13","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Accepted for publication: 9 May 2023 ","text_format":null,"text_level":null,"type":"discarded"}],"string":"[\n {\n \"bbox\": \"[90,91,909,171]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A cross-lagged twin study of emotional symptoms, social isolation and peer victimisation from early adolescence to emerging adulthood \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[124,192,877,237]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Genevieve Morneau-Vaillancourt,1 $\\\\textcircled{1}$ Olakunle Oginni,1,2 $\\\\textcircled{1}$ Elham Assary, D Georgina Krebs,1,3 $\\\\textcircled{1}$ Ellen J. Thompson,1,4 $\\\\textcircled{1}$ Elisavet Palaiologou, Celestine Lockhart,1 $\\\\textcircled{1}$ Louise Arseneault,1 $\\\\textcircled{1}$ and Thalia C. Eley1,5 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,240,926,311]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1Social,Geneticand Developmental PsychiatryCentre,Instituteof Psychiatry,Psychology and Neuroscience,King's College London,London,UK;²DepartmentofMental Health,Obafemi AwolowoUniversity,Ie-Ife,Nigeria;Research Departmentof Clinical,Educationaland HealthPsychology,UniversityCollege London,London,UK; 4Departmentof Twin Research and Genetic EpidemiologyScholof Life Courseand Population Sciences, King's Coege London, London,UK; 5National Institute for Health Research (NIHR) Biomedical Research Centre, South London and Maudsley Hospital, London, UK \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,338,924,611]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Background: Emotional symptoms, such as anxiety and depresive symptoms, are common during adolescence, often persist over time,and can precede the emergence of severe anxiety and depresive disorders. Studies suggest that a vicious cycle of reciprocal infuences between emotional symptoms and interpersonal difculties may explain why some adolescents sufer frompersisting emotional symptoms.However,theroleofdiferent typesofinterpersonal difficulties,suchassocial isolationandpervictimisation,inthesereciproalasociationsisstillunclear.Inaddition, the lack of longitudinal twin studies conducted on emotional symptoms during adolescence means that the genetic and environmental contributions to these relationships during adolescence remain unknown.Methods: Participants $( N = 1 5 , 8 6 9 )$ from the Twins Early Development Study completed self-reports of emotional symptoms,social isolation and peer victimisationat12,16 and 21 years old.A phenotypic cross-lagged model examined reciprocal asociations between variables overtime,and agenetic extensionofthis model examined the aetiologyofthe relationships between variables at eachtimepoint. Results: First,emotional symptoms were reciprocally andindependently associated with both social isolation and peer victimisation over time,indicating that diferent forms of interpersonal dificulties uniquely contributed to emotional symptoms during adolescence and vice versa. Second,early peer victimisation predicted later emotional symptoms via social isolation in mid-adolescence, indicating that social isolation may constitute an intermediate pathway through which peer victimisation predicts longer-term emotional symptoms. Finaly,individual diferences inemotionalsymptoms were mostlyaccountedforbynon-shared environmentalfctors at each timepoint,and both gene-environment and individual-specific environmental mechanisms were involved in therelationships between emotional symptoms and interpersonal diffculties.Conclusions: Our study highlights the necesity to intervene early inadolescence to prevent the escalation of emotional symptoms over time and to consider socialisolation and peer victimisation as importantrisk factors for the long-term persistence of emotional symptoms. Keywords: Anxiety: depression; peer relationships; twins; longitudinal studies. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,649,183,662]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,664,480,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Emotional symptoms,also referred to as internalising problems, comprise a broad range of anxiety and depressive symptoms， such as worry，fear and sadness (Caspi，Houts，Fisher，Danese，& Moffitt,2023). Adolescence is a critical period for the development of emotional symptoms,with approximately $10 \\\\%$ to $2 5 \\\\%$ ofadolescentssuffering from severe clinically relevant anxiety and depressive symptoms (Lu, 2019； Racine et al.， 2021; Tiirikainen，Haravuori，Ranta,Kaltiala-Heino，& Martunen, 2019). Emotional symptoms are associated with detrimental long-term outcomes, including academic disruption and serious mental and physical health problems(Cohen， Edmondson，& Kronish, 2015;Johnson,Dupuis, Piche,Clayborne, & Colman， 2018； López-López et al.， 2020). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,643,924,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Unfortunately,emotional symptoms often remain untreated and show continuity into adulthood, where they can escalate into full-syndrome and recurrent anxietyanddepressivedisorders(Johnson et al., 2018; Kwong et al., 2019; Naicker, Galambos, Zeng，Senthilselvan，& Colman，2013；Wolitzky-Taylor et al., 2O14). It is therefore crucial to identify the mechanisms by which emotional symptoms develop and persist during adolescence,as this will help identify young individuals at risk and inform early intervention strategies. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,799,924,926]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"One pathway contributing to the continuity of emotional symptoms may be negative interpersonal experiences,such as peer victimisation.Peer victimisation affects $1 0 \\\\% { - } 3 5 \\\\%$ of adolescents and is both a precursor and an outcome of emotional symptoms(Cook， Williams， Guerra， Kim，& Sadek, 2010; Modecki, Minchin, Harbaugh, Guerra, & Runions，2014； Moore et al.， 2017； Reijntjes, Kamphuis,Prinzie,& Telch,2O1O).Several studies haveexaminedthebidirectional associations between emotional problems and peer victimisation by using cross-lagged models. Although many sup-port a transactional model where emotional symptoms and peer victimisation mutually reinforce each other over time (e.g. Forrest, Gibson, Halligan, & St Clair,2020; Smith,Hébert,Brendgen,& Blais,2022; Zhu et al.，2022)，others align with a symptomdriven model,where emotional symptoms precede and perhaps evoke peer victimisation (e.g., Barzeva, Richards，Meeus，& Oldehinkel，202O；Lee& Vaillancourt, 2019; Pabian & Vandebosch, 2016). There is also evidence supporting an interpersonal risk model,where peer victimisation predicts later emotional symptoms (Williford et al.,2012). These mixed findings could be explained by the fact that most studies relied on narrow definitions of anxiety and depressive symptoms and evaluated them separately. Yet，adopting a broader definition of emotional symptoms,especially within a longitudinal context, may more accurately depict the experience of adolescents affected by these problems. Anxiety and depressive symptoms often co-occur, are characterised by sequential comorbidity (i.e., where one predicts the emergence of the other), share genetic risks and can be classified under a broader dimension of internalising problems (Caspi et al.，2023； Jacobson & Newman， 2017； Melton, Croarkin, Strawn, & McClintock, 2016; Middeldorp, Cath, van Dyck,& Boomsma, 2005). The few cross-lagged studies that have assessed emotional symptoms broadly (i.e.，both anxiety and depressive symptoms) consistently show that emotional symptoms are reciprocally associated with peer victimisation， thereby supporting the transactional model (e.g.,Boyes,Bowes, Cluver,Ward, & Badcock, 2014; Smith et al., 2022). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,941,926,978]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ 2023TheAuthors.JoualofChildPsychologyndPsychatrypublishdbyohnWiley&SonsLdonehalffAsociationforChildandAdolescent Mental Health. ThisisanoenacsartidertesfteatieComsAtrbutoesechpeitsusistrtiodeprodctio medium, provided the original work is properly cited. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[73,47,551,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Journal of Child Psychology and Psychiatry 64:11 (2023), pp 1569-1582 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[75,921,415,932]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conflict of interest statement: No conflicts declared. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[764,0,1000,43]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates The Association ACAMH for Childand Adolescent Mental Health \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[761,47,924,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"doi:10.1111/jcpp.13847 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[75,8,322,43]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The Journal of Child Psychology and Psychiatry \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[75,78,480,607]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,606,480,932]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In addition,other types of interpersonal experiences during adolescence,such as social isolation, may be involved in these reciprocal associations. Emotionally vulnerable adolescents often withdraw socially because of low mood， loss of interest,or worrying about social situations (Barzeva et al., 2020; Barzeva, Meeus,& Oldehinkel, 2019; Rubin, Coplan, & Bowker,2009). They are also more likely to be rejected or excluded by their peers，who can be reluctant to initiate interactions with adolescents who exhibit negative emotions (Barzeva et al., 2020; Bowker，Nelson，Markovic，& Luster，2013；Oh et al., 20o8). Consequently, socially isolated adolescents are often viewed as vulnerable targets for peer victimisation (Acquah,Topalli,Wilson,Junttila,& Niemi, 2016; Sentse, Prinzie,& Salmivalli, 2017). In turn, their desire not to be exposed to peer victimisation mightlead them to avoid social situations (Hutzell & Payne, 2O12),or peers who are unwilling to affiliate with victims might reject them (Salmivalli, Lagerspetz,Bjorkqvist, Osterman,& Kaukiainen，1996). This cascade of negative social experiences could ultimately aggravate emotional symptoms, as lacking positive peer relationships and opportunities to practise socialising often contributes to emotional symptoms(Barzevaetal., 2020; Matthews et al.,2O16). As such, social isolation may be an intermediate pathway through which emotional symptoms lead to peer victimisation and vice versa. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,80,924,164]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,168,924,577]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Longitudinal twin studies have shed light on the mechanisms underlying these pathways by showing that genetic and environmental factors independently contribute to the associations between emotional symptoms，social isolation and peer victimisation (e.g.，Matthews et al., 2016; Schaefer et al., 2018). Social isolation and peer victimisation do not occur randomly and are both heritable $( 3 0 \\\\%$ to $7 0 \\\\%$ ， suggesting that genetic factors contribute to individual differences in these social difficulties (Fisher et al.，2015；Johansson et al.，2022；Matthews et al.,2016; Veldkamp et al., 2019). Evidence suggests a genetic overlap between emotional symptoms and negative peer experiences,meaning that a genetic disposition for emotional symptoms enhances the risk of being exposed to negative social experiences via gene-environment correlations (Brendgen， Zheng, Vitaro,Dionne,& Boivin,2022). Furthermore,twin studies show that negative peer interactions increase emotional symptoms via non-shared environmental pathways,providing evidence that these associations are unconfounded by pre-existing genetic dispositions or by features of the familial environment shared by both twins (Arseneault et al.， 20o8； Schaefer et al.，2018；Silberg et al.，2016)． Therefore，the mechanisms by which emotional symptoms are related to interpersonal diffculties involve a mix of gene-environmentcorrelationsandindividualspecific environmental influences. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,579,922,889]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Despite these findings,questions remain unanswered.First,most cross-lagged studies on emotional symptoms solely focused on peer victimisation without considering other types of interpersonal difficulties and were conducted over short periods (e.g., 1 or 2 years).As such, it remains unclear whether social isolation acts as an intermediate pathway in the reciprocal associations between emotional symptoms and peer victimisation,and whether these associations persist over an extended period and across life transitions, such as from adolescence and into adulthood. Second, the extent to which genetic and environmental influences contribute to individual differences in emotional symptoms,social isolation and peer victimisation from adolescence to emerging adulthood,as well as to their relationships over this period, remain unknown.Documenting these genetic and environmental influences within a longitudinal perspective could enhance our understanding of the mechanisms underlying the persistence of emotional symptoms and its relationships with interpersonal difficulties over time. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,891,922,932]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The present study aimed to address these gaps by testing (a) the reciprocal phenotypic associations between emotional symptoms,social isolation and peer victimisation from 12 to 21 years old,and (b) the genetic and environmental influences and overlap of emotional symptoms, social isolation and peer victimisation at 12,16 and 21 years old. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,945,922,969]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ 2023 TheAuthors.JoumalofChild Psychologyand Psychiatry publishedbyJohn Wiley&Sons LtdonbehalfofAssociation for Child and Adolescent Mental Health. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[603,49,924,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J Child Psychol Psychiatr 2023; 64(11): 1569-82 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[77,48,398,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1570 Genevieve Morneau-Vaillancourt et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[75,80,480,135]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,166,152,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Methods \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,180,166,193]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Participants \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,202,480,508]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Participants were from the Twins Early Development Study (TEDS),a longitudinal cohort of twins in England and Wales (Lockhart et al.,2023).TEDS recruited more than 13,000 families from a national register of all twins born in England and Wales between 1994 and 1996;and the twins have been followed up into early adulthood.TEDS has assessed the twins' behavioural， cognitive and emotional development from infancy to early adulthood.The present study uses twins' reports of emotional symptoms，social isolation and peer victimisation at 12,16 and 21 years old (Table 1 provides age estimates for each variable).A total of 15,869 participants $5 4 \\\\%$ females) for whom data were available for at least one measure were included in the phenotypic model.Among these participants,only those for whom zygosity was available were included in the genetic model $\\\\scriptstyle { [ N = \\\\ 1 5 , 6 9 5 }$ ，including 2,727 complete monozygotic [MZ],4,837 complete dizygotic [DZ],138 incomplete MZ and 429 incomplete DZ pairs; Table S1 provides a description of twin pairs for each variable).When twins were about 18 months old, zygosity was determined using a parent-reported questionnaire of physical similarity (Price et al.，2Ooo).When unclear, zygosity was determined using DNA testing.Participants who were born in extreme perinatal conditions $N = 2 9 6$ twin pairs or 592 individuals),had severe medical conditions $( N = 9 5$ individuals),or for whom zygosity and/or gender were unknown $( N = 1 0 5$ twin pairs or 210 individuals) were excluded. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,530,247,544]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ethical considerations \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,552,480,575]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ethical approval for TEDS at the 12-，16- and 21-year assessment waves was granted by the King's College London \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,82,924,115]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Research Ethics Committee (reference:05-Q0706-228 and PNM 0910 104)．TEDS collected informed consent from parents or participants prior to each wave of data collection. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,139,591,151]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Measures \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,159,924,279]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Emotional symptoms and social isolation. Twins answered questions from the Strengths and Difficulties Questionnaire (SQD；Goodman，1997) to assess emotional symptoms and social isolation.Participants evaluated how things had been for them over the last three or six months ona three-point scale ( $0 =$ \\\\`not true'; $1 =$ quite true'; $2 =$ very true'). We computed emotional symptoms and social isolation scores by deriving the average of at least three items and multiplying this by the total number of items for each subscale to minimise the impact of missing data. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,280,924,516]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The emotional symptoms subscale includes five items assessing the extent to which participants experience anxiety and depressive symptoms (e.g.，I worry a lot',I am often unhappy，downhearted,or tearful').Emotional symptoms scores ranged from O to 1O.Internal consistency was assessed using ordinal alpha coefficients (Gadermann，Guhn，& Zumbo，2012) and was satisfactory at 12 $( \\\\mathsf { \\\\alpha } ) = \\\\mathsf { \\\\alpha } . 7 9 )$ ，16 $( \\\\alpha = . 7 8 )$ and 21 years old $( \\\\alpha = . 8 6 )$ . Both emotional symptoms and peer problems SDQ subscales show good convergent validity as well as measurement invariance over time (Goodman& Scott, 1999; Speyer,Auyeng,& Murray,2022;Vugteveen,de Bildt,Theunissen,Reijneveld,& Timmerman,2021). They both correlate with similar measures from other widely used scales,such as the Child Behaviour Checklist (CBCL) and the Youth Self-Report (YSR；Achenbach，1991).The SDQ emotional symptoms subscale is strongly associated with anxiety and depression from the CBCL and YSR,and the SDQ peer problems subscale is strongly correlated with social problems from the CBCL and YSR (Goodman & Scott,1999; Vugteveen et al., 2021). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,517,924,574]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We assessed social isolation using four out of the five items of the SDQ peer problems subscale,removing the item ‘Others pick on me or bully me’to avoid overlap with the peer victimisation measure.The other four items assessed the extent to which participants spend time on their own or are socially accepted by peers (e.g.，I am usually on my own.I generally play alone or keep to myself). Scores ranged from O to 8.Internal consistency was satisfactory at 12 $( \\\\mathsf { x } = . 7 0 )$ ，16 $( \\\\mathsf { \\\\alpha } ) = \\\\mathsf { \\\\beta } . 6 7 )$ and 21 years old $( \\\\mathsf { \\\\Omega } \\\\mathsf { \\\\Omega } ) = \\\\mathsf { \\\\Omega } . 6 8 )$ ，as indicated by ordinal alpha coefficients.We also conducted sensitivity analyses using the full five-item peer problems subscale to see whether removing one item would affect the reliability of the scale. Results suggest that it did not,as they were highly similar to results obtained using the four-item social isolation measure (see Table S2, Figure S1 and S2). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[72,609,926,877]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0023/3f1da5d7ce6868a59a0079569f76274586fcd4d66ce1d713ee0bd009e9dde784.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td></td><td></td><td colspan=\\\"2\\\"> MZ Pairs</td><td colspan=\\\"2\\\">DZ Pairs</td><td colspan=\\\"6\\\"></td></tr><tr><td></td><td>N</td><td>Complete</td><td>Incomplete</td><td></td><td>Complete Incomplete</td><td>Age (SD)</td><td>Mean (SD)</td><td>Range</td><td>Skew</td><td></td><td>Kurtosis</td></tr><tr><td>12 years old</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Emotional</td><td>11,681</td><td>2,092</td><td>33</td><td>3,678</td><td>59</td><td>11.28 (.70)</td><td>2.19 (2.06)</td><td></td><td>0-10</td><td>1.02</td><td>3.71</td></tr><tr><td>symptoms Social isolation</td><td>11,675</td><td>2,090</td><td>35</td><td>3,674</td><td>63</td><td>11.28 (.70)</td><td></td><td>1.15 (1.34)</td><td>0-8</td><td>1.51</td><td>5.73</td></tr><tr><td>Peer</td><td>11,729</td><td>2,104</td><td>23</td><td>3,710</td><td>28</td><td>11.28 (.70)</td><td></td><td>3.13 (2.88)</td><td>0-10</td><td>.71</td><td>2.45</td></tr><tr><td>victimisation 16 years old</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Emotional</td><td>10,196</td><td>1,811</td><td>23</td><td>3,209</td><td>66</td><td>16.32 (.68)</td><td>2.75 (2.24)</td><td></td><td>0-10</td><td>.75</td><td>2.99</td></tr><tr><td>symptoms Social isolation</td><td>10,181</td><td>1,804</td><td>30</td><td>3,202</td><td>72</td><td>16.32 (.68)</td><td></td><td>1.47 (1.38)</td><td>0-8</td><td>1.16</td><td>4.44</td></tr><tr><td>Peer</td><td>5,236</td><td>893</td><td>178</td><td>1,416</td><td>416</td><td>16.48 (.27)</td><td></td><td>2.93 (2.72)</td><td>0-10</td><td>.75</td><td>2.70</td></tr><tr><td>victimisation 21 years old</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Emotional</td><td>9,501</td><td>1,271</td><td>1,275</td><td>2,113</td><td>2,454</td><td>22.26 (.91)</td><td>3.61 (2.70)</td><td></td><td>0-10</td><td>.50</td><td>2.29</td></tr><tr><td>symptoms Social isolation</td><td>9,501</td><td>1,270</td><td>1,276</td><td>2,112</td><td>2,453</td><td>22.26 (.91)</td><td></td><td>2.08 (1.68)</td><td>0-8</td><td>.76</td><td>3.23</td></tr><tr><td>Peer victimisation</td><td>8,382</td><td>1,155</td><td>1,343</td><td>1,883</td><td>2,556</td><td>22.84 (.88)</td><td></td><td>1.78 (2.21)</td><td>0-10</td><td>1.26</td><td>4.00</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 1Descriptive statistics for emotional symptoms,socialisolationand peer victimisationat12,16and 21 years old \\\"]\",\n \"table_footnote\": \"[\\\"Emotionalsymptoms refer tosymptoms ofanxietyand depresion.Peer victimisationat16 years old was measured inaseparate web survey that was completed by fewerparticipants forfinancialreasons.Values areforrawscores (Table S4 shows descriptive statistics for transformed scores). DZ,dizygotic twin; MZ,monozygotic twin; $N ,$ number of participants; $S D$ ，standard deviation; Skew, skewness. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[73,49,235,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"doi:10.1111/jcpp.13847 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[70,945,926,969]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ 2023TheAuthors.JoumalofChildPsychologyand Psychiatry publishedbyJohn Wiley&Sons Ltdonbehalfof Association for Child and Adolescent Mental Health. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[468,48,922,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cross-lagged twin study emotional symptoms adolescence 1571 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[75,80,480,198]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,214,480,475]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Peer victimisation. We used the self-reported Multidimensional Peer Victimisation Scale (Mynard & Joseph,2000) to assess peer victimisation at 12,16 and 21 years old. Using a three-point scale $\\\\scriptstyle ( 0 = { \\\\mathrm { \\\\overleftarrow { n o t } } }$ at all'; $1 = \\\\cdot _ { \\\\mathrm { o n c e } } ,$ ； $2 = \\\\cdot _ { \\\\mathrm { m o r e } }$ than once'，participants answered five items about the extent to which they had experienced acts of physical,verbal and social peer victimisation from other students (at 12 and 16 years) or someone excluding their partner or family (at 21 years),in the past year (e.g.，‘How often has someone refused to talk to you?'). The scale focuses specifically on victimisation by peers at 12 and 16 years,whereas the definition of victimisation is broader at 21 years,encompassing victimisation by individuals from wider social networks (e.g.,work colleagues).Despite these differences,the validity and reliability of the Multidimensional Peer Victimisation Scale has been shown to be comparable across developmental periods, including emerging adulthood (Joseph & Stockton，2018)．We computed peer victimisation scores by multiplying the average across at least three items by the total number of items,and scores ranged from O to 1O.Ordinal alpha coefficients indicated satisfactory internal consistency at 12( $\\\\langle \\\\alpha = . 8 5 \\\\rangle$ ，16 $\\\\left( \\\\alpha = . 8 8 \\\\right)$ and 21 years old $( \\\\mathsf { \\\\pmb { x } } = . 8 7 )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,499,220,513]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Statistical analyses \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,521,480,806]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Phenotypic analyses. We first transformed all variables using a square root $+ 1$ transformation to correct for positive skewness.We conducted descriptive statistics and correlations using RStudio (version 2022.2.2.485；RStudio Team，2022). The analysis plan was pre-registered (https://osf.io/sjb5a/? view_only $r =$ lce2374f6e15467ebd2cf626538fbe19). To examine sex differences,we conducted independent samples t-tests for all variables (see Table S3).We found that females showed higher emotional symptoms than males and that males were more victimised by their peers than females at 12,16 and 21 years old.We also found that males were more socially isolated than femalesat12 years old.To see whether these sex differences were reflected in the patterns of associations between variables,we ran a multi-group phenotypic crosslagged model (see Figure S3A; Figure 2B).We found that a cross-lagged modelinwhich parameterswere freely estimated for females and males showed a better fit than a model in which parameters were constrained to equality across sexes. However,these differences were very small in terms of effect sizes and confidence intervals for most estimates overlapped across femalesand males.Therefore,we decided not to test a multi-group genetic cross-lagged model and to report analyses that controlled for age and sex,as is standard practice for twin model fitting (McGue & Bouchard, 1984). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,808,480,925]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To examine the bidirectional phenotypic associations between emotional symptoms, social isolation and peer victimisation at all three timepoints,we specified a crosslagged panel model using the lavaan package (Rosseel, 2012) to obtain structural equation modelling ft indices (using fullinformation maximum likelihood [FIMLl). The cross-lagged model estimated the reciprocal associations between all variables from 12 to 16 years and also from 16 to 21 years. It accounted for the stability across timepoints and preexisting relationships between variables, thereby assessing the prediction of incremental changes in variables over time (Adachi & Willoughby, 2015).We then examined the extent to which social isolation at 16 years mediated the bidirectional associationsbetweenemotionalsymptomsandpeer victimisation by estimating indirect effects and their $9 5 \\\\%$ confidence intervals (CI) using bootstrapping with 1,000 iterations. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,80,924,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,164,922,246]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Asa sensitivity check，and because twin models are estimated by nesting participants within twin pairs,we ran the model in lavaan using one twin per pair who was randomly chosen $( N = 7 , 9 2 6 )$ and then using twin pairs in OpenMx $( N = 8 , 2 6 2$ twin pairs;Boker et al.,2011).Estimates were similar across these models,so we report the results of the model using all participants $( N = 1 5 , 8 6 9 )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,262,924,439]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Genetic analyses. We then examined the genetic and environmental mechanismsunderlying the associations between variables using twin modelling. The twin method compares DZ twins,who share on average $50 \\\\%$ of their segregating genes,to MZ twins,who share $100 \\\\%$ of their genes.Because twins grow up in the same familial environment,contributions from shared environmental factors are expected to increase the similarity between twins.Conversely, non-shared environmental factors unique to each twin are expected to increase differences between twins.The twin method relies on these assumptions to decompose phenotypic variance and covariance into genetic (A),shared (C) and nonshared (E) environmental factors (Rijsdijk & Sham,2002). Non-shared environmental factors (E) also include measurement error. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,441,922,759]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We first ran cross-twin correlations (i.e.,between twins of the same family) and univariate twin models to estimate the extent to whichA,CandE factors contributed to eachvariable individually.We estimated cross-twin cross-trait correlations between variables (e.g.,between trait1 for twin 1 and trait 2 for twin 2) by running a constrained saturated model,where means and variances were constrained to equality across zygosity and birth order while covariances were constrained to be symmetrical for MZ and DZ twin pairs.We then specified a twin extension of the phenotypic cross-lagged model, initially described by Burt，McGue，Krueger，and Iacono (2005). Variances (at 12 years) and residual variances (at 16 and 21 years) were decomposed into ACE components.Genetic and environmental correlations were estimated at each timepoint by examining the extent to which genetic and environmental factors account for the covariance between variables using cross-twin cross-trait correlations.In addition,gene-environment correlations were examined by focusing on the extent to which genetic factors associated with emotional symptoms overlapped with the other two‘environmental’ outcomes,that is social isolation and peer victimisation. Such genetic overlap would indicate that a genetic vulnerability for emotional symptoms increases the risk of experiencing interpersonal difficulties,thereby providing evidence for gene-environment correlations. The genetic cross-lagged model was fitted using FIML and parameter $9 5 \\\\%$ confidence intervals were estimated in OpenMx. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,787,579,800]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Results \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,802,924,857]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Table 1 presents descriptive statistics of raw scores (Table S4 presents descriptive statistics for transformed variables),and Table 2 presents phenotypic correlations between variables. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[519,876,764,889]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Phenotypic cross-lagged model \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,898,924,925]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We examined the bidirectional associations between emotional symptoms, social isolation and peer victimisation from 12 to 21 years old using a phenotypic cross-lagged model (Figure 1). Fit indices indicated a good fit of the model: Root Mean Square Error of Approximation $( \\\\mathrm { R M S E A } ) = . 0 4$ , Comparative Fit Index $\\\\left( \\\\mathrm { C F I } \\\\right) = \\\\ . 9 8$ , Tucker-Lewis Index $( \\\\mathrm { T L I } ) = . 9 1$ Cross-lagged paths were statistically significant for the most part $( \\\\beta = . 0 3$ to .12),except for the crosslagged path from social isolation at 16 years to peer victimisation at 21 years: $\\\\beta = . 0 1$ $1 9 5 \\\\%$ CI: -.02, .03). These cross-lagged associations were observed while accounting for the moderate stability in all variables from 12 to 16 years $( \\\\beta = . 2 6$ to .33) and from 16 to 21 years $( \\\\beta = . 3 0$ to .35)．They also accounted for the pre-existing correlations between variables,which were weak to moderate within each of the timepoints $( r = . 0 3$ to .35). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[68,945,924,969]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ 2023TheAuthors.JoumalofChildPsychologyand Psychiatry publishedbyJohn Wiley&Sons Ltdonbehalfof Association for Child and Adolescent Mental Health. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[603,49,924,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J Child Psychol Psychiatr 2023;64(11): 1569-82 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[77,48,396,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1572 Genevieve Morneau-Vaillancourt et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[517,80,924,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[164,93,357,928]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0452/1d7d06556e26495ad493aa2d2158d0944bbcaf38f5055e109953f8dd1002800b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td></td><td>1.</td><td></td><td>2.</td><td>3.</td><td>4.</td><td>5.</td><td>6.</td><td>7.</td><td>8.</td></tr><tr><td>1. Emotional symptoms 12 years old</td><td>1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>2. Social isolation 12 years old</td><td>.35 (.33, .37)</td><td>1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>3.Peer victimisation 12 years old</td><td>.32 (.31, .34)</td><td></td><td>.27 (.25,28)</td><td>1</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>4.Emotional symptoms 16 years old 5.Social isolation 16 years old</td><td>.34 (.32, .36)</td><td></td><td>.15 (.13, .17)</td><td>.14 (.12, .17)</td><td>1</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>.20 (.17, .22)</td><td></td><td>.30 (.28, .32)</td><td>.16 (.14, .18)</td><td>.34 (.32, .36)</td><td>1</td><td></td><td></td><td></td></tr><tr><td>6.Peer victimisation 16 years old</td><td>.18 (.15, .21)</td><td></td><td>.16 (.13, .19)</td><td>.31 (.29, .34)</td><td>.23 (.20,.25)</td><td>.25 (.22,.28)</td><td>1</td><td>1</td><td></td></tr><tr><td>7. Emotional symptoms 21 years old 8.Social isolation 21 years old</td><td>.25 (.23,.28)</td><td></td><td>.12 (.10, .14) .21 (.19,.23)</td><td>.14 (.12, .16)</td><td>.39 (.37, .41)</td><td>.21 (.19, .24)</td><td>.18 (.15,.22)</td><td></td><td>1</td></tr><tr><td></td><td>.17 (.15, .19)</td><td></td><td></td><td>.14 (.11, .16)</td><td>.20 (.18,.22)</td><td>.35 (.33, .37)</td><td>.20 (.17, .23)</td><td>.38 (.37, .40)</td><td></td></tr><tr><td>9.Peer victimisation 21 years old</td><td>.09 (.06, .11)</td><td></td><td>.10 (.07, .12)</td><td>.18 (.15, .20)</td><td>.13 (.11, .16)</td><td>.11 (.08, .13)</td><td>.33 (.30, .36)</td><td>.21 (.19,.23)</td><td>.12 (.10,.1</td></tr></table>\",\n \"table_caption\": \"[\\\"\\\"]\",\n \"table_footnote\": \"[\\\"\\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[517,307,924,577]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Overall， thephenotypic cross-lagged model showed that emotional symptoms at 12 and 16 years were independently and reciprocally associated with both social isolation and peer victimisation at 16 and 21 years,respectively.We then examined whether some of the cross-lagged paths were stronger than others by comparing models in which the selected parameters were constrained to equality with the unconstrained model in Figure 1 (Table S5 presents these analyses).We found that emotional symptoms equally predicted social isolation and peer victimisation and that conversely,both forms of interpersonal difficulties equally predicted emotional symptoms.In addition,both forms of interpersonal difficulties at 16 years predicted more subsequent emotional symptoms than at 12 years,indicating that social isolation and peer victimisation may carry higher risk for emotional symptoms in mid-adolescence, compared to early adolescence. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,579,924,733]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Notably, the model showed that peer victimisation at 12 years indirectly predicted emotional symptoms at 21 years via social isolation at 16 years: $\\\\beta = . 0 0 4$ $( 9 5 \\\\% \\\\mathrm { ~ C I } = . 0 0 3$ ，.007). This indirect effect explained $3 \\\\%$ of the correlation between peer victimisation at 12 years and emotional symptoms at 21 years $\\\\left( . 0 0 4 / . 1 4 1 = . 0 2 8 \\\\right)$ ． However，we did not observe the reverse sequence of associations,as emotional symptoms at 12 years did not predict peer victimisation at 21 years via social isolation: $\\\\beta = . 0 0 1$ $1 9 5 \\\\%$ $\\\\mathrm { C I } = - . 0 0 2$ , .003). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[519,755,737,769]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Genetic cross-lagged model \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,777,924,932]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Before testing the full genetic cross-lagged model, we inspected the cross-twin correlations and univariate twin models. Cross-twin correlations suggested significant A (i.e.，additive genetic) and $\\\\mathbf { E }$ (nonshared environmental contributions for most variables，as DZ correlations were about half of MZ correlations (i.e.，indicating A influences) and MZ correlations were less than 1 (i.e.，indicating E influences).Univariate twin models confirmed that most variables were accounted for by A and E influences (Table 3, see Table S6 for fit comparisons of univariate twin models).We then specified the genetic cross-lagged model informed by cross-twin cross-trait correlations (Table S7; Burt et al., 2005). We first tested an ACE model and then dropped C parameters that were non-significant or close to 0, one at a time. We compared nested models using likelihood ratio tests. The final model included AE factors for most variables,except for victimisation at 12 years old,which had a significant C component. The genetic model had a better ft than the phenotypic cross-lagged model， suggestingthat adding A and $\\\\mathbf { E }$ components improved model fit (likelihood ratio test, $p < . 0 0 1$ ; Akaike's Information Criterion $[ \\\\mathrm { A I C } ] = 2 3 7 , 4 6 3 . 6 0$ and 234,391.90 for the phenotypic and genetic models，respectively [lower AIC values suggesting better fit]). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,49,235,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"doi:10.1111/jcpp.13847 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[67,945,922,969]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ 2023TheAuthors.JoumalofChildPsychologyand Psychiatry publishedbyJohn Wiley&Sons Ltdon behalf of Association for Child and Adolescent Mental Health. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[468,48,924,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cross-lagged twin study emotional symptoms adolescence 1573 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[82,80,914,400]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 1 Phenotypic cross-lagged panel model. $N = 1 5 , 8 6 9$ .Emotional symptoms refer to symptoms of anxiety and depression.Values on single-headedarrowsarestandardisedpartialregressoncoeficients.Thesecoeficientscanbesquaredtocalculatetheproportionof varianceexplained.Valuesondouble-headedarrows betweenvariableswithintesametimepointarecorelationcoeficients;the correlationsatags16and2areresiduals,dexingtherelationshipetwenvariablesattatagethatisnotexplainedbytiralier association.Solidlines indicate statisticallysignificant parameters.The doted lines indicate non-significant parameters \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0277/3ecc8481be5516b1f95e7ddb6225d5017b5530912474f0f284fb12e1b524a6a6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[72,542,929,718]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0747/f53da1ff011dc2305a2f33053a48074be945dd095cd4481284092320e31862a9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td></td><td>rMZ</td><td>rDZ</td><td>A</td><td>C</td><td>E</td></tr><tr><td>12 years old</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Emotional symptoms</td><td>.40 (.36, .44)</td><td>.20 (.17,.23)</td><td>.40 (.37, .43)</td><td></td><td>.60 (.57, .63)</td></tr><tr><td>Social isolation</td><td>.37 (.33, .41)</td><td>.21 (.18,.24)</td><td>.39 (.36, .42)</td><td></td><td>.61 (.58, .64)</td></tr><tr><td>Peer victimisation</td><td>.57 (.54, .60)</td><td>.33 (.30, .36)</td><td>.45 (.37, .52)</td><td>.11 (.05, .18)</td><td>.44 (.41, .47)</td></tr><tr><td>16 years old</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Emotional symptoms</td><td>.42 (.38, .46)</td><td>.16 (.13, .20)</td><td>.40 (.36, .43)</td><td></td><td>.60 (.57, .64)</td></tr><tr><td>Social isolation</td><td>.41 (.37, .45)</td><td>.20 (.17,.23)</td><td>.41 (.38, .44)</td><td></td><td>.59 (.56, .62)</td></tr><tr><td>Peer victimisation</td><td>.46 (.41, .51)</td><td>.20 (.15,.25)</td><td>.45 (.41, .50)</td><td></td><td>.55 (.50, .59)</td></tr><tr><td>21 years old</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Emotional symptoms</td><td>.34 (.30, .39)</td><td>.19 (.15, .23)</td><td>.35 (.31, .39)</td><td></td><td>.65 (.61, .69)</td></tr><tr><td>Social isolation</td><td>.39 (.35,.43)</td><td>.19 (.16,.23)</td><td>.39 (.35,.42)</td><td></td><td>.61 (.58, .65)</td></tr><tr><td>Peer victimisation</td><td>.29 (.24,.34)</td><td>.14 (.10, .18)</td><td>.29 (.25,.34)</td><td></td><td>.71 (.66,.75)</td></tr></table>\",\n \"table_caption\": \"[\\\"Table3Cross-twincorrlationsandunivarateetimatesforemotioalsymptom,socialisolationandpeervctimisatinat12,16 and 21 years old \\\"]\",\n \"table_footnote\": \"[\\\"Emotional symptoms refer to symptoms of anxiety and depression. $9 5 \\\\\\\\%$ confidence intervals are in parentheses.A,additive genetic factors;C,sardeviromtalactors;Enoaredeioetalctrs;DZ,o-twnorelatiosforgotitwins;, cross-twin correlations for monozygotic twins. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[75,783,480,925]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,783,924,868]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,869,924,925]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The genetic model, presented in Figure 2, shows that A and $\\\\mathrm { E }$ estimates for emotional symptoms and social isolation at 12 years were similar,with E accounting for a larger proportion of variance $( 6 1 \\\\% )$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[68,945,924,969]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ 2023TheAuthors.JoumalofChildPsychologyand Psychiatry publishedbyJohn Wiley&Sons Ltdonbehalfof Association for Child and Adolescent Mental Health. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[603,49,924,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J Child Psychol Psychiatr 2023; 64(11): 1569-82 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[75,48,396,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1574 Genevieve Morneau-Vaillancourt et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[84,80,912,409]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 2 Genetic cross-lagged panel model. $N = 1 5 , 6 9 5 ,$ including 2,727 complete MZ,4,837 complete DZ, 138 incomplete MZ and 429 incomplete DZpairs.Emotional symptomsrefertosymptomsofanxietyanddepresion.Aadditivegeneticfactors;Csharedenvironmentalfactors;Eenvironmentalfactors.Valuesonsingle-eadedarrowsarestandardsedpathestimates;valuesondubleeaded arrowsarecorrelationcoeficients.Solidlines indicatestatisticallysignificantparameters.Thedottedlines indicatenonsignificant parameters.Proportionofvariance explainedbyAandEcomponentscanbeestimatedbysquaring theirrespectivepath estimates \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0109/f35af659f3a3934cfd4bd00f8e38e34cab91c6fafe7855061e1b520bf720fe09.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[75,517,480,898]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"than A $( 3 9 \\\\% )$ . Findings were somewhat different for peer victimisation at 12 years,with A $( 5 0 \\\\% )$ and $\\\\mathtt { E }$ $( 4 4 \\\\% )$ explaining comparable proportions of variance (C,or non-shared environmental factors,explained $6 \\\\%$ of the variance in peer victimisation at 12 years old).At 16 and 21 years,most of the variance was accounted for bynewA $20 \\\\%$ to $3 2 \\\\%$ and $\\\\mathrm { E }$ （ $56 \\\\%$ to $67 \\\\%$ ） components， indicating time-specific genetic and non-shared environmental contributions. The remaining part of the variance in outcome variables was accounted for by previous A and E influences transmitted via stability ( $2 \\\\%$ to $7 \\\\%$ and cross-lagged paths ( $0 1 \\\\%$ to $1 \\\\%$ . Total transmitted contributions accounted for 9 to $1 3 \\\\%$ of the variance in outcomes at 16 and 21 years old,indicating that outcome variables were only partially explained by variables at previous timepoints. These contributions from previous timepoints,along with their genetic and environmental components,are presented in Figure 3 and Table 4. Overall, findings from the genetic cross-lagged model indicate that, for the most part, non-shared environmental influences accounted for the largest proportion of variability in emotional symptoms,social isolation and peer victimisation at all timepoints and that most of the variance in outcome variables at 16 and 21 years was explained by new aetiological influences emerging over time. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,901,480,928]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Genetic and environmental correlations were mostlysignificant， indicatingthatemotional symptoms，social isolation and peer victimisation shared both genetic and environmental risks at each timepoint. Genetic correlations were moderate at 12 years and were weaker at later timepoints ( $_ { r G }$ at 12 years $= . 4 6$ to .60; rG at 16 years $=$ .33 to .48; rG at 21 years $= . 0 0$ to .45). These findings suggest that gene-environment correlations partially account for the relationships between emotional symptoms and both social isolation and peer victimisation. Non-shared environmental correlations were weak but comparable at each timepoint(rEat 12 years $= . 1 2$ to .19；rE at 16 years $\\\\ c = . 0 7$ to .20; rE at 21 years $= . 0 6$ and .27)，indicating that individual-specific environmental pathways modestly accounted for the associations between interpersonal difficulties and emotional symptoms. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,517,924,744]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,772,608,784]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Discussion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,788,924,927]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The present study is the first to use a genetically sensitive design to document the reciprocal associations between emotional symptoms (i.e.，anxiety and depressive symptoms) and two different forms of interpersonal difficulties， social isolation and peer victimisation,over an extended period during adolescence and early adulthood. First,we found that emotional symptoms predicted increases in social isolation and peer victimisation at later timepoints, and that in turn， social isolation andpeer victimisation independently contributed to emotional symptoms over time. We also found a significant longitudinal mediation where social isolation in mid-adolescence partly explained the long-term link between peer victimisation at 12 years and emotional problems at 21 years. Second, twin analyses showed that, for the most part,non-shared environmental influences mostly explained individual differences in all variables during adolescence and early adulthood and that both gene-environment andnon-sharedenvironmentalcorrelations accounted for their associations at each timepoint. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[67,945,922,969]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ 2023TheAuthors.JoumalofChildPsychologyand Psychiatry publishedbyJohn Wiley&Sons Ltdon behalf of Association for Child and Adolescent Mental Health. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[73,49,235,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"doi:10.1111/jcpp.13847 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[468,48,924,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cross-lagged twin study emotional symptoms adolescence 1575 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[84,79,915,431]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure3Proportionofvarianceexplainedinoutcomesfromthegeneticcros-laggedmodelpresentedinFigure2.Emotionalsymptoms refertoanxietyanddepressonsymptoms.Eachbarrepresents thevariance inoutcomevariablesexplainedbygeneticandnon-shared environmentalcontributionsfrompreviousvariables(iecontributionstransmitedovertime).Forexample,thetopbarhowsthe geneticandnon-sharedenvironmentalcontributionstoemotionalsymptomsat16yearsoldfromvariablesat12yearsold.Theleft-hand sideofthebarshowsthegeneticcontributions,andtheright-andsidshowsthenon-sharedenvironmentalcontributions.achcolour representsthepredictingvariablefromwhichthesecontributionsoriginate.Thepinksectionontheeft-andsideofthebarshows,for example,that approximately $4 \\\\\\\\%$ of the variance in emotional symptoms at 16 years old is explained by genetic contributions unique to emotionalsymptomsat12yearsold(theexactestimateisprovidedinTable4).Bycomparison,thepinksectionontheright-andside shows that approximately $6 \\\\\\\\%$ of the variance in emotional symptoms at 16 years old is explained by non-shared environmental factors contributions unique to emotional symptoms at 12 years old. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0762/bef12d863c15a22de167423508befdc26157341acfcc211c05d7b302476f3cf1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[75,595,480,764]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,765,480,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our findings align with prior studies supporting a transactional model of reciprocal influences between emotional symptoms and negative peer interactions (e.g.,Smith et al., 2022),but also extend them by including another aspect of interpersonal difficulties. Prior research has mainly focused on the links between emotional symptoms and peer victimisation without considering that adolescents experiencing anxiety and depressive symptoms may also experience other,maybe less obvious,types of interpersonal difficulties such as social isolation. Our study shows that social isolation carries as much risk as peer victimisation when it comes to exacerbating emotional symptoms during adolescence.As peer relationships become more salient during adolescence, young individuals who show persistent difficulties with social relationships may be viewed as deviating from the peer group norms (Brown & Larson, 2009; Oh et al., 2008;Rubin, Bukowski, & Parker, 2006). This may be the case for emotionally vulnerable adolescents,who often struggle with anxiety in social situations (Nilsen, Karevold, Roysamb,Gustavson,& Mathiesen， 2013；Pickard，Rijsdijk， Happé，& Mandy, 2017). Consequently, these adolescents may miss out on opportunities to create meaningful relationships and gain confidence in their social abilities,which may ultimately exacerbate emotional problems over time (Rubin et al., 2009). Our results also suggest that social isolation and peer victimisation have a greater impact on long-term emotional symptoms if experienced at 16 years,compared to earlier. Experiencing interpersonal difficulties in midadolescence may affect young people's capacity to navigate the transition to early adulthood,a period marking important career and educational choices. As a result,it may be harderfor young individuals who lack the social skills and confidence necessary to establish a supportive social network to maintain emotional well-being during this transition (Nikitin & Freund, 2008). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,595,924,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[68,945,922,969]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ 2023 TheAuthors.JoumalofChildPsychologyand Psychiatry publishedbyJohn Wiley&Sons Ltdonbehalfof Association for Child and Adolescent Mental Health. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[601,49,924,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J Child Psychol Psychiatr 2023; 64(11): 1569-82 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[77,48,396,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1576 Genevieve Morneau-Vaillancourt et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[73,80,480,122]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,80,924,122]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,184,927,907]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0016/0f692444fe2bfb24d1faa5af2b3b55e60c8a707d7d7d94abd6ade40244fa66a1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td></td><td>A</td><td>E</td><td>Total phenotypic variance explained</td></tr><tr><td>Contributions to emotional symptoms at age 16</td><td></td><td></td><td></td></tr><tr><td>Unique effects of</td><td></td><td></td><td></td></tr><tr><td>Stability from emotional symptoms age 12</td><td>.04 (11%)</td><td>.06 (9%)</td><td>.10</td></tr><tr><td>Cross-lagged from social isolation age 12</td><td>.0001 (.04%)</td><td>.0002 (.03%)</td><td>.0003</td></tr><tr><td>Cross-lagged from peer victimisation age 12</td><td>.001 (.1%)</td><td>.0003 (.1%)</td><td>.001</td></tr><tr><td>Common effects of pre-existing correlation with</td><td>.003 (1%)</td><td>.001 (.2%)</td><td>.004</td></tr><tr><td>Social isolation age 12</td><td></td><td></td><td></td></tr><tr><td>Peer victimisation age 12</td><td>.005 (1%)</td><td>.001 (.2%)</td><td>.006</td></tr><tr><td>Residual factors</td><td>.30 (86%)</td><td>.59 (90%)</td><td>.89</td></tr><tr><td>Total AE components</td><td>.35 (100%)</td><td>.65 (100%)</td><td>1.00</td></tr><tr><td>Contributions to social isolation at age 16</td><td></td><td></td><td></td></tr><tr><td>Unique effects of</td><td>.02 (6%)</td><td></td><td></td></tr><tr><td>Stability from social isolation age 12</td><td></td><td>.03 (5%)</td><td>.06</td></tr><tr><td>Cross-lagged from emotional symptoms age 12</td><td>.003 (1%)</td><td>.005 (1%)</td><td>.01</td></tr><tr><td>Cross-lagged from peer victimisation age 12</td><td>.002 (1%)</td><td>.001 (.2%)</td><td>.003</td></tr><tr><td>Common effects of pre-existing correlation with</td><td>.01 (3%)</td><td></td><td></td></tr><tr><td>Emotional symptoms age 12</td><td>.01 (2%)</td><td>.005 (1%)</td><td>.01</td></tr><tr><td>Peer victimisation age 12</td><td></td><td>.002 (.3%)</td><td>.01</td></tr><tr><td>Residual factors</td><td>.31 (88%)</td><td>.60 (93%)</td><td>.91</td></tr><tr><td>Total AE components</td><td>.35 (100%)</td><td>.65 (100%)</td><td>1.00</td></tr><tr><td>Contributions to peer victimisation at age 16</td><td></td><td></td><td></td></tr><tr><td>Unique effects of</td><td>.04 (9%)</td><td>.03 (5%)</td><td></td></tr><tr><td>Stability from peer victimisation age 12</td><td>.002 (1%)</td><td>.003 (1%)</td><td>.07</td></tr><tr><td>Cross-lagged from emotional symptoms age 12</td><td>.002 (1%)</td><td>.003 (1%)</td><td>.005</td></tr><tr><td>Cross-lagged from social isolation age 12</td><td></td><td></td><td>.005</td></tr><tr><td>Common effects of pre-existing correlation with Emotional symptoms age 12</td><td>.01 (2%)</td><td>.003 (.4%)</td><td>.01</td></tr><tr><td></td><td>.01 (2%)</td><td>.002 (.4%)</td><td></td></tr><tr><td>Social isolation age 12</td><td>.33 (85%)</td><td></td><td>.01</td></tr><tr><td>Residual factors</td><td></td><td>.56 (93%)</td><td>.89</td></tr><tr><td>Total AE components</td><td>.39 (100%)</td><td>.61 (100%)</td><td>.99a</td></tr><tr><td>Contributions to emotional symptoms at age 21 Unique effects of</td><td></td><td></td><td></td></tr><tr><td>Stability from emotional symptoms age 16</td><td>.0351 (14%)</td><td>.0680 (9%)</td><td>.1031</td></tr><tr><td>Cross-lagged from social isolation age 16</td><td>.0013 (1%)</td><td>.0025 (.4%)</td><td>.0038</td></tr><tr><td>Cross-lagged from peer victimisation age 16</td><td>.0034 (1%)</td><td>.0059 (1%)</td><td>.0093</td></tr><tr><td>Common effects of pre-existing correlation with</td><td></td><td></td><td></td></tr><tr><td>Social isolation age 16</td><td>.0065 (3%)</td><td>.0053 (1%)</td><td>.0117</td></tr><tr><td>Peer victimisation age 16</td><td>.0073 (3%)</td><td>.0032 (.4%)</td><td>.0105</td></tr><tr><td>Residual factors</td><td>.1980 (79%)</td><td>.6376 (88%)</td><td>.8356</td></tr><tr><td>Total AE components (excluding those at 12 years)</td><td>.2516 (100%)</td><td>.7225 (100%)</td><td>.9741</td></tr><tr><td>Contributions to social isolation at age 21</td><td></td><td></td><td></td></tr><tr><td>Unique effects of</td><td></td><td></td><td></td></tr><tr><td>Stability from social isolation age 16</td><td>.02 (8%)</td><td>.05 (7%)</td><td>.07</td></tr><tr><td>Cross-lagged from emotional symptoms age 16</td><td>.002 (1%)</td><td>.003 (1%)</td><td>.01</td></tr><tr><td>Cross-lagged from peer victimisation age 16</td><td>.01 (2%)</td><td>.01 (1%)</td><td>.01</td></tr><tr><td>Common effects of pre-existing correlation with</td><td></td><td></td><td></td></tr><tr><td>Emotional symptoms age 16</td><td>.01 (2%)</td><td>.01 (1%)</td><td>.01</td></tr><tr><td>Peer victimisation age 16</td><td>.01 (3%)</td><td>.003 (.4%)</td><td>.01</td></tr><tr><td>Residual factors</td><td>.24 (84%)</td><td>.63 (91%)</td><td>.87</td></tr><tr><td>Total AE components (excluding those at 12 years)</td><td>.29 (100%)</td><td>.69 (100%)</td><td>.98</td></tr><tr><td>Contributions to peer victimisation at age 21 Unique effects of</td><td></td><td></td><td></td></tr><tr><td>Stability from peer victimisation age 16</td><td>.03 (14%)</td><td>.06 (8%)</td><td>.09</td></tr><tr><td>Cross-lagged from emotional symptoms age 16</td><td>.001 (.4%)</td><td>.002 (.3%)</td><td>.003</td></tr><tr><td>Cross-lagged from social isolation age 16</td><td>.00003 (.01%)</td><td>.0001 (.01%)</td><td>.0001</td></tr><tr><td>Common effects of pre-existing correlation with</td><td></td><td></td><td></td></tr><tr><td>Emotional symptoms age 16</td><td>.004 (2%)</td><td>.002 (.2%)</td><td>.01</td></tr></table>\",\n \"table_caption\": \"[\\\"Table4 Proportions of genetic and environmental varianceaccounted for in emotional symptoms,social isolation and peer victimisation at 16 and 21 years old \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[850,909,924,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(continues) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,49,235,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"doi:10.1111/jcpp.13847 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[468,48,924,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cross-lagged twin study emotional symptoms adolescence 1577 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[72,944,926,969]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ 2023TheAuthors.JoumalofChildPsychologyand Psychiatry publishedbyJohn Wiley&Sons LtdonbehalfofAssociation for Child and Adolescent Mental Health. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[72,97,926,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0424/1ec62a984c86c48e8aa89e255b91ffab471d3f236747bc4206b8f09acd5d51b9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td></td><td>A</td><td>E</td><td>Total phenotypic variance explained</td></tr><tr><td>Social isolation age 16</td><td>.001 (.3%)</td><td>.0003 (.04%)</td><td>.001</td></tr><tr><td>Residual factors</td><td>.21 (83%)</td><td>.67 (91%)</td><td>.88</td></tr><tr><td>Total AE components (excluding those at 12 years)</td><td>.25 (100%)</td><td>.74 (100%)</td><td>.98</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 4 (continued) \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[72,183,926,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Unique A (i.e.,additive genetic) and $\\\\mathbf { E }$ (non-shared environmental) effects transmitted via stability and cross-lagged paths were calculatedbysquaring theproductoftephenotypicpathbytheAorEpathonthe previous corresponding variable.Forexample, theAeffects transmitedviastabilityofemotionalsymptomsfrom12 to16yearsold werecalculatedbasedonestimatesshownin Figure 2: $( . 3 2 ^ { * } . 6 2 ) ^ { 2 } = . 0 4$ . Common A and $\\\\mathbf { E }$ effects of pre-existing correlations were computed by multiplying by two the product of all paths connecting two variables via prior A or $\\\\mathbf { E }$ influences.For example,the common A effects on emotional symptoms at 16 years duetothe pre-existingcorrelation between emotionalsymptomsand socialisolationwere basedonestimatesshown in Figure 2: $2 ^ { * } ( . 3 2 ^ { * } . 6 2 ^ { * } . 6 0 ^ { * } . 6 2 ^ { * } . 0 2 ) = . 0 0 3$ \\nTo avoid overcrowding the table,contributions tooutcomes atage21years only included efects fromAandEfactorsatage 16, which explains why the total phenotypic variance at age 21 years is smaller than 1. \\nIn parentheses are the percentages of contributions to total A and $\\\\mathbf { E }$ influences of each variable. \\naProportions do not add up to one because peer victimisation at 12 years old had a C component. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,332,480,819]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In addition, our results provide some evidence that social isolation in late adolescence may be an intermediate pathway for the relationship between peer victimisation and emotional symptoms.Peer victimisation in early adolescence predicted later emotional symptoms in emerging adulthood via social isolation， suggesting that adolescents who experience peer victimisation during the first years of secondary school may develop a tendency to withdraw from social interactions (Barzeva et al., 2020; Oh et al.，2Oo8)． In turn，being socially isolated reduces opportunities for socialising and establishing new relationships during the transition to emerging adulthood, thereby leading to more distress and emotional symptoms over time (Oh et al.，2008； Rubin et al.， 2006,2009).However, social isolation did not significantly explain the link between emotional symptoms in early adolescence and peer victimisation in emerging adulthood,as social isolation in mid-adolescence did not significantly predict later peer victimisation. This could be explained by the increased freedom to decide whether and with whom to interact socially in young adulthood (e.g.，at work or university； Nikitin & Freund, 20o8). Young adults may be in a better position to avoid social situations conducive to peer victimisation. Peer victimisation may also be less common once adolescents finish secondary school and are no longer exposed to their peers every day. Therefore, although social isolation in midadolescence was associated with later increases in emotional symptoms, our findings suggest that being socially isolated at 16 years may carry less risk for peer victimisation in emerging adulthood. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,820,480,932]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We further found that non-shared environmental influences accounted for most of individual differences in emotional symptoms,social isolation and peer victimisation and that the magnitude of these influences was consistent across timepoints.Because the cross-lagged model accounted for the stability and prior relationships between variables,non-shared environmental contributions at 16 and 21 years old can be interpreted as explaining incremental change in variables over time. As such，this finding is consistent with longitudinal twin studies showing that change in emotional and social problems is primarily accounted for by environmental influences (Brendgen et al., 2022; Matthews et al., 2022; Nivard et al., 2015). Some of these non-shared environmental contributions could include risk factors with a broader impact on both emotional and social domains such as negative relationships with parents and peers (Kaufman,Kretschmer,Huitsing,& Veenstra, 2019). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,337,924,492]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,493,924,932]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our findings further suggest that both gene-environment non-shared environmental correlations partially accounted for the associationsbetween emotional symptoms, social isolation and peer victi-misation at each timepoint. This finding is in line with studies showing genetic and environmental overlap between emotional symptoms and negative peer experiences but also extends them by providing evidence of these etiological processes at different periods during adolescence and early adulthood (Brendgen et al., 2013, 2022; Matthews et al., 2016; Schaefer et al.,2018； Silberg et al.，2016; Smith et al.,2021). Whereas gene-environment correlations suggest that the risk of experiencing interpersonal difficulties may be traced back to adolescents'genetic predisposition to emotional symptoms,non-shared environmental correlations point to the possibility that the relationships between emotional symptoms and interpersonal difficulties are partly due to direct phenotypic pathways (i.e., unconfounded by genetic and familial vulnerabilities). Interestingly, the geneenvironment correlations were larger at 12 years than at later timepoints,while the magnitude of nonshared environmental correlations was consistent across timepoints.This could be explained by the fact that stabilityin behavioural traits,whichis accounted for in the cross-lagged model，is substantially accounted for by genetic factors (Nivard et al., 2015). Nonetheless, this means that new etiological processes emerge over time, perhaps suggesting that the genetic and non-shared environmental factors responsible for the relationships between emotional symptoms and interpersonal difficulties are not the same at12,16 and 21 years. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[68,945,922,969]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ 2023 TheAuthors.JoumalofChildPsychologyand Psychiatry publishedbyJohn Wiley&Sons Ltdonbehalfof Association for Child and Adolescent Mental Health. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[603,49,924,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"J Child Psychol Psychiatr 2023; 64(11): 1569-82 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[77,48,396,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1578 Genevieve Morneau-Vaillancourt et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[75,80,480,122]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,121,480,664]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our results must be interpreted while considering some limitations. First, the cross-lagged model does not disentangle between- from within-person effects and cannot be used to answer causal inference hypotheses (Hamaker, Kuiper,& Grasman， 2015). Our findings could either reflect between-person processes，where adolescents who manifest more emotional symptoms than their peers are also more victimised, or within-person processes，where adolescents'fluctuations in emotional symptoms lead to more peer victimisation than what they usually experience. We could have examined these within-person associations more specifically by testing a random-intercept cross-lagged model (Hamaker et al., 2015). However, because we aimed to under-stand why some adolescents were at higher risk for emotional symptoms than others and examine the risk factors associated with the persistence of emo-tional symptoms, the cross-lagged panel model was more appropriate. The cross-lagged panel model was also suitable to twin decomposition，as shown in previous studies using the same twin model(e.g., Burt et al., 20o5),and examining the genetic and environ-mental associations between variables was a central aim of our study. Further methodological work would have been required to apply the twin method to a random-intercept cross-lagged model,as this has never been done before. Such methodological work was beyond the scope of the present study. Second, some of the variances in self-reported measures may be due to self-perception biases and may be respon-sible for inflating within-person correlations between variables (i.e.,common method variance) or increas-ing non-shared environmental variance in twin analyses. Third, assessments were separated by a large time interval (i.e.，4-5 years) and this may have affected our capacity to identify the specific developmental periods at which risks were most important. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,692,178,706]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conclusions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[75,708,480,862]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In summary,our study shows that the mechanisms underlying emotional symptoms during adolescence involve reciprocal associations with social isolation and peer victimisation and that both genetic and environmental influences play a role in these mechanisms. If replicated,our findings highlight the necessity to intervene early in adolescence and to pay attention to the different types of interpersonal difficulties that emotionally vulnerable adolescents may experience,including social isolation and peer victimisation,to prevent their exacerbation over time. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[519,86,724,102]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Supporting information \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,103,924,141]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Additional supporting information may be found online in the Supporting Information section at the end of the article: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[515,156,924,413]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Table S1. Number of complete and incomplete twin pairs for each variable. \\nTable S2.Sensitivity correlations between emotional symptoms,peer problems and peer victimisation at 12, 16 and 21 years old. \\nTable S3. Independent sample t-tests to examine sex differences. \\nTable S4. Descriptive statistics for transformed variables at 12,16 and 21 years old. \\nTable S5. Comparison of phenotypic cross-lagged models. \\nTable S6. Fit comparisons for univariate twin models. Table S7. Cross-twin cross-trait correlations, \\nFigure S1.Phenotypic cross-lagged model using the full peer problems scale (sensitivity analysis). \\nFigure S2. Genetic cross-lagged model using the full peer problems scale (sensitivity analysis). \\nFigure S3.Multi-group phenotypic cross-lagged model (sex differences). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,441,690,455]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acknowledgements \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,455,924,755]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The authors gratefully acknowledge the ongoing contribution of the participants in the Twins Early Development Study (TEDS) and their families. TEDSis supported by a program grant from the UK Medical Research Council (MR/V012878/1 and previously MR/ M021475/1)，with additional support from the US National Institutes of Health (AG046938). This study presents independent research part-funded by the National Institute for Health Research (NIHR) Biomedical Research Centre at South London and Maudsley NHS Foundation Trust and King's College London. The views expressed are those of the authors and not necessarily those of the NHS,the NIHR，or the Department of Health.G.M.-V. is supported by postdoctoral fellowships from the Social Sciences and Humanities Research Council of Canada (756-2021- 0516) and Fonds de recherche du Québec Société et Culture (2022-B3Z-297753). For the purposes of open access,the author has applied a Creative Commons Attribution (CC BY) licence to any Accepted Author Manuscript version arising from this submission. The authors have declared that they have no competing or potential conflicts of interest. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,783,658,797]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Correspondence \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,799,924,862]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Thalia C. Eley，Social，Genetic and Developmental Psychiatry Centre, Institute of Psychiatry，Psychology and Neuroscience，King's College London，Memory Lane,London， SE5 8AF，UK; Email: thalia.eley@kcl. ac.uk \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,945,926,969]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\circledcirc$ 2023 TheAuthors.JoumalofChild Psychologyand Psychiatry publishedbyJohn Wiley&Sons LtdonbehalfofAssociation for Child and Adolescent Mental Health. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[73,49,235,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"doi:10.1111/jcpp.13847 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[468,48,924,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cross-lagged twin study emotional symptoms adolescence 1579 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"discarded\"\n },\n {\n \"bbox\": \"[84,129,179,145]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Key points \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[82,151,919,307]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Emotional symptoms,including anxiety and depressive symptoms,and peer victimisation are reciprocally associated during adolescence,and social isolation may explain these bidirectional relationships. \\nBoth genetic and environmental influences account for individual differences in emotional symptoms, social isolation and peer victimisation,as well as for their associations. \\nFindings showed that emotional symptoms were reciprocally and independently associated with social isolation and peer victimisation over time and that social isolation in mid-adolescence partly explained th link between early peer victimisation and later emotional problems. \\nDifferent gene-environment and non-shared environmental correlations explained the associations between emotional symptoms and interpersonal difficulties at each timepoint. \\nThis study underscores the necesity to pay attention to social isolation and peer victimisation during adolescence to prevent the exacerbation of emotional symptoms over time. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,356,169,369]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[72,369,482,928]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"11\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Achenbach，T.M. (1991)．Manual for the Child Behavior Checklist/4-18and 1991 profile. Burlington,VT: University of Vermont, Department of Psychiatry. \\nAcquah, E.O., Topalli, P.Z., Wilson, M.L., Junttila,N.,& Niemi, P.M.(2016).Adolescent loneliness and social anxiety as predictors of bullying victimisation. International Journal of Adolescence and Youth,21,320-331. \\nAdachi,P.,& Willoughby,T. (2015). Interpreting effect sizes when controlling for stability effects in longitudinal autoregressive models: Implications for psychological science. European Journal of Developmental Psychology，12，116- 128. \\nArseneault, L., Milne, B.J., Taylor, A., Adams, F., Delgado, K, Caspi,A.，& Moffitt, T.E. (2008).Being bullied as an environmentally mediated contributing factor to children's internalizing problems. Archives of Pediatrics & Adolescent Medicine,162,145-150. \\nBarzeva, S.A., Meeus,W.H.J.,& Oldehinkel, A.J. (2019). Social withdrawal in adolescence and early adulthood: Measure-ment issues, normative development, and distinct trajectories. Journal of Abnormal Child Psychology, 47, 865-879. \\nBarzeva,S.A.,Richards,J.S.,Meeus,W.H.J.,& Oldehinkel, A.J.(202O). The social withdrawal and social anxiety feedback loop and the role of peer victimization and acceptance in the pathways. Development and Psychopathology,32, 1402-1417. \\nBoker,S., Neale,M., Maes,H.,Wilde,M., Spiegel, M.,Brick,T., ... Fox，J. (2011).OpenMx: An open source extended structural equation modeling framework. Psychometrika, 76, 306-317. \\nBowker, J.C., Nelson,L.J., Markovic,A.,& Luster, S. (2013). Social withdrawal during adolescence and emerging adulthood.In R.J. Coplan & J.C. Bowker (Eds.), The handbook of solitude: Psychological perspectives on social isolation, social withdrawal, and being alone (pp. 167-183). New York: Wiley-Blackwell. \\nBoyes,M.E.,Bowes,L., Cluver,L.D.,Ward, C.L.,& Badcock, N.A. (2014). Bullying victimisation, internalising symptoms, and conduct problems in south African children and adolescents: A longitudinal investigation. Journal of Abnormal Child Psychology, 42, 1313-1324. \\nBrendgen, M., Vitaro,F., Barker, E.D., Girard,A., Dionne, G., Tremblay，R.E.，& Boivin，M.(2013)．Do other people's plights matter? A genetically informed twin study of the role of social context in the link between peer victimization and children's aggression and _depression symptoms. Developmental Psychology, 49, 327-340. \\nBrendgen,M., Zheng,Y., Vitaro,F.,Dionne,G.,& Boivin,M. (2022). Gene-environment interplay linking peer victimiza-tion with adolescents' trajectories of depressive symptoms. Journal of the American Academy of Child & Adolescent Psychiatry,62,261-271. \\nBrown，B.B.，& Larson，J. (2009)．Peer relationships in adolescence.In R.M.Lerner & L. Steinberg (Eds.),Handbook of adolescent psychology. John Wiley & Sons, Inc. \\nBurt, S.A., McGue, M., Krueger,R.F.,& Iacono,W.G. (2005). How are parent-child conflict and childhood externalizing symptoms related over time? Results from a genetically informative cross-lagged study. Development and Psychopathology,17,145-165. \\nCaspi,A.，Houts，R.，Fisher，H.,Danese,A.，& Moffitt，T. (2023). The general factor of psychopathology (p): Choosing among competing models and interpreting p. Clinical Psychological Science. \\nCohen, B.E., Edmondson, D.,& Kronish, I.M. (2015). State of the art review: Depression, stress,anxiety, and cardiovascular disease.American Journal of Hypertension, 28,1295- 1302. \\nCook, C.R., Williams, K.R., Guerra, N.G., Kim, T.E.,& Sadek, S. (201O).Predictors of bullying and victimization in childhood and adolescence: A meta-analytic investigation. School Psychology Quarterly,25,65-83. \\nFisher，H.L.，Caspi，A.，Moffitt，T.E.，Wertz,J.，Gray，R., Newbury, J.,... Arseneault, L. (20i5). Measuring adoles-cents'exposure to victimization: The Environmental Risk (E-risk) Longitudinal Twin Study. Development and Psychopathology,27,1399-1416. \\nForrest,C.L.,Gibson,J.L.,Halligan,S.L.,& St Clair，M.C. (2020). A cross-lagged analysis of emotion regulation, peer problems,and emotional problems in children with and without early language difficulties: Evidence from the Millennium Cohort Study. Journal of Speech, Language, and Hearing Research, 63,1227-1239. \\nGadermann,A.M.,Guhn，M.,& Zumbo,B.D. (2012).Estimating ordinal reliability for Likert-type and ordinal item response data: A conceptual，empirical，and practical guide. Practical Assessment， Research & Evaluation， 17, 1-13. \\nGoodman, R. (1997). The Strengths and Difficulties Questionnaire: A research note. Journal of Child Psychology and Psychiatry, 38, 581-586. \\nGoodman, R., & Scott, S. (1999). Comparing the Strengths and Difficulties Questionnaire and the Child Behavior Checklist: Is_small beautiful? Jourmal of Abnormal Child Psychology, 27, 17-24. \\nHamaker, E.L., Kuiper,R.M.,& Grasman, R.P.P.P. (2015).A critique of the cross-lagged panel model. Psychological Methods,20,102-116. \\nHutzell,K.L.,& Payne,A.A. (2012). The impact of bullying victimization on school avoidance.Youth Violence and Juvenile Justice,10, 370-385. \\nJacobson, N.C., & Newman, M.G. (2017). Anxiety and depression as bidirectional risk factors for one another:A metaanalysis of longitudinal studies.Psychological Bulletin, 143, 1155-1200. \\nJohansson，A.，Huhtamaki，A.，Sainio，M.，Kaljonen，A., Boivin, M.,& Salmivalli, C. (2022). Heritability of bullying and victimization in children and adolescents: Moderation by the KiVa antibullying program. Journal of Clinical Child & Adolescent Psychology, 51, 505-514. \\nJohnson, D., Dupuis, G., Piche,J., Clayborne, Z.,& Colman, I. (2018).Adult mental health outcomes of adolescent depres-sion: A systematic review. Depression and Anxiety, 35, 700- 716. \\nJoseph, S.,& Stockton, H. (2018). The multidimensional peer victimization scale: A systematic review. Aggression and Violent Behavior,42(September-October 2018),96-114. \\nKaufman, T.M.L., Kretschmer, T., Huitsing, G.,& Veenstra, R. (2019). Caught in a vicious cycle? Explaining bidirectional spillover between parent-child relationships and peer victimization. Development and Psychopathology, 32, 11-20. \\nKwong,A.S.F.,López-López,J.A.,Hammerton,G.,Manley,D., Timpson, N.J., Leckie,G.,& Pearson, R.M. (2019). Genetic and environmental risk factors associated with trajectories of depression symptoms from adolescence to young adulthood. JAMA Network Open,2,1-14. \\nLee,K.S.,& Vaillancourt, T. (2019).A four-year prospective study of bullying,anxiety,and disordered eating behavior across early adolescence. Child Psychiatry and Human Development, 50, 815-825. \\nLockhart,C.,Ahmadzadeh,Y.,Breen,G.,Bright,J.,Bristow, S.,Boyd,A.,...Eley,T.C. (2023). Twins Early Development Study (TEDS): A genetically sensitive investigation of mental health outcomes in the mid-twenties. JCPP Advances, e12154. https://doi.0rg/10.1002/jcv2.12154 \\nL6pez-López, J.A.， Kwong, A.S.F.， Washbrook, E.， Pearson, R.M.，Tilling,K.，Fazel,M.S.，... Hammerton，G. (2020). Trajectories of depressive symptoms and adult educational and employment outcomes.BJPsych Open,6,e6. \\nLu,W. (2019). Adolescent depression: National trends,risk factors，and healthcare disparities. American Journal of Health Behavior, 43,181-194. \\nMatthews,T.,Danese,A.,Wertz,J., Odgers,C.L.,Ambler,A., Mofitt，T.E.，& Arseneault,L. (2016). Social isolation, loneliness and depression in young adulthood: A behavioural genetic_analysis. Social Psychiatry and Psychiatric Epidemiology,51,339-348. \\nMatthews,T.,Qualter,P.,Bryan,B.T.,Caspi,A.,Danese,A., Mofitt, T.E.，... Arseneault, L. (2022). The developmental course of loneliness in adolescence: Implications for mental health, educational attainment, and psychosocial functioning.Development and Psychopathology,35,537-546. \\nMcGue,M.,& Bouchard, T.J.(1984).Adjustment of twin data for the effects of age and sex. Behavior Genetics,14,325-343. \\nMelton,T.H.,Croarkin,P.E., Strawn,J.R.,& McClintock,S.M. (2016). Comorbid anxiety and depressive symptoms in children and adolescents: A systematic review and analysis. Journal of Psychiatric Practice, 22, 84-89. \\nMiddeldorp, C.M., Cath,D.C., van Dyck, R.,& Boomsma, D.I. (2005). The co-morbidity of anxiety and depression in the perspective of genetic epidemiology.A review of twin and family studies.Psychological Medicine,35,611-624. \\nModecki，K.，Minchin，J.，Harbaugh，A.，Guerra，N.，& Runions,K. (2014). Bullying prevalence across contexts: A meta-analysis measuring_cyber and traditional bullying. Journal of Adolescent Health, 55, 602-611. \\nMoore,S.，Norman,R.，Suetani, S.， Thomas,H.， Sly, P.，& Scott,J. (2017). Consequences of bullying victimization in childhood and adolescence:A systematic review and metaanalysis. World Journal of Psychiatry, 7, 60-76. \\nMynard, H., & Joseph, S. (2000). Development of the multidimensional peer-victimization scale. Aggressive Behavior, 26,169-178. \\nNaicker,K.，Galambos,N.L., Zeng,Y.，Senthilselvan，A.，& Colman, I. (2013). Social,demographic,and health outcomes in the 10 years following adolescent depression. Journal of Adolescent Health, 52, 533-538. \\nNikitin,J.,& Freund, A.M. (2oo8). The role of social approach and avoidance motives for subjective well-being and the successful transition to adulthood.Applied Psychology,57 (s1), 90-111. \\nNilsen，W.，Karevold，E.，Roysamb，E.，Gustavson，K.，& Mathiesen, K.S. (2013). Social skills and depressive symp-toms across adolescence: Social support as a mediator in girls versus boys. Journal of Adolescence, 36, 11-20. \\nNivard, M.G., Dolan, C.V., Kendler,K.S., Kan,K.J., Willemsen, G.，van Beijsterveldt,C.E.M.，... Boomsma，D.I. (2015). Stability in symptoms of anxiety and depression as a function of genotype and environment: A longitudinal twin study from ages 3 to 63 years. Psychological Medicine, 45, 1039-1049. \\nOh,W.,Rubin,K.H.,Bowker,J.C.,Booth-LaForce,C.,Rose-Krasnor,L.,& Laursen,B. (2008). Trajectories of social withdrawal from middle childhood to early adolescence. Journal of Abnormal Child Psychology, 36, 553-566. \\nPabian, S.,& Vandebosch,H. (2016). An investigation of shortterm longitudinal associations between social anxiety and victimization and perpetration of traditional bullying and cyberbullying. Journal of Youth and Adolescence, 45, 328- 339. \\nPickard,H.,Rijsdijk,F.,Happé,F.,& Mandy,W.(2017).Are social and communication difficulties a risk factor for the development of social anxiety? Journal of the American Academy of Child & Adolescent Psychiatry, 56, 344-351.e3. \\nPrice,T.S., Freeman,B.,Craig,I.,Petrill, S.A.,Ebersole,L.,& Plomin, R. (200o).Infant zygosity_can be assigned by parental report questionnaire data. Twin Research, 3, 129- 133. \\nRacine,N., McArthur,B.A., Cooke,J.E., Eirich,R., Zhu,J.,& Madigan, S. (2021). Global prevalence of depressive and anxiety symptoms in children and adolescents_ during COVID-19:A meta-analysis.JAMA Pediatrics,175,1142- 1150. \\nReijntjes,A.,Kamphuis,J.H.,Prinzie,P.,& Telch,M.J. (2010). Peer victimization and internalizing problems in children: A meta-analysis of longitudinal studies. Child Abuse& Neglect, 34,244-252. \\nRijsdijk,F.V.，& Sham, P.C. (2002). Analytic approaches to twin data using structural equation models.Briefings in Bioinformatics,3,119-133. \\nRosseel，Y. (2012). Lavaan: An R package for structural equation modeling. Journal of Statistical Software,48,1-36. \\nRStudio Team. (2022). RStudio: Integrated development environment for R. Boston,MA: RStudio, PBC. Available from: http://www.rstudio.com/ \\nRubin，K.H.，Bukowski，W.M.，& Parker,J.G.(2006).Peer interactions,relationships,and groups.In N.Eisenberg,W. Damon, & R.M. Lerner (Eds.), Handbook of child psychology: Social, emotional, and personality development (pp. 571- 645). John Wiley & Sons,Inc. \\nRubin，K.H.，Coplan，R.J.，& Bowker，J.C. (2009).Social withdrawal in childhood. Annual Review of Psychology, 60, 141-171. \\nSalmivalli,C.,Lagerspetz,K.,Bjorkqvist,K., Osterman,K.,& Kaukiainen, A. (1996). Bullying as a group process: Participant roles and their relations to social status within the group. Aggressive Behavior, 22, 1-15. \\nSchaefer,J.D.,Moffitt,T.E.,Arseneault,L.,Danese,A.,Fisher, H.L.,Houts,R.,...Caspi,A. (2018).Adolescent victimization and early-adult psychopathology:Approaching causal inference using a longitudinal twin study to rule out noncausal explanations.Clinical Psychological Science,6,352-371. \\nSentse，M.,Prinzie,P.,& Salmivalli,C. (2017). Testing the direction of longitudinal paths between victimization,peer rejection,and different types of internalizing problems in adolescence.Journal of Abnormal Child Psychology，45, 1013-1023. \\nSilberg,J.L.,Copeland,W.,Linker,J.,Moore,A.A.,Roberson-Nay，R.，& York，T.P.(2016)．Psychiatric outcomes of bullying victimization: A study of discordant monozygotic twins.Psychological Medicine,46,1875-1883. \\nSmith,K.,Brendgen,M.,Hébert,M.,Vitaro,F.,Dionne,G.,& Boivin,M. (2021). Links between peer victimization, dating violence victimization and depression in adolescence:A genetically-informed study.Journal of Clinical Child and Adolescent Psychology,1-12,1-12. \\nSmith,K.,Hébert,M.,Brendgen,M.,& Blais,M. (2022).The mediating role of internalizing problems between peer victimization and dating violence victimization:A test of the stress generation hypothesis.Journal of Interpersonal Violence,37,NP11271-NP11295. \\nSpeyer,L.G.,Auyeng,B.,& Murray,A.L. (2022).Longitudinal invariance of the Strengths and Difficulties Questionnaire across ages 4 to 16 in the ALSPAC sample.Assessment. https://doi.0rg/10.1177/10731911221128948 \\nTiirikainen,K.,Haravuori,H.,Ranta,K.,Kaltiala-Heino,R.,& Marttunen,M. (2019).Psychometric properties of the 7-item Generalized Anxiety Disorder scale (GAD-7) in a large representative sample of Finnish adolescents.Psychiatry Research,272,30-35. \\nVeldkamp，S.A.M.，Boomsma，D.I.，de Zeeuw，E.L.，van Beijsterveldt, C.E.M.,Bartels,M.,Dolan,C.V.,& van Bergen, E.(2019).Genetic and environmental influences on diferent forms of bullying perpetration,bullying victimization,and their co-occurrence.Behavior Genetics,49,432-443. \\nVugteveen,J.,de Bildt,A.,Theunissen,M.,Reijneveld, S.A.,& Timmerman,M. (2021).Validity aspects of the Strengths and Difficulties Questionnaire (SDQ) adolescent self-report and parent-report versions among Dutch adolescents. Assessment, 29, 601-616. \\nWilliford,A.,Boulton,A.,Noland,B.,Little,T.D.,Kärnä,A.,& Salmivalli，C. (2012). Effects of the KiVa anti-bullying program on adolescents' depression,anxiety,and perception of peers. Journal of abnormal child psychology,40,289-300. https://doi.0rg/10.1007/s10802-011-9551-1 \\nWolitzky-Taylor,K.,Dour,H., Zinbarg,R.,Mineka,S.,Vrshek-Schallhorn，S.，Epstein，A.，...Craske，M.G． (2014). Experiencing core symptoms of anxiety and unipolar mood disorders in late adolescence predicts disorder onset in early adulthood.Depression and Anxiety,31,207-213. \\nZhu,X.,Griffths,H.,Eisner,M.,Hepp,U.,Ribeaud,D.,& Murray，A.L. (2022)． Developmental relations between bullying victimization and suicidal ideation in middle adolescence and emerging adulthood: Do internalizing problems and substance use mediate their links? 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$^{\\dagger}$ Department of Radiation Oncology, Nanfang Hospital, Southern Medical University, Guangzhou, China; $^{\\ddagger}$ School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu, China; $^{\\S}$ Cancer Clinical Research Center, Sichuan Cancer Hospital & Institute, Sichuan Cancer Center, School of Medicine, University of Electronic Science and Technology of China, Chengdu, China; and $^{\\parallel}$ Department of Radiation Oncology, Sichuan Cancer Hospital & Institute, Sichuan Cancer Center, School of Medicine, University of Electronic Science and Technology of China, Chengdu, China ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,451,446,465]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Received Oct 28, 2021; Accepted for publication Mar 24, 2022 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,496,935,527]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Purpose: We aimed to validate the accuracy and clinical value of a novel semisupervised learning framework for gross tumor volume (GTV) delineation in nasopharyngeal carcinoma. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,527,935,650]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Methods and Materials: Two hundred fifty-eight patients with magnetic resonance imaging data sets were divided into training $(n = 180)$ , validation $(n = 20)$ , and testing $(n = 58)$ cohorts. Ground truth contours of nasopharynx GTV (GTVnx) and node GTV (GTVnd) were manually delineated by 2 experienced radiation oncologists. Twenty percent $(n = 36)$ labeled and $80\\%$ $(n = 144)$ unlabeled images were used to train the model, producing model-generated contours for patients from the testing cohort. Nine experienced experts were invited to revise model-generated GTV in 20 randomly selected patients from the testing cohort. Six junior oncologists were asked to delineate GTV in 12 randomly selected patients from the testing cohort without and with the assistance of the model, and revision degrees were compared under these 2 modes. The Dice similarity coefficient (DSC) was used to quantify the accuracy of the model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,650,935,745]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Results: The model-generated contours showed a high accuracy compared with ground truth contours, with an average DSC score of 0.83 and 0.80 for GTVnx and GTVnd, respectively. There was no significant difference in DSC score between T1-2 and T3-4 patients (0.81 vs 0.83; $P = .223$ ), or between N1-2 and N3 patients (0.80 vs 0.79; $P = .807$ ). The mean revision degree was lower than $10\\%$ in 19 $(95\\%)$ patients for GTVnx and in 16 $(80\\%)$ patients for GTVnd. With assistance of the model, the mean revision degree for GTVnx and GTVnd by junior oncologists was reduced from $25.63\\%$ to $7.75\\%$ and from $21.38\\%$ to $14.44\\%$ , respectively. Meanwhile, the delineating efficiency was improved by over $60\\%$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,817,477,842]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corresponding author: Nianyong Chen, MD, PhD; E-mail: n_ychen@ hotmail.com ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,843,453,868]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Wenjun Liao and Jinlan He made equal contributions to this study. \nDisclosures: none. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,869,477,895]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Data sharing statement: The code for neural network training can be obtained at: https://github.com/HiLab-git/SSL4MIS. Other data generated ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,817,932,843]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"and analyzed during this study can be obtained by contacting the corresponding author with reasonable requirements. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,843,932,869]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Supplementary material associated with this article can be found in the online version at doi:10.1016/j.ijrobp.2022.03.031. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[638,91,935,115]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"INTERNATIONAL JOURNAL OF RADIATION ONCOLOGY $\\cdot$ BIOLOGY $\\cdot$ PHYSICS ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[819,122,934,135]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"www.redjournal.org ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[55,910,435,923]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Int J Radiation Oncol Biol Phys, Vol. 113, No. 4, pp. 893-902, 2022 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[55,923,845,936]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"0360-3016/$ - see front matter © 2022 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[57,936,352,949]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(http://creativecommons.org/licenses/by-nc-nd/4.0/) ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[57,949,298,962]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"https://doi.org/10.1016/j.ijrobp.2022.03.031 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[65,64,935,125]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conclusions: The proposed semisupervised learning-based model showed a high accuracy for delineating GTV of nasopharyngeal carcinoma. It was clinically applicable and could assist junior oncologists to improve GTV contouring accuracy and save contouring time. © 2022 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[64,170,178,186]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[64,206,480,455]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Nasopharyngeal carcinoma (NPC) is one of the most common malignancies in head and neck, and radiation therapy has been the primary effective treatment modality for NPC. $^{1,2}$ Precise delineation of gross target volume (GTV) is substantially important in the process of radiation therapy, especially in the era of intensity modulated radiation therapy. However, GTV contouring for NPC is quite challenging compared with other solid tumors owing to its complex anatomic structures and diverse tumor invasion pathways. $^{3,4}$ The radiation oncologist needs to review multiple types of medical images to determine and delineate lesions slice by slice on images. Furthermore, the margin of tumor is usually hard to determine, which sometimes inevitably depends on personal clinical experience. Therefore, the current manual GTV delineation is time-consuming, laborious, error-prone, $^{5}$ and subject to interobserver and intraobserver heterogeneity. $^{6,7}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[64,455,480,485]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"These factors could affect the accuracy of GTV contouring and may further cause treatment failures in NPC. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[64,486,480,900]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recently, a variety of deep learning models have been proposed to automatically delineate GTV and organs at risk in NPC, such as U-Net networks, and many of them have achieved state-of-the-art performance, thus providing an alternative for manual GTV delineation. $^{8-12}$ However, several limitations exist in these models. The first concern is that previous automatic segmentation models for GTV of NPC were all based on fully supervised learning (SL), which required a large number of manually labeled data for model training. $^{6,8,11}$ For example, manually labeled magnetic resonance imaging (MRI) data sets from more than 700 patients with NPC were used to train the neural network to construct an automatic nasopharynx GTV (GTVnx) segmentation model. $^{6}$ However, it is quite difficult to obtain a large amount of high-quality manually labeled images in clinical practice, as both expertise and time are needed to produce such accurate annotations. Therefore, we have proposed a novel semisupervised learning (SSL) method via an uncertainty rectified pyramid consistency framework for GTV delineation of NPC, $^{13}$ which uses both a small amount of manually labeled data and a large amount of unlabeled data for model development. By using limited (10% or 20%) labeled data to train the uncertainty rectified pyramid consistency network, our model achieved an average Dice score of 0.78 or 0.81 in the testing cohort, which was of comparable performance to that of state-of-the-art models by using full SL. $^{13}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[64,902,480,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Another concern of the deep learning models is the acceptability, utility, and actual benefit in the clinical setting, which is especially critical for medical image ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[520,170,935,307]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"segmentation. However, few studies have reported the clinical value of SL or SSL models, raising our interest in addressing the clinical value of the new SSL model. To achieve this aim, we examined the segmentation performance in different subgroups of patients, invited multicenter experts to evaluate the automatic segmentation results, and asked junior radiation oncologists to use the new model, thus clarifying the clinical acceptance and application of the novel model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[520,351,732,367]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Methods and Materials ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[520,389,875,419]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Data collection and model-generated GTV contours ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[520,439,935,608]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The NPC data sets $(n = 258)$ used were described in our previous study. The inclusion and exclusion criteria were the following: (1) patients who were histologically confirmed as NPC; (2) patients who underwent MRI for nasopharynx and neck before anticancer treatment; (3) the MRI sets included unenhanced T1- and T2-weighted, contrast-enhanced T1-weighted sequences; and (4) the median thickness of the MRI was $1.10\\mathrm{mm}$ (range, $1.00 - 1.25\\mathrm{mm}$ ). The patients were excluded if their images had a low resolution that affected delineation of tumor volume. Details of MRI acquisition are shown in Tables E1 and E2. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[520,610,935,840]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The whole data sets were randomly split into 3 cohorts with a ratio of 7:1:2: a training cohort with 180 patients for model training, a validation cohort with 20 patients for optimizing the model hyperparameters, and a testing cohort with 58 patients to test the performance of the model. In our previous study,[13] we examined the effect of the proportion of labeled data on the performance of our network (Fig. E1). We found that the segmentation accuracy of the network was positively correlated with the proportion of labeled data overall. Therefore, to achieve a balance between the proportion of labeled data and the performance of the network, $20\\%$ $(n = 36)$ labeled and $80\\%$ $(n = 144)$ unlabeled images in the training cohort $(n = 180)$ were selected for network training, thus obtaining model-generated GTV contours for patients in the testing cohort. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[520,842,935,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"All patients in this study were restaged according to the eighth edition of the American Joint Committee on Cancer, and characteristics of these patients are presented in Table E3. There was no significant difference in sex, age, T category, N category, histology, or tumor characteristics between the combined training and validation cohorts and the testing cohort. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,41,90,51]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"894 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[107,41,171,51]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Liao et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[520,41,935,54]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"International Journal of Radiation Oncology $\\cdot$ Biology $\\cdot$ Physics ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[60,69,475,113]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This study was approved by the ethics committee on biomedical research of the hospital and the requirement for informed consent was waived. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[60,143,194,159]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"GTV contouring ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[60,177,475,408]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Two radiation oncologists from our center with more than 10 years of experience in treating patients with NPC were invited to delineate ground truth contours, including GTVnx and node GTV (GTVnd). A third radiation expert with more than 20 years of experience was consulted in cases of disagreement. GTVs (GTVnx and GTVnd) were delineated according to the International Commission on Radiation Units and Measurements report 83,[14] and the positive cervical metastatic lymph node was determined according to the published radiologic criteria.[15] Additionally, to ensure the most precise GTV contours, experts could also refer to other images during contouring when necessary, such as enhanced computed tomography (CT), cervical ultrasound, or positron emission tomography-CT. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[60,408,475,454]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"All manually delineated contours on MRIs in this study were conducted in the ITK-SNAP, an open-source software for image segmentation.[16] ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[60,483,442,499]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Network architecture and its implementation ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[60,516,475,809]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The detailed network architecture is shown in Figure E2. We designed a multiscale prediction 3-dimensional network for semisupervised segmentation GTV of NPC, which took images with a shape of $112^{3}$ as inputs and generated 4 outputs at 4 different scales (the shapes of output 1, 2, 3, 4 were $112^{3}, 56^{3}, 28^{3}, 14^{3}$ , respectively) for each input. For labeled images, this network was supervised by expert annotated labels directly, and a combination loss between cross-entropy loss and Dice loss was used as the supervised loss. A multiscale consistency (unsupervised loss) was used to optimize the network when the inputs were without annotations, where we rescaled all outputs to a fixed resolution $(112^{3})$ and calculated the mean square error between each rescaled output and the average rescaled output. Finally, the semisupervised framework was optimized by a joint loss of the supervised loss and the unsupervised loss. Different from previous deep supervised networks, this framework could use both labeled images and unlabeled images at the same time and lead to better performance. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[60,811,475,949]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In image preprocessing, we first converted the orientation of all images into the standard RAI (right to left, anterior to posterior, inferior to superior in the x-, y-, and z-axes, respectively). The intensity window was clipped at the 0.5 and 99.5 percentiles of the volume voxels and then normalized to zero mean and unit variance. Considering the thickness of images in this data set was very similar, we did not use any spatial interpolation processing to resample the images. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,69,932,192]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In addition, we deposited the code of training, validation, and testing in https:// github.com/HiLab-git/SSL4MIS. Detailed descriptions and the training and inference settings of the proposed network could also be acquired from our previous study. $^{13}$ In this study, we focused on evaluating the potential of the semisupervised delineation method in clinical practice and investigated the efficiency and robustness in a large multicenter users' study. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,212,925,243]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Assessment of the model-generated contours by multicenter experts ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,261,932,584]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Nine radiation oncologists (E1, E2, E3...E9) with more than 10 years' experience in treating NPC constituted the multicenter group, with 3 experts from each of the 3 tertiary hospitals (West China Hospital, Sichuan Cancer Hospital, and Sichuan Provincial People's Hospital). Twenty patients with NPC patients (5 patients in each T category) with the model-generated contours were randomly selected from the testing cohort. Then, each expert needed to revise the incorrect delineations of the GTV contours of all 20 patients. During the revision process, experts were blinded to both the ground truth contours and contours delineated by their counterparts. Moreover, to maximize homogeneity of delineation, the experts were provided with the same image data of each patient and referred to the same GTV contouring principles described previously. Finally, the model-generated GTV contours were compared with their corresponding revised contours to calculate the revision degree (the difference of volumetric overlap) through the formula $(1 - \\text{Dice similarity coefficient} [\\text{DSC}]) \\times \\text{percentage}$ . The flowchart is illustrated in Figure 1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,605,911,636]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The model-assisted junior radiation oncologist GTV delineations ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[517,655,932,949]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Six junior radiation oncologists (R1, R2, R3, . . ., R6) with 1 to 2 years of experience from the same centers (2 oncologists in each center) delineated GTVs of 12 patients randomly selected from the testing cohort in 2 modes. In the first mode without model assistance, the delineation was conducted completely manually from scratch, while in the second mode with model assistance, the oncologists just needed to verify the model-generated GTV and modify the inappropriate delineations when necessary. During the delineation process, these junior radiation oncologists were blinded to contours delineated by their counterparts. Moreover, there was a 4-week interval between 2 modes to reduce the effect of delineation results from the former mode on the latter one. Meanwhile, the delineation time for GTV in each case was recorded. To record the delineation time of junior radiation oncologists more accurately, all junior radiation oncologists were trained to use the contouring software proficiently in advance. And, all junior ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[59,41,276,51]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Volume 113 • Number 4 • 2022 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[647,41,888,51]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Automatic delineation of GTV in NPC ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[904,41,932,51]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"895 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[100,67,902,315]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 1. Flowchart of the study. Abbreviations: DSC = Dice similarity coefficient; GTV = gross tumor volume. \"]","image_footnote":"[]","img_path":"images/0652/73f5950ade712e872bb10d9593351c0b515ce48413aecb34986653c514718999.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[62,368,480,491]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"radiation oncologists were required to review all images and medical history carefully before delineation. After the prework, these oncologists were asked to complete contouring the GTV of a certain patient nonstop, and the delineation time was recorded from the start to the end of the contouring. In addition, the principles of image distribution and contouring were the same as described previously. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,492,480,584]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Finally, the delineated contours without and with the model assistance were compared with the ground truth contours, and the revision degrees were calculated through the formula $(1 - \\mathrm{DSC})\\times$ percentage to evaluate whether the model could improve the accuracy of GTV contouring. The flowchart is shown in Figure 1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,609,239,624]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Performance metrics ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[62,642,480,812]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Metrics to evaluate the performance of segmentation could be roughly divided into 2 categories: the region-based metrics and the distance-based metrics. Therefore, the 2 most widely used metrics, the region-aware $\\mathrm{DSC}^{17}$ and distance-aware average surface distance (ASD), $^{18}$ were applied to evaluate the model performance. DSC was used to evaluate the similarity between 2 contours (\\%) and ASD was used to calculate the average distance between the surfaces of 2 contours (mm). The higher the DSC score (range, 0-1), the better the overlap between 2 contours. The lower the ASD (range, $\\geq 0$ ), the closer the distance between 2 contours in space. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,836,220,853]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Statistical analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[62,871,480,949]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Statistical analysis was performed using SPSS software (version 22.0, IBM). Categorical variables were compared by Pearson $\\chi^2$ test, and numeric variables were compared by $t$ test (2 groups) or variance analysis ( $\\geq 3$ groups). Correlation between DSC or ASD and clinical parameters was analyzed ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[520,368,937,399]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"with Pearson correlation coefficient. A 2-tailed $P$ value $< .05$ was considered significant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[520,429,586,443]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Results ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[520,465,935,498]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Performance of the model in different subgroups of patients ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[520,517,937,640]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Accuracy of the model-generated GTV contours is summarized in Table 1. For GTVnx, the mean DSC score was 0.83 and the mean ASD was $1.48\\mathrm{mm}$ . For GTVnd, the mean DSC was 0.80 and the mean ASD was $2.28\\mathrm{mm}$ . The subsequent visual comparison indicated a high concordance between the expert- and the model-generated GTVnx contours of different T categories (Fig. 2a) and GTVnd contours of different N categories (Fig. 2b). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[520,641,937,809]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the subgroup analysis of GTVnx, comparable DSC scores were obtained between patients with T1-2 and patients with T3-4 (0.81 vs 0.83; $P = .223$ ), while the ASD score of patients with T3-4 was higher than that of patients with T1-2 (1.64 vs 1.16 mm; $P = .036$ ) (Table 1). When the detailed primary tumor sizes were taken into consideration, there was no significant correlation between the GTVnx and DSC score $(P = .225)$ (Fig. E3a), but the ASD score increased with increasing GTVnx, showing a significant positive correlation $(R = 0.51; P < .001)$ (Fig. E3b). Other factors, including age and sex (Table 1), had no effect on contouring accuracy. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[520,811,937,949]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the subgroup analysis of GTVnd, comparable DSC scores were achieved between patients with N1-2 and N3 categories (0.80 vs 0.79; $P = .807$ ), but the ASD score of patients with N1-2 was significantly lower than that of patients with N3 (1.56 vs 3.86 mm; $P = .002$ ) (Table 1). No significant correlation between GTVnd and DSC score ( $P = .293$ ) was found (Fig. E3c). However, the GTVnd was positively associated with the score of ASD ( $R = 0.26$ ; $P = .047$ ) (Fig. E3d). Likewise, DSC and ASD ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,41,93,51]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"896 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[107,41,171,51]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Liao et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[520,41,937,54]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"International Journal of Radiation Oncology $\\cdot$ Biology $\\cdot$ Physics ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[57,87,935,602]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0737/a588a7219c1c67b1c845477dc41381703e56833aeded8e4ee23ccb25a9c3c4e6.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":"<table><tr><td>Variable</td><td>DSC (mean ± SD)</td><td>P value</td><td>ASD (mean ± SD, mm)</td><td>P value</td></tr><tr><td colspan=\"5\">GTVnx</td></tr><tr><td>Total patients</td><td>0.83 ± 0.06</td><td>NA</td><td>1.48 ± 0.82</td><td>NA</td></tr><tr><td>T category</td><td></td><td>.223</td><td></td><td>.036</td></tr><tr><td>T1-2 (n = 20)</td><td>0.81 ± 0.04</td><td></td><td>1.16 ± 0.20</td><td></td></tr><tr><td>T3-4 (n = 38)</td><td>0.83 ± 0.06</td><td></td><td>1.64 ± 0.97</td><td></td></tr><tr><td>Sex</td><td></td><td>.491</td><td></td><td>.923</td></tr><tr><td>Male (n = 41)</td><td>0.83 ± 0.05</td><td></td><td>1.49 ± 0.91</td><td></td></tr><tr><td>Female (n = 17)</td><td>0.82 ± 0.08</td><td></td><td>1.47 ± 0.62</td><td></td></tr><tr><td>Age</td><td></td><td>.489</td><td></td><td>.634</td></tr><tr><td>≤46 (n = 33)</td><td>0.82 ± 0.06</td><td></td><td>1.53 ± 1.03</td><td></td></tr><tr><td>>46 (n = 25)</td><td>0.83 ± 0.05</td><td></td><td>1.42 ± 0.45</td><td></td></tr><tr><td colspan=\"5\">GTVnd</td></tr><tr><td>Total patients</td><td>0.80 ± 0.11</td><td>NA</td><td>2.28 ± 2.66</td><td>NA</td></tr><tr><td>N category</td><td></td><td>.807</td><td></td><td>.002</td></tr><tr><td>N1-2 (n = 40)</td><td>0.80 ± 0.12</td><td></td><td>1.56 ± 1.30</td><td></td></tr><tr><td>N3 (n = 18)</td><td>0.79 ± 0.09</td><td></td><td>3.86 ± 4.04</td><td></td></tr><tr><td>Sex</td><td></td><td>.651</td><td></td><td>.308</td></tr><tr><td>Male (n = 41)</td><td>0.79 ± 0.11</td><td></td><td>2.51 ± 3.01</td><td></td></tr><tr><td>Female (n = 17)</td><td>0.81 ± 0.10</td><td></td><td>1.71 ± 1.56</td><td></td></tr><tr><td>Age</td><td></td><td>.264</td><td></td><td>.300</td></tr><tr><td>≤46 (n = 33)</td><td>0.81 ± 0.09</td><td></td><td>2.59 ± 3.22</td><td></td></tr><tr><td>>46 (n = 25)</td><td>0.78 ± 0.13</td><td></td><td>1.85 ± 1.72</td><td></td></tr><tr><td colspan=\"5\">Abbreviations: ASD = average surface distance; DSC = Dice similarity coefficient; GTV = gross tumor volume; GTVnd = node gross tumor volume; GTVnx = nasopharynx gross tumor volume; NA = not applicable; SD = standard deviation. The median ASD of GTVnx was 1.30 mm (range, 0.66-6.43), and the median ASD of GTVnd was 1.21 mm (range, 0.32-14.64).</td></tr></table>","table_caption":"[\"Table 1 Accuracy comparison for model-generated GTV in different subgroups in the testing cohort \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[55,623,479,654]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"score did not differ between groups of different ages or sexes (Table 1). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,655,479,763]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Given that patients with N3 tended to have more or more widespread metastatic lymph nodes, particular attention was paid to the accuracy of our model in this patient group. In the testing cohort, 18 patients with N3 category had lower neck involvement, but missing segmentation for lower neck lymph nodes occurred in 4 $(22.2\\%)$ patients (Fig. E4a). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,763,480,949]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To analyze the accuracy of the model in different slices of primary tumor, we extracted 7 slices from the beginning slice to the ending slice of the GTVnx continuously and equidistantly in the head-to-foot direction. Before doing this, we averaged 7 sampled slices for each patient in the axial direction and measured the result in each slice; slices 1 and 7 were the first and last of the ground truth in the axial direction. The total slice was 7, as we counted the ground truth of this data set and found that each scan has more than 14 slices with GTVnx annotation in the axial direction. The result showed that the DSC scores at the beginning and ending slices were $0.46 \\pm 0.24$ and $0.47 \\pm 0.26$ , respectively, which ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,623,937,686]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"were significantly lower than other intermediate slices $(P < .001)$ (Fig. E4b). DSC scores of the second, third, fourth, fifth, and sixth slices were $0.84 \\pm 0.09$ , $0.87 \\pm 0.07$ , $0.86 \\pm 0.08$ , $0.83 \\pm 0.14$ , and $0.79 \\pm 0.14$ , respectively (Fig. E4b). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,712,878,732]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Clinical assessment by multicenter experts ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[510,747,934,840]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For GTVnx, the mean revision degree was lower than $10\\%$ in 19 $(95\\%)$ patients (Fig. 3a). In addition, there was no significant difference in the average revision degree of the experts among patients with T1, T2, T3, or T4: $5.29\\% \\pm 3.83\\%$ versus $5.22\\% \\pm 3.30\\%$ versus $5.47\\% \\pm 5.83\\%$ versus $3.82\\% \\pm 3.61\\%$ $(P = .235)$ (Fig. 3b). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,840,935,949]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For GTVnd, the mean revision degree was lower than $5\\%$ in 15 $(75\\%)$ patients. The mean revision degree was between $5\\%$ and $10\\%$ in 1 $(5\\%)$ patient, between $10\\%$ and $20\\%$ in 2 $(10\\%)$ patients, and more than $20\\%$ in 2 $(10\\%)$ patients (Fig. 3c). In addition, among patients with different N categories, the mean degree of revision was lowest in patients with N3 (N1 vs N2 vs N3: $8.78\\% \\pm 12.71\\%$ vs $7.76\\% \\pm 10.13\\%$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[57,38,277,53]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Volume 113 • Number 4 • 2022 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[645,38,890,53]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Automatic delineation of GTV in NPC ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[904,38,934,53]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"897 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[102,64,897,329]","code_body":null,"code_caption":null,"image_caption":"[\"Fig. 2. Visual comparison of the model-generated contours and the ground truth contours in different T categories (A) and N categories (B). Yellow lines denote the expert-delineated ground truth, and green lines denote the model-generated contours. There is a high level of concordance between expert- and model-generated contours. \"]","image_footnote":"[]","img_path":"images/0318/1311f8b9647e715fb97980ac0ea3ccd27bd20de31bc476a2ff0b35a446d064a7.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[102,421,708,604]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0239/2f8da6661c3822ef639eed4db6f10b9eada6cba29a7eeb6581da2ba42540db0d.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[722,468,895,575]","code_body":null,"code_caption":null,"image_caption":"[\"(B) \"]","image_footnote":"[]","img_path":"images/0087/9a9529f7ecab1cd7945d8b1e2b79d4ed78afb56aed1a59efa9b0f6c4ee732941.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[102,605,713,760]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0886/96380eda34e71a97d382f73f1bd42502f9ff6e9ed8078b2e75d13b336266b3f3.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[722,646,895,750]","code_body":null,"code_caption":null,"image_caption":"[\"(D) \",\"Fig. 3. Evaluation of model-generated GTV by multicenter experts. (A) The revision degrees of model-generated GTVnx by multicenter experts. E1 to E9 represents 9 experts; P01 to P20 represents 20 patients randomly selected from the testing cohort. The height of each column represents the revision degree (\\\\%) by each expert (the lower revision degree indicates better overlap between model- and expert-generated contours). (B) Comparison of revision degrees of GTVnx in different T categories (5 patients in each category). The model-generated GTVnx from each patient was revised by these 9 experts. (C) The revision degrees of model-generated GTVnd by multicenter experts. (D) Comparison of revision degrees of GTVnd in different N categories. There were 7, 8, and 5 patients in the N1, N2, and N3 categories, respectively. The model-generated GTVnx from each patient was revised by these 9 experts. The revision degrees of GTVnd in N3 patients were significantly lower than those of GTVnd in N1 and N2 patients. $P$ value was obtained by analysis of variance. Abbreviations: $\\\\mathrm{GTV} =$ gross tumor volume; $\\\\mathrm{GTVnd} =$ node gross tumor volume; $\\\\mathrm{GTVnx} =$ nasopharynx gross tumor volume. \"]","image_footnote":"[]","img_path":"images/0479/de7ea9dfb9a4916df182b52b2c63760d5ebb831fd29e2bc71c68a0160ab70c26.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[62,40,93,53]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"898 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[107,40,173,53]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Liao et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[520,40,937,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"International Journal of Radiation Oncology $\\cdot$ Biology $\\cdot$ Physics ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[55,68,480,100]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"vs $2.56\\% \\pm 2.99\\%$ ; $P = .004$ (Fig. 3d). The visual display of GTVnx modified by 9 experts is presented in Figure E5. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,117,432,168]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Comparison between model-generated and multicenter junior radiation oncologists-generated GTVs ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[55,186,479,324]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The average DSC score of the model-generated GTVnx was $0.85 \\pm 0.03$ (Fig. 4a), while the average DSC scores of GTVnx manually delineated by R1 (junior radiation oncologist 1), R2, R3, R4, R5, and R6 were $0.75 \\pm 0.10$ , $0.70 \\pm 0.12$ , $0.80 \\pm 0.05$ , $0.75 \\pm 0.05$ , $0.69 \\pm 0.10$ , and $0.78 \\pm 0.04$ , respectively. The DSC score of GTVnx generated by the model was significantly higher than that of GTVnx delineated by each junior radiation oncologist (all $P < .05$ ) (Fig. 4a). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[55,324,480,434]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In terms of GTVnd, the average DSC score of the model-generated GTVnd was $0.84 \\pm 0.12$ (Fig. 4b), while the DSC scores of R1, R2, R3, R4, R5, and R6 were $0.80 \\pm 0.15$ , $0.79 \\pm 0.16$ , $0.82 \\pm 0.11$ , $0.78 \\pm 0.15$ , $0.79 \\pm 0.12$ , and $0.74 \\pm 0.11$ , respectively. The DSC score of GTVnd generated by the model was higher than those of GTVnd delineated by 3 junior oncologists (R4, R5, and R6) ( $P < .05$ ). The DSC ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,68,937,113]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"score of the model was still higher than those of another 3 junior radiation oncologists (Fig. 4b), though without significant difference. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[512,134,864,168]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Clinical application by multicenter junior radiation oncologists ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[510,186,935,434]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Having proved that the present model could perform better than junior radiation oncologists, we next investigated whether the model could help them to work. Therefore, we compared the revision degrees of GTV delineation by junior radiation oncologists under 2 modes. The result showed that with the aid of the model, the revision degrees of GTVnx became smaller than those without it among all junior radiation oncologists (all $P < .05$ ) (Table 2). Similarly, with the aid of the model, the revision degrees of GTVnd were much smaller compared with those without it among the 5 junior radiation oncologists ( $P < .05$ ), except R3 ( $P = .191$ ) (Table 2). The representative visual display of GTVnx delineated by 6 junior radiation oncologists under 2 modes is presented in Figure 4c. The assistance of the model increased the accuracy in recognizing the invaded slope, thus increasing the accuracy of GTVnx delineation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[107,463,336,598]","code_body":null,"code_caption":null,"image_caption":"[\"(A) \"]","image_footnote":"[]","img_path":"images/0821/cd5073f104f56c2a17ddaf34860d7ad7028d07c044505e060b6361c79e0e6f66.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[359,474,548,613]","code_body":null,"code_caption":null,"image_caption":"[\"(C) \"]","image_footnote":"[]","img_path":"images/0866/6cb4011c7b93e47c4b382b958a638203613199d90a48b532ac3c97a4bf0706c5.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[555,474,722,613]","code_body":null,"code_caption":null,"image_caption":"[\"Prediction \"]","image_footnote":"[]","img_path":"images/0100/e9dfa262e8adb0508a0976a4ed79795b29167678dbc1e6e8c648b4a20259bdc4.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[725,474,895,613]","code_body":null,"code_caption":null,"image_caption":"[\"Ground truth \"]","image_footnote":"[]","img_path":"images/0922/0a52a3ab5a3df6fdbf4355e59ea90ce9d7a50195fbb99385088fecaf1965bb4f.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[107,622,335,745]","code_body":null,"code_caption":null,"image_caption":"[\"(B) \"]","image_footnote":"[]","img_path":"images/0525/c8aec77e8b894be0bde6195f16d7f48d1dca4e25a1950178b51d1838b3e213f9.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[359,618,550,755]","code_body":null,"code_caption":null,"image_caption":"[\"Axial \"]","image_footnote":"[]","img_path":"images/0363/d14b2d4a83d0517de96edd58401166a392f67502f97c00af8baec74d4f849592.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[555,618,722,756]","code_body":null,"code_caption":null,"image_caption":"[\"Coronal \"]","image_footnote":"[]","img_path":"images/0909/fa52e7bd56d1437c73218eab1a3aad80bb2dde4d79480f291caf0a61ab06ea5a.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[725,618,895,756]","code_body":null,"code_caption":null,"image_caption":"[\"Sagittal \",\"Fig. 4. GTV delineation by junior radiation oncologists with and without model assistance. (A) The DSC score of model-generated GTVnx was significantly higher than those of junior radiation oncologist-generated GTVnx. R1 to R6 represent 6 junior radiation oncologists. The values below the Y axis were DSC scores from junior radiation oncologists or the model. (B) The DSC score of model-generated GTVnd was significantly higher than 3 of 6 junior radiation oncologist-generated GTVnd. $P$ values were obtained by paired $t$ test, $^* < .05$ , $^{**} < .01$ , $^{***} < .001$ in (A) and (B). (C) In the upper column, 4 of 6 junior radiation oncologists failed to accurately delineate the invaded slope (the red arrow) without model assistance. In the lower column, only 1 junior radiation oncologist failed to delineate the invaded slope, and the shape of GTVnx contours also tended to be more consistent with the aid of our model. Yellow lines denote the expert-delineated ground truth, green lines denote the model-generated contours, and other color lines denote junior radiation oncologist-generated contours. Some lines overlapped so they were difficult to distinguish. Abbreviations: DSC = Dice similarity coefficient; GTV = gross tumor volume; GTVnd = node gross tumor volume; GTVnx = nasopharynx gross tumor volume. \"]","image_footnote":"[]","img_path":"images/0571/2448ca8da6aa4755abf1e68850a16d3439a476ea9369c3ac07e8428d81f1373d.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[57,40,277,51]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Volume 113 • Number 4 • 2022 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[645,38,890,53]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Automatic delineation of GTV in NPC ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[904,40,934,51]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"899 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[60,87,935,244]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0019/336b7a3163f2ef5712a43e362a6535d437fa6add429ef35ba9068ca0ab7205de.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":"<table><tr><td rowspan=\"2\">Junior radiation oncologist</td><td colspan=\"3\">GTVnx revision degree, %</td><td colspan=\"3\">GTVnd revision degree, %</td></tr><tr><td>Without model</td><td>With model</td><td>P value</td><td>Without model</td><td>With model</td><td>P value</td></tr><tr><td>R1</td><td>25.45 ± 10.46</td><td>14.69 ± 4.68</td><td><.001</td><td>20.06 ± 14.54</td><td>14.55 ± 13.48</td><td>.008</td></tr><tr><td>R2</td><td>30.15 ± 11.92</td><td>21.36 ± 6.84</td><td>.025</td><td>20.96 ± 15.81</td><td>16.37 ± 13.04</td><td>.035</td></tr><tr><td>R3</td><td>20.41 ± 4.72</td><td>17.44 ± 4.77</td><td>.011</td><td>18.05 ± 11.23</td><td>17.34 ± 11.34</td><td>.191</td></tr><tr><td>R4</td><td>25.41 ± 5.39</td><td>15.71 ± 4.31</td><td><.001</td><td>22.05 ± 15.46</td><td>14.95 ± 11.32</td><td><.001</td></tr><tr><td>R5</td><td>30.74 ± 9.66</td><td>15.72 ± 4.18</td><td><.001</td><td>21.46 ± 12.30</td><td>15.27 ± 10.11</td><td>.003</td></tr><tr><td>R6</td><td>21.61 ± 4.36</td><td>6.11 ± 3.08</td><td><.001</td><td>25.68 ± 11.94</td><td>8.17 ± 7.30</td><td><.001</td></tr></table>","table_caption":"[\"Table 2 Comparison of revision degrees under 2 modes of GTV delineation by junior radiation oncologists \"]","table_footnote":"[\"Abbreviations: GTV = gross tumor volume; GTVnd = node gross tumor volume; GTVnx = nasopharynx gross tumor volume. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[60,300,480,424]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"At last, we analyzed the time spent by the oncologists in delineating GTV. Without assistance from the model, R1 spent an average of 39.8 minutes/case. By contrast, when assisted by the model, the delineation time was substantially reduced to an average value of 14.0 minutes/case, greatly reducing time by $64.8\\%$ . Similarly, the delineation time was reduced by $70.1\\%$ , $61.7\\%$ , $78.3\\%$ , $61.6\\%$ , and $65.9\\%$ for R2, R3, R4, R5, and R6, respectively (all $P < .05$ ) (Table E4). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[60,448,164,465]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Discussion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[60,485,480,716]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In this study, we evaluated the accuracy and application of a new SSL model from the clinical point of view. We first compared the segmentation performance of the model in different subgroups of patients, proving that the model could achieve comparable and high accuracy in delineating GTV for patients with different clinical characteristics. Second, the multicenter experts were invited to evaluate the automatic segmentation results of GTV, further demonstrating that the model-generated contours were clinically acceptable with low revision degrees. Third, the new model performed better than junior radiation oncologists in delineating GTV. With the assistance of the model, contouring accuracy was improved, interobserver variation was reduced, and contouring time was also significantly decreased. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[60,717,480,840]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Advanced radiation techniques such as intensity modulated radiation therapy have considerably improved local control rate of NPC. $^{1,2}$ This clinical benefit is intricately associated with contouring accuracy and precise dose delivery. $^{19}$ However, the processes of GTV delineation are labor-intensive and heterogeneous. $^{20}$ In this context, automatic target volume contouring based on deep learning provides a promising alternative to solve these issues. $^{21,22}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[60,840,480,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In a previous study, we designed a novel segmentation model based on SSL and demonstrated that this method was efficient and precise for GTV contouring of NPC from an algorithmic perspective. In this study, we further examined the performance of the model from the clinical perspective. First of all, in the subgroup analysis, there was no difference in DSC scores and relative volume errors (Supplementary ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[513,300,937,516]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Materials and Table E5) between different T categories or N categories, suggesting that stage or tumor size had little effect on segmentation accuracy of our model. With the increase of T and N, the GTVnd and GTVnx tended to become large. Theoretically, the segmentation network might perform better for targets with larger volume, and it was more difficult to recognize and delineate ones with smaller volume. That's why many recently proposed models have poor segmentation for the optic chiasm,[10,23] a quite small target organ. However, when examining the correlation between tumor size and DSC score or relative volume errors (Supplementary Materials and Fig. E6), we found that both GTVnx and GTVnd had little effect on segmentation accuracy, indicating that the SSL model performed well regardless of tumor volume. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[513,517,937,902]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"However, in terms of ASD, the ASD sore of patients with T3-4 was higher than that of patients with T1-2, and the GTVnx was also positively associated with ASD score. The main reason might be that the tumor shape and margin were more irregular in more advanced NPC. As indicated in its definition, ASD is quite sensitive to outliers, and it is much more challenging to quantify a model's performance in tumors with complex shape and irregular margin.[23] Hence, it was reasonable that the more irregular the tumor, the larger the ASD score. Similar results were reported in other studies.[11] Although the ASD score of GTVnx in T3-4 patients was higher, the mean value of the ASD score (1.64 mm) was much less than the commonly accepted 3-mm margin of systematic and random error in radiation therapy of head and neck cancers,[24] suggesting that the GTVnx predicted by our model was still in high concordance with the ground truth. Unfortunately, the ASD score was more than 3 mm in certain N3 patients, and missing segmentation for lower metastatic neck lymph nodes occurred in 4 N3 patients, in which the missing lymph nodes were mainly located in the level Vb according to the lymph node distribution of NPC.[25] It is suggested that in the future, we need to optimize the algorithm or increase the proportion of patients with lymph nodes located in the level Vb in the training cohort to increase the accuracy of GTVnd contouring in N3 patients. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[513,902,937,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"By examining the DSC score in different GTVnx slices, we found that the accuracy was significantly less at the beginning and ending slices than the other intermediate ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[62,41,93,51]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"900 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[107,41,171,51]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Liao et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[520,41,937,54]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"International Journal of Radiation Oncology $\\cdot$ Biology $\\cdot$ Physics ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[60,69,475,175]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"slices, which was in accord with our common sense. Even the manual delineation of GTVnx often varies greatly at the beginning and ending slices. Therefore, it was acceptable that the GTVnx predicted by the model would also be interfered by the location of slices. This result also suggested a strong concordance between our model and the experts performing GTV delineation manually. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[60,178,475,346]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To further validate this new model, 9 experts from 3 cancer centers were invited to modify the model-generated GTV contours. The mean revision degree was lower than $10\\%$ in 19 $(95\\%)$ patients for GTVnx and in 16 $(80\\%)$ patients for GTVnd, demonstrating that the model-generated contours were highly acceptable, with slight or no revision by these experts. Similar results were also reported in a study in which the average revision degree for GTVnx was less than $20\\%$ in $88.4\\%$ patients. Taken together, this suggests that a deep learning-based contouring model is a powerful tool to achieve accurate target volume delineation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[60,347,475,639]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Given that the targets were often delineated by the junior radiation oncologist first and then modified by the experts, especially in academic institutions, we were concerned how and to what extent the model could help the junior radiation oncologist to work. First, our model outperformed all junior radiation oncologists in delineating GTVnx and GTVnd, which is consistent with the results of another study, in which their model outperformed 4 of 8 radiation oncologists.6 More importantly, with the assistance of our model, a noticeable improvement in the delineation quality of GTVnx and GTVnd was observed. This was reflected in the increase of the DSC and ASD scores. Moreover, with the aid of the model, delineating efficiency was substantially improved, saving the delineation time by as much as over $60\\%$ . Although the delineation differences among different individuals might have a certain effect on the analysis of these results, all of our results confirmed the great potential of our model to assist junior radiation oncologists in improving accuracy as well as saving time in manual delineation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[60,641,475,732]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Considering that the model-generated contours were not significantly superior to those produced by experienced radiation oncologists (6-10 years' experience, as shown in Supplementary Materials and Fig. E7), a comparison was not performed without and with the model assistance for the experienced radiation oncologists in the study. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[60,734,475,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We acknowledge that this study had some limitations. Although we established strict delineation standards and protocols, the ground truth contours were still delineated manually by experts from the same center, with potential subjective variations or even errors. However, it is a problem that also widely exists in other studies that cannot be conquered nowadays.[6,23] Hence, it is quite necessary to further improve the annotation quality and produce a widely acceptable GTV contour data set in the future. Second, the ground truth contours were mainly delineated based on MRI data sets, with which the properties (benign or malignant) of some anatomies, such as small or inflammatory lymph nodes, were difficult to determine. More imaging modalities, such as positron emission tomography/CT or dynamic ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[515,68,932,238]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"contrast-enhanced MRI, could be integrated into GTV delineation in the future. Third, both the building and validation of the deep learning were based on MRI because MRI has superior soft-tissue contrast. Although CT is the most commonly used in treatment planning, GTVs are often produced on MRI and propagated to planning CT by fusing these 2 images in practice. However, it might be more clinically relevant if validation could be performed both in an MRI data set and a CT data set. Additionally, the data set used was relatively small from a single center. As a next step, we plan to collect multicenter images to further validate our model. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,272,628,287]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conclusions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[515,307,932,491]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In summary, we first proposed a novel SSL-based deep learning model to delineate GTVnx and GTVnd in NPC. Different from previous fully SL-based models, the SSL-based model was capable of using limited labeled data to learn efficiently. With subgroup analysis and clinical assessment by multicenter experts, we demonstrated the robustness and accuracy of our model in GTV delineation. It was clinically applicable and could assist junior radiation oncologists in improving contouring accuracy and reducing contouring time. With further improvement of the algorithm, the model could be a promising way to achieve automatic and more accurate GTV delineation in NPC. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[517,527,616,542]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[515,562,932,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. Chen YP, Chan ATC, Le QT, et al. Nasopharyngeal carcinoma. Lancet. 2019;394:64-80. \",\"2. Sun XS, Liu SL, Luo MJ, et al. The association between the development of radiation therapy, image technology, and chemotherapy, and the survival of patients with nasopharyngeal carcinoma: A cohort study from 1990 to 2012. Int J Radiat Oncol Biol Phys 2019;105:581-590. \",\"3. Dubrulle F, Souillard R, Hermans R. Extension patterns of nasopharyngeal carcinoma. Eur Radiol 2007;17:2622-2630. \",\"4. Liang SB, Sun Y, Liu LZ, et al. Extension of local disease in nasopharyngeal carcinoma detected by magnetic resonance imaging: improvement of clinical target volume delineation. Int J Radiat Oncol Biol Phys 2009;75:742-750. \",\"5. Kosmin M, Ledsam J, Romera-Paredes B, et al. Rapid advances in autosegmentation of organs at risk and target volumes in head and neck cancer. Radiother Oncol 2019;135:130-140. \",\"6. Lin L, Dou Q, Jin YM, et al. Deep learning for automated contouring of primary tumor volumes by MRI for nasopharyngeal carcinoma. Radiology 2019;291:677-686. \",\"7. Brouwer CL, Steenbakkers RJ, van den Heuvel E, et al. 3D variation in delineation of head and neck organs at risk. Radiat Oncol 2012;7:32. \",\"8. Men K, Chen X, Zhang Y, et al. Deep deconvolutional neural network for target segmentation of nasopharyngeal cancer in planning computed tomography images. Front Oncol 2017;7:315. \",\"9. Xue X, Qin N, Hao X, et al. Sequential and iterative auto-segmentation of high-risk clinical target volume for radiotherapy of nasopharyngeal carcinoma in planning CT images. Front Oncol 2020;10:1134. \",\"10. Liang S, Tang F, Huang X, et al. Deep-learning-based detection and segmentation of organs at risk in nasopharyngeal carcinoma computed tomographic images for radiotherapy planning. Eur Radiol 2019; 29:1961-1967. \"]","page_idx":"8","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[59,41,276,51]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Volume 113 • Number 4 • 2022 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[649,41,888,51]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Automatic delineation of GTV in NPC ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[904,41,930,51]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"901 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[62,71,482,364]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"11. Li S, Xiao J, He L, et al. The tumor target segmentation of nasopharyngeal cancer in CT images based on deep learning methods. Technol Cancer Res Treat 2019;18:1533033819884561. \",\"12. Zhao L, Lu Z, Jiang J, et al. Automatic nasopharyngeal carcinoma segmentation using fully convolutional networks with auxiliary paths on dual-modality PET-CT images. J Digit Imaging 2019;32:462-470. \",\"13. Luo X, Liao W, Chen J, et al. Efficient Semi-Supervised Gross Target Volume of Nasopharyngeal Carcinoma Segmentation via Uncertainty Rectified Pyramid Consistency[J]. Medical Image Computing and Computer Assisted Intervention (MICCAI) 2021;318-329. \",\"14. Hodapp N. The ICRU report 83: Prescribing, recording and reporting photon-beam intensity-modulated radiation therapy (IMRT). *Strah-lenther Onkol* 2012;188:97-99. \",\"15. van den Brekel MW, Stel HV, Castelijns JA, et al. Cervical lymph node metastasis: Assessment of radiologic criteria. Radiology 1990;177:379-384. \",\"16. Yushkevich PA, Piven J, Hazlett HC, et al. User-guided 3D active contour segmentation of anatomical structures: Significantly improved efficiency and reliability. Neuroimage 2006;31:1116-1128. \",\"17. Carillo V, Cozzarini C, Perna L, et al. Contouring variability of the penile bulb on CT images: Quantitative assessment using a generalized concordance index. Int J Radiat Oncol Biol Phys 2012; 84:841-846. \"]","page_idx":"9","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[519,69,937,364]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"18. Yousefi S, Kehtarnavaz N, Gholipour A. Improved labeling of subcortical brain structures in atlas-based segmentation of magnetic resonance images. IEEE Trans Biomed Eng 2012;59:1808-1817. \",\"19. Jeanneret-Sozzi W, Moeckli R, Valley JF, et al. The reasons for discrepancies in target volume delineation: A SASRO study on head-and-neck and prostate cancers. Strahlenther Onkol 2006;182:450-457. \",\"20. Teguh DN, Levendag PC, Voet PW, et al. Clinical validation of atlas-based auto-segmentation of multiple target volumes and normal tissue (swallowing/mastication) structures in the head and neck. Int J Radiat Oncol Biol Phys 2011;81:950-957. \",\"21. Chang Z. Will AI improve tumor delineation accuracy for radiation therapy? Radiology 2019;291:687-688. \",\"22. Deig CR, Kanwar A, Thompson RF. Artificial intelligence in radiation oncology. Hematol Oncol Clin North Am 2019;33:1095-1104. \",\"23. Tang H, Chen X, Liu Y, et al. Clinically applicable deep learning framework for organs at risk delineation in CT images. Nat Mach Intell 2019;1:480-491. \",\"24. Lee N, Lu JJ. Target Volume Delineation and Field Setup: A Practical Guide for Conformal and Intensity-Modulated Radiation Therapy 2013. \",\"25. Lin L, Lu Y, Wang XJ, et al. Delineation of neck clinical target volume specific to nasopharyngeal carcinoma based on lymph node distribution and the international consensus guidelines. Int J Radiat Oncol Biol Phys 2018;100:891-902. \"]","page_idx":"9","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[62,40,93,53]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"902 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[107,40,173,53]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Liao et al. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[520,40,937,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"International Journal of Radiation Oncology $\\cdot$ Biology $\\cdot$ Physics ","text_format":null,"text_level":null,"type":"header"}],"string":"[\n {\n \"bbox\": \"[57,153,328,174]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PHYSICS CONTRIBUTION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,190,779,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Automatic Delineation of Gross Tumor Volume Based on Magnetic Resonance Imaging by Performing a Novel Semisupervised Learning Framework in Nasopharyngeal Carcinoma \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[881,192,934,232]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0966/7c230d13ac0fe027320d5435e9f3fd4884fe80476eabf2513ba9cfb9e0f424ea.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[55,319,692,350]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Wenjun Liao, MD,\\\\*,† Jinlan He, MD,\\\\* Xiangde Luo, PhD,\\\\* Mengwan Wu, MD,\\\\* Yuanyuan Shen, MD,\\\\* Churong Li, MD,\\\\* Jianghong Xiao, PhD,\\\\* Guotai Wang, PhD,\\\\* and Nianyong Chen, MD, PhD\\\\* \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,360,786,441]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$^{*}$ Department of Radiation Oncology, Cancer Center and State Key Laboratory of Biotherapy, West China Hospital, Sichuan University, Chengdu, China; $^{\\\\dagger}$ Department of Radiation Oncology, Nanfang Hospital, Southern Medical University, Guangzhou, China; $^{\\\\ddagger}$ School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu, China; $^{\\\\S}$ Cancer Clinical Research Center, Sichuan Cancer Hospital & Institute, Sichuan Cancer Center, School of Medicine, University of Electronic Science and Technology of China, Chengdu, China; and $^{\\\\parallel}$ Department of Radiation Oncology, Sichuan Cancer Hospital & Institute, Sichuan Cancer Center, School of Medicine, University of Electronic Science and Technology of China, Chengdu, China \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,451,446,465]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Received Oct 28, 2021; Accepted for publication Mar 24, 2022 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,496,935,527]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Purpose: We aimed to validate the accuracy and clinical value of a novel semisupervised learning framework for gross tumor volume (GTV) delineation in nasopharyngeal carcinoma. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,527,935,650]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Methods and Materials: Two hundred fifty-eight patients with magnetic resonance imaging data sets were divided into training $(n = 180)$ , validation $(n = 20)$ , and testing $(n = 58)$ cohorts. Ground truth contours of nasopharynx GTV (GTVnx) and node GTV (GTVnd) were manually delineated by 2 experienced radiation oncologists. Twenty percent $(n = 36)$ labeled and $80\\\\%$ $(n = 144)$ unlabeled images were used to train the model, producing model-generated contours for patients from the testing cohort. Nine experienced experts were invited to revise model-generated GTV in 20 randomly selected patients from the testing cohort. Six junior oncologists were asked to delineate GTV in 12 randomly selected patients from the testing cohort without and with the assistance of the model, and revision degrees were compared under these 2 modes. The Dice similarity coefficient (DSC) was used to quantify the accuracy of the model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,650,935,745]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Results: The model-generated contours showed a high accuracy compared with ground truth contours, with an average DSC score of 0.83 and 0.80 for GTVnx and GTVnd, respectively. There was no significant difference in DSC score between T1-2 and T3-4 patients (0.81 vs 0.83; $P = .223$ ), or between N1-2 and N3 patients (0.80 vs 0.79; $P = .807$ ). The mean revision degree was lower than $10\\\\%$ in 19 $(95\\\\%)$ patients for GTVnx and in 16 $(80\\\\%)$ patients for GTVnd. With assistance of the model, the mean revision degree for GTVnx and GTVnd by junior oncologists was reduced from $25.63\\\\%$ to $7.75\\\\%$ and from $21.38\\\\%$ to $14.44\\\\%$ , respectively. Meanwhile, the delineating efficiency was improved by over $60\\\\%$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,817,477,842]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Corresponding author: Nianyong Chen, MD, PhD; E-mail: n_ychen@ hotmail.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,843,453,868]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Wenjun Liao and Jinlan He made equal contributions to this study. \\nDisclosures: none. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,869,477,895]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data sharing statement: The code for neural network training can be obtained at: https://github.com/HiLab-git/SSL4MIS. Other data generated \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,817,932,843]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"and analyzed during this study can be obtained by contacting the corresponding author with reasonable requirements. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,843,932,869]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Supplementary material associated with this article can be found in the online version at doi:10.1016/j.ijrobp.2022.03.031. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[638,91,935,115]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"INTERNATIONAL JOURNAL OF RADIATION ONCOLOGY $\\\\cdot$ BIOLOGY $\\\\cdot$ PHYSICS \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[819,122,934,135]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"www.redjournal.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[55,910,435,923]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Int J Radiation Oncol Biol Phys, Vol. 113, No. 4, pp. 893-902, 2022 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[55,923,845,936]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"0360-3016/$ - see front matter © 2022 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[57,936,352,949]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(http://creativecommons.org/licenses/by-nc-nd/4.0/) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[57,949,298,962]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"https://doi.org/10.1016/j.ijrobp.2022.03.031 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[65,64,935,125]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conclusions: The proposed semisupervised learning-based model showed a high accuracy for delineating GTV of nasopharyngeal carcinoma. It was clinically applicable and could assist junior oncologists to improve GTV contouring accuracy and save contouring time. © 2022 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[64,170,178,186]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[64,206,480,455]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Nasopharyngeal carcinoma (NPC) is one of the most common malignancies in head and neck, and radiation therapy has been the primary effective treatment modality for NPC. $^{1,2}$ Precise delineation of gross target volume (GTV) is substantially important in the process of radiation therapy, especially in the era of intensity modulated radiation therapy. However, GTV contouring for NPC is quite challenging compared with other solid tumors owing to its complex anatomic structures and diverse tumor invasion pathways. $^{3,4}$ The radiation oncologist needs to review multiple types of medical images to determine and delineate lesions slice by slice on images. Furthermore, the margin of tumor is usually hard to determine, which sometimes inevitably depends on personal clinical experience. Therefore, the current manual GTV delineation is time-consuming, laborious, error-prone, $^{5}$ and subject to interobserver and intraobserver heterogeneity. $^{6,7}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[64,455,480,485]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"These factors could affect the accuracy of GTV contouring and may further cause treatment failures in NPC. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[64,486,480,900]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recently, a variety of deep learning models have been proposed to automatically delineate GTV and organs at risk in NPC, such as U-Net networks, and many of them have achieved state-of-the-art performance, thus providing an alternative for manual GTV delineation. $^{8-12}$ However, several limitations exist in these models. The first concern is that previous automatic segmentation models for GTV of NPC were all based on fully supervised learning (SL), which required a large number of manually labeled data for model training. $^{6,8,11}$ For example, manually labeled magnetic resonance imaging (MRI) data sets from more than 700 patients with NPC were used to train the neural network to construct an automatic nasopharynx GTV (GTVnx) segmentation model. $^{6}$ However, it is quite difficult to obtain a large amount of high-quality manually labeled images in clinical practice, as both expertise and time are needed to produce such accurate annotations. Therefore, we have proposed a novel semisupervised learning (SSL) method via an uncertainty rectified pyramid consistency framework for GTV delineation of NPC, $^{13}$ which uses both a small amount of manually labeled data and a large amount of unlabeled data for model development. By using limited (10% or 20%) labeled data to train the uncertainty rectified pyramid consistency network, our model achieved an average Dice score of 0.78 or 0.81 in the testing cohort, which was of comparable performance to that of state-of-the-art models by using full SL. $^{13}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[64,902,480,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Another concern of the deep learning models is the acceptability, utility, and actual benefit in the clinical setting, which is especially critical for medical image \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[520,170,935,307]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"segmentation. However, few studies have reported the clinical value of SL or SSL models, raising our interest in addressing the clinical value of the new SSL model. To achieve this aim, we examined the segmentation performance in different subgroups of patients, invited multicenter experts to evaluate the automatic segmentation results, and asked junior radiation oncologists to use the new model, thus clarifying the clinical acceptance and application of the novel model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[520,351,732,367]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Methods and Materials \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[520,389,875,419]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data collection and model-generated GTV contours \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[520,439,935,608]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The NPC data sets $(n = 258)$ used were described in our previous study. The inclusion and exclusion criteria were the following: (1) patients who were histologically confirmed as NPC; (2) patients who underwent MRI for nasopharynx and neck before anticancer treatment; (3) the MRI sets included unenhanced T1- and T2-weighted, contrast-enhanced T1-weighted sequences; and (4) the median thickness of the MRI was $1.10\\\\mathrm{mm}$ (range, $1.00 - 1.25\\\\mathrm{mm}$ ). The patients were excluded if their images had a low resolution that affected delineation of tumor volume. Details of MRI acquisition are shown in Tables E1 and E2. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[520,610,935,840]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The whole data sets were randomly split into 3 cohorts with a ratio of 7:1:2: a training cohort with 180 patients for model training, a validation cohort with 20 patients for optimizing the model hyperparameters, and a testing cohort with 58 patients to test the performance of the model. In our previous study,[13] we examined the effect of the proportion of labeled data on the performance of our network (Fig. E1). We found that the segmentation accuracy of the network was positively correlated with the proportion of labeled data overall. Therefore, to achieve a balance between the proportion of labeled data and the performance of the network, $20\\\\%$ $(n = 36)$ labeled and $80\\\\%$ $(n = 144)$ unlabeled images in the training cohort $(n = 180)$ were selected for network training, thus obtaining model-generated GTV contours for patients in the testing cohort. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[520,842,935,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"All patients in this study were restaged according to the eighth edition of the American Joint Committee on Cancer, and characteristics of these patients are presented in Table E3. There was no significant difference in sex, age, T category, N category, histology, or tumor characteristics between the combined training and validation cohorts and the testing cohort. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,41,90,51]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"894 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[107,41,171,51]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Liao et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[520,41,935,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"International Journal of Radiation Oncology $\\\\cdot$ Biology $\\\\cdot$ Physics \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[60,69,475,113]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This study was approved by the ethics committee on biomedical research of the hospital and the requirement for informed consent was waived. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,143,194,159]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"GTV contouring \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,177,475,408]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Two radiation oncologists from our center with more than 10 years of experience in treating patients with NPC were invited to delineate ground truth contours, including GTVnx and node GTV (GTVnd). A third radiation expert with more than 20 years of experience was consulted in cases of disagreement. GTVs (GTVnx and GTVnd) were delineated according to the International Commission on Radiation Units and Measurements report 83,[14] and the positive cervical metastatic lymph node was determined according to the published radiologic criteria.[15] Additionally, to ensure the most precise GTV contours, experts could also refer to other images during contouring when necessary, such as enhanced computed tomography (CT), cervical ultrasound, or positron emission tomography-CT. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,408,475,454]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"All manually delineated contours on MRIs in this study were conducted in the ITK-SNAP, an open-source software for image segmentation.[16] \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,483,442,499]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Network architecture and its implementation \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,516,475,809]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The detailed network architecture is shown in Figure E2. We designed a multiscale prediction 3-dimensional network for semisupervised segmentation GTV of NPC, which took images with a shape of $112^{3}$ as inputs and generated 4 outputs at 4 different scales (the shapes of output 1, 2, 3, 4 were $112^{3}, 56^{3}, 28^{3}, 14^{3}$ , respectively) for each input. For labeled images, this network was supervised by expert annotated labels directly, and a combination loss between cross-entropy loss and Dice loss was used as the supervised loss. A multiscale consistency (unsupervised loss) was used to optimize the network when the inputs were without annotations, where we rescaled all outputs to a fixed resolution $(112^{3})$ and calculated the mean square error between each rescaled output and the average rescaled output. Finally, the semisupervised framework was optimized by a joint loss of the supervised loss and the unsupervised loss. Different from previous deep supervised networks, this framework could use both labeled images and unlabeled images at the same time and lead to better performance. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,811,475,949]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In image preprocessing, we first converted the orientation of all images into the standard RAI (right to left, anterior to posterior, inferior to superior in the x-, y-, and z-axes, respectively). The intensity window was clipped at the 0.5 and 99.5 percentiles of the volume voxels and then normalized to zero mean and unit variance. Considering the thickness of images in this data set was very similar, we did not use any spatial interpolation processing to resample the images. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,69,932,192]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In addition, we deposited the code of training, validation, and testing in https:// github.com/HiLab-git/SSL4MIS. Detailed descriptions and the training and inference settings of the proposed network could also be acquired from our previous study. $^{13}$ In this study, we focused on evaluating the potential of the semisupervised delineation method in clinical practice and investigated the efficiency and robustness in a large multicenter users' study. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,212,925,243]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Assessment of the model-generated contours by multicenter experts \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,261,932,584]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Nine radiation oncologists (E1, E2, E3...E9) with more than 10 years' experience in treating NPC constituted the multicenter group, with 3 experts from each of the 3 tertiary hospitals (West China Hospital, Sichuan Cancer Hospital, and Sichuan Provincial People's Hospital). Twenty patients with NPC patients (5 patients in each T category) with the model-generated contours were randomly selected from the testing cohort. Then, each expert needed to revise the incorrect delineations of the GTV contours of all 20 patients. During the revision process, experts were blinded to both the ground truth contours and contours delineated by their counterparts. Moreover, to maximize homogeneity of delineation, the experts were provided with the same image data of each patient and referred to the same GTV contouring principles described previously. Finally, the model-generated GTV contours were compared with their corresponding revised contours to calculate the revision degree (the difference of volumetric overlap) through the formula $(1 - \\\\text{Dice similarity coefficient} [\\\\text{DSC}]) \\\\times \\\\text{percentage}$ . The flowchart is illustrated in Figure 1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,605,911,636]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The model-assisted junior radiation oncologist GTV delineations \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,655,932,949]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Six junior radiation oncologists (R1, R2, R3, . . ., R6) with 1 to 2 years of experience from the same centers (2 oncologists in each center) delineated GTVs of 12 patients randomly selected from the testing cohort in 2 modes. In the first mode without model assistance, the delineation was conducted completely manually from scratch, while in the second mode with model assistance, the oncologists just needed to verify the model-generated GTV and modify the inappropriate delineations when necessary. During the delineation process, these junior radiation oncologists were blinded to contours delineated by their counterparts. Moreover, there was a 4-week interval between 2 modes to reduce the effect of delineation results from the former mode on the latter one. Meanwhile, the delineation time for GTV in each case was recorded. To record the delineation time of junior radiation oncologists more accurately, all junior radiation oncologists were trained to use the contouring software proficiently in advance. And, all junior \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[59,41,276,51]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Volume 113 • Number 4 • 2022 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[647,41,888,51]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Automatic delineation of GTV in NPC \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[904,41,932,51]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"895 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[100,67,902,315]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 1. Flowchart of the study. Abbreviations: DSC = Dice similarity coefficient; GTV = gross tumor volume. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0652/73f5950ade712e872bb10d9593351c0b515ce48413aecb34986653c514718999.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[62,368,480,491]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"radiation oncologists were required to review all images and medical history carefully before delineation. After the prework, these oncologists were asked to complete contouring the GTV of a certain patient nonstop, and the delineation time was recorded from the start to the end of the contouring. In addition, the principles of image distribution and contouring were the same as described previously. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,492,480,584]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Finally, the delineated contours without and with the model assistance were compared with the ground truth contours, and the revision degrees were calculated through the formula $(1 - \\\\mathrm{DSC})\\\\times$ percentage to evaluate whether the model could improve the accuracy of GTV contouring. The flowchart is shown in Figure 1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,609,239,624]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Performance metrics \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,642,480,812]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Metrics to evaluate the performance of segmentation could be roughly divided into 2 categories: the region-based metrics and the distance-based metrics. Therefore, the 2 most widely used metrics, the region-aware $\\\\mathrm{DSC}^{17}$ and distance-aware average surface distance (ASD), $^{18}$ were applied to evaluate the model performance. DSC was used to evaluate the similarity between 2 contours (\\\\%) and ASD was used to calculate the average distance between the surfaces of 2 contours (mm). The higher the DSC score (range, 0-1), the better the overlap between 2 contours. The lower the ASD (range, $\\\\geq 0$ ), the closer the distance between 2 contours in space. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,836,220,853]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Statistical analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,871,480,949]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Statistical analysis was performed using SPSS software (version 22.0, IBM). Categorical variables were compared by Pearson $\\\\chi^2$ test, and numeric variables were compared by $t$ test (2 groups) or variance analysis ( $\\\\geq 3$ groups). Correlation between DSC or ASD and clinical parameters was analyzed \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[520,368,937,399]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"with Pearson correlation coefficient. A 2-tailed $P$ value $< .05$ was considered significant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[520,429,586,443]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Results \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[520,465,935,498]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Performance of the model in different subgroups of patients \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[520,517,937,640]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Accuracy of the model-generated GTV contours is summarized in Table 1. For GTVnx, the mean DSC score was 0.83 and the mean ASD was $1.48\\\\mathrm{mm}$ . For GTVnd, the mean DSC was 0.80 and the mean ASD was $2.28\\\\mathrm{mm}$ . The subsequent visual comparison indicated a high concordance between the expert- and the model-generated GTVnx contours of different T categories (Fig. 2a) and GTVnd contours of different N categories (Fig. 2b). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[520,641,937,809]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the subgroup analysis of GTVnx, comparable DSC scores were obtained between patients with T1-2 and patients with T3-4 (0.81 vs 0.83; $P = .223$ ), while the ASD score of patients with T3-4 was higher than that of patients with T1-2 (1.64 vs 1.16 mm; $P = .036$ ) (Table 1). When the detailed primary tumor sizes were taken into consideration, there was no significant correlation between the GTVnx and DSC score $(P = .225)$ (Fig. E3a), but the ASD score increased with increasing GTVnx, showing a significant positive correlation $(R = 0.51; P < .001)$ (Fig. E3b). Other factors, including age and sex (Table 1), had no effect on contouring accuracy. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[520,811,937,949]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the subgroup analysis of GTVnd, comparable DSC scores were achieved between patients with N1-2 and N3 categories (0.80 vs 0.79; $P = .807$ ), but the ASD score of patients with N1-2 was significantly lower than that of patients with N3 (1.56 vs 3.86 mm; $P = .002$ ) (Table 1). No significant correlation between GTVnd and DSC score ( $P = .293$ ) was found (Fig. E3c). However, the GTVnd was positively associated with the score of ASD ( $R = 0.26$ ; $P = .047$ ) (Fig. E3d). Likewise, DSC and ASD \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,41,93,51]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"896 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[107,41,171,51]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Liao et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[520,41,937,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"International Journal of Radiation Oncology $\\\\cdot$ Biology $\\\\cdot$ Physics \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[57,87,935,602]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0737/a588a7219c1c67b1c845477dc41381703e56833aeded8e4ee23ccb25a9c3c4e6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Variable</td><td>DSC (mean ± SD)</td><td>P value</td><td>ASD (mean ± SD, mm)</td><td>P value</td></tr><tr><td colspan=\\\"5\\\">GTVnx</td></tr><tr><td>Total patients</td><td>0.83 ± 0.06</td><td>NA</td><td>1.48 ± 0.82</td><td>NA</td></tr><tr><td>T category</td><td></td><td>.223</td><td></td><td>.036</td></tr><tr><td>T1-2 (n = 20)</td><td>0.81 ± 0.04</td><td></td><td>1.16 ± 0.20</td><td></td></tr><tr><td>T3-4 (n = 38)</td><td>0.83 ± 0.06</td><td></td><td>1.64 ± 0.97</td><td></td></tr><tr><td>Sex</td><td></td><td>.491</td><td></td><td>.923</td></tr><tr><td>Male (n = 41)</td><td>0.83 ± 0.05</td><td></td><td>1.49 ± 0.91</td><td></td></tr><tr><td>Female (n = 17)</td><td>0.82 ± 0.08</td><td></td><td>1.47 ± 0.62</td><td></td></tr><tr><td>Age</td><td></td><td>.489</td><td></td><td>.634</td></tr><tr><td>≤46 (n = 33)</td><td>0.82 ± 0.06</td><td></td><td>1.53 ± 1.03</td><td></td></tr><tr><td>>46 (n = 25)</td><td>0.83 ± 0.05</td><td></td><td>1.42 ± 0.45</td><td></td></tr><tr><td colspan=\\\"5\\\">GTVnd</td></tr><tr><td>Total patients</td><td>0.80 ± 0.11</td><td>NA</td><td>2.28 ± 2.66</td><td>NA</td></tr><tr><td>N category</td><td></td><td>.807</td><td></td><td>.002</td></tr><tr><td>N1-2 (n = 40)</td><td>0.80 ± 0.12</td><td></td><td>1.56 ± 1.30</td><td></td></tr><tr><td>N3 (n = 18)</td><td>0.79 ± 0.09</td><td></td><td>3.86 ± 4.04</td><td></td></tr><tr><td>Sex</td><td></td><td>.651</td><td></td><td>.308</td></tr><tr><td>Male (n = 41)</td><td>0.79 ± 0.11</td><td></td><td>2.51 ± 3.01</td><td></td></tr><tr><td>Female (n = 17)</td><td>0.81 ± 0.10</td><td></td><td>1.71 ± 1.56</td><td></td></tr><tr><td>Age</td><td></td><td>.264</td><td></td><td>.300</td></tr><tr><td>≤46 (n = 33)</td><td>0.81 ± 0.09</td><td></td><td>2.59 ± 3.22</td><td></td></tr><tr><td>>46 (n = 25)</td><td>0.78 ± 0.13</td><td></td><td>1.85 ± 1.72</td><td></td></tr><tr><td colspan=\\\"5\\\">Abbreviations: ASD = average surface distance; DSC = Dice similarity coefficient; GTV = gross tumor volume; GTVnd = node gross tumor volume; GTVnx = nasopharynx gross tumor volume; NA = not applicable; SD = standard deviation. The median ASD of GTVnx was 1.30 mm (range, 0.66-6.43), and the median ASD of GTVnd was 1.21 mm (range, 0.32-14.64).</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 1 Accuracy comparison for model-generated GTV in different subgroups in the testing cohort \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[55,623,479,654]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"score did not differ between groups of different ages or sexes (Table 1). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,655,479,763]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Given that patients with N3 tended to have more or more widespread metastatic lymph nodes, particular attention was paid to the accuracy of our model in this patient group. In the testing cohort, 18 patients with N3 category had lower neck involvement, but missing segmentation for lower neck lymph nodes occurred in 4 $(22.2\\\\%)$ patients (Fig. E4a). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,763,480,949]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To analyze the accuracy of the model in different slices of primary tumor, we extracted 7 slices from the beginning slice to the ending slice of the GTVnx continuously and equidistantly in the head-to-foot direction. Before doing this, we averaged 7 sampled slices for each patient in the axial direction and measured the result in each slice; slices 1 and 7 were the first and last of the ground truth in the axial direction. The total slice was 7, as we counted the ground truth of this data set and found that each scan has more than 14 slices with GTVnx annotation in the axial direction. The result showed that the DSC scores at the beginning and ending slices were $0.46 \\\\pm 0.24$ and $0.47 \\\\pm 0.26$ , respectively, which \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,623,937,686]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"were significantly lower than other intermediate slices $(P < .001)$ (Fig. E4b). DSC scores of the second, third, fourth, fifth, and sixth slices were $0.84 \\\\pm 0.09$ , $0.87 \\\\pm 0.07$ , $0.86 \\\\pm 0.08$ , $0.83 \\\\pm 0.14$ , and $0.79 \\\\pm 0.14$ , respectively (Fig. E4b). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,712,878,732]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Clinical assessment by multicenter experts \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,747,934,840]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For GTVnx, the mean revision degree was lower than $10\\\\%$ in 19 $(95\\\\%)$ patients (Fig. 3a). In addition, there was no significant difference in the average revision degree of the experts among patients with T1, T2, T3, or T4: $5.29\\\\% \\\\pm 3.83\\\\%$ versus $5.22\\\\% \\\\pm 3.30\\\\%$ versus $5.47\\\\% \\\\pm 5.83\\\\%$ versus $3.82\\\\% \\\\pm 3.61\\\\%$ $(P = .235)$ (Fig. 3b). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,840,935,949]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For GTVnd, the mean revision degree was lower than $5\\\\%$ in 15 $(75\\\\%)$ patients. The mean revision degree was between $5\\\\%$ and $10\\\\%$ in 1 $(5\\\\%)$ patient, between $10\\\\%$ and $20\\\\%$ in 2 $(10\\\\%)$ patients, and more than $20\\\\%$ in 2 $(10\\\\%)$ patients (Fig. 3c). In addition, among patients with different N categories, the mean degree of revision was lowest in patients with N3 (N1 vs N2 vs N3: $8.78\\\\% \\\\pm 12.71\\\\%$ vs $7.76\\\\% \\\\pm 10.13\\\\%$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[57,38,277,53]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Volume 113 • Number 4 • 2022 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[645,38,890,53]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Automatic delineation of GTV in NPC \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[904,38,934,53]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"897 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[102,64,897,329]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fig. 2. Visual comparison of the model-generated contours and the ground truth contours in different T categories (A) and N categories (B). Yellow lines denote the expert-delineated ground truth, and green lines denote the model-generated contours. There is a high level of concordance between expert- and model-generated contours. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0318/1311f8b9647e715fb97980ac0ea3ccd27bd20de31bc476a2ff0b35a446d064a7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[102,421,708,604]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0239/2f8da6661c3822ef639eed4db6f10b9eada6cba29a7eeb6581da2ba42540db0d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[722,468,895,575]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(B) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0087/9a9529f7ecab1cd7945d8b1e2b79d4ed78afb56aed1a59efa9b0f6c4ee732941.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[102,605,713,760]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0886/96380eda34e71a97d382f73f1bd42502f9ff6e9ed8078b2e75d13b336266b3f3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[722,646,895,750]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(D) \\\",\\\"Fig. 3. Evaluation of model-generated GTV by multicenter experts. (A) The revision degrees of model-generated GTVnx by multicenter experts. E1 to E9 represents 9 experts; P01 to P20 represents 20 patients randomly selected from the testing cohort. The height of each column represents the revision degree (\\\\\\\\%) by each expert (the lower revision degree indicates better overlap between model- and expert-generated contours). (B) Comparison of revision degrees of GTVnx in different T categories (5 patients in each category). The model-generated GTVnx from each patient was revised by these 9 experts. (C) The revision degrees of model-generated GTVnd by multicenter experts. (D) Comparison of revision degrees of GTVnd in different N categories. There were 7, 8, and 5 patients in the N1, N2, and N3 categories, respectively. The model-generated GTVnx from each patient was revised by these 9 experts. The revision degrees of GTVnd in N3 patients were significantly lower than those of GTVnd in N1 and N2 patients. $P$ value was obtained by analysis of variance. Abbreviations: $\\\\\\\\mathrm{GTV} =$ gross tumor volume; $\\\\\\\\mathrm{GTVnd} =$ node gross tumor volume; $\\\\\\\\mathrm{GTVnx} =$ nasopharynx gross tumor volume. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0479/de7ea9dfb9a4916df182b52b2c63760d5ebb831fd29e2bc71c68a0160ab70c26.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[62,40,93,53]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"898 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[107,40,173,53]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Liao et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[520,40,937,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"International Journal of Radiation Oncology $\\\\cdot$ Biology $\\\\cdot$ Physics \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[55,68,480,100]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"vs $2.56\\\\% \\\\pm 2.99\\\\%$ ; $P = .004$ (Fig. 3d). The visual display of GTVnx modified by 9 experts is presented in Figure E5. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,117,432,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Comparison between model-generated and multicenter junior radiation oncologists-generated GTVs \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,186,479,324]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The average DSC score of the model-generated GTVnx was $0.85 \\\\pm 0.03$ (Fig. 4a), while the average DSC scores of GTVnx manually delineated by R1 (junior radiation oncologist 1), R2, R3, R4, R5, and R6 were $0.75 \\\\pm 0.10$ , $0.70 \\\\pm 0.12$ , $0.80 \\\\pm 0.05$ , $0.75 \\\\pm 0.05$ , $0.69 \\\\pm 0.10$ , and $0.78 \\\\pm 0.04$ , respectively. The DSC score of GTVnx generated by the model was significantly higher than that of GTVnx delineated by each junior radiation oncologist (all $P < .05$ ) (Fig. 4a). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[55,324,480,434]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In terms of GTVnd, the average DSC score of the model-generated GTVnd was $0.84 \\\\pm 0.12$ (Fig. 4b), while the DSC scores of R1, R2, R3, R4, R5, and R6 were $0.80 \\\\pm 0.15$ , $0.79 \\\\pm 0.16$ , $0.82 \\\\pm 0.11$ , $0.78 \\\\pm 0.15$ , $0.79 \\\\pm 0.12$ , and $0.74 \\\\pm 0.11$ , respectively. The DSC score of GTVnd generated by the model was higher than those of GTVnd delineated by 3 junior oncologists (R4, R5, and R6) ( $P < .05$ ). The DSC \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,68,937,113]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"score of the model was still higher than those of another 3 junior radiation oncologists (Fig. 4b), though without significant difference. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[512,134,864,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Clinical application by multicenter junior radiation oncologists \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,186,935,434]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Having proved that the present model could perform better than junior radiation oncologists, we next investigated whether the model could help them to work. Therefore, we compared the revision degrees of GTV delineation by junior radiation oncologists under 2 modes. The result showed that with the aid of the model, the revision degrees of GTVnx became smaller than those without it among all junior radiation oncologists (all $P < .05$ ) (Table 2). Similarly, with the aid of the model, the revision degrees of GTVnd were much smaller compared with those without it among the 5 junior radiation oncologists ( $P < .05$ ), except R3 ( $P = .191$ ) (Table 2). The representative visual display of GTVnx delineated by 6 junior radiation oncologists under 2 modes is presented in Figure 4c. The assistance of the model increased the accuracy in recognizing the invaded slope, thus increasing the accuracy of GTVnx delineation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[107,463,336,598]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(A) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0821/cd5073f104f56c2a17ddaf34860d7ad7028d07c044505e060b6361c79e0e6f66.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[359,474,548,613]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(C) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0866/6cb4011c7b93e47c4b382b958a638203613199d90a48b532ac3c97a4bf0706c5.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[555,474,722,613]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Prediction \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0100/e9dfa262e8adb0508a0976a4ed79795b29167678dbc1e6e8c648b4a20259bdc4.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[725,474,895,613]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Ground truth \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0922/0a52a3ab5a3df6fdbf4355e59ea90ce9d7a50195fbb99385088fecaf1965bb4f.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[107,622,335,745]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"(B) \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0525/c8aec77e8b894be0bde6195f16d7f48d1dca4e25a1950178b51d1838b3e213f9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[359,618,550,755]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Axial \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0363/d14b2d4a83d0517de96edd58401166a392f67502f97c00af8baec74d4f849592.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[555,618,722,756]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Coronal \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0909/fa52e7bd56d1437c73218eab1a3aad80bb2dde4d79480f291caf0a61ab06ea5a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[725,618,895,756]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Sagittal \\\",\\\"Fig. 4. GTV delineation by junior radiation oncologists with and without model assistance. (A) The DSC score of model-generated GTVnx was significantly higher than those of junior radiation oncologist-generated GTVnx. R1 to R6 represent 6 junior radiation oncologists. The values below the Y axis were DSC scores from junior radiation oncologists or the model. (B) The DSC score of model-generated GTVnd was significantly higher than 3 of 6 junior radiation oncologist-generated GTVnd. $P$ values were obtained by paired $t$ test, $^* < .05$ , $^{**} < .01$ , $^{***} < .001$ in (A) and (B). (C) In the upper column, 4 of 6 junior radiation oncologists failed to accurately delineate the invaded slope (the red arrow) without model assistance. In the lower column, only 1 junior radiation oncologist failed to delineate the invaded slope, and the shape of GTVnx contours also tended to be more consistent with the aid of our model. Yellow lines denote the expert-delineated ground truth, green lines denote the model-generated contours, and other color lines denote junior radiation oncologist-generated contours. Some lines overlapped so they were difficult to distinguish. Abbreviations: DSC = Dice similarity coefficient; GTV = gross tumor volume; GTVnd = node gross tumor volume; GTVnx = nasopharynx gross tumor volume. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0571/2448ca8da6aa4755abf1e68850a16d3439a476ea9369c3ac07e8428d81f1373d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[57,40,277,51]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Volume 113 • Number 4 • 2022 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[645,38,890,53]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Automatic delineation of GTV in NPC \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[904,40,934,51]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"899 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[60,87,935,244]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0019/336b7a3163f2ef5712a43e362a6535d437fa6add429ef35ba9068ca0ab7205de.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=\\\"2\\\">Junior radiation oncologist</td><td colspan=\\\"3\\\">GTVnx revision degree, %</td><td colspan=\\\"3\\\">GTVnd revision degree, %</td></tr><tr><td>Without model</td><td>With model</td><td>P value</td><td>Without model</td><td>With model</td><td>P value</td></tr><tr><td>R1</td><td>25.45 ± 10.46</td><td>14.69 ± 4.68</td><td><.001</td><td>20.06 ± 14.54</td><td>14.55 ± 13.48</td><td>.008</td></tr><tr><td>R2</td><td>30.15 ± 11.92</td><td>21.36 ± 6.84</td><td>.025</td><td>20.96 ± 15.81</td><td>16.37 ± 13.04</td><td>.035</td></tr><tr><td>R3</td><td>20.41 ± 4.72</td><td>17.44 ± 4.77</td><td>.011</td><td>18.05 ± 11.23</td><td>17.34 ± 11.34</td><td>.191</td></tr><tr><td>R4</td><td>25.41 ± 5.39</td><td>15.71 ± 4.31</td><td><.001</td><td>22.05 ± 15.46</td><td>14.95 ± 11.32</td><td><.001</td></tr><tr><td>R5</td><td>30.74 ± 9.66</td><td>15.72 ± 4.18</td><td><.001</td><td>21.46 ± 12.30</td><td>15.27 ± 10.11</td><td>.003</td></tr><tr><td>R6</td><td>21.61 ± 4.36</td><td>6.11 ± 3.08</td><td><.001</td><td>25.68 ± 11.94</td><td>8.17 ± 7.30</td><td><.001</td></tr></table>\",\n \"table_caption\": \"[\\\"Table 2 Comparison of revision degrees under 2 modes of GTV delineation by junior radiation oncologists \\\"]\",\n \"table_footnote\": \"[\\\"Abbreviations: GTV = gross tumor volume; GTVnd = node gross tumor volume; GTVnx = nasopharynx gross tumor volume. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[60,300,480,424]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"At last, we analyzed the time spent by the oncologists in delineating GTV. Without assistance from the model, R1 spent an average of 39.8 minutes/case. By contrast, when assisted by the model, the delineation time was substantially reduced to an average value of 14.0 minutes/case, greatly reducing time by $64.8\\\\%$ . Similarly, the delineation time was reduced by $70.1\\\\%$ , $61.7\\\\%$ , $78.3\\\\%$ , $61.6\\\\%$ , and $65.9\\\\%$ for R2, R3, R4, R5, and R6, respectively (all $P < .05$ ) (Table E4). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,448,164,465]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Discussion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,485,480,716]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In this study, we evaluated the accuracy and application of a new SSL model from the clinical point of view. We first compared the segmentation performance of the model in different subgroups of patients, proving that the model could achieve comparable and high accuracy in delineating GTV for patients with different clinical characteristics. Second, the multicenter experts were invited to evaluate the automatic segmentation results of GTV, further demonstrating that the model-generated contours were clinically acceptable with low revision degrees. Third, the new model performed better than junior radiation oncologists in delineating GTV. With the assistance of the model, contouring accuracy was improved, interobserver variation was reduced, and contouring time was also significantly decreased. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,717,480,840]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Advanced radiation techniques such as intensity modulated radiation therapy have considerably improved local control rate of NPC. $^{1,2}$ This clinical benefit is intricately associated with contouring accuracy and precise dose delivery. $^{19}$ However, the processes of GTV delineation are labor-intensive and heterogeneous. $^{20}$ In this context, automatic target volume contouring based on deep learning provides a promising alternative to solve these issues. $^{21,22}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,840,480,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In a previous study, we designed a novel segmentation model based on SSL and demonstrated that this method was efficient and precise for GTV contouring of NPC from an algorithmic perspective. In this study, we further examined the performance of the model from the clinical perspective. First of all, in the subgroup analysis, there was no difference in DSC scores and relative volume errors (Supplementary \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[513,300,937,516]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Materials and Table E5) between different T categories or N categories, suggesting that stage or tumor size had little effect on segmentation accuracy of our model. With the increase of T and N, the GTVnd and GTVnx tended to become large. Theoretically, the segmentation network might perform better for targets with larger volume, and it was more difficult to recognize and delineate ones with smaller volume. That's why many recently proposed models have poor segmentation for the optic chiasm,[10,23] a quite small target organ. However, when examining the correlation between tumor size and DSC score or relative volume errors (Supplementary Materials and Fig. E6), we found that both GTVnx and GTVnd had little effect on segmentation accuracy, indicating that the SSL model performed well regardless of tumor volume. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[513,517,937,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"However, in terms of ASD, the ASD sore of patients with T3-4 was higher than that of patients with T1-2, and the GTVnx was also positively associated with ASD score. The main reason might be that the tumor shape and margin were more irregular in more advanced NPC. As indicated in its definition, ASD is quite sensitive to outliers, and it is much more challenging to quantify a model's performance in tumors with complex shape and irregular margin.[23] Hence, it was reasonable that the more irregular the tumor, the larger the ASD score. Similar results were reported in other studies.[11] Although the ASD score of GTVnx in T3-4 patients was higher, the mean value of the ASD score (1.64 mm) was much less than the commonly accepted 3-mm margin of systematic and random error in radiation therapy of head and neck cancers,[24] suggesting that the GTVnx predicted by our model was still in high concordance with the ground truth. Unfortunately, the ASD score was more than 3 mm in certain N3 patients, and missing segmentation for lower metastatic neck lymph nodes occurred in 4 N3 patients, in which the missing lymph nodes were mainly located in the level Vb according to the lymph node distribution of NPC.[25] It is suggested that in the future, we need to optimize the algorithm or increase the proportion of patients with lymph nodes located in the level Vb in the training cohort to increase the accuracy of GTVnd contouring in N3 patients. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[513,902,937,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"By examining the DSC score in different GTVnx slices, we found that the accuracy was significantly less at the beginning and ending slices than the other intermediate \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[62,41,93,51]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"900 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[107,41,171,51]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Liao et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[520,41,937,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"International Journal of Radiation Oncology $\\\\cdot$ Biology $\\\\cdot$ Physics \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[60,69,475,175]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"slices, which was in accord with our common sense. Even the manual delineation of GTVnx often varies greatly at the beginning and ending slices. Therefore, it was acceptable that the GTVnx predicted by the model would also be interfered by the location of slices. This result also suggested a strong concordance between our model and the experts performing GTV delineation manually. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,178,475,346]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To further validate this new model, 9 experts from 3 cancer centers were invited to modify the model-generated GTV contours. The mean revision degree was lower than $10\\\\%$ in 19 $(95\\\\%)$ patients for GTVnx and in 16 $(80\\\\%)$ patients for GTVnd, demonstrating that the model-generated contours were highly acceptable, with slight or no revision by these experts. Similar results were also reported in a study in which the average revision degree for GTVnx was less than $20\\\\%$ in $88.4\\\\%$ patients. Taken together, this suggests that a deep learning-based contouring model is a powerful tool to achieve accurate target volume delineation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,347,475,639]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Given that the targets were often delineated by the junior radiation oncologist first and then modified by the experts, especially in academic institutions, we were concerned how and to what extent the model could help the junior radiation oncologist to work. First, our model outperformed all junior radiation oncologists in delineating GTVnx and GTVnd, which is consistent with the results of another study, in which their model outperformed 4 of 8 radiation oncologists.6 More importantly, with the assistance of our model, a noticeable improvement in the delineation quality of GTVnx and GTVnd was observed. This was reflected in the increase of the DSC and ASD scores. Moreover, with the aid of the model, delineating efficiency was substantially improved, saving the delineation time by as much as over $60\\\\%$ . Although the delineation differences among different individuals might have a certain effect on the analysis of these results, all of our results confirmed the great potential of our model to assist junior radiation oncologists in improving accuracy as well as saving time in manual delineation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,641,475,732]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Considering that the model-generated contours were not significantly superior to those produced by experienced radiation oncologists (6-10 years' experience, as shown in Supplementary Materials and Fig. E7), a comparison was not performed without and with the model assistance for the experienced radiation oncologists in the study. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,734,475,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We acknowledge that this study had some limitations. Although we established strict delineation standards and protocols, the ground truth contours were still delineated manually by experts from the same center, with potential subjective variations or even errors. However, it is a problem that also widely exists in other studies that cannot be conquered nowadays.[6,23] Hence, it is quite necessary to further improve the annotation quality and produce a widely acceptable GTV contour data set in the future. Second, the ground truth contours were mainly delineated based on MRI data sets, with which the properties (benign or malignant) of some anatomies, such as small or inflammatory lymph nodes, were difficult to determine. More imaging modalities, such as positron emission tomography/CT or dynamic \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[515,68,932,238]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"contrast-enhanced MRI, could be integrated into GTV delineation in the future. Third, both the building and validation of the deep learning were based on MRI because MRI has superior soft-tissue contrast. Although CT is the most commonly used in treatment planning, GTVs are often produced on MRI and propagated to planning CT by fusing these 2 images in practice. However, it might be more clinically relevant if validation could be performed both in an MRI data set and a CT data set. Additionally, the data set used was relatively small from a single center. As a next step, we plan to collect multicenter images to further validate our model. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,272,628,287]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conclusions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[515,307,932,491]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In summary, we first proposed a novel SSL-based deep learning model to delineate GTVnx and GTVnd in NPC. Different from previous fully SL-based models, the SSL-based model was capable of using limited labeled data to learn efficiently. With subgroup analysis and clinical assessment by multicenter experts, we demonstrated the robustness and accuracy of our model in GTV delineation. It was clinically applicable and could assist junior radiation oncologists in improving contouring accuracy and reducing contouring time. With further improvement of the algorithm, the model could be a promising way to achieve automatic and more accurate GTV delineation in NPC. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[517,527,616,542]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[515,562,932,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. Chen YP, Chan ATC, Le QT, et al. Nasopharyngeal carcinoma. Lancet. 2019;394:64-80. \\\",\\\"2. Sun XS, Liu SL, Luo MJ, et al. The association between the development of radiation therapy, image technology, and chemotherapy, and the survival of patients with nasopharyngeal carcinoma: A cohort study from 1990 to 2012. Int J Radiat Oncol Biol Phys 2019;105:581-590. \\\",\\\"3. Dubrulle F, Souillard R, Hermans R. Extension patterns of nasopharyngeal carcinoma. Eur Radiol 2007;17:2622-2630. \\\",\\\"4. Liang SB, Sun Y, Liu LZ, et al. Extension of local disease in nasopharyngeal carcinoma detected by magnetic resonance imaging: improvement of clinical target volume delineation. Int J Radiat Oncol Biol Phys 2009;75:742-750. \\\",\\\"5. Kosmin M, Ledsam J, Romera-Paredes B, et al. Rapid advances in autosegmentation of organs at risk and target volumes in head and neck cancer. Radiother Oncol 2019;135:130-140. \\\",\\\"6. Lin L, Dou Q, Jin YM, et al. Deep learning for automated contouring of primary tumor volumes by MRI for nasopharyngeal carcinoma. Radiology 2019;291:677-686. \\\",\\\"7. Brouwer CL, Steenbakkers RJ, van den Heuvel E, et al. 3D variation in delineation of head and neck organs at risk. Radiat Oncol 2012;7:32. \\\",\\\"8. Men K, Chen X, Zhang Y, et al. Deep deconvolutional neural network for target segmentation of nasopharyngeal cancer in planning computed tomography images. Front Oncol 2017;7:315. \\\",\\\"9. Xue X, Qin N, Hao X, et al. Sequential and iterative auto-segmentation of high-risk clinical target volume for radiotherapy of nasopharyngeal carcinoma in planning CT images. Front Oncol 2020;10:1134. \\\",\\\"10. Liang S, Tang F, Huang X, et al. Deep-learning-based detection and segmentation of organs at risk in nasopharyngeal carcinoma computed tomographic images for radiotherapy planning. Eur Radiol 2019; 29:1961-1967. \\\"]\",\n \"page_idx\": \"8\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[59,41,276,51]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Volume 113 • Number 4 • 2022 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[649,41,888,51]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Automatic delineation of GTV in NPC \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[904,41,930,51]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"901 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[62,71,482,364]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"11. Li S, Xiao J, He L, et al. The tumor target segmentation of nasopharyngeal cancer in CT images based on deep learning methods. Technol Cancer Res Treat 2019;18:1533033819884561. \\\",\\\"12. Zhao L, Lu Z, Jiang J, et al. Automatic nasopharyngeal carcinoma segmentation using fully convolutional networks with auxiliary paths on dual-modality PET-CT images. J Digit Imaging 2019;32:462-470. \\\",\\\"13. Luo X, Liao W, Chen J, et al. Efficient Semi-Supervised Gross Target Volume of Nasopharyngeal Carcinoma Segmentation via Uncertainty Rectified Pyramid Consistency[J]. Medical Image Computing and Computer Assisted Intervention (MICCAI) 2021;318-329. \\\",\\\"14. Hodapp N. The ICRU report 83: Prescribing, recording and reporting photon-beam intensity-modulated radiation therapy (IMRT). *Strah-lenther Onkol* 2012;188:97-99. \\\",\\\"15. van den Brekel MW, Stel HV, Castelijns JA, et al. Cervical lymph node metastasis: Assessment of radiologic criteria. Radiology 1990;177:379-384. \\\",\\\"16. Yushkevich PA, Piven J, Hazlett HC, et al. User-guided 3D active contour segmentation of anatomical structures: Significantly improved efficiency and reliability. Neuroimage 2006;31:1116-1128. \\\",\\\"17. Carillo V, Cozzarini C, Perna L, et al. Contouring variability of the penile bulb on CT images: Quantitative assessment using a generalized concordance index. Int J Radiat Oncol Biol Phys 2012; 84:841-846. \\\"]\",\n \"page_idx\": \"9\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[519,69,937,364]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"18. Yousefi S, Kehtarnavaz N, Gholipour A. Improved labeling of subcortical brain structures in atlas-based segmentation of magnetic resonance images. IEEE Trans Biomed Eng 2012;59:1808-1817. \\\",\\\"19. Jeanneret-Sozzi W, Moeckli R, Valley JF, et al. The reasons for discrepancies in target volume delineation: A SASRO study on head-and-neck and prostate cancers. Strahlenther Onkol 2006;182:450-457. \\\",\\\"20. Teguh DN, Levendag PC, Voet PW, et al. Clinical validation of atlas-based auto-segmentation of multiple target volumes and normal tissue (swallowing/mastication) structures in the head and neck. Int J Radiat Oncol Biol Phys 2011;81:950-957. \\\",\\\"21. Chang Z. Will AI improve tumor delineation accuracy for radiation therapy? Radiology 2019;291:687-688. \\\",\\\"22. Deig CR, Kanwar A, Thompson RF. Artificial intelligence in radiation oncology. Hematol Oncol Clin North Am 2019;33:1095-1104. \\\",\\\"23. Tang H, Chen X, Liu Y, et al. Clinically applicable deep learning framework for organs at risk delineation in CT images. Nat Mach Intell 2019;1:480-491. \\\",\\\"24. Lee N, Lu JJ. Target Volume Delineation and Field Setup: A Practical Guide for Conformal and Intensity-Modulated Radiation Therapy 2013. \\\",\\\"25. Lin L, Lu Y, Wang XJ, et al. Delineation of neck clinical target volume specific to nasopharyngeal carcinoma based on lymph node distribution and the international consensus guidelines. Int J Radiat Oncol Biol Phys 2018;100:891-902. \\\"]\",\n \"page_idx\": \"9\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[62,40,93,53]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"902 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[107,40,173,53]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Liao et al. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[520,40,937,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"International Journal of Radiation Oncology $\\\\cdot$ Biology $\\\\cdot$ Physics \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.1016/j.ijrobp.2022.03.031"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Medicine and Health Sciences"},"sha256":{"kind":"string","value":"53000e0347e136c3b694f691e89c17fff7be29dd2f21ba3eb2cc3e20ff36c6ad"},"title":{"kind":"string","value":"Automatic Delineation of Gross Tumor Volume Based on Magnetic Resonance Imaging by Performing a Novel Semisupervised Learning Framework in Nasopharyngeal Carcinoma"}}},{"rowIdx":69,"cells":{"abstract":{"kind":"string","value":""},"author":{"kind":"string","value":"Sven-Erik Sönksen|Sven Kühn|Mathias Basner|Darius Gerlach|Fabian Hoffmann|Christian Mühl|Jens Tank|Hans-Jürgen Noblé|Katja Akgün|Tjalf Ziemssen|Jens Jordan|Ulrich Limper"},"content_list":{"kind":"list like","value":[{"bbox":"[72,136,924,193]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Brain structure and neurocognitive function in two professional mountaineers during 35 days of severe normobaric hypoxia ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,219,766,241]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sven-Erik Sönksen $^{1}$ | Sven Kühn $^{2}$ | Mathias Basner $^{3}$ | Darius Gerlach $^{4}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,242,793,264]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Fabian Hoffmann $^{4,5}$ | Christian Muhl $^{4}$ | Jens Tank $^{4}$ | Hans-Jürgen Noble $^{1}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,265,739,287]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Katja Akgun $^{6}$ | Tjalf Ziemssen $^{6}$ | Jens Jordan $^{4,7}$ | Ulrich Limper $^{4,8}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,301,880,443]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"$^{1}$ German Air Force Centre of Aerospace Medicine, Fuerstenfeldbruck, Germany \",\"$^{2}$ Department of Radiology, Bundeswehr Central Hospital Koblenz, Koblenz, Germany \",\"$^{3}$ Department of Psychiatry, University of Pennsylvania Perelman School of Medicine, Philadelphia, Pennsylvania, USA \",\"4German Aerospace Center (DLR), Institute of Aerospace Medicine, Cologne, Germany \",\"$^{5}$ Department of Cardiology, University of Cologne, Cologne, Germany \",\"$^{6}$ Center of Clinical Neuroscience, Neurological University Clinic Carl-Gustav Carus, Dresden, Germany \",\"<sup>7</sup>Chair of Aerospace Medicine, University of Cologne, Cologne, Germany \",\"$^{8}$ Department of Anesthesiology and Intensive Care Medicine, Merheim Medical Center, Hospitals of Cologne, University of Witten/Herdecke, Cologne, Germany \"]","page_idx":"0","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[72,476,166,488]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Correspondence ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,489,233,501]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ulrich Limper, Department of ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,503,260,514]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Anesthesiology and Intensive Care ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,516,268,526]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Medicine, Merheim Medical Center, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,528,270,539]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hospitals of Cologne, Ostmerheimer ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,540,295,552]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Strasse 200, D-51109 Cologne, Germany. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,554,218,565]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Email: ulrich.limper@dlr.de ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[72,579,186,590]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Funding information ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,592,263,603]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bundesministerium für Bildung und ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,604,260,616]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Forschung, Grant/Award Number: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,618,299,641]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"50WB1816; Deutsches Zentrum für Luft-und Raumfahrht, Grant/Award Number: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,643,302,654]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"50WB1517, 50WB2119 and Programmatic ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,654,275,667]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"funding of the Institute of Aerospace; ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,668,270,680]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Witten/Herdecke University, Grant/ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,681,230,691]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Award Number: IFF 2020-26 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[324,476,396,491]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[322,497,924,570]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Background and purpose: Animal studies suggest that exposure to severe ambient hypoxia for several days may have beneficial long-term effects on neurodegenerative diseases. Because, the acute risks of exposing human beings to prolonged severe hypoxia on brain structure and function are uncertain, we conducted a pilot study in healthy persons. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[322,572,924,685]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Methods: We included two professional mountaineers (participants A and B) in a 35-day study comprising an acclimatization period and 14 consecutive days with oxygen concentrations between $8\\%$ and $8.8\\%$ . They underwent cerebral magnetic resonance imaging at seven time points and a cognitive test battery covering a spectrum of cognitive domains at 27 time points. We analysed blood neuron specific enolase and neurofilament light chain levels before, during, and after hypoxia. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[322,689,924,837]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Results: In hypoxia, white matter volumes increased (maximum: A, $4.3\\% \\pm 0.9\\%$ ; B, $4.5\\% \\pm 1.9\\%$ ) whilst gray matter volumes (A, $-1.5\\% \\pm 0.8\\%$ ; B, $-2.5\\% \\pm 0.9\\%$ ) and cerebrospinal fluid volumes (A, $-2.7\\% \\pm 2.4\\%$ ; B, $-5.9\\% \\pm 8.2\\%$ ) decreased. Furthermore, the number (A, 11-17; B, 26-126) and volumes (A, 140%; B, 285%) of white matter hyperintensities increased in hypoxia but had returned to baseline after a 3.5-month recovery phase. Diffusion weighted imaging of the white matter indicated cytotoxic edema formation. We did not observe changes in cognitive performance or biochemical brain injury markers. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[72,34,347,47]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Received: 12 May 2022 Accepted: 9 June 2022 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[72,53,208,63]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DOI: 10.1111/ene.15470 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[72,86,322,99]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"SHORT COMMUNICATION ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[794,67,904,79]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"european journal ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[796,79,877,89]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"of neurology ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[796,89,922,94]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"the official journal of the european academy of neurology ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[70,879,326,890]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sven-Erik Sonksen and Sven Kuhn contributed equally. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[70,902,917,928]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[73,930,789,941]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"© 2022 The Authors. European Journal of Neurology published by John Wiley & Sons Ltd on behalf of European Academy of Neurology. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[73,955,104,964]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3112 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[124,955,310,966]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"wileyonlinelibrary.com/journal/ene ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[742,955,922,966]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Eur J Neurol. 2022;29:3112-3116. ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[322,54,926,148]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Discussion: In highly selected healthy individuals, severe sustained normobaric hypoxia over 2 weeks elicited reversible changes in brain morphology without clinically relevant changes in cognitive function or brain injury markers. The finding may pave the way for future translational studies assessing the therapeutic potential of hypoxia in neurodegenerative diseases. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[324,164,416,176]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"KEYWORDS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[322,177,904,205]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"acute mountain sickness, brain recovery, hypoxic limits of the brain, performance, white matter hyperintensities ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[72,245,226,259]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"INTRODUCTION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,278,485,576]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Excessive neuronal oxidative stress and insufficient oxidative stress resistance characteristically occur in the aging brain. This imbalance probably contributes to neurodegenerative diseases such as Alzheimer's disease, Parkinson's disease, multiple sclerosis or amyotrophic lateral sclerosis. Moreover, oxidative stress promotes demyelination, cell cycle arrest and cell death, which may eventually limit neuronal regeneration. The therapeutic potential of hypoxia as a means of controlling oxidative stress has therefore been investigated in human cell and animal studies. In a mouse model of Friedreich's ataxia, breathing $11\\%$ $\\mathrm{O}_2$ diminished ataxia progression [1]. Concordantly, chronic hypoxia ( $11\\%$ $\\mathrm{O}_2$ ) prevented symptom development in a mouse model of the Leigh syndrome, a mitochondrial disease [2]. Yet, translating these findings from animals to patients could be limited by the adverse effects of severe hypoxia on brain tissue integrity and function. Indeed, graded decompression of healthy individuals to an ambient pressure equivalent to the summit of Mount Everest substantially worsened grammatical reasoning [3], psychomotor performance and mental efficiency [4]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,579,485,691]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hypoxic cerebral injury is another concern. Sixteen hours of $12\\%$ normobaric hypoxia led to sodium accumulation in the cerebral extracellular space resulting in ionic edema [5]. Furthermore, cerebral magnetic resonance imaging (MRI) studies showed reversible local vasogenic and cytotoxic white matter edema in mountaineers returning from high altitude [6]. Irreversible damage to white and gray matter was seen after returning from extreme altitude [7]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,696,485,742]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Whether severe normobaric hypoxia, intended for therapeutic purposes, adversely affects cerebral function and integrity in healthy individuals is unknown. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,745,485,826]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In a pilot study applying normobaric hypoxia corresponding to approximately $7000\\mathrm{m}$ altitude for several weeks [8], whether chronic severe hypoxia produces clinical findings in cerebral MRI, changes in blood brain injury biomarkers and a decline in neurocognitive function was assessed in two healthy individuals. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,860,334,874]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"MATERIALS AND METHODS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,895,485,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"At the German Aerospace Center, two healthy professional mountaineers were enrolled in a 35-day normobaric hypoxia study. Participant A (male, 57 years) had extensive experience with ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,246,924,359]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":">8000-m altitude and participant B (female, 49 years) with altitudes up to $6000\\mathrm{m}$ . During the study, arterial partial pressures of oxygen decreased to a minimum of $36\\mathrm{mmHg}$ in participant A and $33\\mathrm{mmHg}$ in participant B [8]. The protocol was approved by the North Rhine Medical Association ethics committee and conducted according to the Declaration of Helsinki (pre-registration DRKS00013772) after written informed consent was obtained. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,361,926,808]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cerebral MRI (3-T Siemens Biograph), with morphological sequences, susceptibility weighted imaging (SWI), arterial angiography and diffusion tensor imaging (DTI), was obtained 1 month before, once during acclimatization, three times in hypoxia, and after 1 and 4 months of recovery. During transfer to and examinations in the MRI facility, hypoxia was maintained by breathing equivalent hypoxic gas mixtures through a face mask from a Douglas bag. Based on T1-weighted magnetization-prepared rapid gradient-echo (MPRAGE) imaging, cerebrospinal fluid and white and gray matter volumes were acquired [9]. The open source FreeSurfer image analysis suite was applied (v7.1.0; http://surfer.nmr.mgh.harvard.edu/). Images were automatically processed with the longitudinal stream in FreeSurfer to extract reliable volume estimates (Appendix S1). White matter hyperintensities were quantified from preprocessed fluid attenuated inversion recovery (FLAIR) images. Subsequently, hyperintensities were assigned to atlas-based brain regions [10]. DTI acquisitions were processed using the ENIGMA-DTI pipeline (http:// enigma.ini.usc.edu), based on the FSL package TBSS [11], to measure average whole-brain fractional anisotropy value and average tract fractional anisotropy values for predefined white matter tracts. Additionally, axial and radial diffusivity and mean diffusivity were measured. Built-in 3D slicer modules were applied to determine basilar skull artery volumes from high-resolution 3D time-of-flight angiography and superior sagittal sinus volumes from an isotropic high-resolution T2-weighted sequence. Morphological sequences as well as SWI were evaluated by two blinded radiologists (Appendix S1). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,812,924,892]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cognitive performance was tested 28 out of 35 study days with the cognition test battery developed for NASA and designed for astronauts. Cognition comprises 10 brief tests covering various cognitive domains (Appendix S1) [12]. Data were corrected for practice and stimulus set difficulty effects prior to analysis [13]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,895,924,942]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The brain injury markers neurofilament light chain and neuron specific enolase were determined in blood samples collected on study days 0, 16, 21, 27, 32 and 35 and after 30 days of recovery. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,26,191,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"BRAIN AND HYPOXIA ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[890,26,922,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3113 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[72,62,157,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RESULTS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,97,485,127]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Both participants experienced high-altitude-like sickness without progression to relevant cerebral edema. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,130,485,276]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Compared with baseline, white matter volumes increased in hypoxia up to $4.3\\%$ in A and $4.5\\%$ in B (95% confidence interval A, 3.4, 5.1; B, 2.6, 6.4). Gray matter volumes did not show consistent changes. Cerebrospinal fluid volumes decreased by $2.7\\%$ in A and by $5.9\\%$ in B during hypoxia (95% confidence interval A -5.1, -0.3; B, -12.1, 2.3). Whereas in both, intracranial arterial volumes increased in hypoxia (maximum: A, +8.1%; B, +5.1%), intracranial venous volumes responded differently with a steady increase until the end of hypoxia in A and a steady decrease in B. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,280,485,377]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The number and volume of white matter hyperintensities increased in hypoxia but rapidly resolved during recovery (Figure 1). Averaged white matter diffusion values indicated an emerging cytotoxic edema component with a maximum during the second hypoxia measurement and partial regression early into the recovery phase. Microhemorrhages were not detected. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,63,924,210]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Under hypoxia, cognitive speed was significantly faster for three tests (digit symbol substitution, line orientation and visual object learning), whilst it was significantly slower for the balloon analog risk test only (Figure 1). Across all domains, speed was significantly faster $(+0.28\\mathrm{SD}, p = 0.028)$ whilst cognitive accuracy exhibited no significant changes $(+0.04\\mathrm{SD}, p = 0.71)$ under hypoxia. Standardized risk-taking propensity on the balloon analog risk task was 0.47 standard deviations higher in the low compared to the high oxygen condition $(+0.27\\mathrm{SD}; p = 0.0921)$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,213,926,377]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hypoxia-associated biochemical signal of brain damage was not detected. Neuron specific enolase, a neuronal damage marker, remained low in both participants in hypoxia (A, normoxia $23.5 \\pm 12.0 \\mu \\mathrm{g} / \\mathrm{l}$ , hypoxia $22.7 \\pm 3.5 \\mu \\mathrm{g} / \\mathrm{l}$ ; B, normoxia $15.4 \\pm 1.1 \\mu \\mathrm{g} / \\mathrm{l}$ , hypoxia $18.2 \\pm 3.0 \\mu \\mathrm{g} / \\mathrm{l}$ ). No obvious reasons were found for the slight, intermittent, neuron specific enolase increase up to $37.3 \\mu \\mathrm{g} / \\mathrm{l}$ in subject A during reoxygenation. Concordantly, neurofilament light chain, a marker for axonal damage, did not increase in hypoxia in either participant (A, normoxia $14.6 \\pm 0.8 \\mathrm{pg} / \\mathrm{ml}$ , hypoxia $13.9 \\pm 1.3 \\mathrm{pg} / \\mathrm{ml}$ ; B, normoxia $8.6 \\pm 0.7 \\mathrm{pg} / \\mathrm{ml}$ , hypoxia $7.2 \\pm 0.9 \\mathrm{pg} / \\mathrm{ml}$ ). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,425,485,597]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0398/4da3ea75598a49118add6d5829f48c0a35df69c355e5517600450daab9540ce1.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[73,608,495,791]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0742/4c2ac3f57b361457e5d0de2fcd177db0e0ca55817351758d1a21caa9e7a12d17.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[517,425,910,597]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0087/a72cceb479445c1b7207696baef4889bf0c13112ee9c1e5bed396770bc576be8.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[517,608,910,787]","code_body":null,"code_caption":null,"image_caption":"[\"FIGURE 1 Results of cerebral magnetic resonance imaging, neurocognitive function testing and biochemical markers of brain injury. (a) Number and volume of white matter hyperintensities over the course of the study (participant A, blue; participant B, orange): BL, baseline; AP, acclimatization; H1-H3, hypoxia; R + 1, recovery after 1 month; R + 4, recovery after 4 months. (b) Representative MRI (T2-FLAIR transversal) sequences showing white matter hyperintensities which emerged in hypoxia. (c) Normobaric hypoxia effects on biochemical markers (participant A, blue; participant B, orange; inspiratory oxygen level, blue area): BL, baseline; AP, acclimatization; H1-H3, hypoxia; R + 1, recovery after 1 month. (d) Normobaric hypoxia effects on cognitive performance shown as the average for both individuals: BART, balloon analog risk test; MP, motor praxis task; AM, abstract matching; ERT, emotion recognition task; NBACK, working memory test; MRT, matrix reasoning test; PVT, psychomotor vigilance test; DSST, digit symbol substitution task; LOT, line orientation test; VOLT, visual object learning test [Colour figure can be viewed at wileyonlinelibrary.com] \"]","image_footnote":"[]","img_path":"images/0011/07589542629c1d57d060d4ee671c6daad5f7dc5f0c2cc2d971e12ad2803f49d6.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[73,26,104,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3114 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[833,26,922,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"SONKSEN ET AL. ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[72,62,194,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DISCUSSION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,97,485,227]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Several weeks of sustained severe normobaric hypoxia did not lead to irreversible structural or functional brain changes. Strikingly, MRI findings in hypoxia, like white matter hyperintensities, were disconnected from cognitive function and from relevant changes in biomarkers of neuronal (neuron specific enolase) and axonal damage (neurofilament light chain). Yet, volume changes of intracranial compartments during hypoxia were observed, suggesting maladaptation on the tissue level. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,230,485,476]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Because hypoxic cerebral vasodilation contributes to cerebral edema formation [14], the large increase in white matter hyperintensity number and volume in our female participant was interpreted as local vasogenic edema formation, probably caused by capillary barrier breakdown. It is speculated that her higher white matter hyperintensity burden at baseline may result from lower white matter integrity, which in turn could increase the susceptibility to hypoxia. However, a gender effect seemed unlikely because a similar phenomenon was also discovered in a male individual after 1 week at $4559\\mathrm{m}$ [6]. White matter seems particularly susceptible for hypoxic stress, which causes an increased serum filament level, a marker for neuro-axial injury, in healthy participants $44\\mathrm{h}$ after ascending actively to $4454\\mathrm{m}$ [15]. After 3 months of recovery, cerebrovascular alterations had regressed completely in our participants, which is in good agreement with the transient nature of hypobaric hypoxia induced cerebral responses [6]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,479,485,691]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our findings confirm that the acclimatized human brain exhibits remarkable hypoxia tolerance. Accuracy of cognitive function was unchanged whilst speed increased slightly but significantly across the domains digit symbol substitution, line orientation and visual object learning performance. The latter can probably be attributed to continued small practice effects beyond the fifteenth repetition of the cognition test [13]. The balloon analog risk test was the only test that exhibited significant slowing and simultaneous non-significant increases in risk taking. Incidentally, impaired grammatical reasoning and an increased number of transposed digits during everyday conversations was also recognized in both participants. Overall our findings suggest that the two experienced mountaineers were able to maintain cognitive function in the face of substantial hypoxia. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,696,485,792]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our participants were professional mountaineers and are therefore not representative for the average population. Additional studies in patients with neurodegenerative disease are required. Initially, such studies should be limited to hypoxia experienced patients, which appears feasible considering that several mountaineers with neurodegenerative diseases have summed Mount Everest. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,795,485,859]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our results suggest that severe sustained normobaric hypoxia is tolerated in highly selected individuals and may pave the way for future translational studies of the therapeutic potential of hypoxia in neurodegenerative diseases [1]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,878,275,890]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"AUTHOR CONTRIBUTIONS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,895,485,959]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Sven-Erik Soenksen: Formal analysis (equal); investigation (equal); methodology (equal); writing - original draft (equal). Sven Kuehn: Formal analysis (equal); investigation (equal); writing - review and editing (equal). Mathias Basner: Formal analysis (equal); ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,63,924,378]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"investigation (equal); methodology (equal); writing - review and editing (equal). Darius Gerlach: Formal analysis (equal); investigation (equal); methodology (equal); writing - review and editing (equal). Fabian Hoffmann: Investigation (equal); methodology (equal); writing - review and editing (equal). Christian Muehl: Formal analysis (equal); investigation (equal); methodology (equal); writing - review and editing (equal). Jens Tank: Conceptualization (equal); investigation (equal); project administration (equal); resources (equal); supervision (equal); writing - review and editing (equal). Hans Juergen Noble: Formal analysis (equal); investigation (equal); methodology (equal); writing - review and editing (equal). Katja Akguen: Conceptualization (supporting); resources (supporting); writing - review and editing (equal). Tjalf Ziemssen: Conceptualization (equal); investigation (equal); resources (equal); writing - review and editing (equal). Jens Jordan: Conceptualization (equal); funding acquisition (equal); supervision (equal); writing - review and editing (equal). Ulrich Limper: Conceptualization (lead); formal analysis (equal); investigation (equal); methodology (equal); writing - original draft (equal); writing - review and editing (equal). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,396,690,407]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ACKNOWLEDGEMENTS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,413,924,526]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The authors would like to thank the participants for their time and patience. This work was supported by the German Aerospace Center (DLR, grant numbers 50WB1517, 50WB2119), the Bundesministerium fuer Bildung und Forschung (BMBF, 50WB1816) and the Faculty of Health of the University of Witten/Herdecke (internal grant program project IFF 2020-26). Open Access funding enabled and organized by Projekt DEAL. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,546,697,558]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CONFLICT OF INTEREST ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,562,924,875]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This study was supported by the programmatic funding of the German Aerospace Center (DLR). Dr Sven-Erik Sonksen has nothing to disclose related to this project. Dr Sven Kuhn has nothing to disclose related to this project. Professor Mathias Basner has nothing to disclose related to this project. Dr Christian Muhl has nothing to disclose related to this project. Dr Darius Gerlach has nothing to disclose related to this project. Dr Fabian Hoffmann received funding from the German Aerospace Center (DLR, 50WB1517) and the Bundesministerium fuer Bildung und Forschung (BMBF, 50WB1816). Professor Jens Tank has nothing to disclose related to this project. Dr Hans-Juergen Noblé has nothing to disclose related to this project. Professor Katja Akgün has nothing to disclose related to this project. Professor Tjalf Ziemssen has nothing to disclose related to this project. Professor Jens Jordan has nothing to disclose related to this project. Dr Ulrich Limper received funding from the internal grant program (project IFF 2020-26) of the Faculty of Health at Witten/Herdecke University, Germany, and from the German Aerospace Center (DLR, 50WB2119) during the conduct of this study. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[510,895,764,907]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DATA AVAILABILITY STATEMENT ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,911,922,957]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The authors declare that anonymized data will be shared from the corresponding author upon reasonable request from any qualified investigator. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,26,191,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"BRAIN AND HYPOXIA ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[890,26,922,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3115 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[72,63,315,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"TRANSPARENCY DECLARATION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,80,485,161]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The lead author affirms that this paper is an honest, accurate and transparent account of the study being reported; that no important aspects of the study have been omitted; and that any discrepancies from the study as planned (and, if relevant, registered) have been explained. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[72,180,238,193]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"TRIAL REGISTRATION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[70,196,485,227]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The study was pre-registered on 16 January 2018 on the German Clinical Trial Registry (DRKS) under ID DRKS00013772. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[72,246,126,258]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ORCID ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[72,262,428,277]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Tjalf Ziemssen $\\oplus$ https://orcid.org/0000-0001-8799-8202 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,278,423,294]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ulrich Limper https://orcid.org/0000-0001-9927-4180 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[72,313,174,324]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"REFERENCES ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,329,485,625]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. Ast T, Meisel JD, Patra S, et al. Hypoxia rescues frataxin loss by restoring iron sulfur cluster biogenesis. Cell. 2019;177(6):1507-21. e16. \",\"2. Jain IH, Zazzeron L, Goli R, et al. Hypoxia as a therapy for mitochondrial disease. Science (New York, NY). 2016;352(6281):54-61. \",\"3. Kennedy RS, Dunlap WP, Banderet LE, Smith MG, Houston CS. Cognitive performance deficits in a simulated climb of Mount Everest: operation Everest II. Aviat Space Environ Med. 1989;60(2):99-104. \",\"4. Abraini JH, Bouquet C, Joulia F, Nicolas M, Kriem B. Cognitive performance during a simulated climb of Mount Everest: implications for brain function and central adaptive processes under chronic hypoxic stress. Pflugers Arch. 1998;436(4):553-559. \",\"5. Biller A, Badde S, Heckel A, et al. Exposure to 16h of normobaric hypoxia induces ionic edema in the healthy brain. Nat Commun. 2021;12(1):5987. \",\"6. Kuhn S, Gerlach D, Noblé HJ, et al. An observational cerebral magnetic resonance imaging study following 7 days at $4554\\\\mathrm{m}$ . High Alt Med Biol. 2019;20(4):407-416. \",\"7. Di Paola M, Bozzali M, Fadda L, Musicco M, Sabatini U, Caltagirone C. Reduced oxygen due to high-altitude exposure relates to atrophy in motor-function brain areas. Eur J Neurol. 2008;15(10):1050-1057. \"]","page_idx":"4","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[512,63,924,370]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"8. Hoffmann F, Limper U, Zaha VG, et al. Evolution of pulmonary hypertension during severe sustained hypoxia. Circulation. 2020;141(18):1504-1506. \",\"9. Reuter M, Schmansky NJ, Rosas HD, Fischl B. Within-subject template estimation for unbiased longitudinal image analysis. Neuroimage. 2012;61(4):1402-1418. \",\"10. McGuire SA, Ryan MC, Sherman PM, et al. White matter and hypoxic hypobaria in humans. Hum Brain Mapp. 2019;40:3165-3173. \",\"11. Smith SM, Jenkinson M, Johansen-Berg H, et al. Tract-based spatial statistics: voxelwise analysis of multi-subject diffusion data. Neuroimage. 2006;31(4):1487-1505. \",\"12. Basner M, Savitt A, Moore TM, et al. Development and validation of the cognition test battery for spaceflight. *Aerosp Med Hum Perform*. 2015;86(11):942-952. \",\"13. Basner M, Hermosillo E, Nasrini J, et al. Cognition test battery: adjusting for practice and stimulus set effects for varying administration intervals in high performing individuals. J Clin Exp Neuropsychol. 2020;42(5):516-529. \",\"14. Wilson MH, Imray CH. The cerebral venous system and hypoxia. J Appl Physiol (1985). 2016;120(2):244-250. \",\"15. Sareban M, Berger MM, Pinter D, et al. Serum neurofilament level increases after ascent to $4559\\\\mathrm{m}$ but is not related to acute mountain sickness. Eur J Neurol. 2021;28(3):1004-1008. \"]","page_idx":"4","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[509,396,732,407]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"SUPPORTING INFORMATION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,413,922,443]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Additional supporting information can be found online in the Supporting Information section at the end of this article. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[529,489,892,570]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"How to cite this article: Sönksen S-E, Kühn S, Basner M, et al. Brain structure and neurocognitive function in two professional mountaineers during 35 days of severe normobaric hypoxia. Eur J Neurol. 2022;29:3112-3116. doi: 10.1111/ene.15470 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[73,26,105,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3116 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[833,26,922,37]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"SONKSEN ET AL. ","text_format":null,"text_level":null,"type":"header"}],"string":"[\n {\n \"bbox\": \"[72,136,924,193]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Brain structure and neurocognitive function in two professional mountaineers during 35 days of severe normobaric hypoxia \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,219,766,241]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sven-Erik Sönksen $^{1}$ | Sven Kühn $^{2}$ | Mathias Basner $^{3}$ | Darius Gerlach $^{4}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,242,793,264]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Fabian Hoffmann $^{4,5}$ | Christian Muhl $^{4}$ | Jens Tank $^{4}$ | Hans-Jürgen Noble $^{1}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,265,739,287]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Katja Akgun $^{6}$ | Tjalf Ziemssen $^{6}$ | Jens Jordan $^{4,7}$ | Ulrich 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\"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"50WB1816; Deutsches Zentrum für Luft-und Raumfahrht, Grant/Award Number: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,643,302,654]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"50WB1517, 50WB2119 and Programmatic \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,654,275,667]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"funding of the Institute of Aerospace; \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,668,270,680]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Witten/Herdecke University, Grant/ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,681,230,691]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Award Number: IFF 2020-26 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[324,476,396,491]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[322,497,924,570]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Background and purpose: Animal studies suggest that exposure to severe ambient hypoxia for several days may have beneficial long-term effects on neurodegenerative diseases. Because, the acute risks of exposing human beings to prolonged severe hypoxia on brain structure and function are uncertain, we conducted a pilot study in healthy persons. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[322,572,924,685]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Methods: We included two professional mountaineers (participants A and B) in a 35-day study comprising an acclimatization period and 14 consecutive days with oxygen concentrations between $8\\\\%$ and $8.8\\\\%$ . They underwent cerebral magnetic resonance imaging at seven time points and a cognitive test battery covering a spectrum of cognitive domains at 27 time points. We analysed blood neuron specific enolase and neurofilament light chain levels before, during, and after hypoxia. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[322,689,924,837]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Results: In hypoxia, white matter volumes increased (maximum: A, $4.3\\\\% \\\\pm 0.9\\\\%$ ; B, $4.5\\\\% \\\\pm 1.9\\\\%$ ) whilst gray matter volumes (A, $-1.5\\\\% \\\\pm 0.8\\\\%$ ; B, $-2.5\\\\% \\\\pm 0.9\\\\%$ ) and cerebrospinal fluid volumes (A, $-2.7\\\\% \\\\pm 2.4\\\\%$ ; B, $-5.9\\\\% \\\\pm 8.2\\\\%$ ) decreased. Furthermore, the number (A, 11-17; B, 26-126) and volumes (A, 140%; B, 285%) of white matter hyperintensities increased in hypoxia but had returned to baseline after a 3.5-month recovery phase. Diffusion weighted imaging of the white matter indicated cytotoxic edema formation. We did not observe changes in cognitive performance or biochemical brain injury markers. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[72,34,347,47]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Received: 12 May 2022 Accepted: 9 June 2022 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[72,53,208,63]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DOI: 10.1111/ene.15470 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[72,86,322,99]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"SHORT COMMUNICATION \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[794,67,904,79]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"european journal \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[796,79,877,89]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"of neurology \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[796,89,922,94]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"the official journal of the european academy of neurology \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[70,879,326,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sven-Erik Sonksen and Sven Kuhn contributed equally. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[70,902,917,928]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[73,930,789,941]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"© 2022 The Authors. European Journal of Neurology published by John Wiley & Sons Ltd on behalf of European Academy of Neurology. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[73,955,104,964]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3112 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[124,955,310,966]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"wileyonlinelibrary.com/journal/ene \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[742,955,922,966]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Eur J Neurol. 2022;29:3112-3116. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[322,54,926,148]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Discussion: In highly selected healthy individuals, severe sustained normobaric hypoxia over 2 weeks elicited reversible changes in brain morphology without clinically relevant changes in cognitive function or brain injury markers. The finding may pave the way for future translational studies assessing the therapeutic potential of hypoxia in neurodegenerative diseases. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[324,164,416,176]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"KEYWORDS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[322,177,904,205]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"acute mountain sickness, brain recovery, hypoxic limits of the brain, performance, white matter hyperintensities \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[72,245,226,259]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"INTRODUCTION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,278,485,576]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Excessive neuronal oxidative stress and insufficient oxidative stress resistance characteristically occur in the aging brain. This imbalance probably contributes to neurodegenerative diseases such as Alzheimer's disease, Parkinson's disease, multiple sclerosis or amyotrophic lateral sclerosis. Moreover, oxidative stress promotes demyelination, cell cycle arrest and cell death, which may eventually limit neuronal regeneration. The therapeutic potential of hypoxia as a means of controlling oxidative stress has therefore been investigated in human cell and animal studies. In a mouse model of Friedreich's ataxia, breathing $11\\\\%$ $\\\\mathrm{O}_2$ diminished ataxia progression [1]. Concordantly, chronic hypoxia ( $11\\\\%$ $\\\\mathrm{O}_2$ ) prevented symptom development in a mouse model of the Leigh syndrome, a mitochondrial disease [2]. Yet, translating these findings from animals to patients could be limited by the adverse effects of severe hypoxia on brain tissue integrity and function. Indeed, graded decompression of healthy individuals to an ambient pressure equivalent to the summit of Mount Everest substantially worsened grammatical reasoning [3], psychomotor performance and mental efficiency [4]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,579,485,691]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hypoxic cerebral injury is another concern. Sixteen hours of $12\\\\%$ normobaric hypoxia led to sodium accumulation in the cerebral extracellular space resulting in ionic edema [5]. Furthermore, cerebral magnetic resonance imaging (MRI) studies showed reversible local vasogenic and cytotoxic white matter edema in mountaineers returning from high altitude [6]. Irreversible damage to white and gray matter was seen after returning from extreme altitude [7]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,696,485,742]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Whether severe normobaric hypoxia, intended for therapeutic purposes, adversely affects cerebral function and integrity in healthy individuals is unknown. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,745,485,826]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In a pilot study applying normobaric hypoxia corresponding to approximately $7000\\\\mathrm{m}$ altitude for several weeks [8], whether chronic severe hypoxia produces clinical findings in cerebral MRI, changes in blood brain injury biomarkers and a decline in neurocognitive function was assessed in two healthy individuals. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,860,334,874]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"MATERIALS AND METHODS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,895,485,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"At the German Aerospace Center, two healthy professional mountaineers were enrolled in a 35-day normobaric hypoxia study. Participant A (male, 57 years) had extensive experience with \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,246,924,359]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \">8000-m altitude and participant B (female, 49 years) with altitudes up to $6000\\\\mathrm{m}$ . During the study, arterial partial pressures of oxygen decreased to a minimum of $36\\\\mathrm{mmHg}$ in participant A and $33\\\\mathrm{mmHg}$ in participant B [8]. The protocol was approved by the North Rhine Medical Association ethics committee and conducted according to the Declaration of Helsinki (pre-registration DRKS00013772) after written informed consent was obtained. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,361,926,808]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cerebral MRI (3-T Siemens Biograph), with morphological sequences, susceptibility weighted imaging (SWI), arterial angiography and diffusion tensor imaging (DTI), was obtained 1 month before, once during acclimatization, three times in hypoxia, and after 1 and 4 months of recovery. During transfer to and examinations in the MRI facility, hypoxia was maintained by breathing equivalent hypoxic gas mixtures through a face mask from a Douglas bag. Based on T1-weighted magnetization-prepared rapid gradient-echo (MPRAGE) imaging, cerebrospinal fluid and white and gray matter volumes were acquired [9]. The open source FreeSurfer image analysis suite was applied (v7.1.0; http://surfer.nmr.mgh.harvard.edu/). Images were automatically processed with the longitudinal stream in FreeSurfer to extract reliable volume estimates (Appendix S1). White matter hyperintensities were quantified from preprocessed fluid attenuated inversion recovery (FLAIR) images. Subsequently, hyperintensities were assigned to atlas-based brain regions [10]. DTI acquisitions were processed using the ENIGMA-DTI pipeline (http:// enigma.ini.usc.edu), based on the FSL package TBSS [11], to measure average whole-brain fractional anisotropy value and average tract fractional anisotropy values for predefined white matter tracts. Additionally, axial and radial diffusivity and mean diffusivity were measured. Built-in 3D slicer modules were applied to determine basilar skull artery volumes from high-resolution 3D time-of-flight angiography and superior sagittal sinus volumes from an isotropic high-resolution T2-weighted sequence. Morphological sequences as well as SWI were evaluated by two blinded radiologists (Appendix S1). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,812,924,892]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cognitive performance was tested 28 out of 35 study days with the cognition test battery developed for NASA and designed for astronauts. Cognition comprises 10 brief tests covering various cognitive domains (Appendix S1) [12]. Data were corrected for practice and stimulus set difficulty effects prior to analysis [13]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,895,924,942]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The brain injury markers neurofilament light chain and neuron specific enolase were determined in blood samples collected on study days 0, 16, 21, 27, 32 and 35 and after 30 days of recovery. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,26,191,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"BRAIN AND HYPOXIA \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[890,26,922,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3113 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[72,62,157,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RESULTS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,97,485,127]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Both participants experienced high-altitude-like sickness without progression to relevant cerebral edema. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,130,485,276]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Compared with baseline, white matter volumes increased in hypoxia up to $4.3\\\\%$ in A and $4.5\\\\%$ in B (95% confidence interval A, 3.4, 5.1; B, 2.6, 6.4). Gray matter volumes did not show consistent changes. Cerebrospinal fluid volumes decreased by $2.7\\\\%$ in A and by $5.9\\\\%$ in B during hypoxia (95% confidence interval A -5.1, -0.3; B, -12.1, 2.3). Whereas in both, intracranial arterial volumes increased in hypoxia (maximum: A, +8.1%; B, +5.1%), intracranial venous volumes responded differently with a steady increase until the end of hypoxia in A and a steady decrease in B. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,280,485,377]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The number and volume of white matter hyperintensities increased in hypoxia but rapidly resolved during recovery (Figure 1). Averaged white matter diffusion values indicated an emerging cytotoxic edema component with a maximum during the second hypoxia measurement and partial regression early into the recovery phase. Microhemorrhages were not detected. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,63,924,210]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Under hypoxia, cognitive speed was significantly faster for three tests (digit symbol substitution, line orientation and visual object learning), whilst it was significantly slower for the balloon analog risk test only (Figure 1). Across all domains, speed was significantly faster $(+0.28\\\\mathrm{SD}, p = 0.028)$ whilst cognitive accuracy exhibited no significant changes $(+0.04\\\\mathrm{SD}, p = 0.71)$ under hypoxia. Standardized risk-taking propensity on the balloon analog risk task was 0.47 standard deviations higher in the low compared to the high oxygen condition $(+0.27\\\\mathrm{SD}; p = 0.0921)$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,213,926,377]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hypoxia-associated biochemical signal of brain damage was not detected. Neuron specific enolase, a neuronal damage marker, remained low in both participants in hypoxia (A, normoxia $23.5 \\\\pm 12.0 \\\\mu \\\\mathrm{g} / \\\\mathrm{l}$ , hypoxia $22.7 \\\\pm 3.5 \\\\mu \\\\mathrm{g} / \\\\mathrm{l}$ ; B, normoxia $15.4 \\\\pm 1.1 \\\\mu \\\\mathrm{g} / \\\\mathrm{l}$ , hypoxia $18.2 \\\\pm 3.0 \\\\mu \\\\mathrm{g} / \\\\mathrm{l}$ ). No obvious reasons were found for the slight, intermittent, neuron specific enolase increase up to $37.3 \\\\mu \\\\mathrm{g} / \\\\mathrm{l}$ in subject A during reoxygenation. Concordantly, neurofilament light chain, a marker for axonal damage, did not increase in hypoxia in either participant (A, normoxia $14.6 \\\\pm 0.8 \\\\mathrm{pg} / \\\\mathrm{ml}$ , hypoxia $13.9 \\\\pm 1.3 \\\\mathrm{pg} / \\\\mathrm{ml}$ ; B, normoxia $8.6 \\\\pm 0.7 \\\\mathrm{pg} / \\\\mathrm{ml}$ , hypoxia $7.2 \\\\pm 0.9 \\\\mathrm{pg} / \\\\mathrm{ml}$ ). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,425,485,597]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0398/4da3ea75598a49118add6d5829f48c0a35df69c355e5517600450daab9540ce1.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[73,608,495,791]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0742/4c2ac3f57b361457e5d0de2fcd177db0e0ca55817351758d1a21caa9e7a12d17.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[517,425,910,597]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0087/a72cceb479445c1b7207696baef4889bf0c13112ee9c1e5bed396770bc576be8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[517,608,910,787]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIGURE 1 Results of cerebral magnetic resonance imaging, neurocognitive function testing and biochemical markers of brain injury. (a) Number and volume of white matter hyperintensities over the course of the study (participant A, blue; participant B, orange): BL, baseline; AP, acclimatization; H1-H3, hypoxia; R + 1, recovery after 1 month; R + 4, recovery after 4 months. (b) Representative MRI (T2-FLAIR transversal) sequences showing white matter hyperintensities which emerged in hypoxia. (c) Normobaric hypoxia effects on biochemical markers (participant A, blue; participant B, orange; inspiratory oxygen level, blue area): BL, baseline; AP, acclimatization; H1-H3, hypoxia; R + 1, recovery after 1 month. (d) Normobaric hypoxia effects on cognitive performance shown as the average for both individuals: BART, balloon analog risk test; MP, motor praxis task; AM, abstract matching; ERT, emotion recognition task; NBACK, working memory test; MRT, matrix reasoning test; PVT, psychomotor vigilance test; DSST, digit symbol substitution task; LOT, line orientation test; VOLT, visual object learning test [Colour figure can be viewed at wileyonlinelibrary.com] \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0011/07589542629c1d57d060d4ee671c6daad5f7dc5f0c2cc2d971e12ad2803f49d6.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[73,26,104,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3114 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[833,26,922,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"SONKSEN ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[72,62,194,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DISCUSSION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,97,485,227]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Several weeks of sustained severe normobaric hypoxia did not lead to irreversible structural or functional brain changes. Strikingly, MRI findings in hypoxia, like white matter hyperintensities, were disconnected from cognitive function and from relevant changes in biomarkers of neuronal (neuron specific enolase) and axonal damage (neurofilament light chain). Yet, volume changes of intracranial compartments during hypoxia were observed, suggesting maladaptation on the tissue level. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,230,485,476]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Because hypoxic cerebral vasodilation contributes to cerebral edema formation [14], the large increase in white matter hyperintensity number and volume in our female participant was interpreted as local vasogenic edema formation, probably caused by capillary barrier breakdown. It is speculated that her higher white matter hyperintensity burden at baseline may result from lower white matter integrity, which in turn could increase the susceptibility to hypoxia. However, a gender effect seemed unlikely because a similar phenomenon was also discovered in a male individual after 1 week at $4559\\\\mathrm{m}$ [6]. White matter seems particularly susceptible for hypoxic stress, which causes an increased serum filament level, a marker for neuro-axial injury, in healthy participants $44\\\\mathrm{h}$ after ascending actively to $4454\\\\mathrm{m}$ [15]. After 3 months of recovery, cerebrovascular alterations had regressed completely in our participants, which is in good agreement with the transient nature of hypobaric hypoxia induced cerebral responses [6]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,479,485,691]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our findings confirm that the acclimatized human brain exhibits remarkable hypoxia tolerance. Accuracy of cognitive function was unchanged whilst speed increased slightly but significantly across the domains digit symbol substitution, line orientation and visual object learning performance. The latter can probably be attributed to continued small practice effects beyond the fifteenth repetition of the cognition test [13]. The balloon analog risk test was the only test that exhibited significant slowing and simultaneous non-significant increases in risk taking. Incidentally, impaired grammatical reasoning and an increased number of transposed digits during everyday conversations was also recognized in both participants. Overall our findings suggest that the two experienced mountaineers were able to maintain cognitive function in the face of substantial hypoxia. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,696,485,792]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our participants were professional mountaineers and are therefore not representative for the average population. Additional studies in patients with neurodegenerative disease are required. Initially, such studies should be limited to hypoxia experienced patients, which appears feasible considering that several mountaineers with neurodegenerative diseases have summed Mount Everest. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,795,485,859]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our results suggest that severe sustained normobaric hypoxia is tolerated in highly selected individuals and may pave the way for future translational studies of the therapeutic potential of hypoxia in neurodegenerative diseases [1]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,878,275,890]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"AUTHOR CONTRIBUTIONS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,895,485,959]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Sven-Erik Soenksen: Formal analysis (equal); investigation (equal); methodology (equal); writing - original draft (equal). Sven Kuehn: Formal analysis (equal); investigation (equal); writing - review and editing (equal). Mathias Basner: Formal analysis (equal); \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,63,924,378]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"investigation (equal); methodology (equal); writing - review and editing (equal). Darius Gerlach: Formal analysis (equal); investigation (equal); methodology (equal); writing - review and editing (equal). Fabian Hoffmann: Investigation (equal); methodology (equal); writing - review and editing (equal). Christian Muehl: Formal analysis (equal); investigation (equal); methodology (equal); writing - review and editing (equal). Jens Tank: Conceptualization (equal); investigation (equal); project administration (equal); resources (equal); supervision (equal); writing - review and editing (equal). Hans Juergen Noble: Formal analysis (equal); investigation (equal); methodology (equal); writing - review and editing (equal). Katja Akguen: Conceptualization (supporting); resources (supporting); writing - review and editing (equal). Tjalf Ziemssen: Conceptualization (equal); investigation (equal); resources (equal); writing - review and editing (equal). Jens Jordan: Conceptualization (equal); funding acquisition (equal); supervision (equal); writing - review and editing (equal). Ulrich Limper: Conceptualization (lead); formal analysis (equal); investigation (equal); methodology (equal); writing - original draft (equal); writing - review and editing (equal). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,396,690,407]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ACKNOWLEDGEMENTS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,413,924,526]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The authors would like to thank the participants for their time and patience. This work was supported by the German Aerospace Center (DLR, grant numbers 50WB1517, 50WB2119), the Bundesministerium fuer Bildung und Forschung (BMBF, 50WB1816) and the Faculty of Health of the University of Witten/Herdecke (internal grant program project IFF 2020-26). Open Access funding enabled and organized by Projekt DEAL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,546,697,558]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CONFLICT OF INTEREST \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,562,924,875]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This study was supported by the programmatic funding of the German Aerospace Center (DLR). Dr Sven-Erik Sonksen has nothing to disclose related to this project. Dr Sven Kuhn has nothing to disclose related to this project. Professor Mathias Basner has nothing to disclose related to this project. Dr Christian Muhl has nothing to disclose related to this project. Dr Darius Gerlach has nothing to disclose related to this project. Dr Fabian Hoffmann received funding from the German Aerospace Center (DLR, 50WB1517) and the Bundesministerium fuer Bildung und Forschung (BMBF, 50WB1816). Professor Jens Tank has nothing to disclose related to this project. Dr Hans-Juergen Noblé has nothing to disclose related to this project. Professor Katja Akgün has nothing to disclose related to this project. Professor Tjalf Ziemssen has nothing to disclose related to this project. Professor Jens Jordan has nothing to disclose related to this project. Dr Ulrich Limper received funding from the internal grant program (project IFF 2020-26) of the Faculty of Health at Witten/Herdecke University, Germany, and from the German Aerospace Center (DLR, 50WB2119) during the conduct of this study. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[510,895,764,907]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DATA AVAILABILITY STATEMENT \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,911,922,957]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The authors declare that anonymized data will be shared from the corresponding author upon reasonable request from any qualified investigator. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,26,191,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"BRAIN AND HYPOXIA \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[890,26,922,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3115 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[72,63,315,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"TRANSPARENCY DECLARATION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,80,485,161]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The lead author affirms that this paper is an honest, accurate and transparent account of the study being reported; that no important aspects of the study have been omitted; and that any discrepancies from the study as planned (and, if relevant, registered) have been explained. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[72,180,238,193]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"TRIAL REGISTRATION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,196,485,227]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The study was pre-registered on 16 January 2018 on the German Clinical Trial Registry (DRKS) under ID DRKS00013772. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[72,246,126,258]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ORCID \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[72,262,428,277]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Tjalf Ziemssen $\\\\oplus$ https://orcid.org/0000-0001-8799-8202 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,278,423,294]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ulrich Limper https://orcid.org/0000-0001-9927-4180 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[72,313,174,324]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"REFERENCES \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,329,485,625]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. Ast T, Meisel JD, Patra S, et al. Hypoxia rescues frataxin loss by restoring iron sulfur cluster biogenesis. Cell. 2019;177(6):1507-21. e16. \\\",\\\"2. Jain IH, Zazzeron L, Goli R, et al. Hypoxia as a therapy for mitochondrial disease. Science (New York, NY). 2016;352(6281):54-61. \\\",\\\"3. Kennedy RS, Dunlap WP, Banderet LE, Smith MG, Houston CS. Cognitive performance deficits in a simulated climb of Mount Everest: operation Everest II. Aviat Space Environ Med. 1989;60(2):99-104. \\\",\\\"4. Abraini JH, Bouquet C, Joulia F, Nicolas M, Kriem B. Cognitive performance during a simulated climb of Mount Everest: implications for brain function and central adaptive processes under chronic hypoxic stress. Pflugers Arch. 1998;436(4):553-559. \\\",\\\"5. Biller A, Badde S, Heckel A, et al. Exposure to 16h of normobaric hypoxia induces ionic edema in the healthy brain. Nat Commun. 2021;12(1):5987. \\\",\\\"6. Kuhn S, Gerlach D, Noblé HJ, et al. An observational cerebral magnetic resonance imaging study following 7 days at $4554\\\\\\\\mathrm{m}$ . High Alt Med Biol. 2019;20(4):407-416. \\\",\\\"7. Di Paola M, Bozzali M, Fadda L, Musicco M, Sabatini U, Caltagirone C. Reduced oxygen due to high-altitude exposure relates to atrophy in motor-function brain areas. Eur J Neurol. 2008;15(10):1050-1057. \\\"]\",\n \"page_idx\": \"4\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[512,63,924,370]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"8. Hoffmann F, Limper U, Zaha VG, et al. Evolution of pulmonary hypertension during severe sustained hypoxia. Circulation. 2020;141(18):1504-1506. \\\",\\\"9. Reuter M, Schmansky NJ, Rosas HD, Fischl B. Within-subject template estimation for unbiased longitudinal image analysis. Neuroimage. 2012;61(4):1402-1418. \\\",\\\"10. McGuire SA, Ryan MC, Sherman PM, et al. White matter and hypoxic hypobaria in humans. Hum Brain Mapp. 2019;40:3165-3173. \\\",\\\"11. Smith SM, Jenkinson M, Johansen-Berg H, et al. Tract-based spatial statistics: voxelwise analysis of multi-subject diffusion data. Neuroimage. 2006;31(4):1487-1505. \\\",\\\"12. Basner M, Savitt A, Moore TM, et al. Development and validation of the cognition test battery for spaceflight. *Aerosp Med Hum Perform*. 2015;86(11):942-952. \\\",\\\"13. Basner M, Hermosillo E, Nasrini J, et al. Cognition test battery: adjusting for practice and stimulus set effects for varying administration intervals in high performing individuals. J Clin Exp Neuropsychol. 2020;42(5):516-529. \\\",\\\"14. Wilson MH, Imray CH. The cerebral venous system and hypoxia. J Appl Physiol (1985). 2016;120(2):244-250. \\\",\\\"15. Sareban M, Berger MM, Pinter D, et al. Serum neurofilament level increases after ascent to $4559\\\\\\\\mathrm{m}$ but is not related to acute mountain sickness. Eur J Neurol. 2021;28(3):1004-1008. \\\"]\",\n \"page_idx\": \"4\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[509,396,732,407]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"SUPPORTING INFORMATION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,413,922,443]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Additional supporting information can be found online in the Supporting Information section at the end of this article. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[529,489,892,570]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"How to cite this article: Sönksen S-E, Kühn S, Basner M, et al. Brain structure and neurocognitive function in two professional mountaineers during 35 days of severe normobaric hypoxia. Eur J Neurol. 2022;29:3112-3116. doi: 10.1111/ene.15470 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,26,105,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3116 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[833,26,922,37]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"SONKSEN ET AL. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.1111/ene.15470"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Life Sciences"},"sha256":{"kind":"string","value":"71e58b13d4fd9edd1146b8571054b7cbc5b9375945c6d8fa4d0e2884b5210c5e"},"title":{"kind":"string","value":"Brain structure and neurocognitive function in two professional mountaineers during 35days of severe normobarichypoxia"}}},{"rowIdx":70,"cells":{"abstract":{"kind":"string","value":""},"author":{"kind":"string","value":"V A Devisilov|N P Frolov"},"content_list":{"kind":"list like","value":[{"bbox":"[60,131,243,144]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PAPER • OPEN ACCESS ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[60,157,914,203]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Investigation of the pressure drop on the filtration and protective baffles of a hydrodynamic vibration filter using ANSYS CFX ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[60,218,672,233]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To cite this article: N P Frolov and V A Devisilov 2021 IOP Conf. 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This paper considers the relationship of the pressure drop at the filtration baffle of a hydrodynamic vibration filter with and without protective baffle for different operation parameters like baffle flowrate and filtration baffle rotation speed. The simulations were done using ANSYS CFX and nonlinear Darcy's law. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,508,882,612]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In different areas of the industry, removal of mechanical impurities from fluids. Filters utilizing vibration, hydrodynamic and centrifugal forces with conventional filtration methods using a porous filtration baffle for removing impurities reduce the loads on the baffle and extend the filter life service time in general [1]. A filter with this construction can also be used as an alternative means of separating fluids with different densities [2-4], especially for small filter flowrate. This paper conducts computer simulation of a filter utilizing hydrodynamic and centrifugal forces for purifying liquids, i.e., a hydrodynamic vibration filter (HVF) [5]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,614,880,673]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The aim of the simulation is to obtain the relationship of the baffle pressure drop vs. the following factors rotation speed of the porous filtration baffle, baffle flowrate for two different filter layouts. Modeling was done using ANSYS CFX. The model did not take into account the increase of the baffle resistance due to the precipitation of impurities. In the model, the working body was water. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,674,880,705]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For simulations, a simplified geometrical model of the HVF was used. Figure 1, 2 show a geometrical and mesh model of the considered filter of two constructions [6]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,65,618,80]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Innovative technologies for environmental protection in the modern world ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[774,65,880,79]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Publishing ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[112,81,882,96]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Conf. Series: Earth and Environmental Science 815 (2021) 012008 doi:10.1088/1755-1315/815/1/012008 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[112,913,198,934]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CC BY ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[198,913,880,935]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[112,938,371,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Published under licence by IOP Publishing Ltd ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[497,940,507,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[206,137,509,350]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0571/62f985a87b5dfb07af0f444c805cbbfe1450c030936e2f683388f0f8f9782ad9.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[534,137,806,349]","code_body":null,"code_caption":null,"image_caption":"[\"b) \",\"1 - bottom tube, 2 - filtration baffle, 3 - outlet tube, 4 - upper tube \"]","image_footnote":"[]","img_path":"images/0085/09a89a8e12e5fb6f3a17c2d38282a2cfb316175a2eb0ea80ad6dc4f2cde80fc9.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[206,408,552,614]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 1. Geometrical (a) and mesh (b) model of the HVF without a protective baffle \"]","image_footnote":"[]","img_path":"images/0922/dab946fe9ef185865cd75422429bdc63de5b891a981cd7715f86d9e8a61116a9.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[552,408,813,614]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 2. Geometrical (a) and mesh (b) model of the HVF with a protective baffle. \"]","image_footnote":"[]","img_path":"images/0649/708a5f47cfedfb871aac474c1392bcbd2301f27dacb5fb927e8c79e0fb72d427.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[112,652,882,698]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For simulation, a free tetrahedral mesh was built [7]. The maximum cell size was $3,5\\mathrm{mm}$ . In the models of a filter without and with a protective baffle, there were 154337 and 279944 tetrahedral cells, accordingly. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,698,880,757]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In a filter, a liquid goes through a bottom tube (1), then passes through a filtration baffle (2), and then the purified water leaves through the outlet tube (3), while some of the fluid is bypassed via the upper tube (4). The working principle of a filter with a protective baffle is the same, but there is a perforated protective baffle with a diameter greater than the filtration baffle. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,758,880,862]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The total HVF flow rate was constant in the simulation studies. In the studies, the bypass fluid volume ratio and filtration baffle rotation speed were varied. The total HVF flow rate was $0.54\\mathrm{m}^3 /\\mathrm{h}$ , the bypass ratio was varied from 5 to $26\\%$ , and the filtration baffle rotation speed was varied from 0 to $2000~\\mathrm{rpm}$ with a step of $100\\mathrm{rpm}$ . In total, 210 simulations were conducted. The coefficient of hydraulic resistance varied as a function of the flowrate over the baffle. For simulating a filtration baffle, nonlinear filtration Darcy's law was used. For theoretical studies of this filter, one can use vortex theories [8] and theories of mass transfer [9, 10]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,863,880,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The mathematical model of the filtration baffle was described in depth in [11, 12]. In the model, the main parameters of the filtration baffle were as follows: pore diameter $-150 \\mu \\mathrm{m}$ , porosity $-0.6$ , baffle area $-0.024 \\mathrm{~m}^2$ (construction without a protective baffle), 0.018 (construction with a protective baffle). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,65,618,79]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Innovative technologies for environmental protection in the modern world ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[774,65,880,79]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Publishing ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[112,81,882,96]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Conf. Series: Earth and Environmental Science 815 (2021) 012008 doi:10.1088/1755-1315/815/1/012008 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[497,940,507,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[112,134,882,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The parameters of the protective baffle were as follows: pore diameter $-500\\mu \\mathrm{m}$ , porosity $-0.36$ , baffle area- $0.024\\mathrm{m}^2$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[136,163,663,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Based on this data, the average fluid flow in the media was calculated: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[482,185,880,214]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nv = \\frac {Q _ {f p}}{S _ {f p} \\cdot \\varepsilon}, \\tag {1}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[112,219,749,235]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $Q_{fp}$ is the filtration baffle flowrate, $\\mathrm{m}^3/\\mathrm{s}$ ; $S_{fp}$ is the baffle area, $\\mathrm{m}^2$ ; $\\varepsilon$ is porosity. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,235,878,264]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Having calculated the average fluid velocity, one should calculate the Reynolds number Re given by: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[478,270,880,296]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\nR e = \\frac {\\rho v D}{\\mu}, \\tag {2}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[112,302,882,349]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $D$ is the characteristic channel diameter, $\\mathrm{m}$ ; $\\rho$ is the fluid density, $\\mathrm{kg} / \\mathrm{m}^3$ ; $\\mu$ is the dynamic viscosity of the fluid, Pa·s. For simulating baffle, the simulation program introduces porosity and the coefficient $\\alpha$ ( $\\mathrm{kg} / \\mathrm{m}^4$ ) given by: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[467,353,880,381]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$$\n\\alpha = \\frac {6 4 \\cdot P \\cdot R e ^ {- 1}}{2 \\varepsilon^ {3} A}, \\tag {3}\n$$","text_format":"latex","text_level":null,"type":"equation"},{"bbox":"[112,387,882,446]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"where $P$ is the perimeter of the wetted area, m; $A$ is the cross-section of the channel of the porous media, $\\mathrm{m}^2$ . The calculated value of the coefficient is put in a corresponding form of simulation software. As seen from the formulas, the coefficient of resistance $\\alpha$ varies with the change of the baffle flowrate. For calculating the coefficient, MathCad 15 was used. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,447,882,492]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For calculating the static pressure drop, at the end of the simulation, the values of the static pressure before the filtration baffle and after it averaged for the whole baffle length were output. Simulation results are shown in figure 3-4. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[213,510,784,770]","code_body":null,"code_caption":null,"image_caption":"[\"Figure 3. Simulation results for a filter without a protective baffle. \"]","image_footnote":"[]","img_path":"images/0799/3a08a67a6bc6a5c7ff1b79993c2674d6a8ec9fefb7ed89a4502c6519b01c5182.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[112,65,618,80]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Innovative technologies for environmental protection in the modern world ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[774,65,882,79]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Publishing ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[112,81,882,96]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Conf. Series: Earth and Environmental Science 815 (2021) 012008 doi:10.1088/1755-1315/815/1/012008 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[497,939,507,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[223,157,779,391]","code_body":null,"code_caption":null,"image_caption":"[\"Pressure drop for the filter with the protective baffle \",\"Figure 4. Simulation results for a filter with a protective baffle. \"]","image_footnote":"[]","img_path":"images/0528/3131f1d1d96fb074c8d985751468233c66de7fbddaf4c95972bae7eecd71c434.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[230,456,771,654]","code_body":null,"code_caption":null,"image_caption":"[\"Pressure drop vs. baffle flowrate \",\"Figure 5. Pressure drop vs. baffle flowrate. \"]","image_footnote":"[]","img_path":"images/0039/c60de3fcca97ea08e88fe934c5d85447159d1f3e8b3914f5d06ac97a5c8679c7.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[112,688,882,749]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"From the simulation results, one can conclude that the pressure drop is a power function of the baffle rotation speed. The filter with a protective baffle enjoys a pressure drop up to 3-4 times greater than that of a filter without a protective baffle. Whether it is justified to use such a construction is to be determined by its efficiency of removing impurities compared to the filter without a protective baffle. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[112,750,882,809]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As seen from figure 5, the pressure drop is a linear function of flowrate, and it is weakly dependent upon the bypass parameter because the ultimate contribution to the pressure drop is made by the baffle rotation speed, and the bypass ratio is to be selected based on other criteria (efficient, the amount of concentrate). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[114,824,206,837]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[112,838,884,897]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"[1] Devisililov V A and Myagkov I A 2012 Izvestia of Samara Scientific Center of the Russian Academy of Sciences 14 860-76 \",\"[2] Stepanova S V, Alekseeva, A A and Khafizova L Y 2020 IOP Conf. Ser.: Mater. Sci. Eng. 791(1) 012076 \"]","page_idx":"4","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[112,65,618,80]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Innovative technologies for environmental protection in the modern world ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[774,65,884,79]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Publishing ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[112,81,884,96]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Conf. Series: Earth and Environmental Science 815 (2021) 012008 doi:10.1088/1755-1315/815/1/012008 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[497,939,507,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[112,80,882,96]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Conf. Series: Earth and Environmental Science 815 (2021) 012008 doi:10.1088/1755-1315/815/1/012008 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[115,133,882,404]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"[3] Alekseeva A A and Stepanova S V 2019 Russ J Gen Chem 89 2763-8 \",\"[4] Shmotkina A N, Shaikhiev I G and Sanatullova Z T 2017 Herald of Kazan Technological University 20 190-4 \",\"[5] Alexandrov A A, Devisilov V A and Sharai E Yu 2020 Herald of the Bauman Moscow State Technical University, Series Natural Sciences 88 32-45 \",\"[6] Alexandrov A A, Devisilov V A, Sharai E Yu and Kiselyova D A 2020 Herald of the Bauman Moscow State Technical University, Series Natural Sciences 2 23-38 \",\"[7] Sharai EYu 2018 Computer modeling of multi-phase flow for solving problems of technosphere safety (Moscow: INFRA-M Publishing) p 128 \",\"[8] Khafizov F Sh, Alexandrov A A, Suschev S P, Abutalipova E M and Khafizov I F 2017 Herald of the Bauman Moscow State Technical University, Series Natural Sciences 1 78-91 \",\"[9] Alexandrov A A, Akatiev V A, Turin M P, Borodina E S and Kochetov O S 2018 Herald of the Bauman Moscow State Technical University, Series Natural Sciences 4 46-58 \",\"[10] Alexandrov A A, Akatiev V A, Turin M P and Borodina E S 2017 Herald of the Bauman Moscow State Technical University, Series Natural Sciences 4 109-21 \",\"[11] Devisilov V A and Sharai E Yu 2010 Izvestia of Samara Scientific Center of the Russian Academy of Sciences 12 2216-22 \",\"[12] Whitaker S 1986 Transport in Porous Media 1 3-25 \"]","page_idx":"5","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[112,65,618,79]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Innovative technologies for environmental protection in the modern world ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[774,65,880,79]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"IOP Publishing ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[497,940,507,948]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 ","text_format":null,"text_level":null,"type":"page_number"}],"string":"[\n {\n \"bbox\": \"[60,131,243,144]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PAPER • OPEN ACCESS \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,157,914,203]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Investigation of the pressure drop on the filtration and protective baffles of a hydrodynamic vibration filter using ANSYS CFX \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,218,672,233]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To cite this article: N P Frolov and V A Devisilov 2021 IOP Conf. Ser.: Earth Environ. Sci. 815 012008 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[60,273,391,287]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"View the article online for updates and enhancements. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[73,790,159,834]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0799/5f2da7930b6dcad6d7687803a41be93927f6dd04322f00180759ed04d9d474d7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n 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\"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Conference Series: Earth and Environmental Science \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[236,942,759,954]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This content was downloaded from IP address 120.77.174.171 on 31/08/2021 at 09:49 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[114,228,828,296]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Investigation of the pressure drop on the filtration and protective baffles of a hydrodynamic vibration filter using ANSYS CFX \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[233,330,478,346]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"N P Frolov* and V A Devisilov \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[231,353,882,384]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ecology and Industrial safety Department, Bauman Moscow State Technical University, 5/1 2nd Baumanskaya str., Moscow 105005, Russia \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[233,397,381,412]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"\\\\*login1619@list.ru \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[231,426,882,480]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract. This paper considers the relationship of the pressure drop at the filtration baffle of a hydrodynamic vibration filter with and without protective baffle for different operation parameters like baffle flowrate and filtration baffle rotation speed. The simulations were done using ANSYS CFX and nonlinear Darcy's law. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,508,882,612]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In different areas of the industry, removal of mechanical impurities from fluids. Filters utilizing vibration, hydrodynamic and centrifugal forces with conventional filtration methods using a porous filtration baffle for removing impurities reduce the loads on the baffle and extend the filter life service time in general [1]. A filter with this construction can also be used as an alternative means of separating fluids with different densities [2-4], especially for small filter flowrate. This paper conducts computer simulation of a filter utilizing hydrodynamic and centrifugal forces for purifying liquids, i.e., a hydrodynamic vibration filter (HVF) [5]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,614,880,673]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The aim of the simulation is to obtain the relationship of the baffle pressure drop vs. the following factors rotation speed of the porous filtration baffle, baffle flowrate for two different filter layouts. Modeling was done using ANSYS CFX. The model did not take into account the increase of the baffle resistance due to the precipitation of impurities. In the model, the working body was water. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,674,880,705]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For simulations, a simplified geometrical model of the HVF was used. Figure 1, 2 show a geometrical and mesh model of the considered filter of two constructions [6]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,65,618,80]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Innovative technologies for environmental protection in the modern world \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[774,65,880,79]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Publishing \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[112,81,882,96]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Conf. Series: Earth and Environmental Science 815 (2021) 012008 doi:10.1088/1755-1315/815/1/012008 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[112,913,198,934]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CC BY \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[198,913,880,935]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[112,938,371,950]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Published under licence by IOP Publishing Ltd \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[497,940,507,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[206,137,509,350]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0571/62f985a87b5dfb07af0f444c805cbbfe1450c030936e2f683388f0f8f9782ad9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[534,137,806,349]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"b) \\\",\\\"1 - bottom tube, 2 - filtration baffle, 3 - outlet tube, 4 - upper tube \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0085/09a89a8e12e5fb6f3a17c2d38282a2cfb316175a2eb0ea80ad6dc4f2cde80fc9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[206,408,552,614]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 1. Geometrical (a) and mesh (b) model of the HVF without a protective baffle \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0922/dab946fe9ef185865cd75422429bdc63de5b891a981cd7715f86d9e8a61116a9.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[552,408,813,614]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 2. Geometrical (a) and mesh (b) model of the HVF with a protective baffle. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0649/708a5f47cfedfb871aac474c1392bcbd2301f27dacb5fb927e8c79e0fb72d427.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[112,652,882,698]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For simulation, a free tetrahedral mesh was built [7]. The maximum cell size was $3,5\\\\mathrm{mm}$ . In the models of a filter without and with a protective baffle, there were 154337 and 279944 tetrahedral cells, accordingly. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,698,880,757]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In a filter, a liquid goes through a bottom tube (1), then passes through a filtration baffle (2), and then the purified water leaves through the outlet tube (3), while some of the fluid is bypassed via the upper tube (4). The working principle of a filter with a protective baffle is the same, but there is a perforated protective baffle with a diameter greater than the filtration baffle. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,758,880,862]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The total HVF flow rate was constant in the simulation studies. In the studies, the bypass fluid volume ratio and filtration baffle rotation speed were varied. The total HVF flow rate was $0.54\\\\mathrm{m}^3 /\\\\mathrm{h}$ , the bypass ratio was varied from 5 to $26\\\\%$ , and the filtration baffle rotation speed was varied from 0 to $2000~\\\\mathrm{rpm}$ with a step of $100\\\\mathrm{rpm}$ . In total, 210 simulations were conducted. The coefficient of hydraulic resistance varied as a function of the flowrate over the baffle. For simulating a filtration baffle, nonlinear filtration Darcy's law was used. For theoretical studies of this filter, one can use vortex theories [8] and theories of mass transfer [9, 10]. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,863,880,909]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The mathematical model of the filtration baffle was described in depth in [11, 12]. In the model, the main parameters of the filtration baffle were as follows: pore diameter $-150 \\\\mu \\\\mathrm{m}$ , porosity $-0.6$ , baffle area $-0.024 \\\\mathrm{~m}^2$ (construction without a protective baffle), 0.018 (construction with a protective baffle). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,65,618,79]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Innovative technologies for environmental protection in the modern world \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[774,65,880,79]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Publishing \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[112,81,882,96]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Conf. Series: Earth and Environmental Science 815 (2021) 012008 doi:10.1088/1755-1315/815/1/012008 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[497,940,507,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"2 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[112,134,882,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The parameters of the protective baffle were as follows: pore diameter $-500\\\\mu \\\\mathrm{m}$ , porosity $-0.36$ , baffle area- $0.024\\\\mathrm{m}^2$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[136,163,663,178]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Based on this data, the average fluid flow in the media was calculated: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[482,185,880,214]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nv = \\\\frac {Q _ {f p}}{S _ {f p} \\\\cdot \\\\varepsilon}, \\\\tag {1}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[112,219,749,235]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $Q_{fp}$ is the filtration baffle flowrate, $\\\\mathrm{m}^3/\\\\mathrm{s}$ ; $S_{fp}$ is the baffle area, $\\\\mathrm{m}^2$ ; $\\\\varepsilon$ is porosity. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,235,878,264]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Having calculated the average fluid velocity, one should calculate the Reynolds number Re given by: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[478,270,880,296]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\nR e = \\\\frac {\\\\rho v D}{\\\\mu}, \\\\tag {2}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[112,302,882,349]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $D$ is the characteristic channel diameter, $\\\\mathrm{m}$ ; $\\\\rho$ is the fluid density, $\\\\mathrm{kg} / \\\\mathrm{m}^3$ ; $\\\\mu$ is the dynamic viscosity of the fluid, Pa·s. For simulating baffle, the simulation program introduces porosity and the coefficient $\\\\alpha$ ( $\\\\mathrm{kg} / \\\\mathrm{m}^4$ ) given by: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[467,353,880,381]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$$\\n\\\\alpha = \\\\frac {6 4 \\\\cdot P \\\\cdot R e ^ {- 1}}{2 \\\\varepsilon^ {3} A}, \\\\tag {3}\\n$$\",\n \"text_format\": \"latex\",\n \"text_level\": null,\n \"type\": \"equation\"\n },\n {\n \"bbox\": \"[112,387,882,446]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"where $P$ is the perimeter of the wetted area, m; $A$ is the cross-section of the channel of the porous media, $\\\\mathrm{m}^2$ . The calculated value of the coefficient is put in a corresponding form of simulation software. As seen from the formulas, the coefficient of resistance $\\\\alpha$ varies with the change of the baffle flowrate. For calculating the coefficient, MathCad 15 was used. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,447,882,492]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For calculating the static pressure drop, at the end of the simulation, the values of the static pressure before the filtration baffle and after it averaged for the whole baffle length were output. Simulation results are shown in figure 3-4. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[213,510,784,770]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Figure 3. Simulation results for a filter without a protective baffle. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0799/3a08a67a6bc6a5c7ff1b79993c2674d6a8ec9fefb7ed89a4502c6519b01c5182.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[112,65,618,80]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Innovative technologies for environmental protection in the modern world \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[774,65,882,79]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Publishing \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[112,81,882,96]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Conf. Series: Earth and Environmental Science 815 (2021) 012008 doi:10.1088/1755-1315/815/1/012008 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[497,939,507,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"3 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[223,157,779,391]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Pressure drop for the filter with the protective baffle \\\",\\\"Figure 4. Simulation results for a filter with a protective baffle. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0528/3131f1d1d96fb074c8d985751468233c66de7fbddaf4c95972bae7eecd71c434.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[230,456,771,654]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Pressure drop vs. baffle flowrate \\\",\\\"Figure 5. Pressure drop vs. baffle flowrate. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0039/c60de3fcca97ea08e88fe934c5d85447159d1f3e8b3914f5d06ac97a5c8679c7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[112,688,882,749]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"From the simulation results, one can conclude that the pressure drop is a power function of the baffle rotation speed. The filter with a protective baffle enjoys a pressure drop up to 3-4 times greater than that of a filter without a protective baffle. Whether it is justified to use such a construction is to be determined by its efficiency of removing impurities compared to the filter without a protective baffle. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,750,882,809]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As seen from figure 5, the pressure drop is a linear function of flowrate, and it is weakly dependent upon the bypass parameter because the ultimate contribution to the pressure drop is made by the baffle rotation speed, and the bypass ratio is to be selected based on other criteria (efficient, the amount of concentrate). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[114,824,206,837]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[112,838,884,897]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"[1] Devisililov V A and Myagkov I A 2012 Izvestia of Samara Scientific Center of the Russian Academy of Sciences 14 860-76 \\\",\\\"[2] Stepanova S V, Alekseeva, A A and Khafizova L Y 2020 IOP Conf. Ser.: Mater. Sci. Eng. 791(1) 012076 \\\"]\",\n \"page_idx\": \"4\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[112,65,618,80]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Innovative technologies for environmental protection in the modern world \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[774,65,884,79]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Publishing \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[112,81,884,96]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Conf. Series: Earth and Environmental Science 815 (2021) 012008 doi:10.1088/1755-1315/815/1/012008 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[497,939,507,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"4 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[112,80,882,96]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Conf. Series: Earth and Environmental Science 815 (2021) 012008 doi:10.1088/1755-1315/815/1/012008 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[115,133,882,404]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"[3] Alekseeva A A and Stepanova S V 2019 Russ J Gen Chem 89 2763-8 \\\",\\\"[4] Shmotkina A N, Shaikhiev I G and Sanatullova Z T 2017 Herald of Kazan Technological University 20 190-4 \\\",\\\"[5] Alexandrov A A, Devisilov V A and Sharai E Yu 2020 Herald of the Bauman Moscow State Technical University, Series Natural Sciences 88 32-45 \\\",\\\"[6] Alexandrov A A, Devisilov V A, Sharai E Yu and Kiselyova D A 2020 Herald of the Bauman Moscow State Technical University, Series Natural Sciences 2 23-38 \\\",\\\"[7] Sharai EYu 2018 Computer modeling of multi-phase flow for solving problems of technosphere safety (Moscow: INFRA-M Publishing) p 128 \\\",\\\"[8] Khafizov F Sh, Alexandrov A A, Suschev S P, Abutalipova E M and Khafizov I F 2017 Herald of the Bauman Moscow State Technical University, Series Natural Sciences 1 78-91 \\\",\\\"[9] Alexandrov A A, Akatiev V A, Turin M P, Borodina E S and Kochetov O S 2018 Herald of the Bauman Moscow State Technical University, Series Natural Sciences 4 46-58 \\\",\\\"[10] Alexandrov A A, Akatiev V A, Turin M P and Borodina E S 2017 Herald of the Bauman Moscow State Technical University, Series Natural Sciences 4 109-21 \\\",\\\"[11] Devisilov V A and Sharai E Yu 2010 Izvestia of Samara Scientific Center of the Russian Academy of Sciences 12 2216-22 \\\",\\\"[12] Whitaker S 1986 Transport in Porous Media 1 3-25 \\\"]\",\n \"page_idx\": \"5\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[112,65,618,79]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Innovative technologies for environmental protection in the modern world \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[774,65,880,79]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"IOP Publishing \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[497,940,507,948]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"5 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.1088/1755-1315/815/1/012008"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Engineering and Manufacturing Science"},"sha256":{"kind":"string","value":"91766d7913abcd12c48ece958e51eed1a7fea5fb032e4f231f7d41aa122eba3b"},"title":{"kind":"string","value":"Investigation of the pressure drop on the filtration and protective baffles of a hydrodynamic vibration filter using ANSYS CFX"}}},{"rowIdx":71,"cells":{"abstract":{"kind":"string","value":"PURPOSE Oncologists are increasingly using molecular profiling to inform personalized patient treatment decisions. Despite its promising utility, the integration of genomic testing into diverse clinical health care settings across geographic settings has been understudied. METHODS We used data from the National Survey of Precision Medicine in Cancer Treatment, a nationally representative sample of practicing US oncologists, to assess the availability of six genomic testing resources, including on-site pathology, contracts with outside laboratories, on-site genetic counselors, internal policies or protocols for using genomic and biomarker testing, electronic medical record alerts, and genomic or molecular tumor boards. We used multivariate logistic regression models to examine differences in the availability of each genomic testing resource by practice type and rurality while adjusting for payer mix and patient volume. RESULTS A larger proportion of multispecialty group and academic practices had genomic testing resources available compared with solo and nonacademic practices. Electronic medical record alerts were the least available resource, whereas contracts with outside laboratories were the most available resource. Compared with urban practices, there were significantly fewer practices located in rural areas that had on-site pathology, on-site genetic counselors, protocols for genomic tests, and molecular tumor boards. CONCLUSION Genomic testing resources varied by practice type and geography among a nationally representative sample of practicing oncologists. This variation has important implications for the development of interventions and policies to support the more equitable delivery of precision oncology to patients with cancer."},"author":{"kind":"string","value":"Gardner, Brittany|Doose, Michelle|Sanchez, Janeth, I|Freedman, Andrew N.|de Moor, Janet S."},"content_list":{"kind":"list like","value":[{"bbox":"[193,56,902,154]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Distribution of Genomic Testing Resources by Oncology Practice and Rurality: A Nationally Representative Study ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[190,159,854,187]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Brittany Gardner, BA<sup>1</sup>; Michelle Doose, PhD, MPH<sup>2</sup>; Janet I. Sanchez, PhD, MPH<sup>2</sup>; Andrew N. Freedman, PhD<sup>3</sup>; and Janet S. de Moor, PhD, MPH<sup>4</sup> ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[190,251,936,297]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PURPOSE Oncologists are increasingly using molecular profiling to inform personalized patient treatment decisions. Despite its promising utility, the integration of genomic testing into diverse clinical health care settings across geographic settings has been understudied. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[190,301,936,393]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"METHODS We used data from the National Survey of Precision Medicine in Cancer Treatment, a nationally representative sample of practicing US oncologists, to assess the availability of six genomic testing resources, including on-site pathology, contracts with outside laboratories, on-site genetic counselors, internal policies or protocols for using genomic and biomarker testing, electronic medical record alerts, and genomic or molecular tumor boards. We used multivariate logistic regression models to examine differences in the availability of each genomic testing resource by practice type and rurality while adjusting for payer mix and patient volume. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[190,398,936,475]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RESULTS A larger proportion of multispecialty group and academic practices had genomic testing resources available compared with solo and nonacademic practices. Electronic medical record alerts were the least available resource, whereas contracts with outside laboratories were the most available resource. Compared with urban practices, there were significantly fewer practices located in rural areas that had on-site pathology, on-site genetic counselors, protocols for genomic tests, and molecular tumor boards. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[190,480,936,526]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CONCLUSION Genomic testing resources varied by practice type and geography among a nationally representative sample of practicing oncologists. This variation has important implications for the development of interventions and policies to support the more equitable delivery of precision oncology to patients with cancer. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[190,531,635,546]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"JCO Precis Oncol 5:1060-1068. © 2021 by American Society of Clinical Oncology ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[191,566,297,579]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"INTRODUCTION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[190,586,558,823]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Oncologists are increasingly using next-generation sequencing (NGS) and gene-expression profiling to inform personalized treatment decisions. Often referred to as precision oncology, these genomic tests include single-gene and multigene tumor panels that require access to advanced pathology, laboratory, personnel, and organizational resources. For example, precision oncology approaches can use NGS on molecularly heterogeneous tumor specimens to inform the use of molecularly targeted therapy, assist in identifying patients who may benefit from less invasive treatments, and reduce overtreatment of others. However, despite the promising utility of precision oncology, the integration of genomic testing into health care settings is challenging. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[190,829,559,906]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Patients with cancer are diagnosed and treated in various health care settings that are diverse in organizational structure.7-9 Growing evidence suggests that the delivery of precision oncology varies by practice structure and other characteristics. For example, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[568,566,936,841]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"oncolologists practicing in nonprofit integrated health systems used genomic testing for colorectal and lung cancer at lower rates than those in hospital-based or single-specialty practices. $^{10}$ Providers that treat more than 50 unique patients per month and have access to a molecular tumor board were more likely to report using NGS tests than providers who did not. $^{2}$ Additionally, community settings may be more likely to experience challenges in delivering guideline-recommended genomic testing, including long turnaround times to receive genomic test results and delayed coordination in the handling of tissue samples. $^{11}$ These findings may be explained by the clinical, financial, and administrative challenges of implementing genomic services into practice. $^{12,13}$ Understanding the availability of resources at the organizational level is a first step to identifying inequities in the utilization of precision oncology. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[568,846,936,906]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Rural oncology practices may also face particular challenges in the delivery of precision oncology. $^{14,15}$ Additionally, rural hospitals face closures, aging infrastructures, payment challenges, workforce shortages, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[58,750,181,826]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Author affiliations and support information (if applicable) appear at the end of this article. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[58,829,181,906]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Accepted on May 11, 2021 and published at ascopubs.org/journal/ p on June 22, 2021: DOI https://doi.org/10. 1200/P0.21.00109 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[825,17,857,40]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"# ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[861,21,989,35]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[196,38,337,52]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PRECISION MEDICINE ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[150,57,183,204]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"sroedr nui8n ","text_format":null,"text_level":null,"type":"aside_text"},{"bbox":"[157,251,183,326]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"# ","text_format":null,"text_level":null,"type":"aside_text"},{"bbox":"[78,932,159,951]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ASCO ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[717,932,936,951]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"JCO® Precision Oncology ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[61,957,90,966]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1060 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[227,970,768,994]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[80,74,145,86]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CONTEXT ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,94,174,108]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Key Objective ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,109,919,185]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Next-generation sequencing is increasingly being used to inform personalized cancer treatment decisions; however, its availability across diverse health care settings and geographic settings is understudied. Does the availability of genomic testing resources—on-site pathology, contracts with outside laboratories, on-site genetic counselors, internal genomic testing policies and protocols, electronic medical record alerts, and genomic or molecular tumor boards—vary by practice type and rural geographic location? ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,186,229,200]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Knowledge Generated ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,202,919,262]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Using a nationally representative sample of oncologists, electronic medical record alerts were the least available resource (16.8%) and contracts with outside laboratories (85.6%) were the most available resource. Overall, a smaller proportion of solo and nonacademic practices and practices located in rural settings had these genomic testing resources compared with multispecialty group and academic practices and practices in urban settings. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,263,152,276]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Relevance ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,277,917,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"These findings highlight opportunities to advance the equitable delivery of personalized medicine for patients with cancer by understanding and addressing the health care delivery context. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,356,527,476]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"quality challenges, and limited access to capital. $^{16}$ Patients with cancer residing in rural areas are also more likely to have limited access to specialty providers and targeted cancer treatments that may be informed by genomic testing. $^{17}$ Although few studies have documented challenges to precision oncology in rural settings, it is unknown whether genomic testing resource availability varies across geographical areas. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,484,527,698]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As the field of precision oncology moves forward, understanding how genomic testing resources and infrastructure vary across diverse clinical contexts and settings (eg, rural and urban and academic and nonacademic) will be informative for building capacity for delivering equitable and evidence-based cancer care.[18] To begin to address the limited knowledge about the organizational contexts that influence the delivery of precision medicine, this study examined the availability of genomic testing resources, including on-site pathology, protocols for genomic test use, electronic medical record (EMR) alerts for genomic tests, and molecular tumor boards, across diverse oncology practice settings using a nationally representative sample of oncologists. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,707,198,719]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"METHODS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,727,376,742]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Data Source and Study Population ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,746,527,884]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We used data from the 2017 National Survey of Precision Medicine in Cancer Treatment, a nationally representative survey of oncologists, hematologists-oncologists, and hematologists. The survey was sponsored by the National Cancer Institute, the National Human Genomic Research Institute, and the American Cancer Society. Survey development, design, weighting, and methods were described in previous work,[2] and all survey questions were self-reported by oncologists. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[537,356,847,386]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Outcome Measure—Availability of Genomic Testing Resources ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[537,391,938,590]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Availability of genomic testing resources was measured using six questions asking about the availability of various genomic testing resources in an oncologist's primary practice: (1) on-site pathology, (2) contracts with outside laboratories to perform tests not available on-site, (3) on-site genetic counselors, (4) internal policies or protocols for use of genomic and biomarker testing, (5) an EMR that alerts providers when a genomic test is recommended for a particular patient or before ordering a particular drug, and (6) genomic or molecular tumor boards. Responses for each question included yes, no, or don't know. We categorized responses as a binary outcome of yes and no (including don't know and missing responses). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[537,604,856,620]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Main Predictors—Practice Type and Rurality ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[535,625,938,884]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Practice type was measured with two survey items. The first question asked oncologists to indicate if their main practice was a solo practice, single-specialty group, multispecialty group, or other type of practice. The second question asked whether their main practice was academically affiliated (yes or no). We then categorized the two responses into six mutually exclusive practice types: (1) solo office academic, (2) single-specialty group academic, (3) multispecialty group academic, (4) solo office nonacademic, (5) single-specialty group nonacademic, and (6) multispecialty group nonacademic. Oncologists whose main practice was other academic or other nonacademic were excluded from the analyses since we could not ascertain the exact type of practice setting. There is no framework for categorizing health systems and practice types.[19] However, we operationalized practice type to capture the potential influence that academic affiliation and practice complexity may have ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[381,21,614,33]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Distribution of Genomic Testing Resources ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[59,934,186,946]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"JCO Precision Oncology ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[907,934,934,945]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1061 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[227,971,768,994]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[126,52,527,113]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"on the availability of cancer resources on the basis of prior literature and definitions from the Compendium of U.S. Health Systems (2016). These definitions are presented in Table 1. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,118,527,210]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Rurality was ascertained using each respondent's practice zip code linked to the 2013 Rural-Urban Continuum Codes to identify RUCC (1-9). Each practice address, including zip code, was obtained from the American Medical Association file and verified by phone. RUCC were categorized as urban (1-3) and rural (4-9).[21] ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,219,433,235]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Covariates—Patient Volume and Payer Mix ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,240,529,378]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Patient volume was identified from one item asking oncologists to indicate how many unique patients they see for evaluation or treatment each month. We dichotomized patient volume as $< 100$ and $\\geq 100$ patients consistent with the literature.[22] Payer mix was identified from one item asking respondents to indicate what proportions of their patients were insured by Medicaid, or self-pay or uninsured. We dichotomized payer mix into $< 10\\%$ and $\\geq 10\\%$ Medicaid, or self-pay or uninsured.[22] ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,386,265,401]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Statistical Analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,406,527,698]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Descriptive statistics are presented to describe the sample and weighted percent of practicing oncologists. We examined the weighted percent of practicing oncologists who reported genomic resources in their practice by their primary practice type (Fig 1). We also examined differences in the proportion of practices with genomic testing resources by geographic area on the basis of RUCC (Fig 2). Chi-square tests were used to determine statistical significance for both figures. Multivariate logistic regression models were conducted to examine differences in each genomic testing resource by practice type and rurality while adjusting for known practice-level confounders (ie, patient volume and payer mix).[23-25] Adjusted odds ratios (aORs) and $95\\%$ CIs for the survey weighted population estimates are presented, and statistical significance for all analyses was determined at the $P < .05$ level (two-sided). Analyses were weighted to account for the probability sampling approach and survey nonresponse. All analyses were performed using SAS version 9.4 (SAS Institute, Cary, NC). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[537,52,604,66]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RESULTS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[535,72,938,210]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Characteristics of the 1,281 oncologists who completed the survey are summarized in Table 2. The mean age of responding oncologists was 50.2 years, and most were male (66.6%) and non-Hispanic White (62.1%). About two thirds (63.1%) of the oncologists' primary practices were academically affiliated. Most respondents practiced in single-specialty (41.8%) and multispecialty (44.8%) group practices. Geographically, almost all (95.7%) practiced in urban settings. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[535,216,938,445]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Genomic testing resources differed by practice setting. In general, a higher percentage of oncologists in multi-specialty group practices and academically affiliated practices reported having genomic testing resources available than oncologists in solo, single-specialty, or nonacademic types of practices (Fig 1). Across all practice types, EMR alerts were the least available type of genomic testing resource and contracts with outside laboratories were the most available resource. Compared with practices located in urban areas, there were significantly fewer practices located in rural areas that had on-site pathology, on-site genetic counselors, protocols for genomic tests, and molecular tumor boards (Fig 2). There were no significant differences between urban and rural settings in the availability of contracts with outside laboratories and EMR alerts. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[537,450,938,496]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Table 3 presents the multivariate regression models of the six genomic testing resources by primary practice type and rurality adjusting for patient volume and payer mix. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[537,506,669,521]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On-Site Pathology ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[535,526,938,618]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A total of $69.9\\%$ of oncologists reported that their primary practice had on-site pathology. In the adjusted model, multispecialty group academic practices were significantly more likely to have on-site pathology compared with other practice types. Onsite pathology did not differ by rurality (aOR, 1.14; $95\\%$ CI, 0.56 to 2.32). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[537,627,797,643]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Contracts With Outside Laboratories ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[535,648,938,695]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A total of $85.6\\%$ of oncologists reported that their primary practice had contracts with outside laboratories. Single-specialty group nonacademic practices (aOR, 0.58; 95% ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[59,746,936,906]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0232/ecf5bb634c5fdab10b2ddd5c8631e9cff6a18d5e8ae53b970d656c51003f514a.jpg","list_items":null,"page_idx":"2","sub_type":null,"table_body":"<table><tr><td>Solo academic practice</td><td>One provider in a single practice that is affiliated with an academic institution</td></tr><tr><td>Single-specialty group academic practice</td><td>More than one provider in a single, academically affiliated practice providing services for one medical specialty</td></tr><tr><td>Multispecialty group academic practice</td><td>More than one provider in a single, academically affiliated practice providing services for more than one medical specialty</td></tr><tr><td>Solo nonacademic practice</td><td>One provider in a single practice that is not affiliated with an academic institution</td></tr><tr><td>Single-specialty group nonacademic practice</td><td>More than one provider in a single practice that is not academically affiliated providing services for one medical specialty</td></tr><tr><td>Multispecialty group nonacademic practice</td><td>More than one provider in a single practice that is not academically affiliated providing services for more than one medical specialty</td></tr></table>","table_caption":"[\"TABLE 1. Oncologists' Practice Types and Operationalized Definitions \",\"Practice Type \",\"Operationalized Definition of Practice Type \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[460,21,537,33]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Gardner et al ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[59,934,90,945]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1062 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[95,933,366,947]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"© 2021 by American Society of Clinical Oncology ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[227,971,768,994]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[297,54,934,402]","code_body":null,"code_caption":null,"image_caption":"[\"FIG 1. Percentage of practices with genomic testing resources by oncology practice type. On the basis of chi-square test: $^{**}P < .01$ , $^{***}P < .001$ . The y-axis represents the weighted percent of oncologists with availability of each resource at their primary practice. EMR, electronic medical record. \"]","image_footnote":"[]","img_path":"images/0730/ced3f1d520de7182ea9ea287f3a72f47004da8042464f1f7a41be604142b5b43.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[126,425,529,519]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cl, 0.40 to 0.90) and solo nonacademic practices (aOR, 0.29; $95\\%$ Cl, 0.13 to 0.65) were significantly less likely to have contracts with outside laboratories compared with multispecialty group academic practices. There were no significant differences in contracts with outside laboratories by rurality (aOR, 2.10; $95\\%$ Cl, 0.83 to 5.31). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[539,425,736,441]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On-Site Genetic Counselors ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[537,453,938,516]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A total of $67.3\\%$ of oncologists reported that their primary practice had on-site genetic counselors. Multispecialty academic groups were significantly more likely to have on-site genetic counselors compared with the other five ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[297,544,936,883]","code_body":null,"code_caption":null,"image_caption":"[\"FIG 2. Percentage of practices with genomic testing resources by rurality. On the basis of chi-square test: $^{*}P < .05$ , $^{***}P < .001$ . The y-axis represents the weighted percent of oncologists with availability of each resource at their primary practice. EMR, electronic medical record. \"]","image_footnote":"[]","img_path":"images/0001/5e4a679fe7d524f05151c90dc137302b29ff6862e2bb303084c4ae3284df563c.jpg","list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[381,20,616,34]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Distribution of Genomic Testing Resources ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[59,934,188,947]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"JCO Precision Oncology ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[907,934,936,945]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1063 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[229,971,768,994]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[59,80,527,687]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0285/9d3220b85c3851fc714dbf90414350e0757781e36dd9db5cf017a437caf02d24.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":"<table><tr><td>Characteristic</td><td>N (weighted %)</td></tr><tr><td>Provider</td><td></td></tr><tr><td>Age (mean), years</td><td>50.16</td></tr><tr><td>30-39</td><td>271 (21.65)</td></tr><tr><td>40-49</td><td>391 (31.13)</td></tr><tr><td>50-59</td><td>304 (23.53)</td></tr><tr><td>≥ 60</td><td>315 (23.69)</td></tr><tr><td>Sex</td><td></td></tr><tr><td>Female</td><td>353 (33.43)</td></tr><tr><td>Male</td><td>928 (66.57)</td></tr><tr><td>Race and ethnicity</td><td></td></tr><tr><td>Non-Hispanic White</td><td>804 (62.13)</td></tr><tr><td>Asian</td><td>362 (29.39)</td></tr><tr><td>Black or African American</td><td>27 (2.12)</td></tr><tr><td>Hispanic</td><td>64 (4.82)</td></tr><tr><td>Practice</td><td></td></tr><tr><td>Primary practice affiliated with academic institution</td><td></td></tr><tr><td>No</td><td>478 (36.94)</td></tr><tr><td>Yes</td><td>796 (63.06)</td></tr><tr><td>Practice category</td><td></td></tr><tr><td>Solo</td><td>55 (4.30)</td></tr><tr><td>Single-specialty group</td><td>544 (41.83)</td></tr><tr><td>Multispecialty group</td><td>566 (44.78)</td></tr><tr><td>Others</td><td>110 (9.09)</td></tr><tr><td>Practice type</td><td></td></tr><tr><td>Solo academic</td><td>16 (1.18)</td></tr><tr><td>Single-specialty group academic</td><td>258 (20.58)</td></tr><tr><td>Multispecialty group academic</td><td>430 (33.96)</td></tr><tr><td>Solo nonacademic</td><td>39 (3.14)</td></tr><tr><td>Single-specialty group nonacademic</td><td>282 (21.13)</td></tr><tr><td>Multispecialty group nonacademic</td><td>136 (11.05)</td></tr><tr><td>Rurality</td><td></td></tr><tr><td>Urban</td><td>1,224 (95.65)</td></tr><tr><td>Rural</td><td>57 (4.35)</td></tr></table>","table_caption":"[\"TABLE 2. Characteristics of Oncologists and Their Primary Practice Setting in the United States ${\\\\left( \\\\mathrm{N} = 1,{281}\\\\right) }^{\\\\mathrm{a}}$ \"]","table_footnote":"[\"aPercentages weighted to account for complex survey design and nonresponse. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[126,758,527,819]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"practice types. The odds of having on-site genetic counselors were $70\\%$ lower among rural practices compared with urban practices (aOR, 0.30; $95\\%$ CI, 0.14 to 0.63), which was statistically significant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,826,390,838]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Internal Protocols for Genomic Tests ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,846,527,907]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A total of $48.1\\%$ of oncologists reported that their primary practice had internal protocols for genomic tests. Multispecialty group academic practices were significantly more likely to have internal protocols for genomic tests compared ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[537,52,936,127]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"with all other practices, including multispecialty group nonacademic practices (aOR, 0.24; $95\\%$ CI, 0.15 to 0.36). There were no significant differences in internal protocols for genomic tests by rurality (aOR, 0.54; $95\\%$ CI, 0.27 to 1.07). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[537,136,756,150]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"EMR Alerts for Genomic Tests ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[535,156,936,278]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A total of $16.8\\%$ of oncologists reported that their primary practice had EMR alerts for genomic tests. The odds of having EMR alerts in single-specialty group academic practice were significantly lower compared with multispecialty group academic practices (aOR, 0.52; $95\\%$ CI, 0.32 to 0.83). There were no significant differences in EMR alerts for genomic testing by other practice types and rurality (aOR, 0.46; $95\\%$ CI, 0.16 to 1.31). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[537,287,708,301]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Molecular Tumor Board ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[535,306,936,445]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A total of $36.3\\%$ of oncologists reported that their primary practice had a molecular tumor board. Multispecialty group academic practices were significantly more likely to have molecular tumor boards compared with all other practice types, including multispecialty group nonacademic practices (aOR, 0.14; $95\\%$ CI, 0.08 to 0.24). Rural practices had $80\\%$ lower odds of having molecular tumor boards compared with practices in urban settings (aOR, 0.20; $95\\%$ CI, 0.06 to 0.64), which was statistically significant. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[537,453,628,466]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DISCUSSION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[535,473,936,734]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Oncologists are increasingly using molecular profiling to inform cancer treatment decisions. Our study suggests that there are variations in the availability of genomic testing resources by the oncology practice type and geographic setting. Across practice types, contracts with outside laboratories and on-site pathology were the most commonly available genomic testing resources. Yet on-site pathology was notably less available in rural practice settings. These resources are often used to provide standard of care in most health care settings, with the latter being a cost-effective approach for providing immediate and accurate identification of tumor staging that can be used for the development of cancer treatment plans.[26,27] Overall, our findings highlight an opportunity to advance the delivery of personalized medicine for patients with cancer by understanding the health care setting context of genomic testing resource availability. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[535,739,936,906]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conversely, protocols for genomic testing, molecular tumor boards, and EMR alerts were the least available genomic resources across all practice types. Established policies and protocols for the use of genomic tests are a critical step for assisting providers with testing platform selection and decision making. However, research has shown that the establishment of such protocols and processes is challenging, requiring organizational buy-in, capital, resources, and involvement of multiple levels of clinical and non-clinical stakeholders.[3,28] Although multidisciplinary molecular tumor boards can optimize the translation of ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[460,21,537,33]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Gardner et al ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[59,934,90,945]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1064 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[95,933,366,947]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"© 2021 by American Society of Clinical Oncology ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[227,971,768,994]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[59,66,926,414]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0312/6093785f04a38915006bbe538a22ce2a2ea78c865b676b2183a064ce37ade163.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":"<table><tr><td rowspan=\"2\">Characteristic</td><td>On-Site Pathology</td><td>Contracts With Outside Laboratories</td><td>On-Site Genetic Counselors</td></tr><tr><td>aOR (95% CI)</td><td>aOR (95% CI)</td><td>aOR (95% CI)</td></tr><tr><td colspan=\"4\">Primary practice type</td></tr><tr><td>Multispecialty group academic</td><td>Ref</td><td>Ref</td><td>Ref</td></tr><tr><td>Multispecialty group nonacademic</td><td>0.23 (0.14 to 0.38)</td><td>0.92 (0.50 to 1.72)</td><td>0.16 (0.10 to 0.26)</td></tr><tr><td>Single-specialty group academic</td><td>0.39 (0.26 to 0.60)</td><td>0.83 (0.52 to 1.32)</td><td>0.32 (0.22 to 0.49)</td></tr><tr><td>Single-specialty group nonacademic</td><td>0.1 (0.07 to 0.15)</td><td>0.58 (0.40 to 0.90)</td><td>0.12 (0.08 to 0.18)</td></tr><tr><td>Solo academic</td><td>0.06 (0.02 to 0.18)</td><td>0.32 (0.10 to 1.04)</td><td>0.08 (0.03 to 0.27)</td></tr><tr><td>Solo nonacademic</td><td>0.02 (0.01 to 0.05)</td><td>0.29 (0.13 to 0.65)</td><td>0.01 (0.003 to 0.04)</td></tr><tr><td colspan=\"4\">Rurality</td></tr><tr><td>Urban</td><td>Ref</td><td>Ref</td><td>Ref</td></tr><tr><td>Rural</td><td>1.14 (0.56 to 2.32)</td><td>2.1 (0.83 to 5.31)</td><td>0.3 (0.14 to 0.63)</td></tr><tr><td colspan=\"4\">Patient volume</td></tr><tr><td>< 100</td><td>Ref</td><td>Ref</td><td>Ref</td></tr><tr><td>≥ 100</td><td>0.67 (0.50 to 0.91)</td><td>0.77 (0.54 to 1.10)</td><td>0.82 (0.61 to 1.09)</td></tr><tr><td colspan=\"4\">Payer mix</td></tr><tr><td>Self-pay or uninsured, or \nMedicaid < 10%</td><td>Ref</td><td>Ref</td><td>Ref</td></tr><tr><td>Self-pay or uninsured, or \nMedicaid ≥ 10%</td><td>1.4 (1.02 to 1.93)</td><td>0.89 (0.61 to 1.31)</td><td>1.11 (0.81 to 1.52)</td></tr></table>","table_caption":"[\"TABLE 3. Multivariate Regression Models of Genomic Testing Resources by Primary Practice Type and Rurality \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[58,421,926,785]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0454/246b9a822a029884e03626c73d780746ede532f68815ae6a5b82088ba9dc4ab7.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":"<table><tr><td></td><td colspan=\"2\">Internal Policies and Protocols for \nGenomic Tests</td><td>EMR Alerts for Genomic Tests</td><td>Genomic or Molecular Tumor Board</td></tr><tr><td></td><td>aOR (95% CI)</td><td>aOR (95% CI)</td><td>aOR (95% CI)</td><td></td></tr><tr><td colspan=\"5\">Primary practice type</td></tr><tr><td>Multispecialty group academic</td><td>Ref</td><td>Ref</td><td>Ref</td><td></td></tr><tr><td>Multispecialty group nonacademic</td><td>0.24 (0.15 to 0.36)</td><td>0.65 (0.37 to 1.13)</td><td>0.14 (0.08 to 0.24)</td><td></td></tr><tr><td>Single-specialty group academic</td><td>0.48 (0.35 to 0.67)</td><td>0.52 (0.32 to 0.83)</td><td>0.35 (0.25 to 0.49)</td><td></td></tr><tr><td>Single-specialty group nonacademic</td><td>0.19 (0.14 to 0.27)</td><td>0.84 (0.56 to 1.25)</td><td>0.07 (0.04 to 0.11)</td><td></td></tr><tr><td>Solo academic</td><td>0.13 (0.03 to 0.53)</td><td>0.76 (0.16 to 3.59)</td><td>0.12 (0.03 to 0.62)</td><td></td></tr><tr><td>Solo nonacademic</td><td>0.11 (0.05 to 0.28)</td><td>0.72 (0.26 to 1.99)</td><td>0.16 (0.07 to 0.40)</td><td></td></tr><tr><td colspan=\"5\">Rurality</td></tr><tr><td>Urban</td><td>Ref</td><td>Ref</td><td>Ref</td><td></td></tr><tr><td>Rural</td><td>0.54 (0.27 to 1.07)</td><td>0.46 (0.16 to 1.31)</td><td>0.2 (0.06 to 0.64)</td><td></td></tr><tr><td colspan=\"5\">Patient volume</td></tr><tr><td>< 100</td><td>Ref</td><td>Ref</td><td>Ref</td><td></td></tr><tr><td>≥ 100</td><td>0.79 (0.61 to 1.01)</td><td>1.02 (0.74 to 1.41)</td><td>0.7 (0.53 to 0.92)</td><td></td></tr><tr><td colspan=\"5\">Payer mix</td></tr><tr><td>Self-pay or uninsured, or \nMedicaid < 10%</td><td>Ref</td><td>Ref</td><td>Ref</td><td></td></tr><tr><td>Self-pay or uninsured, or \nMedicaid ≥ 10%</td><td>0.75 (0.57 to 0.99)</td><td>1.14 (0.79 to 1.65)</td><td>0.94 (0.68 to 1.28)</td><td></td></tr></table>","table_caption":"[]","table_footnote":"[\"Abbreviations: aOR, adjusted odds ratio; EMR, electronic medical record; ref, reference. \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[126,829,527,905]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"complex NGS information into appropriate targeted therapy options for patients with cancer, $^{12}$ their development and integration within health care systems require additional institutional resources and capacity, which may not be available in smaller practices. $^{29,30}$ In clinical settings, ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[537,829,936,905]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"information technologies and software are often used to offer automated clinical decision support to providers, but in our study, EMR alerts were the least available resource across all practice types. This could reflect the possible lack of interoperability between EMR software used across ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[380,21,614,33]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Distribution of Genomic Testing Resources ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[59,934,186,947]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"JCO Precision Oncology ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[907,934,936,945]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1065 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[227,971,768,994]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[126,52,527,144]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"practice settings. $^{31}$ As more clinical practices move toward interoperable and standardized electronic records, EMRs offer a modifiable point of intervention to efficiently integrate genomic testing processes (eg, provider reminders to use genomic testing and standardized genomic testing information) and facilitate shared decision making. $^{32,33}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,148,529,514]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Our study found that the availability of genomic testing resources varied by practice type. A larger proportion of practices that were a specialty group and/or academically affiliated had genomic testing resources compared with solo oncology practices and nonacademic practices. A smaller proportion of solo practices and nonacademic practices had any of the six genomic testing resources compared with specialty group and academic practices. For example, solo nonacademic practices had $89\\%$ lower odds of having protocols for genomic tests compared with multispecialty group academic practices. These findings may suggest that solo, small practices face limited infrastructure, capital, staff, and other resources, whereas specialized clinical expertise and advanced technology may be concentrated in academic medical centers.[34] To address this gap, academic medical centers and larger health systems have been actively involved in targeted efforts to bring precision medicine to diverse health care settings.[35,36] For example, collaboration through community-academic partnerships and the development of decentralized, online resources have provided infrastructure for many patients in nonacademic practice settings to benefit from precision oncology advances.[35-37] ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[126,520,529,810]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Genomic testing resources also varied between urban and rural practices, such that a smaller proportion of rural practices had on-site genetic counselors, internal protocols for genomic tests, on-site pathology, and molecular tumor boards compared with urban practices. These findings are consistent with previous studies indicating limited availability of cancer care services in rural settings.[17] Further exacerbating limited access to genomic testing resources in rural areas are the financial pressures of the competitive market and lower patient volumes that have led to the closure of many small and/or rural hospitals.[38] Interventions that leverage policy changes that expand access to care may help address this critical gap in access to specialized care for rural patients with cancer. For example, recent changes to Medicare increase reimbursement for telehealth services, which expands access to much needed specialty care in rural communities.[39] Telehealth technologies, including virtual molecular tumor boards and appointments with genetic counselors, can be leveraged to ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[537,52,936,81]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"transcend the travel barriers that rural patients with cancer face in accessing genomic testing resources. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[537,86,938,500]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This study is not without limitations. The cooperation rate $(38\\%)$ for the National Survey of Precision Medicine in Cancer Treatment is lower than that of previous physician surveys on the topic of precision medicine.[40,41] Additionally, our findings reflect the availability of genomic testing resources in 2017 but the state of precision medicine is rapidly evolving. For example, in 2018, the Centers for Medicare & Medicaid Services announced the coverage of Food and Drug Administration-approved NGS tests for patients with advanced-stage cancer.[42] These broadscale policy decisions are ongoing and affect the cost, value, and availability of precision medicine resources.[43,44] Nonetheless, this study uses the most currently available data. Additionally, for the outcome measure, availability of genomic testing resources, don't know responses were included in the same category as no responses because they could not confirm the availability of a resource. This may cause the no category to be artificially inflated. The survey did not collect patient-level data (eg, clinical outcomes and cancer types for which oncologists used genomic testing resources), which is an area for future investigations. Regarding the statistical analysis, the number of oncologists practicing in rural areas was low, which can limit conclusions that can be drawn from this analysis. Future studies should oversample oncologists in rural areas and other underserved areas to better understand care delivery in underserved settings. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[537,505,938,810]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Despite these limitations, the survey used in this study is a unique resource for understanding access to precision oncology, especially for underserved populations. To our knowledge, this is the first study to examine the availability of genomic testing resources by practice type and geography among a nationally representative sample of oncologists. Our findings suggest that the availability of genomic testing resources vary by organizational structure and geographic location of where oncologists practice. This variation may affect the equitable access to and utilization of precision medicine for patients with cancer. These findings warrant further investigation to delineate the extent to which genomic testing utilization and patient outcomes are influenced by the availability of genomic resources at the health system level. Additionally, exploring ways to facilitate effective integration between smaller oncology practices and academic or integrated health systems will increase the accessibility of targeted cancer treatments to patients in diverse populations and ensure more equitable distribution of precision oncology resources. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[128,841,224,854]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"AFFILIATIONS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[128,856,520,882]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"<sup>1</sup>Division of Cancer Control and Population Sciences, National Cancer Institute, Rockville, MD ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[537,838,936,902]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$^{2}$ Healthcare Delivery Research Program, Division of Cancer Control and Population Sciences, National Cancer Institute, Rockville, MD $^{3}$ Epidemiology and Genomics Research Program, Division of Cancer Control and Population Sciences, National Cancer Institute, Rockville, MD ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[460,21,537,33]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Gardner et al ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[59,934,90,945]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1066 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[95,933,368,947]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"© 2021 by American Society of Clinical Oncology ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[227,971,768,994]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[128,52,527,76]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Healthcare Delivery Research Program, Division of Cancer Control and Population Sciences, National Cancer Institute, Rockville, MD ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[128,93,309,106]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CORRESPONDING AUTHOR ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,109,523,148]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Brittany Gardner, BA, Division of Cancer Control and Population Sciences, National Cancer Institute, 9609 Medical Center Dr, 4E556, Rockville, MD 20850; e-mail: brittany.gardner@.nih.gov. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[128,163,217,176]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DISCLAIMER ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,179,527,269]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This article was prepared as part of some of the authors' (J.S.D., A.N.F., and J.I.S.) official duties as employees of the US federal government. The findings and conclusions in this report are those of the authors and do not necessarily represent the official position of the National Cancer Institute. The funder had no role in the design of the study; the collection, analysis, and interpretation of the data; the writing of the manuscript; or the decision to submit the manuscript for publication. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[128,283,282,297]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PRIOR PRESENTATION ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,300,525,338]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Presented at the American Association for Cancer Research—The Science of Cancer Health Disparities in Racial/Ethnic Minorities and the Medically Underserved, Virtual Conference, October 2-4, 2020. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[128,354,196,367]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"SUPPORT ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,370,527,433]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This research was conducted by federal employees of the National Cancer Institute and the National Institutes of Health. Specific funding was not provided for this research. The survey on which this research is based was funded by the National Institutes of Health under contract number HHSN2612010000861 to RTI International. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[128,450,323,462]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"DATA SHARING STATEMENT ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[126,465,515,491]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The data underlying this article cannot be shared publicly due to the privacy of individuals who participated in the study. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[539,52,715,65]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"AUTHOR CONTRIBUTIONS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[539,67,936,182]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"Conception and design: Brittany Gardner, Michelle Doose, Janet I. Sanchez, Janet S. de Moor \",\"Provision of study materials or patients: Andrew N. Freedman, Janet S. de Moor \",\"Collection and assembly of data: Andrew N. Freedman, Janet S. de Moor \\nData analysis and interpretation: All authors \",\"Manuscript writing: All authors \",\"Final approval of manuscript: All authors \",\"Accountable for all aspects of the work: All authors \"]","page_idx":"7","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[537,198,914,226]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"AUTHORS' DISCLOSURES OF POTENTIAL CONFLICTS OF INTEREST ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[537,228,936,318]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The following represents disclosure information provided by the authors of this manuscript. All relationships are considered compensated unless otherwise noted. Relationships are self-held unless noted. I = Immediate Family Member, Inst = My Institution. Relationships may not relate to the subject matter of this manuscript. For more information about ASCO's conflict of interest policy, please refer to www.asco.org/rwc or ascopubs.org/po/author-center. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[537,319,936,356]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Open Payments is a public database containing information reported by companies about payments made to US-licensed physicians (Open Payments). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[539,364,633,375]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Janet S. de Moor ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[539,378,667,390]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Employment: Biogen (1) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[539,391,797,402]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Stock and Other Ownership Interests: Biogen (I) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[539,404,792,415]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Travel, Accommodations, Expenses: Biogen (1) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[539,423,837,436]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"No other potential conflicts of interest were reported. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[539,452,676,465]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ACKNOWLEDGMENT ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[537,468,936,506]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We want to give special thanks to Dr Sallie Weaver and Dr Shobha Srinivasan at the National Cancer Institute in the Division of Cancer Control and Population Sciences for editing assistance and mentorship. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[128,544,220,558]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"REFERENCES ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[128,563,936,904]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. Collins FS, Varmus H: A new initiative on precision medicine. N Engl J Med 372:793-795, 2015 \",\"2. Freedman AN, Klabunde CN, Wiant K, et al: Use of next-generation sequencing tests to guide cancer treatment: Results from a nationally representative survey of oncologists in the United States. JCO Precis Oncol 7:1-13, 2018 \",\"3. Oishi SM, Marshall N, Hamilton AB, et al: Assessing multilevel determinants of adoption and implementation of genomic medicine: An organizational mixed-methods approach. Genet Med 17:919-926, 2015 \",\"4. El-Deiry WS, Goldberg RM, Lenz HJ, et al: The current state of molecular testing in the treatment of patients with solid tumors, 2019. CA Cancer J Clin 69:305-343, 2019 \",\"5. Kwa M, Makris A, Esteva FJ: Clinical utility of gene-expression signatures in early stage breast cancer. Nat Rev Clin Oncol 14:595-610, 2017 \",\"6. Manolio TA, Chisholm RL, Ozenberger B, et al: Implementing genomic medicine in the clinic: The future is here. Genet Med 15:258-267, 2013 \",\"7. Heeringa J, Mutti A, Furukawa MF, et al: Horizontal and vertical integration of health care providers: A framework for understanding various provider organizational structures. Int J Integr Care 20:2, 2020 \",\"8. Kirkwood MK, Hanley A, Bruinooge SS, et al: The state of oncology practice in America, 2018: Results of the ASCO practice census survey. J Oncol Pract 14:e412-e420, 2018 \",\"9. National Cancer Institute: NCI designated cancer centers. https://www.cancer.gov/research/infrastructure/cancer-centers \",\"10. Gray SW, Kim B, Sholl L, et al: Medical oncologists' experiences in using genomic testing for lung and colorectal cancer care. JCO Oncol Pract 13:e185-e196, 2017 \",\"11. Gutierrez ME, Choi K, Lanman RB, et al: Genomic profiling of advanced non-small cell lung cancer in community settings: Gaps and opportunities. Clin Lung Cancer 18:651-659, 2017 \",\"12. Levit LA, Kim ES, McAneny BL, et al: Implementing precision medicine in community-based oncology programs: Three models. JCO Oncol Pract 15:325-329, 2019 \",\"13. Sperber NR, Carpenter JS, Cavallari LH, et al: Challenges and strategies for implementing genomic services in diverse settings: Experiences from the Implementing Genomics In pracTicE (IGNITE) network. BMC Med Genomics 10:35, 2017 \",\"14. Fogleman AJ, Zahnd WE, Lipka AE, et al: Knowledge, attitudes, and perceived barriers towards genetic testing across three rural Illinois communities. J Community Genet 10:417-423, 2019 \",\"15. Salloum RG, George TJ, Silver N, et al: Rural-urban and racial-ethnic differences in awareness of direct-to-consumer genetic testing. BMC Public Health 18:277, 2018 \",\"16. Slonim AD, See H, Slonim S: Challenges confronting rural hospitals accentuated during COVID-19. J Biomed Res 34:397-409, 2020 \"]","page_idx":"7","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[380,21,614,33]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Distribution of Genomic Testing Resources ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[59,934,186,946]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"JCO Precision Oncology ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[905,934,936,945]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1067 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[227,971,768,994]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[130,52,936,604]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"17. Charlton M, Schlichting J, Chioreso C, et al: Challenges of rural cancer care in the United States. Oncology (Williston Park) 29:633-640, 2015 \",\"18. Horowitz CR, Orlando LA, Slavotinek AM, et al: The genomic medicine integrative research framework: A conceptual framework for conducting genomic medicine research. Am J Hum Genet 104:1088-1096, 2019 \",\"19. Pina IL, Cohen PD, Larson DB, et al: A framework for describing health care delivery organizations and systems. Am J Public Health 105:670-679, 2015 \",\"20. Agency for Healthcare Research and Quality: Defining Health Systems, 2017. https://www.ahrq.gov/chsp/chsp-reports/resources-for-understanding-health-systems/defining-health-systems.html \",\"21. United States Department of Agriculture: Rural Urban Continuum Codes Documentation, 2019. https://www.ers.usda.gov/data-products/rural-urban-continuum-codes/documentation/ \",\"22. Yabroff KR, Zhao J, de Moor JS, et al: Factors associated with oncologist discussions of the costs of genomic testing and related treatments. J Natl Cancer Inst 112:498-506, 2020 \",\"23. Amato L, Fusco D, Acamora A, et al: Volume and health outcomes: Evidence from systematic reviews and from evaluation of Italian hospital data. Epidemiol Prev 41:1-128, 2017 (suppl 2) \",\"24. Healy MA, Pradarelli JC, Krell RW, et al: Insurance status and hospital payer mix are linked with variation in metastatic site resection in patients with advanced colorectal cancers. Dis Colon Rectum 59:1047-1054, 2016 \",\"25. Ihemelandu C, Zheng C, Hall E, et al: Multimorbidity and access to major cancer surgery at high-volume hospitals in a regionalized era. Am J Surg 211:697-702, 2016 \",\"26. Nasuti JF, Gupta PK, Baloch ZW: Diagnostic value and cost-effectiveness of on-site evaluation of fine-needle aspiration specimens: Review of 5,688 cases. Diagn Cytopathol 27:1-4, 2002 \",\"27. Selhi PK, Tyagi R, Bansal P, et al: Hepatic fine-needle aspiration cytology: The role of rapid on-site evaluation in the assessment of hepatic lesions. Turk J Gastroenterol 29:442-447, 2018 \",\"28. Amendola LM, Berg JS, Horowitz CR, et al: The clinical sequencing evidence-generating research consortium: Integrating genomic sequencing in diverse and medically underserved populations. Am J Hum Genet 103:319-327, 2018 \",\"29. Luchini C, Lawlor RT, Milella M, et al: Molecular tumor boards in clinical practice. Trends Cancer 6:738-744, 2020 \",\"30. Van Der Velden DL, Van Herpen CML, Van Laarhoven HWM, et al: Molecular tumor boards: Current practice and future needs. Ann Oncol 28:3070-3075, 2017 \",\"31. Reisman M: EHRs: The challenge of making electronic data usable and interoperable. P T 42:572-575, 2017 \",\"32. Warner JL, Jain SK, Levy MA: Integrating cancer genomic data into electronic health records. Genome Med 8:113, 2016 \",\"33. Williams MS, Taylor CO, Walton NA, et al: Genomic information for clinicians in the electronic health record: Lessons learned from the clinical genome resource project and the electronic medical records and genomics network. Front Genet 10:1059, 2019 \",\"34. Lee D, Fiack KJ, Knapp KM: A profile of solo/two-physician practices. J Health Hum Serv Adm 36:297-322, 2014 \",\"35. Kaplan B, Caddle-Steele C, Chisholm G, et al: A culture of understanding: Reflections and suggestions from a genomics research community board. Prog Community Health Partnersh 11:161-165, 2017 \",\"36. Melas M, Subbiah S, Saadat S, et al: The community oncology and academic medical center alliance in the age of precision medicine: Cancer genetics and genomics considerations. J Clin Med 9:2125, 2020 \",\"37. Dzau VJ, Ackerly DC, Sutton-Wallace P, et al: The role of academic health science systems in the transformation of medicine. Lancet 375:949-953, 2010 \",\"38. Kaufman BG, Thomas SR, Randolph RK, et al: The rising rate of rural hospital closures. J Rural Health 32:35-43, 2016 \",\"39. United States Department of Health and Human Services: Rural action plan. https://www.hhs.gov/sites/default/files/hhs-rural-action-plan.pdf \",\"40. Gray SW, Hicks-Courant K, Cronin A, et al: Physicians' attitudes about multiplex tumor genomic testing. J Clin Oncol 32:1317-1323, 2014 \",\"41. Wideroff L, Freedman AN, Olson L, et al: Physician use of genetic testing for cancer susceptibility: Results of a national survey. Cancer Epidemiol Biomarkers Prev 12:295-303, 2003 \",\"42. Centers for Medicare & Medicaid Services: Decision memo for next generation sequencing (NGS) for medicare beneficiaries with advanced cancer (CAG00450N). https://www.cms.gov/medicare-coverage-database/details/nca-decision-memo.aspx?NCAId=290 \",\"43. Burris HA, Saltz LB, Yu PP: Assessing the value of next-generation sequencing tests in a dynamic environment. Am Soc Clin Oncol Ed Book 38:139-146, 2018 \",\"44. Klein RD: Current policy challenges in genomic medicine. Clin Chem 66:61-67, 2020 \"]","page_idx":"8","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[460,21,537,31]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Gardner et al ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[61,934,90,945]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1068 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[95,934,366,946]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"© 2021 by American Society of Clinical Oncology ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[229,971,768,994]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. ","text_format":null,"text_level":null,"type":"footer"}],"string":"[\n {\n \"bbox\": \"[193,56,902,154]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Distribution of Genomic Testing Resources by Oncology Practice and Rurality: A Nationally Representative Study \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[190,159,854,187]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Brittany Gardner, BA<sup>1</sup>; Michelle Doose, PhD, MPH<sup>2</sup>; Janet I. Sanchez, PhD, MPH<sup>2</sup>; Andrew N. Freedman, PhD<sup>3</sup>; and Janet S. de Moor, PhD, MPH<sup>4</sup> \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[190,251,936,297]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PURPOSE Oncologists are increasingly using molecular profiling to inform personalized patient treatment decisions. Despite its promising utility, the integration of genomic testing into diverse clinical health care settings across geographic settings has been understudied. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[190,301,936,393]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"METHODS We used data from the National Survey of Precision Medicine in Cancer Treatment, a nationally representative sample of practicing US oncologists, to assess the availability of six genomic testing resources, including on-site pathology, contracts with outside laboratories, on-site genetic counselors, internal policies or protocols for using genomic and biomarker testing, electronic medical record alerts, and genomic or molecular tumor boards. We used multivariate logistic regression models to examine differences in the availability of each genomic testing resource by practice type and rurality while adjusting for payer mix and patient volume. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[190,398,936,475]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RESULTS A larger proportion of multispecialty group and academic practices had genomic testing resources available compared with solo and nonacademic practices. Electronic medical record alerts were the least available resource, whereas contracts with outside laboratories were the most available resource. Compared with urban practices, there were significantly fewer practices located in rural areas that had on-site pathology, on-site genetic counselors, protocols for genomic tests, and molecular tumor boards. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[190,480,936,526]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CONCLUSION Genomic testing resources varied by practice type and geography among a nationally representative sample of practicing oncologists. This variation has important implications for the development of interventions and policies to support the more equitable delivery of precision oncology to patients with cancer. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[190,531,635,546]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JCO Precis Oncol 5:1060-1068. © 2021 by American Society of Clinical Oncology \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[191,566,297,579]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"INTRODUCTION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[190,586,558,823]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Oncologists are increasingly using next-generation sequencing (NGS) and gene-expression profiling to inform personalized treatment decisions. Often referred to as precision oncology, these genomic tests include single-gene and multigene tumor panels that require access to advanced pathology, laboratory, personnel, and organizational resources. For example, precision oncology approaches can use NGS on molecularly heterogeneous tumor specimens to inform the use of molecularly targeted therapy, assist in identifying patients who may benefit from less invasive treatments, and reduce overtreatment of others. However, despite the promising utility of precision oncology, the integration of genomic testing into health care settings is challenging. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[190,829,559,906]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Patients with cancer are diagnosed and treated in various health care settings that are diverse in organizational structure.7-9 Growing evidence suggests that the delivery of precision oncology varies by practice structure and other characteristics. For example, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[568,566,936,841]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"oncolologists practicing in nonprofit integrated health systems used genomic testing for colorectal and lung cancer at lower rates than those in hospital-based or single-specialty practices. $^{10}$ Providers that treat more than 50 unique patients per month and have access to a molecular tumor board were more likely to report using NGS tests than providers who did not. $^{2}$ Additionally, community settings may be more likely to experience challenges in delivering guideline-recommended genomic testing, including long turnaround times to receive genomic test results and delayed coordination in the handling of tissue samples. $^{11}$ These findings may be explained by the clinical, financial, and administrative challenges of implementing genomic services into practice. $^{12,13}$ Understanding the availability of resources at the organizational level is a first step to identifying inequities in the utilization of precision oncology. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[568,846,936,906]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Rural oncology practices may also face particular challenges in the delivery of precision oncology. $^{14,15}$ Additionally, rural hospitals face closures, aging infrastructures, payment challenges, workforce shortages, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[58,750,181,826]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Author affiliations and support information (if applicable) appear at the end of this article. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[58,829,181,906]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Accepted on May 11, 2021 and published at ascopubs.org/journal/ p on June 22, 2021: DOI https://doi.org/10. 1200/P0.21.00109 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[825,17,857,40]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"# \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[861,21,989,35]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Check for updates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[196,38,337,52]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PRECISION MEDICINE \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[150,57,183,204]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"sroedr nui8n \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"aside_text\"\n },\n {\n \"bbox\": \"[157,251,183,326]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"# \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"aside_text\"\n },\n {\n \"bbox\": \"[78,932,159,951]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ASCO \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[717,932,936,951]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JCO® Precision Oncology \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[61,957,90,966]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1060 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[227,970,768,994]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[80,74,145,86]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CONTEXT \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,94,174,108]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Key Objective \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,109,919,185]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Next-generation sequencing is increasingly being used to inform personalized cancer treatment decisions; however, its availability across diverse health care settings and geographic settings is understudied. Does the availability of genomic testing resources—on-site pathology, contracts with outside laboratories, on-site genetic counselors, internal genomic testing policies and protocols, electronic medical record alerts, and genomic or molecular tumor boards—vary by practice type and rural geographic location? \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,186,229,200]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Knowledge Generated \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,202,919,262]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Using a nationally representative sample of oncologists, electronic medical record alerts were the least available resource (16.8%) and contracts with outside laboratories (85.6%) were the most available resource. Overall, a smaller proportion of solo and nonacademic practices and practices located in rural settings had these genomic testing resources compared with multispecialty group and academic practices and practices in urban settings. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,263,152,276]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Relevance \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,277,917,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"These findings highlight opportunities to advance the equitable delivery of personalized medicine for patients with cancer by understanding and addressing the health care delivery context. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,356,527,476]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"quality challenges, and limited access to capital. $^{16}$ Patients with cancer residing in rural areas are also more likely to have limited access to specialty providers and targeted cancer treatments that may be informed by genomic testing. $^{17}$ Although few studies have documented challenges to precision oncology in rural settings, it is unknown whether genomic testing resource availability varies across geographical areas. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,484,527,698]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As the field of precision oncology moves forward, understanding how genomic testing resources and infrastructure vary across diverse clinical contexts and settings (eg, rural and urban and academic and nonacademic) will be informative for building capacity for delivering equitable and evidence-based cancer care.[18] To begin to address the limited knowledge about the organizational contexts that influence the delivery of precision medicine, this study examined the availability of genomic testing resources, including on-site pathology, protocols for genomic test use, electronic medical record (EMR) alerts for genomic tests, and molecular tumor boards, across diverse oncology practice settings using a nationally representative sample of oncologists. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,707,198,719]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"METHODS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,727,376,742]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data Source and Study Population \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,746,527,884]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We used data from the 2017 National Survey of Precision Medicine in Cancer Treatment, a nationally representative survey of oncologists, hematologists-oncologists, and hematologists. The survey was sponsored by the National Cancer Institute, the National Human Genomic Research Institute, and the American Cancer Society. Survey development, design, weighting, and methods were described in previous work,[2] and all survey questions were self-reported by oncologists. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,356,847,386]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Outcome Measure—Availability of Genomic Testing Resources \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,391,938,590]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Availability of genomic testing resources was measured using six questions asking about the availability of various genomic testing resources in an oncologist's primary practice: (1) on-site pathology, (2) contracts with outside laboratories to perform tests not available on-site, (3) on-site genetic counselors, (4) internal policies or protocols for use of genomic and biomarker testing, (5) an EMR that alerts providers when a genomic test is recommended for a particular patient or before ordering a particular drug, and (6) genomic or molecular tumor boards. Responses for each question included yes, no, or don't know. We categorized responses as a binary outcome of yes and no (including don't know and missing responses). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,604,856,620]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Main Predictors—Practice Type and Rurality \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[535,625,938,884]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Practice type was measured with two survey items. The first question asked oncologists to indicate if their main practice was a solo practice, single-specialty group, multispecialty group, or other type of practice. The second question asked whether their main practice was academically affiliated (yes or no). We then categorized the two responses into six mutually exclusive practice types: (1) solo office academic, (2) single-specialty group academic, (3) multispecialty group academic, (4) solo office nonacademic, (5) single-specialty group nonacademic, and (6) multispecialty group nonacademic. Oncologists whose main practice was other academic or other nonacademic were excluded from the analyses since we could not ascertain the exact type of practice setting. There is no framework for categorizing health systems and practice types.[19] However, we operationalized practice type to capture the potential influence that academic affiliation and practice complexity may have \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[381,21,614,33]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Distribution of Genomic Testing Resources \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[59,934,186,946]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JCO Precision Oncology \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[907,934,934,945]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1061 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[227,971,768,994]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[126,52,527,113]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"on the availability of cancer resources on the basis of prior literature and definitions from the Compendium of U.S. Health Systems (2016). These definitions are presented in Table 1. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,118,527,210]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Rurality was ascertained using each respondent's practice zip code linked to the 2013 Rural-Urban Continuum Codes to identify RUCC (1-9). Each practice address, including zip code, was obtained from the American Medical Association file and verified by phone. RUCC were categorized as urban (1-3) and rural (4-9).[21] \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,219,433,235]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Covariates—Patient Volume and Payer Mix \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,240,529,378]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Patient volume was identified from one item asking oncologists to indicate how many unique patients they see for evaluation or treatment each month. We dichotomized patient volume as $< 100$ and $\\\\geq 100$ patients consistent with the literature.[22] Payer mix was identified from one item asking respondents to indicate what proportions of their patients were insured by Medicaid, or self-pay or uninsured. We dichotomized payer mix into $< 10\\\\%$ and $\\\\geq 10\\\\%$ Medicaid, or self-pay or uninsured.[22] \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,386,265,401]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Statistical Analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,406,527,698]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Descriptive statistics are presented to describe the sample and weighted percent of practicing oncologists. We examined the weighted percent of practicing oncologists who reported genomic resources in their practice by their primary practice type (Fig 1). We also examined differences in the proportion of practices with genomic testing resources by geographic area on the basis of RUCC (Fig 2). Chi-square tests were used to determine statistical significance for both figures. Multivariate logistic regression models were conducted to examine differences in each genomic testing resource by practice type and rurality while adjusting for known practice-level confounders (ie, patient volume and payer mix).[23-25] Adjusted odds ratios (aORs) and $95\\\\%$ CIs for the survey weighted population estimates are presented, and statistical significance for all analyses was determined at the $P < .05$ level (two-sided). Analyses were weighted to account for the probability sampling approach and survey nonresponse. All analyses were performed using SAS version 9.4 (SAS Institute, Cary, NC). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,52,604,66]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RESULTS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[535,72,938,210]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Characteristics of the 1,281 oncologists who completed the survey are summarized in Table 2. The mean age of responding oncologists was 50.2 years, and most were male (66.6%) and non-Hispanic White (62.1%). About two thirds (63.1%) of the oncologists' primary practices were academically affiliated. Most respondents practiced in single-specialty (41.8%) and multispecialty (44.8%) group practices. Geographically, almost all (95.7%) practiced in urban settings. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[535,216,938,445]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Genomic testing resources differed by practice setting. In general, a higher percentage of oncologists in multi-specialty group practices and academically affiliated practices reported having genomic testing resources available than oncologists in solo, single-specialty, or nonacademic types of practices (Fig 1). Across all practice types, EMR alerts were the least available type of genomic testing resource and contracts with outside laboratories were the most available resource. Compared with practices located in urban areas, there were significantly fewer practices located in rural areas that had on-site pathology, on-site genetic counselors, protocols for genomic tests, and molecular tumor boards (Fig 2). There were no significant differences between urban and rural settings in the availability of contracts with outside laboratories and EMR alerts. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,450,938,496]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Table 3 presents the multivariate regression models of the six genomic testing resources by primary practice type and rurality adjusting for patient volume and payer mix. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,506,669,521]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On-Site Pathology \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[535,526,938,618]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A total of $69.9\\\\%$ of oncologists reported that their primary practice had on-site pathology. In the adjusted model, multispecialty group academic practices were significantly more likely to have on-site pathology compared with other practice types. Onsite pathology did not differ by rurality (aOR, 1.14; $95\\\\%$ CI, 0.56 to 2.32). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,627,797,643]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Contracts With Outside Laboratories \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[535,648,938,695]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A total of $85.6\\\\%$ of oncologists reported that their primary practice had contracts with outside laboratories. Single-specialty group nonacademic practices (aOR, 0.58; 95% \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[59,746,936,906]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0232/ecf5bb634c5fdab10b2ddd5c8631e9cff6a18d5e8ae53b970d656c51003f514a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Solo academic practice</td><td>One provider in a single practice that is affiliated with an academic institution</td></tr><tr><td>Single-specialty group academic practice</td><td>More than one provider in a single, academically affiliated practice providing services for one medical specialty</td></tr><tr><td>Multispecialty group academic practice</td><td>More than one provider in a single, academically affiliated practice providing services for more than one medical specialty</td></tr><tr><td>Solo nonacademic practice</td><td>One provider in a single practice that is not affiliated with an academic institution</td></tr><tr><td>Single-specialty group nonacademic practice</td><td>More than one provider in a single practice that is not academically affiliated providing services for one medical specialty</td></tr><tr><td>Multispecialty group nonacademic practice</td><td>More than one provider in a single practice that is not academically affiliated providing services for more than one medical specialty</td></tr></table>\",\n \"table_caption\": \"[\\\"TABLE 1. Oncologists' Practice Types and Operationalized Definitions \\\",\\\"Practice Type \\\",\\\"Operationalized Definition of Practice Type \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[460,21,537,33]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Gardner et al \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[59,934,90,945]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1062 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[95,933,366,947]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"© 2021 by American Society of Clinical Oncology \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[227,971,768,994]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[297,54,934,402]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIG 1. Percentage of practices with genomic testing resources by oncology practice type. On the basis of chi-square test: $^{**}P < .01$ , $^{***}P < .001$ . The y-axis represents the weighted percent of oncologists with availability of each resource at their primary practice. EMR, electronic medical record. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0730/ced3f1d520de7182ea9ea287f3a72f47004da8042464f1f7a41be604142b5b43.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[126,425,529,519]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cl, 0.40 to 0.90) and solo nonacademic practices (aOR, 0.29; $95\\\\%$ Cl, 0.13 to 0.65) were significantly less likely to have contracts with outside laboratories compared with multispecialty group academic practices. There were no significant differences in contracts with outside laboratories by rurality (aOR, 2.10; $95\\\\%$ Cl, 0.83 to 5.31). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[539,425,736,441]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On-Site Genetic Counselors \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,453,938,516]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A total of $67.3\\\\%$ of oncologists reported that their primary practice had on-site genetic counselors. Multispecialty academic groups were significantly more likely to have on-site genetic counselors compared with the other five \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[297,544,936,883]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIG 2. Percentage of practices with genomic testing resources by rurality. On the basis of chi-square test: $^{*}P < .05$ , $^{***}P < .001$ . The y-axis represents the weighted percent of oncologists with availability of each resource at their primary practice. EMR, electronic medical record. \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0001/5e4a679fe7d524f05151c90dc137302b29ff6862e2bb303084c4ae3284df563c.jpg\",\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[381,20,616,34]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Distribution of Genomic Testing Resources \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[59,934,188,947]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JCO Precision Oncology \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[907,934,936,945]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1063 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[229,971,768,994]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[59,80,527,687]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0285/9d3220b85c3851fc714dbf90414350e0757781e36dd9db5cf017a437caf02d24.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td>Characteristic</td><td>N (weighted %)</td></tr><tr><td>Provider</td><td></td></tr><tr><td>Age (mean), years</td><td>50.16</td></tr><tr><td>30-39</td><td>271 (21.65)</td></tr><tr><td>40-49</td><td>391 (31.13)</td></tr><tr><td>50-59</td><td>304 (23.53)</td></tr><tr><td>≥ 60</td><td>315 (23.69)</td></tr><tr><td>Sex</td><td></td></tr><tr><td>Female</td><td>353 (33.43)</td></tr><tr><td>Male</td><td>928 (66.57)</td></tr><tr><td>Race and ethnicity</td><td></td></tr><tr><td>Non-Hispanic White</td><td>804 (62.13)</td></tr><tr><td>Asian</td><td>362 (29.39)</td></tr><tr><td>Black or African American</td><td>27 (2.12)</td></tr><tr><td>Hispanic</td><td>64 (4.82)</td></tr><tr><td>Practice</td><td></td></tr><tr><td>Primary practice affiliated with academic institution</td><td></td></tr><tr><td>No</td><td>478 (36.94)</td></tr><tr><td>Yes</td><td>796 (63.06)</td></tr><tr><td>Practice category</td><td></td></tr><tr><td>Solo</td><td>55 (4.30)</td></tr><tr><td>Single-specialty group</td><td>544 (41.83)</td></tr><tr><td>Multispecialty group</td><td>566 (44.78)</td></tr><tr><td>Others</td><td>110 (9.09)</td></tr><tr><td>Practice type</td><td></td></tr><tr><td>Solo academic</td><td>16 (1.18)</td></tr><tr><td>Single-specialty group academic</td><td>258 (20.58)</td></tr><tr><td>Multispecialty group academic</td><td>430 (33.96)</td></tr><tr><td>Solo nonacademic</td><td>39 (3.14)</td></tr><tr><td>Single-specialty group nonacademic</td><td>282 (21.13)</td></tr><tr><td>Multispecialty group nonacademic</td><td>136 (11.05)</td></tr><tr><td>Rurality</td><td></td></tr><tr><td>Urban</td><td>1,224 (95.65)</td></tr><tr><td>Rural</td><td>57 (4.35)</td></tr></table>\",\n \"table_caption\": \"[\\\"TABLE 2. Characteristics of Oncologists and Their Primary Practice Setting in the United States ${\\\\\\\\left( \\\\\\\\mathrm{N} = 1,{281}\\\\\\\\right) }^{\\\\\\\\mathrm{a}}$ \\\"]\",\n \"table_footnote\": \"[\\\"aPercentages weighted to account for complex survey design and nonresponse. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[126,758,527,819]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"practice types. The odds of having on-site genetic counselors were $70\\\\%$ lower among rural practices compared with urban practices (aOR, 0.30; $95\\\\%$ CI, 0.14 to 0.63), which was statistically significant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,826,390,838]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Internal Protocols for Genomic Tests \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,846,527,907]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A total of $48.1\\\\%$ of oncologists reported that their primary practice had internal protocols for genomic tests. Multispecialty group academic practices were significantly more likely to have internal protocols for genomic tests compared \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,52,936,127]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"with all other practices, including multispecialty group nonacademic practices (aOR, 0.24; $95\\\\%$ CI, 0.15 to 0.36). There were no significant differences in internal protocols for genomic tests by rurality (aOR, 0.54; $95\\\\%$ CI, 0.27 to 1.07). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,136,756,150]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"EMR Alerts for Genomic Tests \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[535,156,936,278]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A total of $16.8\\\\%$ of oncologists reported that their primary practice had EMR alerts for genomic tests. The odds of having EMR alerts in single-specialty group academic practice were significantly lower compared with multispecialty group academic practices (aOR, 0.52; $95\\\\%$ CI, 0.32 to 0.83). There were no significant differences in EMR alerts for genomic testing by other practice types and rurality (aOR, 0.46; $95\\\\%$ CI, 0.16 to 1.31). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,287,708,301]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Molecular Tumor Board \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[535,306,936,445]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A total of $36.3\\\\%$ of oncologists reported that their primary practice had a molecular tumor board. Multispecialty group academic practices were significantly more likely to have molecular tumor boards compared with all other practice types, including multispecialty group nonacademic practices (aOR, 0.14; $95\\\\%$ CI, 0.08 to 0.24). Rural practices had $80\\\\%$ lower odds of having molecular tumor boards compared with practices in urban settings (aOR, 0.20; $95\\\\%$ CI, 0.06 to 0.64), which was statistically significant. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,453,628,466]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DISCUSSION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[535,473,936,734]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Oncologists are increasingly using molecular profiling to inform cancer treatment decisions. Our study suggests that there are variations in the availability of genomic testing resources by the oncology practice type and geographic setting. Across practice types, contracts with outside laboratories and on-site pathology were the most commonly available genomic testing resources. Yet on-site pathology was notably less available in rural practice settings. These resources are often used to provide standard of care in most health care settings, with the latter being a cost-effective approach for providing immediate and accurate identification of tumor staging that can be used for the development of cancer treatment plans.[26,27] Overall, our findings highlight an opportunity to advance the delivery of personalized medicine for patients with cancer by understanding the health care setting context of genomic testing resource availability. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[535,739,936,906]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conversely, protocols for genomic testing, molecular tumor boards, and EMR alerts were the least available genomic resources across all practice types. Established policies and protocols for the use of genomic tests are a critical step for assisting providers with testing platform selection and decision making. However, research has shown that the establishment of such protocols and processes is challenging, requiring organizational buy-in, capital, resources, and involvement of multiple levels of clinical and non-clinical stakeholders.[3,28] Although multidisciplinary molecular tumor boards can optimize the translation of \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[460,21,537,33]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Gardner et al \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[59,934,90,945]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1064 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[95,933,366,947]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"© 2021 by American Society of Clinical Oncology \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[227,971,768,994]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[59,66,926,414]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0312/6093785f04a38915006bbe538a22ce2a2ea78c865b676b2183a064ce37ade163.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=\\\"2\\\">Characteristic</td><td>On-Site Pathology</td><td>Contracts With Outside Laboratories</td><td>On-Site Genetic Counselors</td></tr><tr><td>aOR (95% CI)</td><td>aOR (95% CI)</td><td>aOR (95% CI)</td></tr><tr><td colspan=\\\"4\\\">Primary practice type</td></tr><tr><td>Multispecialty group academic</td><td>Ref</td><td>Ref</td><td>Ref</td></tr><tr><td>Multispecialty group nonacademic</td><td>0.23 (0.14 to 0.38)</td><td>0.92 (0.50 to 1.72)</td><td>0.16 (0.10 to 0.26)</td></tr><tr><td>Single-specialty group academic</td><td>0.39 (0.26 to 0.60)</td><td>0.83 (0.52 to 1.32)</td><td>0.32 (0.22 to 0.49)</td></tr><tr><td>Single-specialty group nonacademic</td><td>0.1 (0.07 to 0.15)</td><td>0.58 (0.40 to 0.90)</td><td>0.12 (0.08 to 0.18)</td></tr><tr><td>Solo academic</td><td>0.06 (0.02 to 0.18)</td><td>0.32 (0.10 to 1.04)</td><td>0.08 (0.03 to 0.27)</td></tr><tr><td>Solo nonacademic</td><td>0.02 (0.01 to 0.05)</td><td>0.29 (0.13 to 0.65)</td><td>0.01 (0.003 to 0.04)</td></tr><tr><td colspan=\\\"4\\\">Rurality</td></tr><tr><td>Urban</td><td>Ref</td><td>Ref</td><td>Ref</td></tr><tr><td>Rural</td><td>1.14 (0.56 to 2.32)</td><td>2.1 (0.83 to 5.31)</td><td>0.3 (0.14 to 0.63)</td></tr><tr><td colspan=\\\"4\\\">Patient volume</td></tr><tr><td>< 100</td><td>Ref</td><td>Ref</td><td>Ref</td></tr><tr><td>≥ 100</td><td>0.67 (0.50 to 0.91)</td><td>0.77 (0.54 to 1.10)</td><td>0.82 (0.61 to 1.09)</td></tr><tr><td colspan=\\\"4\\\">Payer mix</td></tr><tr><td>Self-pay or uninsured, or \\nMedicaid < 10%</td><td>Ref</td><td>Ref</td><td>Ref</td></tr><tr><td>Self-pay or uninsured, or \\nMedicaid ≥ 10%</td><td>1.4 (1.02 to 1.93)</td><td>0.89 (0.61 to 1.31)</td><td>1.11 (0.81 to 1.52)</td></tr></table>\",\n \"table_caption\": \"[\\\"TABLE 3. Multivariate Regression Models of Genomic Testing Resources by Primary Practice Type and Rurality \\\"]\",\n \"table_footnote\": \"[]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[58,421,926,785]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0454/246b9a822a029884e03626c73d780746ede532f68815ae6a5b82088ba9dc4ab7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td></td><td colspan=\\\"2\\\">Internal Policies and Protocols for \\nGenomic Tests</td><td>EMR Alerts for Genomic Tests</td><td>Genomic or Molecular Tumor Board</td></tr><tr><td></td><td>aOR (95% CI)</td><td>aOR (95% CI)</td><td>aOR (95% CI)</td><td></td></tr><tr><td colspan=\\\"5\\\">Primary practice type</td></tr><tr><td>Multispecialty group academic</td><td>Ref</td><td>Ref</td><td>Ref</td><td></td></tr><tr><td>Multispecialty group nonacademic</td><td>0.24 (0.15 to 0.36)</td><td>0.65 (0.37 to 1.13)</td><td>0.14 (0.08 to 0.24)</td><td></td></tr><tr><td>Single-specialty group academic</td><td>0.48 (0.35 to 0.67)</td><td>0.52 (0.32 to 0.83)</td><td>0.35 (0.25 to 0.49)</td><td></td></tr><tr><td>Single-specialty group nonacademic</td><td>0.19 (0.14 to 0.27)</td><td>0.84 (0.56 to 1.25)</td><td>0.07 (0.04 to 0.11)</td><td></td></tr><tr><td>Solo academic</td><td>0.13 (0.03 to 0.53)</td><td>0.76 (0.16 to 3.59)</td><td>0.12 (0.03 to 0.62)</td><td></td></tr><tr><td>Solo nonacademic</td><td>0.11 (0.05 to 0.28)</td><td>0.72 (0.26 to 1.99)</td><td>0.16 (0.07 to 0.40)</td><td></td></tr><tr><td colspan=\\\"5\\\">Rurality</td></tr><tr><td>Urban</td><td>Ref</td><td>Ref</td><td>Ref</td><td></td></tr><tr><td>Rural</td><td>0.54 (0.27 to 1.07)</td><td>0.46 (0.16 to 1.31)</td><td>0.2 (0.06 to 0.64)</td><td></td></tr><tr><td colspan=\\\"5\\\">Patient volume</td></tr><tr><td>< 100</td><td>Ref</td><td>Ref</td><td>Ref</td><td></td></tr><tr><td>≥ 100</td><td>0.79 (0.61 to 1.01)</td><td>1.02 (0.74 to 1.41)</td><td>0.7 (0.53 to 0.92)</td><td></td></tr><tr><td colspan=\\\"5\\\">Payer mix</td></tr><tr><td>Self-pay or uninsured, or \\nMedicaid < 10%</td><td>Ref</td><td>Ref</td><td>Ref</td><td></td></tr><tr><td>Self-pay or uninsured, or \\nMedicaid ≥ 10%</td><td>0.75 (0.57 to 0.99)</td><td>1.14 (0.79 to 1.65)</td><td>0.94 (0.68 to 1.28)</td><td></td></tr></table>\",\n \"table_caption\": \"[]\",\n \"table_footnote\": \"[\\\"Abbreviations: aOR, adjusted odds ratio; EMR, electronic medical record; ref, reference. \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[126,829,527,905]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"complex NGS information into appropriate targeted therapy options for patients with cancer, $^{12}$ their development and integration within health care systems require additional institutional resources and capacity, which may not be available in smaller practices. $^{29,30}$ In clinical settings, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,829,936,905]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"information technologies and software are often used to offer automated clinical decision support to providers, but in our study, EMR alerts were the least available resource across all practice types. This could reflect the possible lack of interoperability between EMR software used across \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[380,21,614,33]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Distribution of Genomic Testing Resources \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[59,934,186,947]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JCO Precision Oncology \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[907,934,936,945]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1065 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[227,971,768,994]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[126,52,527,144]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"practice settings. $^{31}$ As more clinical practices move toward interoperable and standardized electronic records, EMRs offer a modifiable point of intervention to efficiently integrate genomic testing processes (eg, provider reminders to use genomic testing and standardized genomic testing information) and facilitate shared decision making. $^{32,33}$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,148,529,514]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Our study found that the availability of genomic testing resources varied by practice type. A larger proportion of practices that were a specialty group and/or academically affiliated had genomic testing resources compared with solo oncology practices and nonacademic practices. A smaller proportion of solo practices and nonacademic practices had any of the six genomic testing resources compared with specialty group and academic practices. For example, solo nonacademic practices had $89\\\\%$ lower odds of having protocols for genomic tests compared with multispecialty group academic practices. These findings may suggest that solo, small practices face limited infrastructure, capital, staff, and other resources, whereas specialized clinical expertise and advanced technology may be concentrated in academic medical centers.[34] To address this gap, academic medical centers and larger health systems have been actively involved in targeted efforts to bring precision medicine to diverse health care settings.[35,36] For example, collaboration through community-academic partnerships and the development of decentralized, online resources have provided infrastructure for many patients in nonacademic practice settings to benefit from precision oncology advances.[35-37] \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,520,529,810]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Genomic testing resources also varied between urban and rural practices, such that a smaller proportion of rural practices had on-site genetic counselors, internal protocols for genomic tests, on-site pathology, and molecular tumor boards compared with urban practices. These findings are consistent with previous studies indicating limited availability of cancer care services in rural settings.[17] Further exacerbating limited access to genomic testing resources in rural areas are the financial pressures of the competitive market and lower patient volumes that have led to the closure of many small and/or rural hospitals.[38] Interventions that leverage policy changes that expand access to care may help address this critical gap in access to specialized care for rural patients with cancer. For example, recent changes to Medicare increase reimbursement for telehealth services, which expands access to much needed specialty care in rural communities.[39] Telehealth technologies, including virtual molecular tumor boards and appointments with genetic counselors, can be leveraged to \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,52,936,81]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"transcend the travel barriers that rural patients with cancer face in accessing genomic testing resources. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,86,938,500]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This study is not without limitations. The cooperation rate $(38\\\\%)$ for the National Survey of Precision Medicine in Cancer Treatment is lower than that of previous physician surveys on the topic of precision medicine.[40,41] Additionally, our findings reflect the availability of genomic testing resources in 2017 but the state of precision medicine is rapidly evolving. For example, in 2018, the Centers for Medicare & Medicaid Services announced the coverage of Food and Drug Administration-approved NGS tests for patients with advanced-stage cancer.[42] These broadscale policy decisions are ongoing and affect the cost, value, and availability of precision medicine resources.[43,44] Nonetheless, this study uses the most currently available data. Additionally, for the outcome measure, availability of genomic testing resources, don't know responses were included in the same category as no responses because they could not confirm the availability of a resource. This may cause the no category to be artificially inflated. The survey did not collect patient-level data (eg, clinical outcomes and cancer types for which oncologists used genomic testing resources), which is an area for future investigations. Regarding the statistical analysis, the number of oncologists practicing in rural areas was low, which can limit conclusions that can be drawn from this analysis. Future studies should oversample oncologists in rural areas and other underserved areas to better understand care delivery in underserved settings. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,505,938,810]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Despite these limitations, the survey used in this study is a unique resource for understanding access to precision oncology, especially for underserved populations. To our knowledge, this is the first study to examine the availability of genomic testing resources by practice type and geography among a nationally representative sample of oncologists. Our findings suggest that the availability of genomic testing resources vary by organizational structure and geographic location of where oncologists practice. This variation may affect the equitable access to and utilization of precision medicine for patients with cancer. These findings warrant further investigation to delineate the extent to which genomic testing utilization and patient outcomes are influenced by the availability of genomic resources at the health system level. Additionally, exploring ways to facilitate effective integration between smaller oncology practices and academic or integrated health systems will increase the accessibility of targeted cancer treatments to patients in diverse populations and ensure more equitable distribution of precision oncology resources. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[128,841,224,854]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"AFFILIATIONS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[128,856,520,882]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"<sup>1</sup>Division of Cancer Control and Population Sciences, National Cancer Institute, Rockville, MD \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,838,936,902]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$^{2}$ Healthcare Delivery Research Program, Division of Cancer Control and Population Sciences, National Cancer Institute, Rockville, MD $^{3}$ Epidemiology and Genomics Research Program, Division of Cancer Control and Population Sciences, National Cancer Institute, Rockville, MD \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[460,21,537,33]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Gardner et al \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[59,934,90,945]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1066 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[95,933,368,947]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"© 2021 by American Society of Clinical Oncology \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[227,971,768,994]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[128,52,527,76]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Healthcare Delivery Research Program, Division of Cancer Control and Population Sciences, National Cancer Institute, Rockville, MD \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[128,93,309,106]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CORRESPONDING AUTHOR \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,109,523,148]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Brittany Gardner, BA, Division of Cancer Control and Population Sciences, National Cancer Institute, 9609 Medical Center Dr, 4E556, Rockville, MD 20850; e-mail: brittany.gardner@.nih.gov. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[128,163,217,176]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DISCLAIMER \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,179,527,269]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This article was prepared as part of some of the authors' (J.S.D., A.N.F., and J.I.S.) official duties as employees of the US federal government. The findings and conclusions in this report are those of the authors and do not necessarily represent the official position of the National Cancer Institute. The funder had no role in the design of the study; the collection, analysis, and interpretation of the data; the writing of the manuscript; or the decision to submit the manuscript for publication. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[128,283,282,297]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PRIOR PRESENTATION \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,300,525,338]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Presented at the American Association for Cancer Research—The Science of Cancer Health Disparities in Racial/Ethnic Minorities and the Medically Underserved, Virtual Conference, October 2-4, 2020. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[128,354,196,367]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"SUPPORT \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,370,527,433]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This research was conducted by federal employees of the National Cancer Institute and the National Institutes of Health. Specific funding was not provided for this research. The survey on which this research is based was funded by the National Institutes of Health under contract number HHSN2612010000861 to RTI International. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[128,450,323,462]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DATA SHARING STATEMENT \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[126,465,515,491]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The data underlying this article cannot be shared publicly due to the privacy of individuals who participated in the study. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[539,52,715,65]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"AUTHOR CONTRIBUTIONS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[539,67,936,182]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"Conception and design: Brittany Gardner, Michelle Doose, Janet I. Sanchez, Janet S. de Moor \\\",\\\"Provision of study materials or patients: Andrew N. Freedman, Janet S. de Moor \\\",\\\"Collection and assembly of data: Andrew N. Freedman, Janet S. de Moor \\\\nData analysis and interpretation: All authors \\\",\\\"Manuscript writing: All authors \\\",\\\"Final approval of manuscript: All authors \\\",\\\"Accountable for all aspects of the work: All authors \\\"]\",\n \"page_idx\": \"7\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[537,198,914,226]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"AUTHORS' DISCLOSURES OF POTENTIAL CONFLICTS OF INTEREST \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,228,936,318]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The following represents disclosure information provided by the authors of this manuscript. All relationships are considered compensated unless otherwise noted. Relationships are self-held unless noted. I = Immediate Family Member, Inst = My Institution. Relationships may not relate to the subject matter of this manuscript. For more information about ASCO's conflict of interest policy, please refer to www.asco.org/rwc or ascopubs.org/po/author-center. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,319,936,356]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Open Payments is a public database containing information reported by companies about payments made to US-licensed physicians (Open Payments). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[539,364,633,375]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Janet S. de Moor \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[539,378,667,390]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Employment: Biogen (1) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[539,391,797,402]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Stock and Other Ownership Interests: Biogen (I) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[539,404,792,415]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Travel, Accommodations, Expenses: Biogen (1) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[539,423,837,436]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"No other potential conflicts of interest were reported. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[539,452,676,465]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ACKNOWLEDGMENT \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[537,468,936,506]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We want to give special thanks to Dr Sallie Weaver and Dr Shobha Srinivasan at the National Cancer Institute in the Division of Cancer Control and Population Sciences for editing assistance and mentorship. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[128,544,220,558]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"REFERENCES \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[128,563,936,904]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. Collins FS, Varmus H: A new initiative on precision medicine. N Engl J Med 372:793-795, 2015 \\\",\\\"2. Freedman AN, Klabunde CN, Wiant K, et al: Use of next-generation sequencing tests to guide cancer treatment: Results from a nationally representative survey of oncologists in the United States. JCO Precis Oncol 7:1-13, 2018 \\\",\\\"3. Oishi SM, Marshall N, Hamilton AB, et al: Assessing multilevel determinants of adoption and implementation of genomic medicine: An organizational mixed-methods approach. Genet Med 17:919-926, 2015 \\\",\\\"4. El-Deiry WS, Goldberg RM, Lenz HJ, et al: The current state of molecular testing in the treatment of patients with solid tumors, 2019. CA Cancer J Clin 69:305-343, 2019 \\\",\\\"5. Kwa M, Makris A, Esteva FJ: Clinical utility of gene-expression signatures in early stage breast cancer. Nat Rev Clin Oncol 14:595-610, 2017 \\\",\\\"6. Manolio TA, Chisholm RL, Ozenberger B, et al: Implementing genomic medicine in the clinic: The future is here. Genet Med 15:258-267, 2013 \\\",\\\"7. Heeringa J, Mutti A, Furukawa MF, et al: Horizontal and vertical integration of health care providers: A framework for understanding various provider organizational structures. Int J Integr Care 20:2, 2020 \\\",\\\"8. Kirkwood MK, Hanley A, Bruinooge SS, et al: The state of oncology practice in America, 2018: Results of the ASCO practice census survey. J Oncol Pract 14:e412-e420, 2018 \\\",\\\"9. National Cancer Institute: NCI designated cancer centers. https://www.cancer.gov/research/infrastructure/cancer-centers \\\",\\\"10. Gray SW, Kim B, Sholl L, et al: Medical oncologists' experiences in using genomic testing for lung and colorectal cancer care. JCO Oncol Pract 13:e185-e196, 2017 \\\",\\\"11. Gutierrez ME, Choi K, Lanman RB, et al: Genomic profiling of advanced non-small cell lung cancer in community settings: Gaps and opportunities. Clin Lung Cancer 18:651-659, 2017 \\\",\\\"12. Levit LA, Kim ES, McAneny BL, et al: Implementing precision medicine in community-based oncology programs: Three models. JCO Oncol Pract 15:325-329, 2019 \\\",\\\"13. Sperber NR, Carpenter JS, Cavallari LH, et al: Challenges and strategies for implementing genomic services in diverse settings: Experiences from the Implementing Genomics In pracTicE (IGNITE) network. BMC Med Genomics 10:35, 2017 \\\",\\\"14. Fogleman AJ, Zahnd WE, Lipka AE, et al: Knowledge, attitudes, and perceived barriers towards genetic testing across three rural Illinois communities. J Community Genet 10:417-423, 2019 \\\",\\\"15. Salloum RG, George TJ, Silver N, et al: Rural-urban and racial-ethnic differences in awareness of direct-to-consumer genetic testing. BMC Public Health 18:277, 2018 \\\",\\\"16. Slonim AD, See H, Slonim S: Challenges confronting rural hospitals accentuated during COVID-19. J Biomed Res 34:397-409, 2020 \\\"]\",\n \"page_idx\": \"7\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[380,21,614,33]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Distribution of Genomic Testing Resources \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[59,934,186,946]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"JCO Precision Oncology \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[905,934,936,945]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1067 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[227,971,768,994]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[130,52,936,604]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"17. Charlton M, Schlichting J, Chioreso C, et al: Challenges of rural cancer care in the United States. Oncology (Williston Park) 29:633-640, 2015 \\\",\\\"18. Horowitz CR, Orlando LA, Slavotinek AM, et al: The genomic medicine integrative research framework: A conceptual framework for conducting genomic medicine research. Am J Hum Genet 104:1088-1096, 2019 \\\",\\\"19. Pina IL, Cohen PD, Larson DB, et al: A framework for describing health care delivery organizations and systems. Am J Public Health 105:670-679, 2015 \\\",\\\"20. Agency for Healthcare Research and Quality: Defining Health Systems, 2017. https://www.ahrq.gov/chsp/chsp-reports/resources-for-understanding-health-systems/defining-health-systems.html \\\",\\\"21. United States Department of Agriculture: Rural Urban Continuum Codes Documentation, 2019. https://www.ers.usda.gov/data-products/rural-urban-continuum-codes/documentation/ \\\",\\\"22. Yabroff KR, Zhao J, de Moor JS, et al: Factors associated with oncologist discussions of the costs of genomic testing and related treatments. J Natl Cancer Inst 112:498-506, 2020 \\\",\\\"23. Amato L, Fusco D, Acamora A, et al: Volume and health outcomes: Evidence from systematic reviews and from evaluation of Italian hospital data. Epidemiol Prev 41:1-128, 2017 (suppl 2) \\\",\\\"24. Healy MA, Pradarelli JC, Krell RW, et al: Insurance status and hospital payer mix are linked with variation in metastatic site resection in patients with advanced colorectal cancers. Dis Colon Rectum 59:1047-1054, 2016 \\\",\\\"25. Ihemelandu C, Zheng C, Hall E, et al: Multimorbidity and access to major cancer surgery at high-volume hospitals in a regionalized era. Am J Surg 211:697-702, 2016 \\\",\\\"26. Nasuti JF, Gupta PK, Baloch ZW: Diagnostic value and cost-effectiveness of on-site evaluation of fine-needle aspiration specimens: Review of 5,688 cases. Diagn Cytopathol 27:1-4, 2002 \\\",\\\"27. Selhi PK, Tyagi R, Bansal P, et al: Hepatic fine-needle aspiration cytology: The role of rapid on-site evaluation in the assessment of hepatic lesions. Turk J Gastroenterol 29:442-447, 2018 \\\",\\\"28. Amendola LM, Berg JS, Horowitz CR, et al: The clinical sequencing evidence-generating research consortium: Integrating genomic sequencing in diverse and medically underserved populations. Am J Hum Genet 103:319-327, 2018 \\\",\\\"29. Luchini C, Lawlor RT, Milella M, et al: Molecular tumor boards in clinical practice. Trends Cancer 6:738-744, 2020 \\\",\\\"30. Van Der Velden DL, Van Herpen CML, Van Laarhoven HWM, et al: Molecular tumor boards: Current practice and future needs. Ann Oncol 28:3070-3075, 2017 \\\",\\\"31. Reisman M: EHRs: The challenge of making electronic data usable and interoperable. P T 42:572-575, 2017 \\\",\\\"32. Warner JL, Jain SK, Levy MA: Integrating cancer genomic data into electronic health records. Genome Med 8:113, 2016 \\\",\\\"33. Williams MS, Taylor CO, Walton NA, et al: Genomic information for clinicians in the electronic health record: Lessons learned from the clinical genome resource project and the electronic medical records and genomics network. Front Genet 10:1059, 2019 \\\",\\\"34. Lee D, Fiack KJ, Knapp KM: A profile of solo/two-physician practices. J Health Hum Serv Adm 36:297-322, 2014 \\\",\\\"35. Kaplan B, Caddle-Steele C, Chisholm G, et al: A culture of understanding: Reflections and suggestions from a genomics research community board. Prog Community Health Partnersh 11:161-165, 2017 \\\",\\\"36. Melas M, Subbiah S, Saadat S, et al: The community oncology and academic medical center alliance in the age of precision medicine: Cancer genetics and genomics considerations. J Clin Med 9:2125, 2020 \\\",\\\"37. Dzau VJ, Ackerly DC, Sutton-Wallace P, et al: The role of academic health science systems in the transformation of medicine. Lancet 375:949-953, 2010 \\\",\\\"38. Kaufman BG, Thomas SR, Randolph RK, et al: The rising rate of rural hospital closures. J Rural Health 32:35-43, 2016 \\\",\\\"39. United States Department of Health and Human Services: Rural action plan. https://www.hhs.gov/sites/default/files/hhs-rural-action-plan.pdf \\\",\\\"40. Gray SW, Hicks-Courant K, Cronin A, et al: Physicians' attitudes about multiplex tumor genomic testing. J Clin Oncol 32:1317-1323, 2014 \\\",\\\"41. Wideroff L, Freedman AN, Olson L, et al: Physician use of genetic testing for cancer susceptibility: Results of a national survey. Cancer Epidemiol Biomarkers Prev 12:295-303, 2003 \\\",\\\"42. Centers for Medicare & Medicaid Services: Decision memo for next generation sequencing (NGS) for medicare beneficiaries with advanced cancer (CAG00450N). https://www.cms.gov/medicare-coverage-database/details/nca-decision-memo.aspx?NCAId=290 \\\",\\\"43. Burris HA, Saltz LB, Yu PP: Assessing the value of next-generation sequencing tests in a dynamic environment. Am Soc Clin Oncol Ed Book 38:139-146, 2018 \\\",\\\"44. Klein RD: Current policy challenges in genomic medicine. Clin Chem 66:61-67, 2020 \\\"]\",\n \"page_idx\": \"8\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[460,21,537,31]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Gardner et al \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[61,934,90,945]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"1068 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[95,934,366,946]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"© 2021 by American Society of Clinical Oncology \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[229,971,768,994]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Downloaded from ascopubs.org by 218.70.106.126 on December 22, 2023 from 218.070.106.126 Copyright © 2023 American Society of Clinical Oncology. All rights reserved. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.1200/po.21.00109"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"en"},"sci_category":{"kind":"string","value":"Medicine and Health Sciences"},"sha256":{"kind":"string","value":"57454066d82c8ffeed38438a64f9f8e5471ab7b54d6b210199cdc0be0ea64fcb"},"title":{"kind":"string","value":"Distribution of Genomic Testing Resources by Oncology Practice and Rurality: A Nationally Representative Study"}}},{"rowIdx":72,"cells":{"abstract":{"kind":"string","value":""},"author":{"kind":"string","value":"Andrade, Mônica Viegas|Noronha, Kenya|Cardoso, Clareci Silva|Oliveira, Cláudia Di Lorenzo|Calazans, Júlia Almeida|Souza, Michelle Nepomuceno"},"content_list":{"kind":"list like","value":[{"bbox":"[67,126,670,215]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Análise da concordância entre as informações reportadas pelas mães e dos cartões de vacina dascriçças no Brasil (2013 e 2015) ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[65,224,596,288]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Analysis of the agreement between information on child vaccination reported by mothers and that on the children's vaccination cards in Brazil (2013-2015) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,303,653,331]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Mónica Viegas Andrade<sup>1</sup>, Kenya Noronha<sup>1</sup>, Clareci Silva Cardoso<sup>2</sup>, Cláudia Di Lorenzo Oliveira<sup>2</sup>, Júlia Almeida Calazans<sup>1</sup>, Michelle Nepomuceno Souza<sup>1</sup> ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,337,702,358]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$^{1}$ Faculdade de Ciências Econômicas - Centro de Desenvolvimento e Planejamento Regional (Cedeplar), Universidade Federal de Minas Gerais (UFMG) - Belo Horizonte (MG), Brasil ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[67,360,574,372]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$^{2}$ Departamento de Medicina, Universidade Federal de Sao Joao del-Rei - Sao Joao del-Rei (MG), Brasil ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,399,677,436]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Comounar Andrade MV, Noronha K, Cardoso CS, Oliveira CL, Calazans JA, Souza MN. Analise da concordancia entre as informacoes reportadas pelas mares e dos cartoes de vacina das crianças no Brasil (2013 e 2015). Cad Saude Colet, 2021;29esp.:40-50. https://doi.org/10.1590/1414-462X202199010302 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[67,462,126,476]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Resumo ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[65,481,715,520]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Introdução: Este trabalho discute a possibídale de'utilização de informações sobre a vacinação infantil reportada pelas mães em inquisitados domiciliares para a construção de指示ores de cobertura vacINAL no Brasil. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,522,714,549]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Objetivo: Avaliar o potencial das informações declaradas pelas mães sobre a imunização dascriçanas em inquisitços domicíliares como uma fonte para o calculo do indicator de cobertura vacinal. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,549,714,602]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Método: Analisaram-se os指示ores de confiabilidade (preciso) e validade (concordância) entre as informações disponveis nos cartões de vacina dascriçças menores de 2 anos e as informações reportadas pelas mães em pesquisas domicíliares realizadas nos anos de 2013 e 2015 no[municipio de Santo Antônio do Monte, Minas Gerais. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,604,714,644]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Resultados: O principal resultado minha que, em um(contexto de alta cobertura, as mães tendiam a informar sobre a vacinação de seuis filhos de forma aleatoria, ou sera, informações reportadas pelas mães aparevam baixa validade. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,645,714,684]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conclusao: Embora a coleta de informacoes dos cartoes de vacina das crianças sera um procedimento mais custoso, esta e a forma mais adequada e confiavel de se mensurar a cobertura vacinal nocontexto Brasileiro. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,689,470,703]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Palavras-chave: vacinas; estudos de validação; autorreportado. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[67,717,129,730]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Abstract ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[65,735,715,762]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Background: This study discusses the possibility of using information on child vaccination reported by mothers through household surveys to construct indicators of immunization coverage in Brazil. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,762,714,789]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Objective: Evaluate the potential of information reported by mothers on the immunization of children in household surveys as a source to calculate a vaccination coverage indicator. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,790,714,817]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Method: The indicators of reliability (accuracy) and validity (concordance) between the information available on the vaccination cards of children under two years of age and that reported by mothers in ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[757,824,828,844]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0373/1ddba5d1646efac7041e98f979a7f93a49d0930372d3802cfae38e77fa8fd3df.jpg","list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[757,848,919,934]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Este é um artigo(publicado em acesso aberto (Open Access) sob a licença Creative Commons Attribution, que permite uso, distribuição e reprodução em qualcer meio, sem restricções desde que o trabalho original sera corretamente citado. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[67,31,90,54]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"a ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[141,32,168,52]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CC ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[67,90,174,103]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ARTIGO ORIGINAL ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[754,31,929,93]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"cadernos Saude Coletiva ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[815,98,929,109]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ISSN 2358-291X (Online) ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[65,848,423,858]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Trabalho realizao na/no/em Universidade Federal de Minas Gerais ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[65,860,460,871]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Correspondência: Mônia Viegas Andrade. E-mail: mviegas123@gmail.com ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[65,872,702,905]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Fonte de Financiamento: Organização Pan-Americana da Saúde (OPAS), Conselho Nacional de Secretários Municipais de Saúde (CONASS), FUNDação DE APOIO À PESQUISA DO ESTADO DE MINAS GERAIS (FAPEMIG)/PROGRAMA DE PESQUISA PARA O SUS (PPSUS)/EDITAL 14/2013 PROCESSO APQ0355913. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[67,906,272,915]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conflito de Interesses: nada a declarar. ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[67,916,366,928]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recebido em: Ago. 04, 2019. Aprovado em: Dez. 26, 2019 ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[68,944,85,954]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"40 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[514,943,929,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cad. Saude Colet., 2021;29(esp.): | https://doi.org/10.1590/1414-462X202199010302 ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[282,85,927,112]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"two household surveys conducted in 2013 and 2015 in the municipality of Santo Antônio do Monte, state of Minas Gerais, Brazil. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[282,114,929,168]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Results: The main result shows that, in a context of high coverage, the mothers tended to inform randomly about the vaccination of their children, that is, the information reported by mothers presented low validity. Conclusion: Although collecting information from children's vaccination cards is a more costly procedure, it is the most adequate and reliable way to measure vaccination coverage in the Brazilian context. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[284,172,598,185]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Keywords: vaccine; validation studies; self-report. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[284,208,386,225]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"INTRODUÇÃO ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[284,228,927,303]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Descuão do Plano Nacional de Imunização em 1973, a imunização infantil se destáca por ser uma dasunisticas de saúde mais bem-sucedidas no Brasil, alcancando uma cobertura proxima da universalidade para as vacinas preconizadas pelo Ministério da Saúde para crianças menores de 5 anos de idade<sup>1,2</sup>. Para que o monitoramento das metas de vacinação sera feito de forma eficiente, são necessários bons indicadores de cobertura vincinal. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[282,305,929,577]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"No Brasil, o indicatoricial de cobertura vacinal é obtido por meio da razão entre o número de doses aplicadas de determinada vacina e a estimativa da population na idade espécífica da vaciação<sup>3,4,5</sup> Por sua vez, o número de doses aplicadas é calculado com base nos boletins mensais emitidos pelos postos de vaciação. Esse indicator de cobertura vacinal é intuitivo, direto, de simples interpretação e pode ser calculado para diversas unidas geograficas e periodos no tempo, fazendo com que ele sera amplamente utilizes na avaliação da cobertura. Contudo, muitas vezes esse indicator pode fornecer estimativas imprecisas, visto que ele está passivel de erros tanto no registrar das doses aplicadas quando nas estimativas populaciones<sup>5,6,7</sup> No calculo das doses aplicadas, são contabilizadas todas as doses efetivamente recebidas, independentelemente da idade em que foram aplicadas. Desse modo, crianças que tomam doses fora da idade adequada são contabilizadas no numerator de cobertura vacinal. O denominador también é passivel deerro, uma vez que as estimativas populacionesis, principalmente para crianças de 0 a 2 anos,也是非常PRESENTAMeros em decorrência da subenumeração dos registrados de nascimento no pais<sup>8</sup>. Como resultado, o indicator de cobertura vacinal, na maior parte das vezes, pode estar superestimado, especialmente em municípios menores. Por exemplo, em Minas Gerais, em 2015, a cobertura vacinal calculada pelo Ministerio da Saude para as vacinas BCG, hepatite B, rotavirus, pneumonia, meningocócica e triplice viral está acima de $100\\%$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[282,578,929,699]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Uma forma alternatively de averir a cobertura vacinal é por meio de pesquisas domiciliares. Nesse caso,两大 informações podem ser investigadas: (1) a declaração das mães ou responsaveis sobre a vacinação das crianças; e (2) averification do cartão de vacina. Avantagem do uso de pesquisas domiciliares é a possibídia de avaliar a cobertura vacinal considerando atributos socieônômicos e demógraficos das famílias das crianças. Do ponto de vista de política Pública, é fundamental entender o alcance dos programas de imunização e identificar os grupos populaciónais mais vulnárveis. O indicator oficial do Ministerio da Saúde está permiteanalises em que o menor nível de desaggregação é um citizenshipo. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[282,699,929,791]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A verificacao direta do cartao de vacina ja tem sido realizada para o Brasil6,8,9,10,11. Em 2007, um estado mais amplo incluindo todas as capais foi realizao para averir a real cobertura vacinal em crianças meores de 18 mezes de idade pelo grupo Vaccine Coverage Survey $2007^{12}$ . Segundo esta pesquisa, a cobertura para as crianças da amostra era de $83\\%$ ,送去 a cobertura maior nos grupos com menor renda familiar, evidenciando uma desigualdade favoravel na cobertura vacinal entre os grupos sococonomicos mais baixos. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[284,791,927,852]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Nas pesquisas domiciliares, a forma mais fácil de averir a cobertura vacinal é por intermédiação da informação reportada pelas mães. Pesquisas de ambito nacional, como a Pesquisa Nacional de Saúde (PNS/IBGE) $^{13}$ , realizada em 2013, investigam a imunização das crianças menores de 1 ano a partir de informations declaradas pelas mães. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[284,853,927,912]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"O objetivo de este estudo foi, portanto, avaliar o potencial das informações declaradas pelas mães sobre a imunização das crianças em inquecretos domicíliares como uma fonte para o calculo do indicator de cobertura vacinal. Para tanto, a declaração das mães sobre a imunização de seuços foi comparada com o registro dos cartões de vacina dessas crianças coletados em ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[70,45,173,81]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"cadernos\nSaude Coletiva ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[537,39,927,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Concordância entre informações reportadas pelas mães e dos cartões de vacina ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[68,945,226,954]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cad. Saude Colet., 2021;29(esp.): ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[914,945,927,953]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"41 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[65,85,712,116]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"una subpoena de campo realizada no municipio de Santo Antonio do Monte (Minas Gerais, Brasil) nos anos de 2013 e 2015. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,116,715,177]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Diversos estudos buscam validar as informações sobre cobertura vacinal reportadas em pesquisas domicíliares a partir de registros administrativos e médicos $^{14-28}$ . A maior desses artigos avalia as informações atentes à cobertura das vacinas contra gripe (influenza) e pneumonia nos Estados Unidos, Reino Unido, Espanha, Franca, Itália e Austrácia. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,177,715,298]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"De forma geral, esses estudos muito que a cobertura vacinal, calculada a partir das informações declaradas, está muitoproxima aos registrados oficial para a vacina contra influenza,送去 que a concordança entre as两大 fontes de informacao varia entre $95\\%^{15,18,28}$ e $98\\%^{17,24,27}$ . Japara a vacina contra pneumonia, a concordancia varia entre $64\\%^{28}$ e $90\\%^{20}$ . Lu et al.25 e Boakye et al.26 analisaram os dados da National Immunization Survey-Teen (NIS-Teen) para a cobertura vacinal de adolescentes americanos de 13 a 17 anos para as vacinas contra gripe (influenza) e HPV, respectivamente. Nos dos estudos, entre os adolescentes com registro clnico, $86\\%$ dos pais affirmaram que seu filho havia tomado a vacina. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,298,715,404]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Brown et al. $^{16}$ compararamDados relacionados à vacina contra influenza para 3.072 crianças americanas, no estudo de New Vaccine Surveillance Network. Essa pesquisa questionou os pais das crianças sobre o recebimento de vacina contra influenza e conferiu a resposta com oprofessional provedor da vacina.Os resultadosencionados no estudo做不到 que os pais tendiam a responder corretamente sobre o status vacinal das crianças (92% de sensibilitadede $82\\%$ de especificidade),porém a concordencia entre as respostas diminuía com o aumento do tempo après o recebimento da vacina. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,404,715,571]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Uma série de estudos vem ressaltando también a importante das caracteristicas sociodemograficas e econômicas como fatores determinantes da concordância entre a informação reportada e os registros oficials<sup>15,21,22,29</sup>. AsMulheres tendem a responder sobre suas vacinas de forma mais similar aos registros oficials do que os homens. Por及其他ado, a concordância de crianças e idosos tends a ser menor que a concordância de jovens adultos. Martin et al.<sup>21</sup>也是非常asnstraramqueindividualoscommajorccessoao servicosbasicosde saude témmaioreschancedes responder sobereu statusdevacinaçãodemaneira similaraoresignadosoficiais.Rolnick et al.<sup>22</sup>ressaltamqueacordânciaentreasinformaçõesdeclaradas e os registros administrativos e clínicos varia expressivamenteentreasvacinas.A vacina contra hepatite A tem as menores taxas de concordância,uponas vacinascontra tétano,HPV e gripeapursematleadivastaxasde concordância. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,572,715,679]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ainda que a literatura a respeito da validação das informações declaradas sobre cobertura vacinal está vasta, esse campo de estudo não está esgotado. O presente artigo se coloca como uma importante contribuição, uma vez que busca validar as informações de cobertura vacinal para um grupo atéOOTInedito na litteratura:as crianças de ate 2 anos de idade.No Brasil, a maior das vacinas é ministrada na primeira infancia, de modo que as informações reportadas pelas mães em pesquisas domiciliares podem se configurar como um importante mecanismo de monitoramento da cobertura vacinal no pais. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[67,699,137,715]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"MÉTODO ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[67,742,196,757]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Fonte de dados ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[65,762,715,852]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Este estudo realizou as informações fornecidas por um inquisito domiciliar realizado emodos recortes transversais independentes no municipio de Santo Antonio do Monte, em Minas Gerais, no ambito da Pesquisa de Avaliação do Laboratório de Inovações no Atença às Condições Crónicas (LIACC) na Atença Primária à Saúde<sup>30</sup>. Essa pesquisa avaliou as mudanças em diversos indicadores de saúde antes, no decorrer e(before da intervenção do LIACC para individuos com hipertensão, diabetes, gestantes e crianças menores de 2 anos. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,853,715,914]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"As datas de realização das entrevistas eram referentes aos periodos antes e.after do processo de intervenção do LIACC no@municipio. Aserviewas respondidas em 2013 diziam respeito ascriçanas nascidas entre janeiro de 2011 e dezembro de 2012, enquanto as entrevistas respondidas em 2015 se referiam ascriçanas nascidas entre janeiro e dezembro de 2014. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[67,42,460,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Concordância entre informações reportadas pelas mães e dos cartões de vacina ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[821,44,929,83]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"cademos\nSaüde Coletiva ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[68,945,84,954]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"42 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[768,944,929,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cad. Saude Colet., 2021;29 Esp.: ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[282,85,929,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"O tamanho amostral foi definido a partir do número médio anual de nascimento ocorridos no municipio de Santo Antônio do Monte, registrados no Sistema de Informação sobre Nascidos Vivos (SINASC). A amostra final foi representativa da população de crianças menores de 2 anos do municipio, sendo investigadas 185 crianças em 2013 e 155 crianças em 2015. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[282,163,929,286]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Primeiramente, perguntou-se às vezes se a criança havia tornado pelo menos uma dose de cada vacina analisada: BCG-ID, hepatite B, antipólico, tetravalente, rotavirus, pneumocócica, meningocócica, triplice viral e febre amarela. Em seguida, as quando responderam se a criançainha cartão de vacina e se ele estava presente no momento da在网络. Esses cartões foram fotografados peloscretivestadores com o consentimento das vezes, e a elaboração do banco de dados com tais informações teve dupla checagem para a garantia da qualida informacional. Optou-se por travalear comigo não uma das doses para que fosse possivel realizar a comparação com as informações reportadas pelas vezes. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[282,286,929,378]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Osindicadores foram construuidos apenas para as crianças que tinkham a idade completa preconizada pelo calendario vacinal do Ministerio da Saude, com una janela de tempo de dosmeses. Essa janela é necessaria,pois,muitas vezes,a crianca não tomou a vacina na idade exata por estar com algo problema de saude ou por indisponibilitadedefcontinuidade do proprietary systema31 Não ha na literatura um accordo sobre o tempo valido de espera para a proxima dose da vacina6,por isso tal estrategia fou adotada sem um parametro predefinido. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[282,379,929,500]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"O calendário vacinal do Ministerio da Saúde sempre passa por alterações no que tange ao número de doses e a idade de cada uma das vacinas. Este trabalho utilizeou o calendário vigente até o ano de 2015, uma vez que as crianças analisadas foram imunizadas a partir das normas do calendário antigio. Ressalta-se que, durante o ano de 2012, a vacina tetravalente foi substituira pela pentavalente, e em muitos cartões de vacina analisados o carimbo ou a anotação que validava a aplicação da vacina VINHA com uma observação de que a criança nãoinha tomado a tetravalente, mas sim a pentavalente. Nesses casos, o trabalho considerou a criança imune pela tetravalente. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[282,501,929,624]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Esta Pesquisa foi avaliada e aprovada pelo Comité de Ética em Pesquisa da UFSJ/Campus Centro-Oeste Dona Lindu, sob o Parecer n° 369.942, no ano de 2013. Os dados individuais coletados se tornaram anônimos e não apareceram em nenhum relatório ou(publicação,sendo resguardado o direito ao sigilo. Após a Pesquisa, os dados foram arquivos e não foram realizados para及其他 fim, de acordo com os princípios do art. $6^{\\circ}$ da Lei Geral de Proteção de Dados<sup>32</sup>, em especial o da segurança e transparência, e o art. $7^{\\circ}$ da referida lei, quando ao tratamento dos dados. Informações completeness sobre a Pesquisa está disponível no relatório técnico entrega à Organização Pan-Americana de Saude em novembro de 2016. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[284,646,885,664]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Indicadores de confiabilidad e validade das informacoes autodeclaradas ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[282,667,929,728]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Os registros dos cartões de vacina foram considerados como o grupo de referencia para a avaliação da qualidade das informações autodeclaradas das vezes. As consistências entre as两大 fontes de dados foram analisadas a partir de vez指示ores amplamente realizados em estudos epidemiológicos<sup>33</sup>: ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[282,728,929,912]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. Sensibilitidade: proportao de crianças que foram vacinadas segudo o relato materno entre aquelas vacinadas com base nos cartoes de vacina. \",\"2. Especialidade: proporcão de crianças que não foram vacinadas conforme o relato materno entre aquelas não vacinadas com base nos cartões de vacina. \",\"3. Valor preditivo positivo:proporcão de crianças que foram vacinadas com base nos cartoes de vacina entre aquelas vacinadas de acordo com o relato das mães. \",\"4. Valor preditivo negativo: proportao de crianças que nao foram vacinadas com base nos cartoes de vacina entre aquelas nao vacinadas conforme o relato das mães. \",\"5. Taxa de concordância geral: proporcão de concordância entre as respostas das mães e os registros nos cartões. \",\"6. Úndice Kappa: taxa de concordância geral, ponderada pela concordância esperada. Esse Úndice, usualmente, varia entre 0 e 1, de tal forma que, quanto maior for seu valor, maior é \"]","page_idx":"3","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[70,45,173,81]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"cadernos\nSaude Coletiva ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[537,39,927,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Concordância entre informações reportadas pelas mães e dos cartões de vacina ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[68,944,228,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cad. Saude Colet., 2021;29(esp.): ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[914,945,927,953]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"43 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[100,85,715,148]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"a concordância entre as两大 fontes de informação. Contudo, em algumas circunstâncias, é possível observar um indice Kappa negativo, quando a concordância observada é menor que a concordância esperada. Valores negativos baixos (até -0,1) são usualmente interpretados como não concordância<sup>34</sup>. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,148,717,226]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Parte da consonança verificada pela taxa de concordância geral pode ser aleatoria, e não em decorrência da concordância entre as两大 fontes de informação. Assim, o indice Kappa é estatistically superior à taxa de concordância geral, uma vez que, ao ponderar a taxa de concordância geral pela taxa de concordância esperada, é possível obter a concordância liquida, para às vezes daquela que seria esperada ao acaso. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,249,268,264]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"RESULTADOS E DISCUSSAO ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[65,268,717,440]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Realizaram-se entreprises com 340 mães de crianças de ate 2 anos de idade nos anos de 2013 e 2015. A maioria dessas mãesinha entre 20 e 35 anos (76,47%), com pelo menos o ensino fundamental completo (59,42%) e pertencente à classe de riqueza C (56,64%),utilizing os parâmetros do Critério Brasil da Associacao Brasileira de Empresa de Pesquisa de $2014^{35}$ Todas as mães declararam que a criançainha cartao de vacina. Contudo, 42 crianças foram excluidas da analise, uma vez que 33 mães não deixaram oscretivestadores verem o cartao de vacina da criança, 8 mães affirmaram que a criança ja teve outro cartao de vacina e 1 mãenão sabia se a criança ja havia sido outro cartao. Dessa forma, a analise da cobertura vacinal considerou 298 crianças, as quais representavam $88\\%$ da amostra inicial. Não houve diferencas socioeconomicas estatisticamente significativas entre as mães das crianças da amostra inicial e o subGrupo selecionado. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,441,717,551]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Tabela 1 apareça as taxas de cobertura vacinal para as crianças de até 2 anos de idade no[municipio de Santo Antônio do Monte nos anos de 2013 e 2015,considerando as informações reportadas pelas mães e os registros presentes nos cartões de vacina. Em primeiro lugar,notou-se elevada cobertura no[municipio para as vacinas preconizadas no primeiro ano de vida,independente da informaçãoutilizada.Com base nas informações dos cartões, $100\\%$ das crianças tomaram pelo menos uma dose de BCG-ID,antipólio,pneumocócica, meningocócica e febre amarela. Para as demais,a cobertura vacinal foi de $99\\%$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[67,609,715,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0994/248fe639b3a00c78435826d985381338408365cbcf007c6578fcf62484290f52.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":"<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">Cobertura \nvacinal – mãe</td><td colspan=\"2\">Cobertura \nvacinal – \ncartões</td><td colspan=\"2\">Teste de média</td><td>Total de crianças \nelegíveis à vacina</td></tr><tr><td>n</td><td>%</td><td>n</td><td>%</td><td>p-valor</td><td>Significância</td><td>n</td></tr><tr><td>BCG-ID</td><td>283</td><td>98,26</td><td>288</td><td>100</td><td>0,025</td><td>**</td><td>288</td></tr><tr><td>Hepatite B</td><td>285</td><td>98,28</td><td>288</td><td>99,31</td><td>0,254</td><td>NS</td><td>290</td></tr><tr><td>Antipólio</td><td>275</td><td>98,57</td><td>279</td><td>100</td><td>0,045</td><td>**</td><td>279</td></tr><tr><td>Tetravalente</td><td>261</td><td>93,21</td><td>279</td><td>99,64</td><td>0</td><td>***</td><td>280</td></tr><tr><td>Rotavirus</td><td>268</td><td>96,4</td><td>277</td><td>99,64</td><td>0,006</td><td>***</td><td>278</td></tr><tr><td>Pneumocócica</td><td>241</td><td>87,32</td><td>276</td><td>100</td><td>0</td><td>***</td><td>276</td></tr><tr><td>Meningocócica</td><td>229</td><td>82,37</td><td>278</td><td>100</td><td>0</td><td>***</td><td>278</td></tr><tr><td>Tríplice viral</td><td>150</td><td>97,4</td><td>152</td><td>98,7</td><td>0,41</td><td>NS</td><td>154</td></tr><tr><td>Febre amarela</td><td>158</td><td>96,34</td><td>164</td><td>100</td><td>0,013</td><td>**</td><td>164</td></tr></table>","table_caption":"[\"Tabela 1. Cobertura vacinal declarada pelas mães e registrarada no cartão de vacina para crianças menores de 2 anos, Santo Antonio do Monte, Minas Gerais, 2013 e 2015 \"]","table_footnote":"[\"Fonte: Pesquisa SAMONTE (2013 e 2015); Nota: *** Estatisticamente significativo a 1%; ** Estatisticamente significativo a 5%; * Estatisticamente significativo a 10%; NS não estatisticamente significativo \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[67,42,460,54]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Concordância entre informações reportadas pelas mães e dos cartões de vacina ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[821,44,929,83]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"cademos Saude Coletiva ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[68,945,84,954]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"44 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[768,944,929,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cad. Saude Colet., 2021;29(esp.): ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[284,85,929,192]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A cobertura vacinal estimada com base nas respostas das mães foi elevada, mas foi menor do que a estimada de acordo com os registros dos cartões. As diferencias entre a resposta da maior e a informação do cartão foram estatistically significativas para todas as vacinas, exceto para hepatite B e triplice viral. As maiores diferencias foram observadas para as vacinas tetravalente, pneumocócica e meningocócica (Tabela 1). Essas vacinas, como mencionado na metodologia, passaram por mudanças no calendário vacinal, o que pode levar a um maior desconhecimento delas por parte das mães. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[284,195,929,346]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Algumas mudanças que ocorreram no calendário vacinal preconizzato pelo Ministerio da Saude podem, em alguma medida, explicar a subdeclaracao pelas mães. A vacina tetravalente foi substituidade no ano de 2012 pela pentavalente que protege contra difteria, tétano, coqueluche, Haemophilus influenzae tipo B e hepatite B. Essa mudança pode justIFICAR a subestimação de resposta para hepatite B e tetravalente. No caso das vacinas pneumocócica e meningocócica, a subestimação pode estar associada à inclusão recente dessas两大 vacinas no calendario vacinal, que passaram a ser ofertadas somente a partir de 2010. Possivelmente as mães tiveram dificuldades de identificar a imunização da criança por causa da pouca familiaridade e da baixa difusão de informações sobre essas vacinas. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[284,349,929,549]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Tabela 2 apareça os indices de sensibilitadé e especialidade entre as informações declaradas pelas mães e as informações registradas nos cartões de vacina das crianças menores de 2 anos em Santo Antônio do Monte. De forma geral, entre as crianças com registrar no cartão, houve um percentual expressivo de mães que declaresram que a criança tomou a vacina. Para as vacinas BCG-ID, hepatite B, antipólico, triplice virale e febre amarela, a sensibilitadé foi maior que $97\\%$ . Por及其他, a proporcão de mães que declaresram que a criança tomou a vacinas pneumocócica e meningocócica entre aquelas com registrar no cartão foi de $88\\%$ e $83\\%$ respectively, indicando que, para essas vacinas, houve uma menor concordância entre mães e cartões. Por及其他, em todos os casos em que não havia registrar no cartão, as mães declaresram que a criançainha tomado a vacina, ou seja, a especialidade foi igual a zero para todas as vacinas. Desse modo, observa-se uma fragilidade na responça das mães que reportaram que a criança sempre recebeu a vacina. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[282,631,929,885]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0934/973c161c25e82ce1b0dec65c168c7635310263510db7c548d45f2de9273d30d0.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":"<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Crianças com registrar no cartão</td><td colspan=\"2\">Sensibilidade</td><td rowspan=\"2\">Crianças sem registrar no cartão</td><td colspan=\"2\">Especialidade</td><td rowspan=\"2\">Crianças elegíveis à vacina</td></tr><tr><td>n</td><td>%</td><td>n</td><td>%</td></tr><tr><td>BCG-ID</td><td>288</td><td>283</td><td>98.26</td><td>0</td><td>0</td><td>0.00</td><td>288</td></tr><tr><td>Hepatite B</td><td>288</td><td>283</td><td>98.26</td><td>2</td><td>0</td><td>0.00</td><td>290</td></tr><tr><td>Antipólio</td><td>294</td><td>290</td><td>98.64</td><td>0</td><td>0</td><td>0.00</td><td>294</td></tr><tr><td>Tetravalente</td><td>290</td><td>271</td><td>93.45</td><td>1</td><td>0</td><td>0.00</td><td>291</td></tr><tr><td>Rotavírus</td><td>284</td><td>274</td><td>96.48</td><td>1</td><td>0</td><td>0.00</td><td>285</td></tr><tr><td>Pneumocérica</td><td>285</td><td>250</td><td>87.72</td><td>0</td><td>0</td><td>0.00</td><td>285</td></tr><tr><td>Meningocérica</td><td>286</td><td>237</td><td>82.87</td><td>0</td><td>0</td><td>0.00</td><td>286</td></tr><tr><td>Tríplice viral</td><td>168</td><td>164</td><td>97.62</td><td>22</td><td>0</td><td>0.00</td><td>190</td></tr><tr><td>Febre amarela</td><td>211</td><td>205</td><td>97.16</td><td>2</td><td>0</td><td>0.00</td><td>213</td></tr></table>","table_caption":"[\"Tabela 2. Sensibilitad e especificidade para a concordancia entre as informacoes declaradas pelas mares e as informacoes registradas no cartao de vacina para crianças meores de 2 anos, Santo Antônio do Monte, Minas Gerais, 2013 e 2015 \"]","table_footnote":"[\"Fonte: Pesquisa SAMONTE (2013 e 2015); Nota: *** Estatisticamente significativo a 1%; ** Estatisticamente significativo a 5%; * Estatisticamente significativo a 10%; NS Não estatisticamente significativo \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[70,45,173,81]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"cadernos\nSaude Coletiva ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[537,39,927,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Concordância entre informações reportadas pelas mães e dos cartões de vacina ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[68,945,228,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cad. Saude Colet., 2021;29(esp.): ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[914,945,927,954]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"45 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[68,87,712,208]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Assim, quando a criança não tomava uma vacina espécífica, aõe tendia a affirmar que a criança ainha tornado. Esse padrão de resposa provavelmente tenha sido influenciado pela elevada cobertura vacinal observada no@municipio. Como a cobertura foi proxima de $100\\%$ as mães, em geral, tendiam a reportar que as crianças estavam com todas as vacinas em dia. É interessante ressaltar o caso que ocorreu com a vacina triplice viral. Entre 2012 e 2014, segundo o relato dos professionais de sauède Santo Antonio, o@municipio passou por problemas de abastecimento dessa vacina, de modo que um percentual significativo das crianças não a tomou. Mesmo assim, todas as mães affirmaram que seuis filhos tomaram a vacina triplice viral. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[68,210,712,332]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Tabela 3 apareça os values previdos positivos e negativos para a concordância entre as informações declaradas pelas mães e as informações nos cartões de vacina das crianças menores de 2 anos em Santo Antonio do Monte. Para todas as vacinas, com exceço da triplice viral, constatou-se que, para praticamente todas as criançasURT ja havia declarado que elas tomaram a vacina, havia registrar tambem no cartão,uma vez que o valor preditivo positivo foi maior que $99\\%$ .Por及其他lado,entre as criançasURT ja havia declarado que elas não tinkham tomando a vacina, todas tinkham registrar no cartao,de modo que o valor preditivo negativo foi igual a zero para todas as vacinas. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[67,402,714,670]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0028/c1f503861a84e0e9d69bf621862864a1f23ae7a125b7054c8142f4253e0fe76a.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":"<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Afirmação a partir do relato materno</td><td colspan=\"2\">PPV</td><td rowspan=\"2\">Negação a partir do relato materno</td><td colspan=\"2\">NPV</td><td rowspan=\"2\">Crianças elegíveis à vacina</td></tr><tr><td>n</td><td>%</td><td>n</td><td>%</td></tr><tr><td>BCG-ID</td><td>283</td><td>283</td><td>100.00</td><td>5</td><td>0</td><td>0.00</td><td>288</td></tr><tr><td>Hepatite B</td><td>285</td><td>283</td><td>99.30</td><td>5</td><td>0</td><td>0.00</td><td>290</td></tr><tr><td>Antipólio</td><td>290</td><td>290</td><td>100.00</td><td>4</td><td>0</td><td>0.00</td><td>294</td></tr><tr><td>Tetravalente</td><td>272</td><td>271</td><td>99.63</td><td>19</td><td>0</td><td>0.00</td><td>291</td></tr><tr><td>Rotavírus</td><td>275</td><td>274</td><td>99.64</td><td>10</td><td>0</td><td>0.00</td><td>285</td></tr><tr><td>Pneumocérica</td><td>250</td><td>250</td><td>100.00</td><td>35</td><td>0</td><td>0.00</td><td>285</td></tr><tr><td>Meningocérica</td><td>237</td><td>237</td><td>100.00</td><td>49</td><td>0</td><td>0.00</td><td>286</td></tr><tr><td>Tríplice viral</td><td>186</td><td>164</td><td>88.17</td><td>4</td><td>0</td><td>0.00</td><td>190</td></tr><tr><td>Febre amarela</td><td>207</td><td>205</td><td>99.03</td><td>6</td><td>0</td><td>0.00</td><td>213</td></tr></table>","table_caption":"[\"Tabela 3. Valor preditivo positivo e valor preditivo negativo para a concordância entre as informações declaradas pelas mães e as informações registradas no cartão de vacina para crianças menores de 2 anos, Santo Antônio do Monte, Minas Gerais, 2013 e 2015 \"]","table_footnote":"[\"Fonte: Pesquisa SAMONTE (2013 e 2015); Nota: *** Estatisticamente significativo a 1%; ** Estatisticamente significativo a 5%; * Estatisticamente significativo a 10%; NS não estatisticamente significativo \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[68,725,712,818]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A Tabela 4 apareça a taxa de concordância geral e o Úndice Kappa para a concordância entre as informações declaradas pelas mães e as informações nos cartões de vacina das crianças menores de 2 anos em Santo Antônio do Monte. Notou-se um altoível de concordância para todas as vacinas, embora existisse expressiva variação entre os típos. A maior taxa de concordância foi observada para a BCG-ID (98%), o que era esperado, uma vez que esta vacina é tomada em geral na maternidade, e a menor foi a meningocócica (83%). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[68,819,712,912]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Embora a taxa de concordância geral estivesse bastante elevada, esse indicator não foi suficiente para garantir a validade da informação das mães, uma vez que seu valor foi fortemente determinado pela concordância entre as respostas positivas das mães e os registros no cartão. Como a especialidade e o valor preditivo negativo foram igualais a zero, ao ponderar a concordância observada pela concordância esperada, foi obtido o indice Kappa proxies a zero para praticamente todas as vacinas, o que significica que a concordância percebida entre as ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[68,42,458,54]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Concordância entre informações reportadas pelas mães e dos cartões de vacina ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[821,44,927,81]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"cademos\nSaüde Coletiva ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[68,945,84,954]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"46 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[769,945,927,954]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cad. Saude Colet., 2021;29(esp.): ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[282,129,929,382]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/1012/15f1838c10bb48d9d8ddc3445d938e6de39e78a16286ae1f43f6be4235e78238.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":"<table><tr><td></td><td>Número de mazes cartão em concordância</td><td>Taxa de concordância geral</td><td>Índice Kappa</td><td>Crianças elegíves à vacina</td></tr><tr><td>BCG-ID</td><td>283</td><td>98.26</td><td>0.00</td><td>288</td></tr><tr><td>Hepatite B</td><td>283</td><td>97.59</td><td>-0.01</td><td>290</td></tr><tr><td>Antipólio</td><td>290</td><td>98.64</td><td>0.00</td><td>294</td></tr><tr><td>Tetravalente</td><td>271</td><td>93.13</td><td>-0.01</td><td>291</td></tr><tr><td>Rotavírus</td><td>274</td><td>96.14</td><td>-0.01</td><td>285</td></tr><tr><td>Pneumocérica</td><td>250</td><td>87.72</td><td>0.00</td><td>285</td></tr><tr><td>Meningocérica</td><td>237</td><td>82.87</td><td>0.00</td><td>286</td></tr><tr><td>Tríplice viral</td><td>164</td><td>86.32</td><td>-0.04</td><td>190</td></tr><tr><td>Febre amarela</td><td>205</td><td>96.24</td><td>-0.01</td><td>213</td></tr></table>","table_caption":"[\"Tabela 4. Taxa de concordância geral e Índice Kappa para a concordância entre as informações declaradas pelas mães e as informações registradas no cartão de vacina para crianças menores de 2 anos, Santo Antônio do Monte, Minas Gerais, 2013 e 2015 \"]","table_footnote":"[\"Fonte: Pesquisa SAMONTE (2013 e 2015) \"]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[284,418,927,462]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"respostas das mães e o registrar no cartão foi aleatoria. No caso específico em analise, como a cobertura vacinal era muito elevada, os dados sugerem que as mães tendiam sempre a reportar que os hilhos tomaram a vacina. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[284,464,927,601]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"O Grácfo 1 analisa a discordância entre as两大 fontes de informacao. No total, 194 mães reportaram todas as vacinas em concordância com as informacoes dos cartoes e 94 crianças tiveram informacao reportada emleo menos uma vacina em discordancia comas informacoes registradas nos cartoes de vacina,sendo que a maior das mães reportaram atetres vacinas em divergencia com os cartoes. Ademais,as mães que nao reportaram de forma correta a vacinação de seu filhos não diferiam em termos sococonomicos das demais mães entwickidas no que se refere àidade, classe de riqueza e escolaridade. Esse achado é importante,pois sugere que o desconhecimento da informacao pelas mães não está relacionado a caracteristicas especialicas delas. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[302,633,917,852]","code_body":null,"code_caption":null,"image_caption":"[\"Fonte: Pesquina SAMONTE, 2013 e 2015. \",\"Gráfico 1. Número de discordâncias entre as informações declaradas pelas mães e as informações registradas no cartão de vacinas para cada criança menor de 02 anos – Santo Antônio do Monte, 2013 e 2015 \"]","image_footnote":"[]","img_path":"images/0867/dfcb07c19c30a80d699ffaa88ee877220e4c340f57e636f63174f465861c3638.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[70,45,173,81]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"cadernos\nSaude Coletiva ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[539,39,927,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Concordância entre informações reportadas pelas mães e dos cartões de vacina ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[68,945,226,954]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cad. Saude Colet., 2021;29(esp.): ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[914,945,927,953]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"47 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[65,85,250,102]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CONSIDERACOES FINAIS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[65,105,715,168]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Este trabalho analisou a possibidade de'utilização de informações sobre a vacinação infantil reportadas pelas mães para a construção de指示ores de cobertura vincinal em um@municipio brasilheiro de poucojo porte. O estado é inédito no Brasil, uma vez que são escassos os蹼 doscheckados de informações domicíares sobre a cobertura vincinal infantil. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,171,715,234]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A principal limitação do estado é se referir a uma populatione exigência de um@mancipio de pouco no Brasil,sendo,portanto,um estudo de caso.Adespeito da particularidade do municiao, o estudo gera novas evidências para politicas de monitoramento da cobertura vacinal no Brasil. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,235,715,380]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Os Resultados nostraram que, em umcontexto de alta cobertura, as mães tendiam a informar sobre a vacinação de seuis filhos de forma aleatoria. A evidência de Santo Antônio do Monte minha que a informação reportada pelas mãesinha pouca validade para mensurar a cobertura vincinal. Estecontexto difere dos demais estudos que buscaram validar informações autorreportadas sobre vacinas para outros grupos populacionais. Como a cobertura vincinal era alta, as mães provavelmente tinkham baixo conheimento especialico sobre cada vincina, nãosendocapazesdeidentificarquaisestavamforodocumentoadequado.Essesresultados está emconsonânciacom os achados de Brown et al.16,que ja evidenciarama fragilidade das informações reportadas pelas mãesno que tange à vacinação dascriçanas. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,380,715,445]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A baixa concordância das informações das mães ficou mais evidente para a vacina triplice viral. Segundo relato dos professionis de saúde, entre os anos de 2012 e 2014, o@municão de Santo Antonio do Monte apareceu problemas de abastecimento dessa vacina. Ainda assim, todas as mães declararam que seu filhos haviam tornado esta vacina. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,446,715,557]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A não concordança entre as informações declaradas pelas mães e as informações dos cartões de vacina pode ser gerada por causa do viés de的记忆a das mães<sup>29</sup>. Contudo, a partir dos resultadospresentados neste estudo, não se pode affirmar que a não concordança sera decorrente de um viés de的记忆a ou de falta de的记忆ão por parte das mães. Cordeiro et al.<sup>36</sup>, ao analisarem o esquema vicinal de crianças assistidas em uma Unidade Básica de Saúde no[municipio]de Jaboatão dos Guararapes, Pernambuco, evidenciaram a falta de conheção das mães e ressaltaram a importânca de praticas educativas na atençao primária. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,559,715,654]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"O Brasil alcâncôu, com o Sistema Único de Saúde, elevada cobertura vacinal infantil. A poliomielite, a varíola e o sarampo foram erradicados, e a incidência de diversas outras doenças infecções, como febre amarela, difteria, tétano neonatal, coqueluche, meningite, caxumba e rubéola, tem sido mantida em eles relativamente baixos. No município de Santo Antônio do Monte, assim como para a maioría dos municípios brasilieiros, foi observada universalização da cobertura vacinal para as crianças de 0 a 2 anos. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,656,715,718]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Contudo, nos últimos anos, o número de famílias que decidem não vacinar seuis filhos vem crescendo expressivamente em diversos páises do mundo37,38. Ainda que o movimento antivacinha sera muito pouco no Brasil, ele vem ganhando cada vez mais notorialidade39, especialmente entre casais de alto nível de escolaridade e residentes em和地区 urbanas. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,721,715,800]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Considerando que os indicatoros calculados de cobertura vacinal por meio de informações administrativas podem superestimar esta cobertura, a inclusão de informations sobre imunização em pesquisas domicílieares pode ser um instrumento promissor para o monitoramento continuo das metas de vacinação, assim como para a identificação de heterogeneidade de cobertura entreroupos populacionesis. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[65,800,715,912]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Embora a coleta de informações dos cartões de vacina das crianças está oprocesso mais custoso para obter informações de imunização, este estudo indica que esta forma sera a mais adequada de se mensurar a cobertura vacinal nocontextobrasileiro em comparação—as informações declaradas pelas mães. Garantir que a cobertura vacinal continue universal e que doencas infecciosas até então erradicadas no Brasil não ressurjam é um desafio em termos de saude Pública e deve ser encarada como uma das prioridades da Atença Primária à Saude no País. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[67,42,460,54]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Concordância entre informações reportadas pelas mães e dos cartões de vacina ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[821,44,929,83]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"cademos\nSaüde Coletiva ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[68,945,85,954]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"48 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[768,944,929,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cad. Saude Colet., 2021;29 Esp.: ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[282,85,389,100]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"REFERÉNCIAS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[282,107,929,912]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"1. Victora C, Aquino E, Leal MC, Monteiro C, Barros F, Szwarcwald C. Maternal and child health in Brazil: progress and challenges. Lancet. 2011;377(9780):1863-76. http://dx.doi.org/10.1016/S0140-6736(11)60138-4. \",\"2. Palm J, Travassos C, Almeida C, Bahia L, Macinko J. The Brazilian health system: history, advances, and challenges. Lancet. 2011;377(9779):1778-97. http://dx.doi.org/10.1016/S0140-6736(11)60054-8. PMID:21561655. \",\"3. Brasil. Ministerio da Saude, Secretaria de Vigilancia em Saude. Programa Nacional de Imunizacoes: 30 anos. Série C. Brasília: Ministerio da Saude; 2003. \",\"4. Brasil. Ministério da Saúde, SVS-PNI. Departamento de Informática do SUS - DATASUS [Internet]. 2016 [citado em 2016 Out 1]. Disponível em: http://datasus.saude.gov.br. \",\"5. Teixeira AMDS, Mota ELA. Denominadores para o calculo de coberturas vacinais: um estudo das bases de dados para estimar a populatione menor de um ano de idade. Epidemiol Serv Saude. 2010;19(3):187-203. http://dx.doi.org/10.5123/S1679-49742010000300002. \",\"6. Moraes J, Ribeiro M, Simões O, Castro P, Barata R. Qual é a cobertura vacinal real?. Epidemiol Serv Saude. 2003;12(3):147-53. \",\"7. Moraes J, Ribeiro M. Desigualdades sociais e cobertura vacinal. Rev Bras Epidemiol. 2008;11(Supl 1):113-24. \",\"8. Pedraza D. Qualidade do Sistema de Informacoes sobre Nascidos Vivos. Cienc Saude Colet. 2012;17(10):2729-37. \",\"9. Miranda A, Scheibel I, Tavares M, Takeda S. Avaliação da cobertura vacinalo do esquema Basicsico para o primeiro ano de vida. Rev Saude Pública. 1995;29(3):208-14. http://dx.doi.org/10.1590/S0034-89101995000300008. \",\"10. Brasil. Ministério da Saúde, Departamento de Ciência e Tecnologia. PNDS 2006 - Pesquisa Nacional de Demografia e Saúde da Criança e da Mulher. Relatório Tecnico. Brasília: Ministério da Saúde; 2008. \",\"11. Yokokura, AVCP, Silva, AAMD, Bernardes, ACF, Lamy, F Fo, Alves, MTSSDB, Cabra, NAL, et al. Cobertura vacinal e fatores associados ao esquema vacinal Basics incomplete aos 12 mezes de idade, São Luís, Maranhão, Brasil, 2006. Cad Saude Pública. 2013;29(3):522-34. \",\"12. Barata R, Ribeiro M, Moraes J, Flannery B. Socioeconomic inequalities and vaccination coverage: results of an immunisation coverage survey in 27 Brazilian capitals, 2007-2008. J Epidemiol Community Health. 2012;66:934-41. \",\"13. Brasil. Pesquisa Nacional de Saude 2013: Percepção do estado de saude, estilos de vida e doenças crônicas. Rio de Janeiro: IBGE; 2014. \",\"14. Bayas JM, Izquierdo C, Ruiz L, Sintes X, Sousa D, Celorrio JM, et al. Validity of self-reported pneumococcal vaccination status in the elderly in Spain. Vaccine. 2009;27(34):4560-4. http://dx.doi.org/10.1016/j.vaccine.2009.05.057. PMID:19508910. \",\"15. Irving SA, Donahue JG, Shay DK, Ellis-Coyle TL, Belongia EA. Evaluation of self-reported and registry-based influenza vaccination status in a Wisconsin cohort. Vaccine. 2009;27(47):6546-9. http://dx.doi.org/10.1016/j.vaccine.2009.08.050. PMID:19729083. \",\"16. Brown C, Clayton-Boswell H, Chaves SS, Prill MM, Iwane MK, Szilagyi PG, et al. Validity of parental report of influenza vaccination in young children seeking medical care. Vaccine. 2011;29(51):9488-92. http://dx.doi.org/10.1016/j.vaccine.2011.10.023. PMID:22015394. \",\"17. King JP, McLean HQ, Belongia EA. Validation of self-reported influenza vaccination in the current and prior season. Influenza Other Respir Viruses. 2018;12(6):808-13. http://dx.doi.org/10.1111/irv.12593. PMID:30028081. \",\"18. Llupià A, García-Basteiro AL, Mena G, Ríos J, Puig J, Bayas JM, et al. Vaccination behaviour influences self-report of influenza vaccination status: a cross-sectional study among health care workers. PLoS One. 2012;7(7):e39496. http://dx.doi.org/10.1371/journal.pone.0039496. PMID:22808039. \",\"19. Louergue P, Pulcini C, Massin S, Bernhard M, Fonteneau L, Levy-Bruhl D, et al. Validity of self-reported vaccination status among French healthcare students. Clin Microbiol Infect. 2014;20(12):O1152-4. http://dx.doi.org/10.1111/1469-0691.12759. PMID:25040583. \",\"20. MacDonald R, Baken L, Nelson A, Nichol KL. Validation of self-report of influenza and pneumococcal vaccination status in elderly outpatients. Am J Prev Med. 1999;16(3):173-7. http://dx.doi.org/10.1016/S0749-3797(98)00159-7. PMID:10198654. \"]","page_idx":"9","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[68,44,174,81]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"cademos\nSaude Coletiva ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[537,37,929,50]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Concordância entre informações reportadas pelas mães e dos cartões de vacina ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[68,944,228,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cad. Saude Colet., 2021;29Esp.: ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[914,945,929,954]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"49 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[67,85,715,858]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"21. Martin LM, Leff M, Calonge N, Garrett C, Nelson DE. Validation of self-reported chronic conditions and health services in a managed care population. Am J Prev Med. 2000;18(3):215-8. http://dx.doi.org/10.1016/S0749-3797(99)00158-0. PMID:10722987. \",\"22. Rolnick SJ, Parker ED, Nordin JD, Hedblom BD, Wei F, Kerby T, et al. Self-report compared to electronic medical record across eight adult vaccines: do results vary by demographic factors? Vaccine. 2013;31(37):3928-35. http://dx.doi.org/10.1016/j.vaccine.2013.06.041. PMID:23806243. \",\"23. Shenson D, DiMartino D, Bolen J, Campbell M, Lu PJ, Singleton JA. Validation of self-reported pneumococcal vaccination in behavioral risk factor surveillance surveys: experience from the sickness prevention achieved through regional collaboration (SPARC) program. Vaccine. 2005;23(8):1015-20. http://dx.doi.org/10.1016/j.vaccine.2004.07.039. PMID:15620474. \",\"24. Skull SA, Andrews RM, Byrnes GB, Kelly HA, Nolan TM, Brown GV, et al. Validity of self-reported influenza and pneumococcal vaccination status among a cohort of hospitalized elderly inpatients. Vaccine. 2007;25(25):4775-83. http://dx.doi.org/10.1016/j.vaccine.2007.04.015. PMID:17499402. \",\"25. Lu PJ, Dorell C, Yankey D, Santibanez TA, Singleton JA. A comparison of parent and provider reported influenza vaccination status of adolescents. Vaccine. 2012;30(22):3278-85. http://dx.doi.org/10.1016/j.vaccine.2012.03.015. PMID:22445809. \",\"26. Boakye EA, Tobo BB, Osazuwa-Peters N, Mohammed KA, Geneus CJ, Schootman M. A comparison of parent-and provider-reported human papillomavirus vaccination of adolescents. Am J Prev Med. 2017;52(6):742-52. http://dx.doi.org/10.1016/j.amepre.2016.10.016. PMID:27890518. \",\"27. Zimmerman RK, Raymund M, Janosky JE, Nowalk MP, Fine MJ. Sensitivity and specificity of patient self-report of influenza and pneumococcal polysaccharide vaccinations among elderly outpatients in diverse patient care strata. Vaccine. 2003;21(13-14):1486-91. http://dx.doi.org/10.1016/S0264-410X(02)00700-4. PMID:12615445. \",\"28. Mangtani P, Shah A, Roberts JA. Validation of influenza and pneumococcal vaccine status in adults based on self-report. Epidemiol Infect. 2007;135(1):139-43. http://dx.doi.org/10.1017/S0950268806006479. PMID:16740194. \",\"29. Gordon NP, Wortley PM, Singleton JA, Lin TY, Bardenheier BH. Race/ethnicity and validity of self-reported pneumococcal vaccination. BMC Public Health. 2008;8(1):227. http://dx.doi.org/10.1186/1471-2458-8-227. PMID:18598363. \",\"30. Andrade MV, Noronha K, Cardoso CS, Oliveira CD, Calazans JA, Souza MN. Challenges and lessons from a primary care intervention in a Brazilian municipality. Rev Saude Publica. 2019;53:45. http://dx.doi.org/10.11606/S1518-8787.2019053000457. PMID:31066823. \",\"31. Domingues CMAS, Teixeira AMDS. Coberturas vacinais e adoções imunopreveníveis no Brasil no periodo 1982-2012: avanços e desafiros do Programa Nacional de Imunizações. Epidemiol Serv Saude. 2013;22(1):9-27. http://dx.doi.org/10.5123/S1679-49742013000100002. \",\"32. Brasil. Lei n° 13.709, de 14 de ago do 2018. Lei Geral de Protecao de Dados Pessoa (LGPD). Diario Oficial da Uniao; Brasilia; 2018. \",\"33. Landis JR, Koch GG. The measurement of observer agreement for categorical data. Biometrics. 1977;33(1):159-74. http://dx.doi.org/10.2307/2529310. PMID:843571. \",\"34. McHugh ML. Interrater reliability: the kappa statistic. Biochem Med. 2012;22(3):276-82. http://dx.doi.org/10.11613/BM.2012.031.PMid:23092060. \",\"35. Associacao Brasileira de Empresas de Pesquisa. Critério de classificacao economica Brasil. Sao Paulo: ABEP; 2008. \",\"36. Cordeiro EL, Silva LSR, Urquiza JL, Nascimento MA, Silva RM, Souza GCS, et al. conheicao das mães sobre o esquema vacinal de seu filhos assistidos em uma unidade basia de saude. Brazilian Journal of Health Review. 2019;2(1):644-60. \",\"37. Poland GA, Jacobson RM. Understanding those who do not understand: a brief review of the anti-vaccine movement. Vaccine. 2001;19(17-19):2440-5. http://dx.doi.org/10.1016/S0264-410X(00)00469-2. PMID:11257375. \",\"38. Hornsey MJ, Harris EA, Fielding KS. The psychological roots of anti-vaccination attitudes: a 24-nation investigation. Health Psychol. 2018;37(4):307-15. http://dx.doi.org/10.1037/hea0000586.PMid:29389158. \",\"39. Barbieri CLA, Couto MT. Decision-making on childhood vaccination by highly educated parents. Rev Saude Publica. 2015;49:18. http://dx.doi.org/10.1590/S0034-8910.2015049005149. PMID:25830870. \"]","page_idx":"10","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[67,42,460,55]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Concordância entre informações reportadas pelas mães e dos cartões de vacina ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[821,44,863,52]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"cademos ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[823,52,927,81]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Saude Coletiva ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[68,945,84,954]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"50 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[768,944,929,955]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cad. Saude Colet., 2021;29 Esp.: ","text_format":null,"text_level":null,"type":"footer"}],"string":"[\n {\n \"bbox\": \"[67,126,670,215]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Análise da concordância entre as informações reportadas pelas mães e dos cartões de vacina dascriçças no Brasil (2013 e 2015) \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,224,596,288]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Analysis of the agreement between information on child vaccination reported by mothers and that on the children's vaccination cards in Brazil (2013-2015) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,303,653,331]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Mónica Viegas Andrade<sup>1</sup>, Kenya Noronha<sup>1</sup>, Clareci Silva Cardoso<sup>2</sup>, Cláudia Di Lorenzo Oliveira<sup>2</sup>, Júlia Almeida Calazans<sup>1</sup>, Michelle Nepomuceno Souza<sup>1</sup> \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,337,702,358]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$^{1}$ Faculdade de Ciências Econômicas - Centro de Desenvolvimento e Planejamento Regional (Cedeplar), Universidade Federal de Minas Gerais (UFMG) - Belo Horizonte (MG), Brasil \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[67,360,574,372]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$^{2}$ Departamento de Medicina, Universidade Federal de Sao Joao del-Rei - Sao Joao del-Rei (MG), Brasil \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,399,677,436]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Comounar Andrade MV, Noronha K, Cardoso CS, Oliveira CL, Calazans JA, Souza MN. Analise da concordancia entre as informacoes reportadas pelas mares e dos cartoes de vacina das crianças no Brasil (2013 e 2015). Cad Saude Colet, 2021;29esp.:40-50. https://doi.org/10.1590/1414-462X202199010302 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[67,462,126,476]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Resumo \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,481,715,520]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Introdução: Este trabalho discute a possibídale de'utilização de informações sobre a vacinação infantil reportada pelas mães em inquisitados domiciliares para a construção de指示ores de cobertura vacINAL no Brasil. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,522,714,549]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Objetivo: Avaliar o potencial das informações declaradas pelas mães sobre a imunização dascriçanas em inquisitços domicíliares como uma fonte para o calculo do indicator de cobertura vacinal. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,549,714,602]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Método: Analisaram-se os指示ores de confiabilidade (preciso) e validade (concordância) entre as informações disponveis nos cartões de vacina dascriçças menores de 2 anos e as informações reportadas pelas mães em pesquisas domicíliares realizadas nos anos de 2013 e 2015 no[municipio de Santo Antônio do Monte, Minas Gerais. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,604,714,644]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Resultados: O principal resultado minha que, em um(contexto de alta cobertura, as mães tendiam a informar sobre a vacinação de seuis filhos de forma aleatoria, ou sera, informações reportadas pelas mães aparevam baixa validade. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,645,714,684]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conclusao: Embora a coleta de informacoes dos cartoes de vacina das crianças sera um procedimento mais custoso, esta e a forma mais adequada e confiavel de se mensurar a cobertura vacinal nocontexto Brasileiro. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,689,470,703]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Palavras-chave: vacinas; estudos de validação; autorreportado. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[67,717,129,730]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Abstract \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,735,715,762]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Background: This study discusses the possibility of using information on child vaccination reported by mothers through household surveys to construct indicators of immunization coverage in Brazil. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,762,714,789]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Objective: Evaluate the potential of information reported by mothers on the immunization of children in household surveys as a source to calculate a vaccination coverage indicator. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,790,714,817]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Method: The indicators of reliability (accuracy) and validity (concordance) between the information available on the vaccination cards of children under two years of age and that reported by mothers in \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[757,824,828,844]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0373/1ddba5d1646efac7041e98f979a7f93a49d0930372d3802cfae38e77fa8fd3df.jpg\",\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[757,848,919,934]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Este é um artigo(publicado em acesso aberto (Open Access) sob a licença Creative Commons Attribution, que permite uso, distribuição e reprodução em qualcer meio, sem restricções desde que o trabalho original sera corretamente citado. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[67,31,90,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"a \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[141,32,168,52]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CC \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[67,90,174,103]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ARTIGO ORIGINAL \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[754,31,929,93]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"cadernos Saude Coletiva \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[815,98,929,109]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"ISSN 2358-291X (Online) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[65,848,423,858]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Trabalho realizao na/no/em Universidade Federal de Minas Gerais \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[65,860,460,871]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Correspondência: Mônia Viegas Andrade. E-mail: mviegas123@gmail.com \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[65,872,702,905]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Fonte de Financiamento: Organização Pan-Americana da Saúde (OPAS), Conselho Nacional de Secretários Municipais de Saúde (CONASS), FUNDação DE APOIO À PESQUISA DO ESTADO DE MINAS GERAIS (FAPEMIG)/PROGRAMA DE PESQUISA PARA O SUS (PPSUS)/EDITAL 14/2013 PROCESSO APQ0355913. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[67,906,272,915]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conflito de Interesses: nada a declarar. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[67,916,366,928]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recebido em: Ago. 04, 2019. Aprovado em: Dez. 26, 2019 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_footnote\"\n },\n {\n \"bbox\": \"[68,944,85,954]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"40 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[514,943,929,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cad. Saude Colet., 2021;29(esp.): | https://doi.org/10.1590/1414-462X202199010302 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[282,85,927,112]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"two household surveys conducted in 2013 and 2015 in the municipality of Santo Antônio do Monte, state of Minas Gerais, Brazil. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[282,114,929,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Results: The main result shows that, in a context of high coverage, the mothers tended to inform randomly about the vaccination of their children, that is, the information reported by mothers presented low validity. Conclusion: Although collecting information from children's vaccination cards is a more costly procedure, it is the most adequate and reliable way to measure vaccination coverage in the Brazilian context. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[284,172,598,185]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Keywords: vaccine; validation studies; self-report. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[284,208,386,225]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"INTRODUÇÃO \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[284,228,927,303]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Descuão do Plano Nacional de Imunização em 1973, a imunização infantil se destáca por ser uma dasunisticas de saúde mais bem-sucedidas no Brasil, alcancando uma cobertura proxima da universalidade para as vacinas preconizadas pelo Ministério da Saúde para crianças menores de 5 anos de idade<sup>1,2</sup>. Para que o monitoramento das metas de vacinação sera feito de forma eficiente, são necessários bons indicadores de cobertura vincinal. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[282,305,929,577]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"No Brasil, o indicatoricial de cobertura vacinal é obtido por meio da razão entre o número de doses aplicadas de determinada vacina e a estimativa da population na idade espécífica da vaciação<sup>3,4,5</sup> Por sua vez, o número de doses aplicadas é calculado com base nos boletins mensais emitidos pelos postos de vaciação. Esse indicator de cobertura vacinal é intuitivo, direto, de simples interpretação e pode ser calculado para diversas unidas geograficas e periodos no tempo, fazendo com que ele sera amplamente utilizes na avaliação da cobertura. Contudo, muitas vezes esse indicator pode fornecer estimativas imprecisas, visto que ele está passivel de erros tanto no registrar das doses aplicadas quando nas estimativas populaciones<sup>5,6,7</sup> No calculo das doses aplicadas, são contabilizadas todas as doses efetivamente recebidas, independentelemente da idade em que foram aplicadas. Desse modo, crianças que tomam doses fora da idade adequada são contabilizadas no numerator de cobertura vacinal. O denominador también é passivel deerro, uma vez que as estimativas populacionesis, principalmente para crianças de 0 a 2 anos,也是非常PRESENTAMeros em decorrência da subenumeração dos registrados de nascimento no pais<sup>8</sup>. Como resultado, o indicator de cobertura vacinal, na maior parte das vezes, pode estar superestimado, especialmente em municípios menores. Por exemplo, em Minas Gerais, em 2015, a cobertura vacinal calculada pelo Ministerio da Saude para as vacinas BCG, hepatite B, rotavirus, pneumonia, meningocócica e triplice viral está acima de $100\\\\%$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[282,578,929,699]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Uma forma alternatively de averir a cobertura vacinal é por meio de pesquisas domiciliares. Nesse caso,两大 informações podem ser investigadas: (1) a declaração das mães ou responsaveis sobre a vacinação das crianças; e (2) averification do cartão de vacina. Avantagem do uso de pesquisas domiciliares é a possibídia de avaliar a cobertura vacinal considerando atributos socieônômicos e demógraficos das famílias das crianças. Do ponto de vista de política Pública, é fundamental entender o alcance dos programas de imunização e identificar os grupos populaciónais mais vulnárveis. O indicator oficial do Ministerio da Saúde está permiteanalises em que o menor nível de desaggregação é um citizenshipo. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[282,699,929,791]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A verificacao direta do cartao de vacina ja tem sido realizada para o Brasil6,8,9,10,11. Em 2007, um estado mais amplo incluindo todas as capais foi realizao para averir a real cobertura vacinal em crianças meores de 18 mezes de idade pelo grupo Vaccine Coverage Survey $2007^{12}$ . Segundo esta pesquisa, a cobertura para as crianças da amostra era de $83\\\\%$ ,送去 a cobertura maior nos grupos com menor renda familiar, evidenciando uma desigualdade favoravel na cobertura vacinal entre os grupos sococonomicos mais baixos. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[284,791,927,852]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Nas pesquisas domiciliares, a forma mais fácil de averir a cobertura vacinal é por intermédiação da informação reportada pelas mães. Pesquisas de ambito nacional, como a Pesquisa Nacional de Saúde (PNS/IBGE) $^{13}$ , realizada em 2013, investigam a imunização das crianças menores de 1 ano a partir de informations declaradas pelas mães. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[284,853,927,912]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"O objetivo de este estudo foi, portanto, avaliar o potencial das informações declaradas pelas mães sobre a imunização das crianças em inquecretos domicíliares como uma fonte para o calculo do indicator de cobertura vacinal. Para tanto, a declaração das mães sobre a imunização de seuços foi comparada com o registro dos cartões de vacina dessas crianças coletados em \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[70,45,173,81]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"cadernos\\nSaude Coletiva \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[537,39,927,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Concordância entre informações reportadas pelas mães e dos cartões de vacina \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[68,945,226,954]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cad. Saude Colet., 2021;29(esp.): \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[914,945,927,953]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"41 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[65,85,712,116]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"una subpoena de campo realizada no municipio de Santo Antonio do Monte (Minas Gerais, Brasil) nos anos de 2013 e 2015. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,116,715,177]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Diversos estudos buscam validar as informações sobre cobertura vacinal reportadas em pesquisas domicíliares a partir de registros administrativos e médicos $^{14-28}$ . A maior desses artigos avalia as informações atentes à cobertura das vacinas contra gripe (influenza) e pneumonia nos Estados Unidos, Reino Unido, Espanha, Franca, Itália e Austrácia. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,177,715,298]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"De forma geral, esses estudos muito que a cobertura vacinal, calculada a partir das informações declaradas, está muitoproxima aos registrados oficial para a vacina contra influenza,送去 que a concordança entre as两大 fontes de informacao varia entre $95\\\\%^{15,18,28}$ e $98\\\\%^{17,24,27}$ . Japara a vacina contra pneumonia, a concordancia varia entre $64\\\\%^{28}$ e $90\\\\%^{20}$ . Lu et al.25 e Boakye et al.26 analisaram os dados da National Immunization Survey-Teen (NIS-Teen) para a cobertura vacinal de adolescentes americanos de 13 a 17 anos para as vacinas contra gripe (influenza) e HPV, respectivamente. Nos dos estudos, entre os adolescentes com registro clnico, $86\\\\%$ dos pais affirmaram que seu filho havia tomado a vacina. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,298,715,404]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Brown et al. $^{16}$ compararamDados relacionados à vacina contra influenza para 3.072 crianças americanas, no estudo de New Vaccine Surveillance Network. Essa pesquisa questionou os pais das crianças sobre o recebimento de vacina contra influenza e conferiu a resposta com oprofessional provedor da vacina.Os resultadosencionados no estudo做不到 que os pais tendiam a responder corretamente sobre o status vacinal das crianças (92% de sensibilitadede $82\\\\%$ de especificidade),porém a concordencia entre as respostas diminuía com o aumento do tempo après o recebimento da vacina. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,404,715,571]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Uma série de estudos vem ressaltando también a importante das caracteristicas sociodemograficas e econômicas como fatores determinantes da concordância entre a informação reportada e os registros oficials<sup>15,21,22,29</sup>. AsMulheres tendem a responder sobre suas vacinas de forma mais similar aos registros oficials do que os homens. Por及其他ado, a concordância de crianças e idosos tends a ser menor que a concordância de jovens adultos. Martin et al.<sup>21</sup>也是非常asnstraramqueindividualoscommajorccessoao servicosbasicosde saude témmaioreschancedes responder sobereu statusdevacinaçãodemaneira similaraoresignadosoficiais.Rolnick et al.<sup>22</sup>ressaltamqueacordânciaentreasinformaçõesdeclaradas e os registros administrativos e clínicos varia expressivamenteentreasvacinas.A vacina contra hepatite A tem as menores taxas de concordância,uponas vacinascontra tétano,HPV e gripeapursematleadivastaxasde concordância. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,572,715,679]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ainda que a literatura a respeito da validação das informações declaradas sobre cobertura vacinal está vasta, esse campo de estudo não está esgotado. O presente artigo se coloca como uma importante contribuição, uma vez que busca validar as informações de cobertura vacinal para um grupo atéOOTInedito na litteratura:as crianças de ate 2 anos de idade.No Brasil, a maior das vacinas é ministrada na primeira infancia, de modo que as informações reportadas pelas mães em pesquisas domiciliares podem se configurar como um importante mecanismo de monitoramento da cobertura vacinal no pais. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[67,699,137,715]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"MÉTODO \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[67,742,196,757]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Fonte de dados \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,762,715,852]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Este estudo realizou as informações fornecidas por um inquisito domiciliar realizado emodos recortes transversais independentes no municipio de Santo Antonio do Monte, em Minas Gerais, no ambito da Pesquisa de Avaliação do Laboratório de Inovações no Atença às Condições Crónicas (LIACC) na Atença Primária à Saúde<sup>30</sup>. Essa pesquisa avaliou as mudanças em diversos indicadores de saúde antes, no decorrer e(before da intervenção do LIACC para individuos com hipertensão, diabetes, gestantes e crianças menores de 2 anos. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,853,715,914]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"As datas de realização das entrevistas eram referentes aos periodos antes e.after do processo de intervenção do LIACC no@municipio. Aserviewas respondidas em 2013 diziam respeito ascriçanas nascidas entre janeiro de 2011 e dezembro de 2012, enquanto as entrevistas respondidas em 2015 se referiam ascriçanas nascidas entre janeiro e dezembro de 2014. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[67,42,460,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Concordância entre informações reportadas pelas mães e dos cartões de vacina \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[821,44,929,83]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"cademos\\nSaüde Coletiva \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[68,945,84,954]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"42 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[768,944,929,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cad. Saude Colet., 2021;29 Esp.: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[282,85,929,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"O tamanho amostral foi definido a partir do número médio anual de nascimento ocorridos no municipio de Santo Antônio do Monte, registrados no Sistema de Informação sobre Nascidos Vivos (SINASC). A amostra final foi representativa da população de crianças menores de 2 anos do municipio, sendo investigadas 185 crianças em 2013 e 155 crianças em 2015. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[282,163,929,286]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Primeiramente, perguntou-se às vezes se a criança havia tornado pelo menos uma dose de cada vacina analisada: BCG-ID, hepatite B, antipólico, tetravalente, rotavirus, pneumocócica, meningocócica, triplice viral e febre amarela. Em seguida, as quando responderam se a criançainha cartão de vacina e se ele estava presente no momento da在网络. Esses cartões foram fotografados peloscretivestadores com o consentimento das vezes, e a elaboração do banco de dados com tais informações teve dupla checagem para a garantia da qualida informacional. Optou-se por travalear comigo não uma das doses para que fosse possivel realizar a comparação com as informações reportadas pelas vezes. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[282,286,929,378]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Osindicadores foram construuidos apenas para as crianças que tinkham a idade completa preconizada pelo calendario vacinal do Ministerio da Saude, com una janela de tempo de dosmeses. Essa janela é necessaria,pois,muitas vezes,a crianca não tomou a vacina na idade exata por estar com algo problema de saude ou por indisponibilitadedefcontinuidade do proprietary systema31 Não ha na literatura um accordo sobre o tempo valido de espera para a proxima dose da vacina6,por isso tal estrategia fou adotada sem um parametro predefinido. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[282,379,929,500]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"O calendário vacinal do Ministerio da Saúde sempre passa por alterações no que tange ao número de doses e a idade de cada uma das vacinas. Este trabalho utilizeou o calendário vigente até o ano de 2015, uma vez que as crianças analisadas foram imunizadas a partir das normas do calendário antigio. Ressalta-se que, durante o ano de 2012, a vacina tetravalente foi substituira pela pentavalente, e em muitos cartões de vacina analisados o carimbo ou a anotação que validava a aplicação da vacina VINHA com uma observação de que a criança nãoinha tomado a tetravalente, mas sim a pentavalente. Nesses casos, o trabalho considerou a criança imune pela tetravalente. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[282,501,929,624]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Esta Pesquisa foi avaliada e aprovada pelo Comité de Ética em Pesquisa da UFSJ/Campus Centro-Oeste Dona Lindu, sob o Parecer n° 369.942, no ano de 2013. Os dados individuais coletados se tornaram anônimos e não apareceram em nenhum relatório ou(publicação,sendo resguardado o direito ao sigilo. Após a Pesquisa, os dados foram arquivos e não foram realizados para及其他 fim, de acordo com os princípios do art. $6^{\\\\circ}$ da Lei Geral de Proteção de Dados<sup>32</sup>, em especial o da segurança e transparência, e o art. $7^{\\\\circ}$ da referida lei, quando ao tratamento dos dados. Informações completeness sobre a Pesquisa está disponível no relatório técnico entrega à Organização Pan-Americana de Saude em novembro de 2016. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[284,646,885,664]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Indicadores de confiabilidad e validade das informacoes autodeclaradas \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[282,667,929,728]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Os registros dos cartões de vacina foram considerados como o grupo de referencia para a avaliação da qualidade das informações autodeclaradas das vezes. As consistências entre as两大 fontes de dados foram analisadas a partir de vez指示ores amplamente realizados em estudos epidemiológicos<sup>33</sup>: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[282,728,929,912]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. Sensibilitidade: proportao de crianças que foram vacinadas segudo o relato materno entre aquelas vacinadas com base nos cartoes de vacina. \\\",\\\"2. Especialidade: proporcão de crianças que não foram vacinadas conforme o relato materno entre aquelas não vacinadas com base nos cartões de vacina. \\\",\\\"3. Valor preditivo positivo:proporcão de crianças que foram vacinadas com base nos cartoes de vacina entre aquelas vacinadas de acordo com o relato das mães. \\\",\\\"4. Valor preditivo negativo: proportao de crianças que nao foram vacinadas com base nos cartoes de vacina entre aquelas nao vacinadas conforme o relato das mães. \\\",\\\"5. Taxa de concordância geral: proporcão de concordância entre as respostas das mães e os registros nos cartões. \\\",\\\"6. Úndice Kappa: taxa de concordância geral, ponderada pela concordância esperada. Esse Úndice, usualmente, varia entre 0 e 1, de tal forma que, quanto maior for seu valor, maior é \\\"]\",\n \"page_idx\": \"3\",\n \"sub_type\": \"text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[70,45,173,81]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"cadernos\\nSaude Coletiva \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[537,39,927,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Concordância entre informações reportadas pelas mães e dos cartões de vacina \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[68,944,228,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cad. Saude Colet., 2021;29(esp.): \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[914,945,927,953]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"43 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[100,85,715,148]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"a concordância entre as两大 fontes de informação. Contudo, em algumas circunstâncias, é possível observar um indice Kappa negativo, quando a concordância observada é menor que a concordância esperada. Valores negativos baixos (até -0,1) são usualmente interpretados como não concordância<sup>34</sup>. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,148,717,226]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Parte da consonança verificada pela taxa de concordância geral pode ser aleatoria, e não em decorrência da concordância entre as两大 fontes de informação. Assim, o indice Kappa é estatistically superior à taxa de concordância geral, uma vez que, ao ponderar a taxa de concordância geral pela taxa de concordância esperada, é possível obter a concordância liquida, para às vezes daquela que seria esperada ao acaso. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,249,268,264]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"RESULTADOS E DISCUSSAO \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,268,717,440]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Realizaram-se entreprises com 340 mães de crianças de ate 2 anos de idade nos anos de 2013 e 2015. A maioria dessas mãesinha entre 20 e 35 anos (76,47%), com pelo menos o ensino fundamental completo (59,42%) e pertencente à classe de riqueza C (56,64%),utilizing os parâmetros do Critério Brasil da Associacao Brasileira de Empresa de Pesquisa de $2014^{35}$ Todas as mães declararam que a criançainha cartao de vacina. Contudo, 42 crianças foram excluidas da analise, uma vez que 33 mães não deixaram oscretivestadores verem o cartao de vacina da criança, 8 mães affirmaram que a criança ja teve outro cartao de vacina e 1 mãenão sabia se a criança ja havia sido outro cartao. Dessa forma, a analise da cobertura vacinal considerou 298 crianças, as quais representavam $88\\\\%$ da amostra inicial. Não houve diferencas socioeconomicas estatisticamente significativas entre as mães das crianças da amostra inicial e o subGrupo selecionado. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,441,717,551]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Tabela 1 apareça as taxas de cobertura vacinal para as crianças de até 2 anos de idade no[municipio de Santo Antônio do Monte nos anos de 2013 e 2015,considerando as informações reportadas pelas mães e os registros presentes nos cartões de vacina. Em primeiro lugar,notou-se elevada cobertura no[municipio para as vacinas preconizadas no primeiro ano de vida,independente da informaçãoutilizada.Com base nas informações dos cartões, $100\\\\%$ das crianças tomaram pelo menos uma dose de BCG-ID,antipólio,pneumocócica, meningocócica e febre amarela. Para as demais,a cobertura vacinal foi de $99\\\\%$ \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[67,609,715,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0994/248fe639b3a00c78435826d985381338408365cbcf007c6578fcf62484290f52.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=\\\"2\\\"></td><td colspan=\\\"2\\\">Cobertura \\nvacinal – mãe</td><td colspan=\\\"2\\\">Cobertura \\nvacinal – \\ncartões</td><td colspan=\\\"2\\\">Teste de média</td><td>Total de crianças \\nelegíveis à vacina</td></tr><tr><td>n</td><td>%</td><td>n</td><td>%</td><td>p-valor</td><td>Significância</td><td>n</td></tr><tr><td>BCG-ID</td><td>283</td><td>98,26</td><td>288</td><td>100</td><td>0,025</td><td>**</td><td>288</td></tr><tr><td>Hepatite B</td><td>285</td><td>98,28</td><td>288</td><td>99,31</td><td>0,254</td><td>NS</td><td>290</td></tr><tr><td>Antipólio</td><td>275</td><td>98,57</td><td>279</td><td>100</td><td>0,045</td><td>**</td><td>279</td></tr><tr><td>Tetravalente</td><td>261</td><td>93,21</td><td>279</td><td>99,64</td><td>0</td><td>***</td><td>280</td></tr><tr><td>Rotavirus</td><td>268</td><td>96,4</td><td>277</td><td>99,64</td><td>0,006</td><td>***</td><td>278</td></tr><tr><td>Pneumocócica</td><td>241</td><td>87,32</td><td>276</td><td>100</td><td>0</td><td>***</td><td>276</td></tr><tr><td>Meningocócica</td><td>229</td><td>82,37</td><td>278</td><td>100</td><td>0</td><td>***</td><td>278</td></tr><tr><td>Tríplice viral</td><td>150</td><td>97,4</td><td>152</td><td>98,7</td><td>0,41</td><td>NS</td><td>154</td></tr><tr><td>Febre amarela</td><td>158</td><td>96,34</td><td>164</td><td>100</td><td>0,013</td><td>**</td><td>164</td></tr></table>\",\n \"table_caption\": \"[\\\"Tabela 1. Cobertura vacinal declarada pelas mães e registrarada no cartão de vacina para crianças menores de 2 anos, Santo Antonio do Monte, Minas Gerais, 2013 e 2015 \\\"]\",\n \"table_footnote\": \"[\\\"Fonte: Pesquisa SAMONTE (2013 e 2015); Nota: *** Estatisticamente significativo a 1%; ** Estatisticamente significativo a 5%; * Estatisticamente significativo a 10%; NS não estatisticamente significativo \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[67,42,460,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Concordância entre informações reportadas pelas mães e dos cartões de vacina \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[821,44,929,83]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"cademos Saude Coletiva \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[68,945,84,954]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"44 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[768,944,929,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cad. Saude Colet., 2021;29(esp.): \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[284,85,929,192]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A cobertura vacinal estimada com base nas respostas das mães foi elevada, mas foi menor do que a estimada de acordo com os registros dos cartões. As diferencias entre a resposta da maior e a informação do cartão foram estatistically significativas para todas as vacinas, exceto para hepatite B e triplice viral. As maiores diferencias foram observadas para as vacinas tetravalente, pneumocócica e meningocócica (Tabela 1). Essas vacinas, como mencionado na metodologia, passaram por mudanças no calendário vacinal, o que pode levar a um maior desconhecimento delas por parte das mães. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[284,195,929,346]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Algumas mudanças que ocorreram no calendário vacinal preconizzato pelo Ministerio da Saude podem, em alguma medida, explicar a subdeclaracao pelas mães. A vacina tetravalente foi substituidade no ano de 2012 pela pentavalente que protege contra difteria, tétano, coqueluche, Haemophilus influenzae tipo B e hepatite B. Essa mudança pode justIFICAR a subestimação de resposta para hepatite B e tetravalente. No caso das vacinas pneumocócica e meningocócica, a subestimação pode estar associada à inclusão recente dessas两大 vacinas no calendario vacinal, que passaram a ser ofertadas somente a partir de 2010. Possivelmente as mães tiveram dificuldades de identificar a imunização da criança por causa da pouca familiaridade e da baixa difusão de informações sobre essas vacinas. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[284,349,929,549]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Tabela 2 apareça os indices de sensibilitadé e especialidade entre as informações declaradas pelas mães e as informações registradas nos cartões de vacina das crianças menores de 2 anos em Santo Antônio do Monte. De forma geral, entre as crianças com registrar no cartão, houve um percentual expressivo de mães que declaresram que a criança tomou a vacina. Para as vacinas BCG-ID, hepatite B, antipólico, triplice virale e febre amarela, a sensibilitadé foi maior que $97\\\\%$ . Por及其他, a proporcão de mães que declaresram que a criança tomou a vacinas pneumocócica e meningocócica entre aquelas com registrar no cartão foi de $88\\\\%$ e $83\\\\%$ respectively, indicando que, para essas vacinas, houve uma menor concordância entre mães e cartões. Por及其他, em todos os casos em que não havia registrar no cartão, as mães declaresram que a criançainha tomado a vacina, ou seja, a especialidade foi igual a zero para todas as vacinas. Desse modo, observa-se uma fragilidade na responça das mães que reportaram que a criança sempre recebeu a vacina. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[282,631,929,885]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0934/973c161c25e82ce1b0dec65c168c7635310263510db7c548d45f2de9273d30d0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=\\\"2\\\"></td><td rowspan=\\\"2\\\">Crianças com registrar no cartão</td><td colspan=\\\"2\\\">Sensibilidade</td><td rowspan=\\\"2\\\">Crianças sem registrar no cartão</td><td colspan=\\\"2\\\">Especialidade</td><td rowspan=\\\"2\\\">Crianças elegíveis à vacina</td></tr><tr><td>n</td><td>%</td><td>n</td><td>%</td></tr><tr><td>BCG-ID</td><td>288</td><td>283</td><td>98.26</td><td>0</td><td>0</td><td>0.00</td><td>288</td></tr><tr><td>Hepatite B</td><td>288</td><td>283</td><td>98.26</td><td>2</td><td>0</td><td>0.00</td><td>290</td></tr><tr><td>Antipólio</td><td>294</td><td>290</td><td>98.64</td><td>0</td><td>0</td><td>0.00</td><td>294</td></tr><tr><td>Tetravalente</td><td>290</td><td>271</td><td>93.45</td><td>1</td><td>0</td><td>0.00</td><td>291</td></tr><tr><td>Rotavírus</td><td>284</td><td>274</td><td>96.48</td><td>1</td><td>0</td><td>0.00</td><td>285</td></tr><tr><td>Pneumocérica</td><td>285</td><td>250</td><td>87.72</td><td>0</td><td>0</td><td>0.00</td><td>285</td></tr><tr><td>Meningocérica</td><td>286</td><td>237</td><td>82.87</td><td>0</td><td>0</td><td>0.00</td><td>286</td></tr><tr><td>Tríplice viral</td><td>168</td><td>164</td><td>97.62</td><td>22</td><td>0</td><td>0.00</td><td>190</td></tr><tr><td>Febre amarela</td><td>211</td><td>205</td><td>97.16</td><td>2</td><td>0</td><td>0.00</td><td>213</td></tr></table>\",\n \"table_caption\": \"[\\\"Tabela 2. Sensibilitad e especificidade para a concordancia entre as informacoes declaradas pelas mares e as informacoes registradas no cartao de vacina para crianças meores de 2 anos, Santo Antônio do Monte, Minas Gerais, 2013 e 2015 \\\"]\",\n \"table_footnote\": \"[\\\"Fonte: Pesquisa SAMONTE (2013 e 2015); Nota: *** Estatisticamente significativo a 1%; ** Estatisticamente significativo a 5%; * Estatisticamente significativo a 10%; NS Não estatisticamente significativo \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[70,45,173,81]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"cadernos\\nSaude Coletiva \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[537,39,927,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Concordância entre informações reportadas pelas mães e dos cartões de vacina \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[68,945,228,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cad. Saude Colet., 2021;29(esp.): \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[914,945,927,954]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"45 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[68,87,712,208]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Assim, quando a criança não tomava uma vacina espécífica, aõe tendia a affirmar que a criança ainha tornado. Esse padrão de resposa provavelmente tenha sido influenciado pela elevada cobertura vacinal observada no@municipio. Como a cobertura foi proxima de $100\\\\%$ as mães, em geral, tendiam a reportar que as crianças estavam com todas as vacinas em dia. É interessante ressaltar o caso que ocorreu com a vacina triplice viral. Entre 2012 e 2014, segundo o relato dos professionais de sauède Santo Antonio, o@municipio passou por problemas de abastecimento dessa vacina, de modo que um percentual significativo das crianças não a tomou. Mesmo assim, todas as mães affirmaram que seuis filhos tomaram a vacina triplice viral. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[68,210,712,332]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Tabela 3 apareça os values previdos positivos e negativos para a concordância entre as informações declaradas pelas mães e as informações nos cartões de vacina das crianças menores de 2 anos em Santo Antonio do Monte. Para todas as vacinas, com exceço da triplice viral, constatou-se que, para praticamente todas as criançasURT ja havia declarado que elas tomaram a vacina, havia registrar tambem no cartão,uma vez que o valor preditivo positivo foi maior que $99\\\\%$ .Por及其他lado,entre as criançasURT ja havia declarado que elas não tinkham tomando a vacina, todas tinkham registrar no cartao,de modo que o valor preditivo negativo foi igual a zero para todas as vacinas. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[67,402,714,670]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/0028/c1f503861a84e0e9d69bf621862864a1f23ae7a125b7054c8142f4253e0fe76a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td rowspan=\\\"2\\\"></td><td rowspan=\\\"2\\\">Afirmação a partir do relato materno</td><td colspan=\\\"2\\\">PPV</td><td rowspan=\\\"2\\\">Negação a partir do relato materno</td><td colspan=\\\"2\\\">NPV</td><td rowspan=\\\"2\\\">Crianças elegíveis à vacina</td></tr><tr><td>n</td><td>%</td><td>n</td><td>%</td></tr><tr><td>BCG-ID</td><td>283</td><td>283</td><td>100.00</td><td>5</td><td>0</td><td>0.00</td><td>288</td></tr><tr><td>Hepatite B</td><td>285</td><td>283</td><td>99.30</td><td>5</td><td>0</td><td>0.00</td><td>290</td></tr><tr><td>Antipólio</td><td>290</td><td>290</td><td>100.00</td><td>4</td><td>0</td><td>0.00</td><td>294</td></tr><tr><td>Tetravalente</td><td>272</td><td>271</td><td>99.63</td><td>19</td><td>0</td><td>0.00</td><td>291</td></tr><tr><td>Rotavírus</td><td>275</td><td>274</td><td>99.64</td><td>10</td><td>0</td><td>0.00</td><td>285</td></tr><tr><td>Pneumocérica</td><td>250</td><td>250</td><td>100.00</td><td>35</td><td>0</td><td>0.00</td><td>285</td></tr><tr><td>Meningocérica</td><td>237</td><td>237</td><td>100.00</td><td>49</td><td>0</td><td>0.00</td><td>286</td></tr><tr><td>Tríplice viral</td><td>186</td><td>164</td><td>88.17</td><td>4</td><td>0</td><td>0.00</td><td>190</td></tr><tr><td>Febre amarela</td><td>207</td><td>205</td><td>99.03</td><td>6</td><td>0</td><td>0.00</td><td>213</td></tr></table>\",\n \"table_caption\": \"[\\\"Tabela 3. Valor preditivo positivo e valor preditivo negativo para a concordância entre as informações declaradas pelas mães e as informações registradas no cartão de vacina para crianças menores de 2 anos, Santo Antônio do Monte, Minas Gerais, 2013 e 2015 \\\"]\",\n \"table_footnote\": \"[\\\"Fonte: Pesquisa SAMONTE (2013 e 2015); Nota: *** Estatisticamente significativo a 1%; ** Estatisticamente significativo a 5%; * Estatisticamente significativo a 10%; NS não estatisticamente significativo \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[68,725,712,818]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A Tabela 4 apareça a taxa de concordância geral e o Úndice Kappa para a concordância entre as informações declaradas pelas mães e as informações nos cartões de vacina das crianças menores de 2 anos em Santo Antônio do Monte. Notou-se um altoível de concordância para todas as vacinas, embora existisse expressiva variação entre os típos. A maior taxa de concordância foi observada para a BCG-ID (98%), o que era esperado, uma vez que esta vacina é tomada em geral na maternidade, e a menor foi a meningocócica (83%). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[68,819,712,912]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Embora a taxa de concordância geral estivesse bastante elevada, esse indicator não foi suficiente para garantir a validade da informação das mães, uma vez que seu valor foi fortemente determinado pela concordância entre as respostas positivas das mães e os registros no cartão. Como a especialidade e o valor preditivo negativo foram igualais a zero, ao ponderar a concordância observada pela concordância esperada, foi obtido o indice Kappa proxies a zero para praticamente todas as vacinas, o que significica que a concordância percebida entre as \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[68,42,458,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Concordância entre informações reportadas pelas mães e dos cartões de vacina \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[821,44,927,81]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"cademos\\nSaüde Coletiva \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[68,945,84,954]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"46 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[769,945,927,954]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cad. Saude Colet., 2021;29(esp.): \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[282,129,929,382]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": \"images/1012/15f1838c10bb48d9d8ddc3445d938e6de39e78a16286ae1f43f6be4235e78238.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": \"<table><tr><td></td><td>Número de mazes cartão em concordância</td><td>Taxa de concordância geral</td><td>Índice Kappa</td><td>Crianças elegíves à vacina</td></tr><tr><td>BCG-ID</td><td>283</td><td>98.26</td><td>0.00</td><td>288</td></tr><tr><td>Hepatite B</td><td>283</td><td>97.59</td><td>-0.01</td><td>290</td></tr><tr><td>Antipólio</td><td>290</td><td>98.64</td><td>0.00</td><td>294</td></tr><tr><td>Tetravalente</td><td>271</td><td>93.13</td><td>-0.01</td><td>291</td></tr><tr><td>Rotavírus</td><td>274</td><td>96.14</td><td>-0.01</td><td>285</td></tr><tr><td>Pneumocérica</td><td>250</td><td>87.72</td><td>0.00</td><td>285</td></tr><tr><td>Meningocérica</td><td>237</td><td>82.87</td><td>0.00</td><td>286</td></tr><tr><td>Tríplice viral</td><td>164</td><td>86.32</td><td>-0.04</td><td>190</td></tr><tr><td>Febre amarela</td><td>205</td><td>96.24</td><td>-0.01</td><td>213</td></tr></table>\",\n \"table_caption\": \"[\\\"Tabela 4. Taxa de concordância geral e Índice Kappa para a concordância entre as informações declaradas pelas mães e as informações registradas no cartão de vacina para crianças menores de 2 anos, Santo Antônio do Monte, Minas Gerais, 2013 e 2015 \\\"]\",\n \"table_footnote\": \"[\\\"Fonte: Pesquisa SAMONTE (2013 e 2015) \\\"]\",\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"table\"\n },\n {\n \"bbox\": \"[284,418,927,462]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"respostas das mães e o registrar no cartão foi aleatoria. No caso específico em analise, como a cobertura vacinal era muito elevada, os dados sugerem que as mães tendiam sempre a reportar que os hilhos tomaram a vacina. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[284,464,927,601]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"O Grácfo 1 analisa a discordância entre as两大 fontes de informacao. No total, 194 mães reportaram todas as vacinas em concordância com as informacoes dos cartoes e 94 crianças tiveram informacao reportada emleo menos uma vacina em discordancia comas informacoes registradas nos cartoes de vacina,sendo que a maior das mães reportaram atetres vacinas em divergencia com os cartoes. Ademais,as mães que nao reportaram de forma correta a vacinação de seu filhos não diferiam em termos sococonomicos das demais mães entwickidas no que se refere àidade, classe de riqueza e escolaridade. Esse achado é importante,pois sugere que o desconhecimento da informacao pelas mães não está relacionado a caracteristicas especialicas delas. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[302,633,917,852]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Fonte: Pesquina SAMONTE, 2013 e 2015. \\\",\\\"Gráfico 1. Número de discordâncias entre as informações declaradas pelas mães e as informações registradas no cartão de vacinas para cada criança menor de 02 anos – Santo Antônio do Monte, 2013 e 2015 \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0867/dfcb07c19c30a80d699ffaa88ee877220e4c340f57e636f63174f465861c3638.jpg\",\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[70,45,173,81]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"cadernos\\nSaude Coletiva \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[539,39,927,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Concordância entre informações reportadas pelas mães e dos cartões de vacina \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[68,945,226,954]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cad. Saude Colet., 2021;29(esp.): \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[914,945,927,953]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"47 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[65,85,250,102]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CONSIDERACOES FINAIS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,105,715,168]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Este trabalho analisou a possibidade de'utilização de informações sobre a vacinação infantil reportadas pelas mães para a construção de指示ores de cobertura vincinal em um@municipio brasilheiro de poucojo porte. O estado é inédito no Brasil, uma vez que são escassos os蹼 doscheckados de informações domicíares sobre a cobertura vincinal infantil. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,171,715,234]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A principal limitação do estado é se referir a uma populatione exigência de um@mancipio de pouco no Brasil,sendo,portanto,um estudo de caso.Adespeito da particularidade do municiao, o estudo gera novas evidências para politicas de monitoramento da cobertura vacinal no Brasil. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,235,715,380]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Os Resultados nostraram que, em umcontexto de alta cobertura, as mães tendiam a informar sobre a vacinação de seuis filhos de forma aleatoria. A evidência de Santo Antônio do Monte minha que a informação reportada pelas mãesinha pouca validade para mensurar a cobertura vincinal. Estecontexto difere dos demais estudos que buscaram validar informações autorreportadas sobre vacinas para outros grupos populacionais. Como a cobertura vincinal era alta, as mães provavelmente tinkham baixo conheimento especialico sobre cada vincina, nãosendocapazesdeidentificarquaisestavamforodocumentoadequado.Essesresultados está emconsonânciacom os achados de Brown et al.16,que ja evidenciarama fragilidade das informações reportadas pelas mãesno que tange à vacinação dascriçanas. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,380,715,445]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A baixa concordância das informações das mães ficou mais evidente para a vacina triplice viral. Segundo relato dos professionis de saúde, entre os anos de 2012 e 2014, o@municão de Santo Antonio do Monte apareceu problemas de abastecimento dessa vacina. Ainda assim, todas as mães declararam que seu filhos haviam tornado esta vacina. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,446,715,557]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A não concordança entre as informações declaradas pelas mães e as informações dos cartões de vacina pode ser gerada por causa do viés de的记忆a das mães<sup>29</sup>. Contudo, a partir dos resultadospresentados neste estudo, não se pode affirmar que a não concordança sera decorrente de um viés de的记忆a ou de falta de的记忆ão por parte das mães. Cordeiro et al.<sup>36</sup>, ao analisarem o esquema vicinal de crianças assistidas em uma Unidade Básica de Saúde no[municipio]de Jaboatão dos Guararapes, Pernambuco, evidenciaram a falta de conheção das mães e ressaltaram a importânca de praticas educativas na atençao primária. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,559,715,654]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"O Brasil alcâncôu, com o Sistema Único de Saúde, elevada cobertura vacinal infantil. A poliomielite, a varíola e o sarampo foram erradicados, e a incidência de diversas outras doenças infecções, como febre amarela, difteria, tétano neonatal, coqueluche, meningite, caxumba e rubéola, tem sido mantida em eles relativamente baixos. No município de Santo Antônio do Monte, assim como para a maioría dos municípios brasilieiros, foi observada universalização da cobertura vacinal para as crianças de 0 a 2 anos. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,656,715,718]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Contudo, nos últimos anos, o número de famílias que decidem não vacinar seuis filhos vem crescendo expressivamente em diversos páises do mundo37,38. Ainda que o movimento antivacinha sera muito pouco no Brasil, ele vem ganhando cada vez mais notorialidade39, especialmente entre casais de alto nível de escolaridade e residentes em和地区 urbanas. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,721,715,800]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Considerando que os indicatoros calculados de cobertura vacinal por meio de informações administrativas podem superestimar esta cobertura, a inclusão de informations sobre imunização em pesquisas domicílieares pode ser um instrumento promissor para o monitoramento continuo das metas de vacinação, assim como para a identificação de heterogeneidade de cobertura entreroupos populacionesis. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[65,800,715,912]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Embora a coleta de informações dos cartões de vacina das crianças está oprocesso mais custoso para obter informações de imunização, este estudo indica que esta forma sera a mais adequada de se mensurar a cobertura vacinal nocontextobrasileiro em comparação—as informações declaradas pelas mães. Garantir que a cobertura vacinal continue universal e que doencas infecciosas até então erradicadas no Brasil não ressurjam é um desafio em termos de saude Pública e deve ser encarada como uma das prioridades da Atença Primária à Saude no País. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[67,42,460,54]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Concordância entre informações reportadas pelas mães e dos cartões de vacina \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[821,44,929,83]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"cademos\\nSaüde Coletiva \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[68,945,85,954]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"48 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[768,944,929,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cad. Saude Colet., 2021;29 Esp.: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[282,85,389,100]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"REFERÉNCIAS \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[282,107,929,912]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"1. Victora C, Aquino E, Leal MC, Monteiro C, Barros F, Szwarcwald C. Maternal and child health in Brazil: progress and challenges. Lancet. 2011;377(9780):1863-76. http://dx.doi.org/10.1016/S0140-6736(11)60138-4. \\\",\\\"2. Palm J, Travassos C, Almeida C, Bahia L, Macinko J. The Brazilian health system: history, advances, and challenges. Lancet. 2011;377(9779):1778-97. http://dx.doi.org/10.1016/S0140-6736(11)60054-8. PMID:21561655. \\\",\\\"3. Brasil. Ministerio da Saude, Secretaria de Vigilancia em Saude. Programa Nacional de Imunizacoes: 30 anos. Série C. Brasília: Ministerio da Saude; 2003. \\\",\\\"4. Brasil. Ministério da Saúde, SVS-PNI. Departamento de Informática do SUS - DATASUS [Internet]. 2016 [citado em 2016 Out 1]. Disponível em: http://datasus.saude.gov.br. \\\",\\\"5. Teixeira AMDS, Mota ELA. Denominadores para o calculo de coberturas vacinais: um estudo das bases de dados para estimar a populatione menor de um ano de idade. Epidemiol Serv Saude. 2010;19(3):187-203. http://dx.doi.org/10.5123/S1679-49742010000300002. \\\",\\\"6. Moraes J, Ribeiro M, Simões O, Castro P, Barata R. Qual é a cobertura vacinal real?. Epidemiol Serv Saude. 2003;12(3):147-53. \\\",\\\"7. Moraes J, Ribeiro M. Desigualdades sociais e cobertura vacinal. Rev Bras Epidemiol. 2008;11(Supl 1):113-24. \\\",\\\"8. Pedraza D. Qualidade do Sistema de Informacoes sobre Nascidos Vivos. Cienc Saude Colet. 2012;17(10):2729-37. \\\",\\\"9. Miranda A, Scheibel I, Tavares M, Takeda S. Avaliação da cobertura vacinalo do esquema Basicsico para o primeiro ano de vida. Rev Saude Pública. 1995;29(3):208-14. http://dx.doi.org/10.1590/S0034-89101995000300008. \\\",\\\"10. Brasil. Ministério da Saúde, Departamento de Ciência e Tecnologia. PNDS 2006 - Pesquisa Nacional de Demografia e Saúde da Criança e da Mulher. Relatório Tecnico. Brasília: Ministério da Saúde; 2008. \\\",\\\"11. Yokokura, AVCP, Silva, AAMD, Bernardes, ACF, Lamy, F Fo, Alves, MTSSDB, Cabra, NAL, et al. Cobertura vacinal e fatores associados ao esquema vacinal Basics incomplete aos 12 mezes de idade, São Luís, Maranhão, Brasil, 2006. Cad Saude Pública. 2013;29(3):522-34. \\\",\\\"12. Barata R, Ribeiro M, Moraes J, Flannery B. Socioeconomic inequalities and vaccination coverage: results of an immunisation coverage survey in 27 Brazilian capitals, 2007-2008. J Epidemiol Community Health. 2012;66:934-41. \\\",\\\"13. Brasil. Pesquisa Nacional de Saude 2013: Percepção do estado de saude, estilos de vida e doenças crônicas. Rio de Janeiro: IBGE; 2014. \\\",\\\"14. Bayas JM, Izquierdo C, Ruiz L, Sintes X, Sousa D, Celorrio JM, et al. Validity of self-reported pneumococcal vaccination status in the elderly in Spain. Vaccine. 2009;27(34):4560-4. http://dx.doi.org/10.1016/j.vaccine.2009.05.057. PMID:19508910. \\\",\\\"15. Irving SA, Donahue JG, Shay DK, Ellis-Coyle TL, Belongia EA. Evaluation of self-reported and registry-based influenza vaccination status in a Wisconsin cohort. Vaccine. 2009;27(47):6546-9. http://dx.doi.org/10.1016/j.vaccine.2009.08.050. PMID:19729083. \\\",\\\"16. Brown C, Clayton-Boswell H, Chaves SS, Prill MM, Iwane MK, Szilagyi PG, et al. Validity of parental report of influenza vaccination in young children seeking medical care. Vaccine. 2011;29(51):9488-92. http://dx.doi.org/10.1016/j.vaccine.2011.10.023. PMID:22015394. \\\",\\\"17. King JP, McLean HQ, Belongia EA. Validation of self-reported influenza vaccination in the current and prior season. Influenza Other Respir Viruses. 2018;12(6):808-13. http://dx.doi.org/10.1111/irv.12593. PMID:30028081. \\\",\\\"18. Llupià A, García-Basteiro AL, Mena G, Ríos J, Puig J, Bayas JM, et al. Vaccination behaviour influences self-report of influenza vaccination status: a cross-sectional study among health care workers. PLoS One. 2012;7(7):e39496. http://dx.doi.org/10.1371/journal.pone.0039496. PMID:22808039. \\\",\\\"19. Louergue P, Pulcini C, Massin S, Bernhard M, Fonteneau L, Levy-Bruhl D, et al. Validity of self-reported vaccination status among French healthcare students. Clin Microbiol Infect. 2014;20(12):O1152-4. http://dx.doi.org/10.1111/1469-0691.12759. PMID:25040583. \\\",\\\"20. MacDonald R, Baken L, Nelson A, Nichol KL. Validation of self-report of influenza and pneumococcal vaccination status in elderly outpatients. Am J Prev Med. 1999;16(3):173-7. http://dx.doi.org/10.1016/S0749-3797(98)00159-7. PMID:10198654. \\\"]\",\n \"page_idx\": \"9\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[68,44,174,81]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"cademos\\nSaude Coletiva \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[537,37,929,50]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Concordância entre informações reportadas pelas mães e dos cartões de vacina \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[68,944,228,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cad. Saude Colet., 2021;29Esp.: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[914,945,929,954]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"49 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[67,85,715,858]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"21. Martin LM, Leff M, Calonge N, Garrett C, Nelson DE. Validation of self-reported chronic conditions and health services in a managed care population. Am J Prev Med. 2000;18(3):215-8. http://dx.doi.org/10.1016/S0749-3797(99)00158-0. PMID:10722987. \\\",\\\"22. Rolnick SJ, Parker ED, Nordin JD, Hedblom BD, Wei F, Kerby T, et al. Self-report compared to electronic medical record across eight adult vaccines: do results vary by demographic factors? Vaccine. 2013;31(37):3928-35. http://dx.doi.org/10.1016/j.vaccine.2013.06.041. PMID:23806243. \\\",\\\"23. Shenson D, DiMartino D, Bolen J, Campbell M, Lu PJ, Singleton JA. Validation of self-reported pneumococcal vaccination in behavioral risk factor surveillance surveys: experience from the sickness prevention achieved through regional collaboration (SPARC) program. Vaccine. 2005;23(8):1015-20. http://dx.doi.org/10.1016/j.vaccine.2004.07.039. PMID:15620474. \\\",\\\"24. Skull SA, Andrews RM, Byrnes GB, Kelly HA, Nolan TM, Brown GV, et al. Validity of self-reported influenza and pneumococcal vaccination status among a cohort of hospitalized elderly inpatients. Vaccine. 2007;25(25):4775-83. http://dx.doi.org/10.1016/j.vaccine.2007.04.015. PMID:17499402. \\\",\\\"25. Lu PJ, Dorell C, Yankey D, Santibanez TA, Singleton JA. A comparison of parent and provider reported influenza vaccination status of adolescents. Vaccine. 2012;30(22):3278-85. http://dx.doi.org/10.1016/j.vaccine.2012.03.015. PMID:22445809. \\\",\\\"26. Boakye EA, Tobo BB, Osazuwa-Peters N, Mohammed KA, Geneus CJ, Schootman M. A comparison of parent-and provider-reported human papillomavirus vaccination of adolescents. Am J Prev Med. 2017;52(6):742-52. http://dx.doi.org/10.1016/j.amepre.2016.10.016. PMID:27890518. \\\",\\\"27. Zimmerman RK, Raymund M, Janosky JE, Nowalk MP, Fine MJ. Sensitivity and specificity of patient self-report of influenza and pneumococcal polysaccharide vaccinations among elderly outpatients in diverse patient care strata. Vaccine. 2003;21(13-14):1486-91. http://dx.doi.org/10.1016/S0264-410X(02)00700-4. PMID:12615445. \\\",\\\"28. Mangtani P, Shah A, Roberts JA. Validation of influenza and pneumococcal vaccine status in adults based on self-report. Epidemiol Infect. 2007;135(1):139-43. http://dx.doi.org/10.1017/S0950268806006479. PMID:16740194. \\\",\\\"29. Gordon NP, Wortley PM, Singleton JA, Lin TY, Bardenheier BH. Race/ethnicity and validity of self-reported pneumococcal vaccination. BMC Public Health. 2008;8(1):227. http://dx.doi.org/10.1186/1471-2458-8-227. PMID:18598363. \\\",\\\"30. Andrade MV, Noronha K, Cardoso CS, Oliveira CD, Calazans JA, Souza MN. Challenges and lessons from a primary care intervention in a Brazilian municipality. Rev Saude Publica. 2019;53:45. http://dx.doi.org/10.11606/S1518-8787.2019053000457. PMID:31066823. \\\",\\\"31. Domingues CMAS, Teixeira AMDS. Coberturas vacinais e adoções imunopreveníveis no Brasil no periodo 1982-2012: avanços e desafiros do Programa Nacional de Imunizações. Epidemiol Serv Saude. 2013;22(1):9-27. http://dx.doi.org/10.5123/S1679-49742013000100002. \\\",\\\"32. Brasil. Lei n° 13.709, de 14 de ago do 2018. Lei Geral de Protecao de Dados Pessoa (LGPD). Diario Oficial da Uniao; Brasilia; 2018. \\\",\\\"33. Landis JR, Koch GG. The measurement of observer agreement for categorical data. Biometrics. 1977;33(1):159-74. http://dx.doi.org/10.2307/2529310. PMID:843571. \\\",\\\"34. McHugh ML. Interrater reliability: the kappa statistic. Biochem Med. 2012;22(3):276-82. http://dx.doi.org/10.11613/BM.2012.031.PMid:23092060. \\\",\\\"35. Associacao Brasileira de Empresas de Pesquisa. Critério de classificacao economica Brasil. Sao Paulo: ABEP; 2008. \\\",\\\"36. Cordeiro EL, Silva LSR, Urquiza JL, Nascimento MA, Silva RM, Souza GCS, et al. conheicao das mães sobre o esquema vacinal de seu filhos assistidos em uma unidade basia de saude. Brazilian Journal of Health Review. 2019;2(1):644-60. \\\",\\\"37. Poland GA, Jacobson RM. Understanding those who do not understand: a brief review of the anti-vaccine movement. Vaccine. 2001;19(17-19):2440-5. http://dx.doi.org/10.1016/S0264-410X(00)00469-2. PMID:11257375. \\\",\\\"38. Hornsey MJ, Harris EA, Fielding KS. The psychological roots of anti-vaccination attitudes: a 24-nation investigation. Health Psychol. 2018;37(4):307-15. http://dx.doi.org/10.1037/hea0000586.PMid:29389158. \\\",\\\"39. Barbieri CLA, Couto MT. Decision-making on childhood vaccination by highly educated parents. Rev Saude Publica. 2015;49:18. http://dx.doi.org/10.1590/S0034-8910.2015049005149. PMID:25830870. \\\"]\",\n \"page_idx\": \"10\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[67,42,460,55]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Concordância entre informações reportadas pelas mães e dos cartões de vacina \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[821,44,863,52]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"cademos \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[823,52,927,81]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Saude Coletiva \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[68,945,84,954]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"50 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[768,944,929,955]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cad. Saude Colet., 2021;29 Esp.: \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n }\n]","isConversational":false},"doi":{"kind":"string","value":"10.1590/1414-462x202199010302"},"is_oa":{"kind":"bool","value":true,"string":"true"},"language":{"kind":"string","value":"pt"},"sci_category":{"kind":"string","value":"Medicine and Health Sciences"},"sha256":{"kind":"string","value":"3e7662a9afcc0f8dd34a931ffdd207912ba4a321502b48b43ed33a4a81707714"},"title":{"kind":"string","value":"Análise da concordância entre as informações reportadas pelas mães e dos cartões de vacina das crianças no Brasil (2013 e 2015)"}}},{"rowIdx":73,"cells":{"abstract":{"kind":"string","value":"Repeated exposure to psychostimulants such as methamphetamine (METH) induces neuronal adaptations in the mesocorticolimbic dopamine system, including the ventral tegmental area (VTA). These changes lead to persistently enhanced neuronal activity causing increased dopamine release and addictive phenotypes. A factor contributing to increased dopaminergic activity in this system appears to be reduced GABAB receptor-mediated neuronal inhibition in the VTA. Dephosphorylation of serine 783 (Ser783) of the GABAB2 subunit by protein phosphatase 2A (PP2A) appears to trigger the downregulation GABAB receptors in psychostimulant-addicted rodents. Therefore, preventing the interaction of GABAB receptors with PP2A using an interfering peptide is a promising strategy to restore GABAB receptor-mediated neuronal inhibition. We have previously developed an interfering peptide (PP2A-Pep) that inhibits the GABAB receptors/PP2A interaction and thereby restores receptor expression under pathological conditions. Here, we tested the hypothesis that restoration of GABAB receptor expression in the VTA of METH addicted mice reduce addictive phenotypes. We found that the expression of GABAB receptors was significantly reduced in the VTA and nucleus accumbens but not in the hippocampus and somatosensory cortex of METH-addicted mice. Infusion of PP2A-Pep into the VTA of METH-addicted mice restored GABAB receptor expression in the VTA and inhibited METH-induced locomotor sensitization as assessed in the open field test. Moreover, administration of PP2A-Pep into the VTA also reduced drug seeking behavior in the conditioned place preference test. These observations underscore the importance of VTA GABAB receptors in controlling addictive phenotypes. Furthermore, this study illustrates the value of interfering peptides targeting diseases-related protein-protein interactions as an alternative approach for a potential development of selective therapeutic interventions."},"author":{"kind":"string","value":"Mohammad Hleihil|Dietmar Benke"},"content_list":{"kind":"list like","value":[{"bbox":"[80,143,105,159]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0992/acc6bafecea79aec229a67bafe80c48930a0b71066d4004adc645aa5f4abaa2d.jpg","list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[109,146,206,156]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Check for updates 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These changes lead to persistently enhanced neuronal activity causing increased dopamine release and addictive phenotypes. A factor contributing to increased dopaminergic activity in this system appears to be reduced $\\mathrm{GABA_B}$ receptor-mediated neuronal inhibition in the VTA. Dephosphorylation of serine 783 (Ser783) of the $\\mathrm{GABA_{B2}}$ subunit by protein phosphatase 2A (PP2A) appears to trigger the downregulation $\\mathrm{GABA_B}$ receptors in psychostimulant-addicted rodents. Therefore, preventing the interaction of $\\mathrm{GABA_B}$ receptors with PP2A using an interfering peptide is a promising strategy to restore $\\mathrm{GABA_B}$ receptor-mediated neuronal inhibition. We have previously developed an interfering peptide (PP2A-Pep) that inhibits the $\\mathrm{GABA_B}$ receptors/PP2A interaction and thereby restores receptor expression under pathological conditions. Here, we tested the hypothesis that restoration of $\\mathrm{GABA_B}$ receptor expression in the VTA of METH addicted mice reduce addictive phenotypes. We found that the expression of $\\mathrm{GABA_B}$ receptors was significantly reduced in the VTA and nucleus accumbens but not in the hippocampus and somatosensory cortex of METH-addicted mice. Infusion of PP2A-Pep into the VTA of METH-addicted mice restored $\\mathrm{GABA_B}$ receptor expression in the VTA and inhibited METH-induced locomotor sensitization as assessed in the open field test. Moreover, administration of PP2A-Pep into the VTA also reduced drug seeking behavior in the conditioned place preference test. These observations underscore the importance of VTA $\\mathrm{GABA_B}$ receptors in controlling addictive phenotypes. 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","text_format":null,"text_level":null,"type":"text"},{"bbox":"[364,152,421,162]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"KEYWORDS ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[363,167,917,192]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"addiction, GABA receptor, interfering peptide, methamphetamine, protein phosphatase 2A, ventral tegmental area (VTA) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,256,216,273]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[77,290,489,475]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Drug addiction is a complex chronic disease with biological, behavioral, and social components. Psychostimulants such as cocaine, amphetamine, and methamphetamine (METH) are highly addictive drugs, whose abuse can lead to serious health problems and significant healthcare costs (May et al., 2020; Cheron and Kerchove d'Exaerde, 2021). For instance, in 2018, 2.3 million employees receiving employer-sponsored insurance in the USA (representing $1.4\\%$ of the total population of employees receiving employer-sponsored insurance in the USA) were diagnosed with substance use disorders with total annual medical cost of $35.3 billion (Li M. et al., 2023). Currently, there is no effective pharmacological therapy available to cure patients from addiction. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,476,489,719]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Addiction to psychostimulants results from repeated and prolonged actions of the drug on the brain inducing long-lasting plastic neuronal adaptations (Ostroumov and Dani, 2018; Fischer et al., 2021). The effects of psychostimulants are mediated via enhanced dopamine release in the mesocorticolimbic dopamine pathway. The mesocorticolimbic circuit is involved in many cognitive processes including reward, motivation, learning, and fear (Volman et al., 2013; Baik, 2020). A key structure triggering addictive behavior is the ventral tegmental area (VTA) located in the midbrain. Acute effects of psychostimulants, such as feelings of euphoria, pleasure, and reward, are mediated via increased activity of VTA dopamine neurons projecting to the nucleus accumbens, amygdala, and medial prefrontal cortex (Nestler, 2005). Repeated and chronic consumption of psychostimulants induces plastic neuronal adaptations resulting in permanently increased dopamine release in the mesocorticolimbic dopamine pathway. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,720,490,919]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"One important factor controlling neuronal excitability and neurotransmitter release is the $\\mathrm{GABA_B}$ receptor, which is expressed in most neurons at axon terminals, dendritic and somatic sites (Chalifoux and Carter, 2011). $\\mathrm{GABA_B}$ receptors are heterodimeric assemblies composed of $\\mathrm{GABA_{B1}}$ and $\\mathrm{GABA_{B2}}$ subunits and mediate their effects by activating Gi/o proteins (Jones et al., 1998; Kaupmann et al., 1998; White et al., 1998). Their most prominent actions in the brain are the activation of G protein-coupled inwardly rectifying potassium channels (GIRK or $K_{ir}3$ ) at postsynaptic locations to reduce neuronal excitability (Gahwiler and Brown, 1985; Luscher et al., 1997) and the inhibition of voltage-gated $\\mathrm{Ca^{2+}}$ channels at presynaptic sites to reduce transmitter release (Mintz and Bean, 1993; Santos et al., 1995; Chen and van den Pol, 1998). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[504,253,919,649]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\mathrm{GABA_B}$ receptors are well expressed in the mesocorticolimbic dopamine system, where they control neuronal activity and the release of GABA, glutamate and dopamine (Lalive and Luscher, 2016). Accordingly, pharmacological activation of $\\mathrm{GABA_B}$ receptors by the selective agonist baclofen reduced several addictive behaviors in preclinical studies such as the development and expression of psychostimulant-induced motor sensitization (Bartoletti et al., 2004, 2005; Cedillo et al., 2019), drug seeking (Li et al., 2001), drug self-administration (Roberts and Brebner, 2000; Ranaldi and Poeggel, 2002; Brebner et al., 2005) and drug-induced cognitive impairment (Arai et al., 2009; Mizoguchi and Yamada, 2011). Early clinical studies reported some positive effects of baclofen in reducing psychostimulant consumption and drug craving (Ling et al., 1998; Shoptaw et al., 2003; Heinzerling et al., 2006). There are clinical evidences for the benefit of baclofen for the treatment of alcohol use disorders (Agabio et al., 2023), but its use is controversial (de Beaurepaire et al., 2018; Leggio and Litten, 2021). Currently, baclofen is approved for the treatment of alcohol addiction in France (Garbutt, 2020; Rolland et al., 2020). One problem of baclofen at high dosage are numerous side effects such as sedation, dizziness, fatigue, insomnia, headache, paresthesia, tinnitus, and restlessness (Rolland et al., 2018). The wide range of adverse side effects might reflect the fact that systemic application of baclofen activates all $\\mathrm{GABA_B}$ receptors in the body and not only in the neuronal pathways related to the addicted state. This might generally restrict the broad clinical application of baclofen and need for more specific interventions. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[504,650,919,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"There is considerable evidence that $\\mathrm{GABA_B}$ receptor-mediated inhibition is reduced in brain structures of the mesocorticolimbic system after psychostimulant treatment, diminishing inhibitory control and enhancing neuronal excitability (Arora et al., 2011; Padgett et al., 2012; Hearing et al., 2013; Sharpe et al., 2015; Munoz et al., 2016). This has been attributed to downregulation of either $\\mathrm{GABA_B}$ receptors and GIRK channels or both (Arora et al., 2011; Padgett et al., 2012; Hearing et al., 2013; Munoz et al., 2016; Lu et al., 2023). Aberrant downregulation of $\\mathrm{GABA_B}$ receptors is a response to sustained activity of glutamate receptors, which dysregulates $\\mathrm{GABA_B}$ receptor trafficking from the constitutive recycling pathway to lysosomal degradation (Guetg et al., 2010; Maier et al., 2010; Terunuma et al., 2010). Key steps in this aberrant regulation are the phosphorylation of Ser867 in the $\\mathrm{GABA_{B1}}$ subunit mediated by $\\mathrm{Ca^{2+}}$ -calmodulin dependent protein kinase IIβ (CaMKIIβ) (Guetg et al., 2010; Zemoura et al., 2019) and dephosphorylation of Ser783 in $\\mathrm{GABA_{B2}}$ by protein phosphatase 2A (PP2A) (Terunuma et al., 2010; Hleihil et al., 2022). Downregulation of cell surface and total ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,38,178,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hleihil and Benke ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[752,38,917,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10.3389/fnmol.2024.1347228 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[78,948,280,959]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Molecular Neuroscience ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[489,948,507,958]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"02 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[835,948,915,959]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[77,87,487,187]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\mathrm{GABA_B}$ receptors in the nucleus accumbens after repeated cocaine administration involved the activation of CaMKII (Lu et al., 2023), whereas selective loss of cell surface receptors was observed in the mPFC and VTA after cocaine and methamphetamine exposure, respectively, due to PP2A-mediated dephosphorylation of Ser783 in $\\mathrm{GABA_{B2}}$ (Padgett et al., 2012; Hearing et al., 2013). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,187,489,401]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recently, we developed an interfering peptide (PP2A-Pep) that inhibits the interaction of $\\mathrm{GABA_B}$ receptors with PP2A (Hleihil et al., 2022). This peptide restored downregulated $\\mathrm{GABA_B}$ receptor expression after an ischemic insult, normalized neuronal overexcitation and inhibited progressive neuronal death. Since $\\mathrm{GABA_B}$ receptor expression and function appears to be reduced in the VTA after METH application, we tested in the present study the hypothesis whether restoring $\\mathrm{GABA_B}$ receptor expression in the VTA of METH-addicted mice, by blocking the interaction of $\\mathrm{GABA_B}$ receptors with PP2A, would reduce phenotypes associated with addiction. For this we infused PP2A-Pep into the VTA of METH-addicted mice and tested for $\\mathrm{GABA_B}$ receptor expression using immunohistochemistry, locomotor sensitization in the open field test (OFT) and drug seeking behavior in the conditioned place preference test (CPP). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,435,321,451]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Materials and methods ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[78,468,157,483]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Animals ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[77,499,489,612]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Experiments were performed using 8-10 weeks old adult male and female C57BL/6 mice. Mice were either purchased from Envigo (Netherlands) or bred at the Laboratory Animal Service Center of the University of Zurich (Switzerland). The mice were group-housed up to five in filter-top cages with a standard $12/12$ -h light/dark cycle and food and water available ad libitum. All experiments were approved by the veterinary office of the Canton of Zurich (license ZH164/2020). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,645,369,662]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Interfering peptide (PP2A-Pep) ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[77,677,489,917]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The interfering peptide (PP2A-Pep, YTIWMPENPRPGTP CDIFTNSRGKRASNGGGGRRRRRRRRFQFTQNQKKEDSKTS TSV) used in this study inhibits the interaction of PP2A with $\\mathrm{GABA}_B$ receptors (Hleihil et al., 2022). It contains a sequence derived from the intracellularly located C-terminal domain of $\\mathrm{GABA}_{B2}$ (red characters) and was rendered cell permeable by tagging it at the N-terminus with a peptide sequence derived from the Rabies virus glycoprotein followed by 9 arginine residues (gray characters). An inactive control peptide (Ctrl-Pep) contained the same amino acids but in a random sequence (except for the cell penetrating peptide sequence, YTIWMPENPRPGTPCDIFTNSRGKRASNGGGGRRRRRRRRQ KFSVNTFQEKDTKSQTS). The peptides were additionally tagged with biotin to permit their detection by immunohistochemical staining using AlexaFluor 488-conjugated streptavidin. The peptides were custom-synthesized by Pepmic Co., Ltd., Suzhou, China. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,87,880,104]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cannulation and peptide administration ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[505,118,917,417]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cannula implantations were conducted using a stereotaxic frame (David Kopf Instruments and NeuroStar) under general anesthesia using continuous isoflurane inhalation via a nose mask (induction: $5\\%$ at $1\\mathrm{l / min}$ , surgery: $1.5\\%$ in a mixture of $\\mathrm{O}_2$ and air at $0.4 - 0.6\\mathrm{l / min}$ ). The body temperature of the mice was kept at $36.5\\pm 0.5^{\\circ}\\mathrm{C}$ using a feed-back controlled heating mat with a rectal temperature probe. Eyes were protected from drying using vitamin A ointment. For analgesia, mice received a subcutaneous injection of buprenorphine $(0.1\\mathrm{mg / kg})$ 20-30 min before surgery and an additional subcutaneous injection of lidocaine $(2\\mathrm{mg / kg})$ /bupivacaine $(1\\mathrm{mg / kg})$ at the site of incision about $5\\mathrm{min}$ before surgery. Before exposing the skull, the incision site was cleaned with $70\\%$ ethanol. A unilateral cannula implantation was performed using a 26G guide cannula (#C315GS-5/SPC, Bilaney Consultants, Germany). A hole was drilled through the skull above the VTA (anterior-posterior $-3.3$ ; medial-lateral $-0.9$ , relative to bregma). The guide cannula was placed through the hole above the VTA with a $10^{\\circ}$ angle (dorsal-ventral $-4.25$ ). The cannula was held in place using dental cement. All mice received post-surgical analgesia for additional 2 days $(10\\mathrm{mg / kg}$ carprofen, s. c. twice per day). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[505,417,917,531]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Peptides (0.8 μl PP2A-Pep, 100 μg/ml or 0.8 μl Ctrl-Pep, 100 μg/ml) were administered into the VTA with a rate of 100 nl/min using an automatic pump (PHD ULTRA Nanomite, Harvard Apparatus) connected by PE tubing which was attached to an injection cannula (#C315DCS-4/SPC, Bilaney Consultants, Germany) with 0.25 mm projection from the tip of the guiding cannula. Following injections, the cannula was left in place for additional 5 min to allow peptide infusion. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,560,658,576]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Behavioral tests ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[505,590,917,719]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Seven days prior to any experiment, the mice were habituated to the experimenter by gentle handling in the housing facility. At least $1\\mathrm{h}$ prior to the experiment, mice were transferred to the experimental room (50 lux dim light). All tests were performed in the light cycle. The respective experimental apparatus was cleaned with $70\\%$ ethanol before testing a mouse to avoid any sensory stimulation related to the previous mouse tested. In this study we used non-contingent drug administration as it ensures the expression of robust addictive phenotypes. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[505,733,680,747]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Open field test (OFT) ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[505,749,917,891]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The OFT was used to test for METH-induced locomotor sensitization (Good and Radcliffe, 2011; Jing et al., 2014). The locomotor activity was monitored in the open-field test chamber $(32\\mathrm{cm}\\times 32\\mathrm{cm}\\times 23\\mathrm{cm})$ with an infrared detection system under dim light conditions (50 lux). Mice were transferred from the housing facility to the experimental room $1\\textrm{h}$ prior to the experiment. All mice received i.p. injection (saline or METH) and immediately placed into the open field chamber $(32\\mathrm{cm}\\times 32\\mathrm{cm}\\times 23\\mathrm{cm})$ , where their activity was recorded for $30\\mathrm{min}$ . Then the mice were returned to their home cages. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[505,892,917,919]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"To induce methamphetamine (METH) addiction, mice were injected alternatively with either saline or methamphetamine ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,38,178,48]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hleihil and Benke ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[754,38,917,48]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10.3389/fnmol.2024.1347228 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[80,948,280,958]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Molecular Neuroscience ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[490,948,507,957]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"03 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[835,948,915,959]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[80,87,487,340]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"(METH, $2\\mathrm{mg / kg}$ , i. p., dissolved in saline) on every other day for 10 days (METH injections on day 1, 3, 5, 7 and 9; saline injections on day 2, 4, 6, 8, 10). Even a single METH injection has been shown to induce robust and long-lasting locomotor sensitization (Jing et al., 2014). We verified the development of locomotor sensitization in a subgroup of mice by daily open field tests. The control group received solely saline injections for 10 days. One day after the last saline injection (day 11), mice were microinfused with the peptides dissolved in saline $(0.8\\mu l$ PP2A-Pep, $100~\\mu \\mathrm{g / ml}$ or $0.8\\mu l$ Ctrl-Pep, $100~\\mu \\mathrm{g / ml})$ via the implanted cannula and then placed back to their home cage. The next day (day 12), the mice were injected with a challenging dose of METH $(1\\mathrm{mg / kg},$ i. p., dissolved in saline) and immediately placed into the open-field test chamber to allow free exploration for $30\\mathrm{min}$ . The mice were monitored by a digital camera placed above the center of the arena. From each session, $10\\mathrm{min}$ of the recordings were extracted for analysis with the EthoVision XT 15 software (Noldus, Wageningen, Netherlands). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,354,374,368]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conditioned place preference (CPP) ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[80,370,487,508]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"CCP was used to test drug seeking behavior (Cunningham et al., 2006). CPP was conducted in a two-compartment plexiglass apparatus (size of each compartment: $20\\mathrm{cm} \\times 20\\mathrm{cm} \\times 15\\mathrm{cm}$ ), in which the two conditioning compartments were connected through a short tunnel ( $10\\mathrm{cm}$ ) (Figure 5). The two conditioning chambers were visually and tactile distinct. Chamber 1 consisted of white walls with black strips and a plain white floor with gray cross lines. Chamber 2 consisted of black walls and a rough textured plexiglass floor. The behavior of the mice was recorded by a digital camera placed above the CCP device. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,511,487,651]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The CPP procedure consisted of three phases (Figure 5): habituation and baseline preference (day 1), conditioning phase (days 2-6) and preference tests (days 7 and day 8). In the habituation/preference evaluation (day 1), mice were placed in the tunnel connecting the two chambers and allowed to freely explore all compartments for $30\\mathrm{min}$ . This allowed to assign a preferred compartment for each mouse to conduct a biased CPP experiment: the mice received saline injections in their initially preferred chamber (saline-paired chamber) and METH injections in the non-preferred compartment (METH-paired chamber). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,652,487,777]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"During the conditioning phase (day 2-6), access was restricted to the saline-paired or METH-paired compartment. In the METH group, mice were injected with METH $(2\\mathrm{mg / kg}$ i.p., dissolved in saline) and immediately placed into the METH-paired chamber for $30\\mathrm{min}$ and then returned to the home cage. Six hours later, the mice were injected with saline and placed into the saline-paired chamber for $30\\mathrm{min}$ . For the control saline group, the same procedure was conducted but mice were injected with two saline doses. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,778,487,917]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On the test day (day 7), the preference test was performed $24\\mathrm{h}$ after the last conditioning trial by placing the mouse in the tunnel connecting the two compartments and allowing to freely explore all chambers for $10\\mathrm{min}$ . One hour after the test, mice were microinfused with the peptides dissolved in saline via the implanted cannula $(0.8\\mu l$ PP2A-Pep, $100~\\mu \\mathrm{g / ml}$ or $0.8\\mu l$ Ctrl-Pep, $100~\\mu \\mathrm{g / ml})$ and then placed back into their home cage. The timeline for peptide treatment was selected according to Li J. Y. et al. (2023), which predominantly affects memory reconsolidation. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,87,915,173]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"On day 8, the preference test was conducted again for $10\\mathrm{min}$ to test for the peptide effect. The recordings were analyzed with the EthoVision XT 15 software (Noldus, Wageningen, Netherlands) to extract the CPP data. The CPP preference score was calculated as a difference between the relative amount of time spent in the saline-paired chamber to the time spent in the METH-paired chamber. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,200,610,217]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Antibodies ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,233,915,303]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Primary antibodies: rabbit anti-GABAB2N [1:250, custom-made by GeneScript, (Benke et al., 2002)], rabbit anti-GFAP (1:500, Agilent Technologies # Z0334), guinea pig anti-NeuN (1:500, Synaptic systems # 266004), rabbit anti-Iba1 (1:500, WAKO Chemicals # 019-19741). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,304,915,374]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Secondary antibodies: donkey anti-rabbit AlexaFluor Plus 555 (1:2000, Thermo Fisher Scientific # A32794), goat anti-guinea pig AlexaFluor 647 (1:1000, Jackson ImmunoResearch # 706-605-148), and AlexaFluor 488-conjugated streptavidin (1:500, Jackson ImmunoResearch # 016-540-084). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,403,727,437]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Tissue preparation and immunohistochemistry ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,451,915,564]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A loss of $\\mathrm{GABA_B}$ receptor immunofluorescence staining under ischemic conditions always correlated with a similar reduction of receptor protein signals in western blots and with diminished $\\mathrm{GABA_B}$ receptor mediated inhibition (see e.g., Zemoura et al., 2019; Hleihil et al., 2021, 2022; Bhat et al., 2022; Balakrishnan et al., 2023). Therefore, we assume that a reduction in $\\mathrm{GABA_B}$ receptor immunofluorescence staining in this study indicates diminished $\\mathrm{GABA_B}$ receptor mediated inhibition. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,565,915,678]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Immediately after the last behavioral test, mice were transcranially perfused with ice-cold oxygenated ACSF (125 mM NaCl, 2.5 mM KCl, 1.25 mM $\\mathrm{NaH_2PO_4}$ , 25 mM $\\mathrm{NaHCO_3}$ , 1 mM $\\mathrm{MgCl_2}$ , 2 mM $\\mathrm{CaCl_2}$ , and 25 mM glucose, pH 7.4, pH 7.4), for 2 min. Brains were then immediately dissected and cut in the medial midline. One of the hemispheres were then fixed in $4\\%$ PFA overnight, followed by incubation in $30\\%$ sucrose/PBS for $24\\mathrm{h}$ at $4^{\\circ}\\mathrm{C}$ for immunohistochemical staining. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,678,915,917]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For immunohistochemistry, free floating $40~\\mu \\mathrm{m}$ thick coronal sections were prepared using a sliding microtome (HM400; Microm) and stored in antifreeze solution (50 mM Na-phosphate buffer, pH 7.4, 1.6 M Glucose, 20 mM ethylene glycol, 6 mM sodium azide) at $-20^{\\circ}\\mathrm{C}$ for further analysis. After rinsing once in PBS, sections were incubated overnight at $4^{\\circ}\\mathrm{C}$ with primary antibody (or AlexaFluor 488-conjugated streptavidin for peptide detection) prepared in Tris-Triton solution [50 mM Tris pH 7.4, 150 mM NaCl containing $0.2\\%$ Triton X-100 and $5\\%$ normal donkey serum (NDS)]. After 3 washes for 10 min with Tris-Triton without NDS, sections were incubated with secondary antibody diluted in Tris-Triton containing $2\\%$ NDS for $1\\textrm{h}$ at room temperature. Following 3 washes for 10 min with Tris-Triton without NDS, brains were mounted onto glass slides (SuperFrost Plus, Thermo Fisher Scientific), dried and coversliped with DAKO fluorescence mounting medium supplemented with DAPI (Fluoroshield with DAPI, #F6057, Merk). 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For imaging of the brain areas, a stacks of 10 optical sections at $1\\mu \\mathrm{m}$ z-spacing were taken using a plan-apochromat $10\\mathrm{x} / 0.45$ objective in sequential mode with a resolution of $1,024\\times 1,024$ pixels. The laser intensity and the detector gain were adjusted to values that avoid signal saturation and all images of one experiment was imaged with the same settings in one continuous session. For images on the cellular level, 5 optical sections with $0.7\\mu \\mathrm{m}$ z-spacing were taken using a $40\\mathrm{x} / 1.4$ planapochromat oil differential interference contrast objective with $0.5\\mathrm{x}$ digital zoom. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,244,489,330]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For quantification of $\\mathrm{GABA}_B$ receptor expression, all images belonging to an experiment were recorded with identical settings. Analysis was performed after merging the optical planes into a single image and an area of interest was defined. The mean fluorescence values were determined using ImageJ (Schindelin et al., 2012). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,331,489,374]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The original images recorded were too dim to be shown directly. Therefore, we enhanced the brightness of the images to make the staining more easily visible to the reader. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,406,250,422]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Statistical analysis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[77,437,490,594]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The data is shown as means $\\pm$ SD. Immunohistochemical data was analyzed by two-way analysis of variance with type III sum of square calculations for an unbalanced data design (ANOVA, Figures 2, 3) using GraphPad Prism 8.4.3, followed by Tukey's multiple comparison test as indicated in the respective figure legends. Behavioral experiments were analyzed by two-way AVOVA with type III sum of square calculations for an unbalanced data design (Figures 4, 5D) and two-tailed unpaired $t$ -test (Figure 5C). All data sets passed tests for normal distribution and homoscedasticity. Differences were considered statistically significant when $p < 0.05$ . ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,627,159,643]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Results ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[78,659,426,678]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Neuron-specific uptake of PP2A-Pep ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[77,690,490,919]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For testing the effect of PP2A-Pep on METH-addicted mice, the peptide was unilaterally infused into the VTA via an implanted canula (Figure 1A). Unilateral infusion had been shown to cover both sides of the VTA (Harada et al., 2021). To largely restrict uptake of PP2A-Pep (as well as the control peptide Ctrl-Pep) into neurons, it contained at the N-terminus a cell-penetrating peptide sequence derived from the Rabies virus glycoprotein (Kumar et al., 2007). We previously showed that peptides containing this sequence are selectively taken up by neurons via a receptor-mediated mechanism (Balakrishnan et al., 2023). PP2A-Pep was indeed taken up by neurons after infusion into the VTA as shown by co-immunofluorescence staining for the peptide, neurons (NeuN antibody) and nuclei (DAPI) (Figures 1B, C). The peptide was not taken up by glia cells as shown by co-staining for astrocytes with GFAP antibodies (Figure 1B) and microglia by co-staining with Iba1 antibodies (Figure 1C). 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(A) Infusion site and spread of PP2A-Pep. Right: Brain section stained for the infused PP2A-Pep using AlexaFluor 488-conjugated streptavidin (green) and DAPI (blue). The placement of the canula is indicated (white lines). Left: Location of section in the mouse brain atlas of Paxinos and Franklin (2001). Scale bar: $500\\\\mu m$ . (B) PP2A-Pep was selectively taken up by neurons but not by astrocytes. Brain sections containing the VTA were stained for the infused PP2A-Pep (green), Neurons (NeuN, red), astrocytes (GFAP, magenta) and nuclei (DAPI, blue). Scale bar: $20\\\\mu m$ . (C) PP2A-Pep was selectively taken up by neurons but not by microglia. Brain sections containing the VTA were stained for the infused PP2A-Pep (green), Neurons (NeuN, red), microglia (Iba1, magenta) and nuclei (DAPI, blue). Scale bar: $20\\\\mu m$ \"]","img_path":"images/0757/e315d040a3e05407dd289f1d37fb9ebfccd727d03172e48afb9899c719286247.jpg","list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[505,812,890,862]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"$\\mathrm{GABA}_B$ receptor expression was reduced in the VTA of METH-addicted mice but normalized after PP2A-Pep infusion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[505,877,917,921]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"First, we analyzed whether $\\mathrm{GABA}_B$ receptor expression in the VTA was reduced in METH-addicted mice. 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Mice received multiple METH/saline (METH-addicted mice) or saline/saline (control mice) injections and were then either infused into the VTA with PP2A-Pep or the inactive Ctrl-Pep. About $24\\\\mathrm{h}$ after peptide application the mice were analyzed for $\\\\mathrm{GABA_B}$ receptor expression using $\\\\mathrm{GABA_{B2}}$ antibodies in the VTA (A), nucleus accumbens (B), hippocampus (C), and motor cortex (D) on a regional level. (A) $\\\\mathrm{GABA_B}$ receptor expression was reduced in the VTA of METH-addicted mice and restored after infusion of PP2A-Pep. Left: Representative images display peptide uptake (green) and $\\\\mathrm{GABA_B}$ receptor expression (red) in the VTA (scale bar: $100~\\\\mu \\\\mathrm{m}$ ). Right: Quantification of fluorescence intensities in the area stained for the peptide (the data was normalized to the mean immunofluorescence value of Ctrl-Pep infused into the VTA of saline treated control mice). The data represents the mean $\\\\pm$ SD of 9-10 mice per condition. Statistical analysis: Two-way ANOVA with Tukey's multiple comparison test (ns, $p > 0.05$ ). (B) $\\\\mathrm{GABA_B}$ expression was reduced in the nucleus accumbens of METH-addicted mice but unaffected by PP2A-Pep infusion into the VTA. Left: Representative images (upper row: low magnification images, scale bar $200~\\\\mu \\\\mathrm{m}$ ; lower row, high magnification images, scale bar $20~\\\\mu \\\\mathrm{m}$ ). Right: Quantification of fluorescence intensities (the data was normalized to the mean immunofluorescence value of control mice). The data represents the mean $\\\\pm$ SD of 8-10 mice per condition. Statistical analysis: Two-way ANOVA with Tukey's multiple comparison test (ns, $p > 0.05$ ). (C) $\\\\mathrm{GABA_B}$ expression remained unaffected in the hippocampus of METH-addicted mice. Left: Representative images (upper row: low magnification images, scale bar $100~\\\\mu \\\\mathrm{m}$ ; lower row, high magnification images, scale bar $20~\\\\mu \\\\mathrm{m}$ ). Right: Quantification of fluorescence intensities (the data was normalized to the mean immunofluorescence value of control mice). The data represents the mean $\\\\pm$ SD of 5-7 mice per condition. Statistical analysis: Two-way ANOVA with Tukey's multiple comparison test (ns, $p > 0.05$ ). (D) $\\\\mathrm{GABA_B}$ expression in the motor cortex remained unaffected in METH-addicted mice. Left: Representative images (upper row: low magnification images, scale bar $200~\\\\mu \\\\mathrm{m}$ ; lower row, high magnification images, scale bar $20~\\\\mu \\\\mathrm{m}$ ). Right: Quantification of fluorescence intensities (the data was normalized to the mean immunofluorescence of control mice). The data represents the mean $\\\\pm$ SD of 7-8 mice per condition. Statistical analysis: Two-way ANOVA with Tukey's multiple comparison test (ns, $p > 0.05$ ). \"]","img_path":"images/0123/d2535f8f4c9a81d71a0f47144363ca520cb04cabf54d0231a612ae285ed4c5f3.jpg","list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[80,38,179,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hleihil and Benke ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[752,38,915,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10.3389/fnmol.2024.1347228 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[80,948,280,958]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Molecular Neuroscience ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[489,948,507,958]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"06 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[835,948,915,959]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[119,102,606,281]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0006/c2ee56d217293bfe66fc138b7b89b5f81f4a6af9ad102bfaa0783e21bc373f7d.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[611,112,873,282]","code_body":null,"code_caption":null,"image_caption":"[\"FIGURE 3 \"]","image_footnote":"[\"PP2A-Pep normalized $\\\\mathrm{GABA_B}$ receptor expression in the somata of VTA neurons of METH-addicted mice. METH-addicted and control mice were infused into the VTA either with PP2A-Pep or the inactive Ctrl-Pep. About $24\\\\mathrm{h}$ after peptide application, the mice were analyzed for $\\\\mathrm{GABA_B}$ receptor expression in the soma of individual neurons that had taken up the peptide using $\\\\mathrm{GABA_{B2}}$ antibodies. Left: Representative images displaying peptide uptake (green) and $\\\\mathrm{GABA_B}$ receptor expression (orange) in VTA neurons (scale bar: $20~\\\\mu \\\\mathrm{m}$ ). Right: Quantification of fluorescence intensities (the data was normalized to the mean immunofluorescence value of Ctrl-Pep infused into the VTA of saline treated control mice). The data represents the mean $\\\\pm$ SD of 15 neurons per mouse derived from 6-8 mice per condition. The symbols in the graph display the mean values of all neurons analyzed in a single mouse. Statistical analysis: Two-way ANOVA with Tukey's multiple comparison test (ns, $p > 0.05$ ). \"]","img_path":"images/0030/32f06784f4158fb54f3342b06408b27355364f14c582d6bc05e31c8d66ed8854.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[186,432,478,472]","code_body":null,"code_caption":null,"image_caption":"[\"A OFT procedure \",\"C \"]","image_footnote":"[]","img_path":"images/0823/e41cc29b183deca5394cc96f9602a3ffdf0c3ec61c0dc178fe93fade942452d0.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[512,451,643,542]","code_body":null,"code_caption":null,"image_caption":"[\"Saline-treated mice +Ctrl-Pep \"]","image_footnote":"[]","img_path":"images/0063/ca2e86567f00e0e6f98d91b3cf88afa7de9c6353089772dd376b6d9be881676b.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[652,451,813,542]","code_body":null,"code_caption":null,"image_caption":"[\"Saline treated mice +PP2A-Pep \"]","image_footnote":"[]","img_path":"images/0691/6796dd4734876dcc1c1d6c35287ec020e1f6a34fbb2a19c24a6dec53f7251059.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[176,498,478,678]","code_body":null,"code_caption":null,"image_caption":"[\"B \"]","image_footnote":"[]","img_path":"images/0364/3aff6ab6256ec1f7e7602616f589686b067c461775b6ac1e1727ee6953a246a8.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[510,575,640,665]","code_body":null,"code_caption":null,"image_caption":"[\"METH-treated mice +Ctrl-Pep \"]","image_footnote":"[]","img_path":"images/0075/d6b085d10f8f5d8379383724bd7862e8c2be4d810714399e50f4e5be532fe275.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[652,575,813,665]","code_body":null,"code_caption":null,"image_caption":"[\"METH-treated mice +PP2A-Pep \"]","image_footnote":"[\"FIGURE 4 PP2A-Pep infusion into the VTA reduced METH-induced locomotor sensitization in the OFT. (A) Schematic outline of the OFT procedure. Mice received alternating METH and saline injections for 10 days to induce METH addiction. The control group received solely saline injections. One day after the last saline injection (day 11), the interfering peptide (PP2A-Pep) or the control peptide (Ctrl-Pep) was infused into the VTA via the implanted cannula. The next day, the mice were injected with a challenge dose of METH ( $1\\\\mathrm{mg / kg}$ i.p.) and immediately placed into the open-field test chamber to allow free exploration for $30\\\\mathrm{min}$ . (B) METH-addicted mice exhibited increased locomotor activity, which was reduced to normal levels after PP2A-Pep injection into the VTA. Application of the control peptide (Ctrl-Pep) had no effect. The data represents the mean $\\\\pm$ SD of 7-11 mice per condition. Statistical analysis: Two-way ANOVA with Tukey's multiple comparison test (ns, $p > 0.05$ ). (C) Heat maps depicting the frequencies of locations visited by mice in the open field arena under the different experimental conditions (representative images). \"]","img_path":"images/0452/e81dc07ee7219726401d625c584ad8f63a4c5be4c89016b167aae70811241b35.jpg","list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[77,816,490,919]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"the inactive control peptide (Ctrl-Pep) infused into the VTA of saline-injected control mice and METH-addicted mice. The area of the VTA containing neurons that displayed staining for the peptide was then analyzed for $\\mathrm{GABA}_B$ receptor expression using antibodies directed against the $\\mathrm{GABA}_{B2}$ subunit. As expected, we detected a considerable reduction of $\\mathrm{GABA}_B$ receptor expression in the VTA in METH-addicted mice (Figure 2A). We then tested ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[505,816,919,919]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"whether inhibiting the interaction of $\\mathrm{GABA_B}$ receptors with PP2A would restore $\\mathrm{GABA_B}$ receptor expression. Indeed, infusion of PP2A-Pep into the VTA of METH-addicted mice normalized $\\mathrm{GABA_B}$ receptor expression as tested about $24\\mathrm{h}$ after peptide application (Figure 2A). Infusion of PP2A-Pep into the VTA of non-addicted control mice had no effect on the expression of $\\mathrm{GABA_B}$ receptors. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,38,179,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hleihil and Benke ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[752,38,917,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10.3389/fnmol.2024.1347228 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[78,948,280,958]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Molecular Neuroscience ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[490,948,507,958]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"07 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[835,948,915,959]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[77,87,487,187]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"One of the main projection areas of the VTA involved in addictive phenotypes is the nucleus accumbens. Like the VTA, $\\mathrm{GABA_B}$ receptor expression was significantly reduced in the nucleus accumbens but remained unaffected by infusion of PP2A-Pep into the VTA (Figure 2B). By contrast, $\\mathrm{GABA_B}$ receptor expression in the hippocampus (Figure 2C) and motor cortex (Figure 2D) was not significantly affected in METH-addicted mice. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,187,487,300]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Next, we analyzed $\\mathrm{GABA_B}$ receptor expression in the VTA on the cellular level. For this, we quantified immunofluorescence intensities for $\\mathrm{GABA_{B2}}$ in the somata of peptide positive VTA neurons. As observed on the more global, VTA wide level, $\\mathrm{GABA_B}$ receptor expression in the soma of individual VTA neurons was downregulated in METH-addicted mice (Figure 3). Infusion of PP2A-Pep into the VTA normalized $\\mathrm{GABA_B}$ receptor expression in the somata of neurons in METH-addicted mice (Figure 3). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,300,487,370]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Taken together, these results indicate that $\\mathrm{GABA_B}$ receptors were downregulated in the VTA and nucleus accumbens of METH-addicted mice. Infusion of PP2A-Pep into the VTA restored receptor expression to normal levels in the VTA but did not affect receptor expression in the nucleus accumbens. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,398,458,430]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PP2A-Pep infusion into the VTA reduced METH-induced locomotor sensitization ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[77,445,487,629]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Locomotor sensitization is a hallmark of psychostimulant-induced addiction (Tirelli et al., 2003). To test whether restoring $\\mathrm{GABA}_B$ receptor expression in the VTA affects locomotor sensitization, mice were injected with METH and saline for ten days followed by infusion of PP2A-Pep or Ctrl-Pep into the VTA. Locomotor sensitization was then tested the next day in the open field arena (Figure 4A). METH-addicted mice showed increased locomotor activity, which was reduced to levels of non-addicted, saline injected, control mice that were infused with the non-active control peptide (Ctrl-Pep, Figures 4B, C). Infusion of PP2A-Pep into the VTA of non-addicted control mice did not affect locomotor activity (Figures 4B, C). Thus, PP2A-Pep effectively inhibited increased locomotor in METH-addicted mice. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,657,411,707]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"PP2A-Pep infusion into the VTA of METH-addicted mice reduced drug seeking behavior ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[77,721,487,917]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The effect of PP2A-Pep on drug seeking behavior of METH-addicted mice was evaluated in the CPP test (Bardo and Bevins, 2000; McKendrick and Graziane, 2020). For the CPP test, we used a biased design in which the preference of the mice for one of the two compartments was determined before starting the conditioning phase (a schematic outline of the device and the experimental procedure is shown in Figures 5A, B). During the conditioning phase, the METH group received METH injections in the initially non-preferred compartment, followed by saline injections $6\\mathrm{h}$ later in the opposite compartment. Mice in the saline control group received injections in the preferred compartment, followed by a second saline injection $6\\mathrm{h}$ later in the opposite compartment. One day after the conditioning phase (day 7), mice were subjected to the CPP test and $1\\mathrm{h}$ thereafter the mice were microinfused ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[504,87,917,258]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"either with PP2A-Pep or Ctrl-Pep via the implanted cannula and returned to the home cage. A second preference test was performed the next day. As expected, METH-conditioned mice spent more time in the METH-paired compartment, indicating drug seeking behavior (Figure 5C). Infusion of PP2A-Pep into the VTA of METH-conditioned mice reduced the drug seeking behavior as the mice explored both compartments almost equally. Infusion of the inactive control peptide into the VTA of METH-conditioned mice was without effect as was infusion of PP2A-Pep into the VTA of saline control mice. These results indicate that a single administration of PP2A-Pep into the VTA of METH-addicted mice diminishes drug seeking behavior. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,290,623,306]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Discussion ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[504,322,917,406]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"This study indicates that repeated METH treatment in mice downregulates $\\mathrm{GABA_B}$ receptor expression in the VTA. Infusion of an interfering peptide that inhibits the interaction of $\\mathrm{GABA_B}$ receptors with PP2A into the VTA of METH-treated mice restored the expression of $\\mathrm{GABA_B}$ receptors. This diminished METH-induced locomotor sensitization and drug seeking behavior. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[504,407,917,705]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Excessive neuronal excitation triggers the downregulation of $\\mathrm{GABA_B}$ receptors by increasing their lysosomal degradation on the expense of recycling the receptors back to the plasma membrane (Guetg et al., 2010; Maier et al., 2010; Terunuma et al., 2010). Two key steps are involved in this pathway: CaMKII $\\beta$ -mediated phosphorylation of the $\\mathrm{GABA_B1}$ subunit at Ser867 (Guetg et al., 2010; Zemoura et al., 2019; Balakrishnan et al., 2023) and PP2A-dependent dephosphorylation of the $\\mathrm{GABA_B2}$ subunit at Ser783 (Terunuma et al., 2010; Hleihil et al., 2022). Elevated intracellular $\\mathrm{Ca^{2+}}$ levels due to enhanced neuronal activity activates CaMKII $\\beta$ to induce phosphorylation of Ser867 in $\\mathrm{GABA_B1}$ (Maier et al., 2010). Recent evidence suggests that CaMKII $\\beta$ not directly phosphorylate $\\mathrm{GABA_B1}$ but activates ERK1/2 which in turn phosphorylate Ser867 and the adjacent Thr872 (Bhat et al., 2023). Phosphorylation of $\\mathrm{GABA_B1}$ is required to enable K63-linked ubiquitination of the receptor by the E3-ubiquitin ligase MIB-2, which serves as a signal for their lysosomal degradation (Zemoura et al., 2019). In addition to CaMKII $\\beta$ , activation of PP2A under these conditions dephosphorylates Ser783 in $\\mathrm{GABA_B2}$ . This inhibits fast recycling of the receptors and directs them into the degradation pathway controlled by CaMKII $\\beta$ (Terunuma et al., 2010; Hleihil et al., 2022). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[504,707,917,876]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In the present study we found that in most neurons of the VTA analyzed, $\\mathrm{GABA_B}$ receptors were downregulated after repetitive METH treatment in mice. Likewise, we found a similar reduction of $\\mathrm{GABA_B}$ receptors in the nucleus accumbent, a main projection area of the VTA neurons associated with drug addiction. This finding is well supported by a recent study indicating reduced $\\mathrm{GABA_B}$ receptor expression in the nucleus accumbens of cocaine-addicted rats involving CaMKII dependent phosphorylation of the receptors (Lu et al., 2023). By contrast, $\\mathrm{GABA_B}$ receptor expression in the hippocampus and motor cortex remained unaffected, reassuring that the changes in immunostaining we detected were not caused by factors unrelated to the treatment. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[504,877,917,919]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In all our previous studies, a loss of $\\mathrm{GABA_B}$ receptor immunofluorescence staining always correlated with a similar reduction of receptor protein signals in western blots and ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[78,38,178,48]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hleihil and Benke ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[752,38,915,48]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10.3389/fnmo1.2024.1347228 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[78,948,280,958]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Molecular Neuroscience ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[489,948,505,957]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"08 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[835,948,915,959]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[127,103,258,149]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0087/de1d90cd04a2c12345b378e33e7934a3b22f7895630d49cf88c1691244306915.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[273,103,436,225]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0987/c2930ea68d66626a695a46a16535bf66256ae2c4952949fe34fbcab52cfa3376.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[127,160,245,217]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/1022/e18aa7feb750289712a7c76504d9d6cf0bcaf37a328fc17c2900412cb944c6ed.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[127,227,436,354]","code_body":null,"code_caption":null,"image_caption":"[]","image_footnote":"[]","img_path":"images/0194/f11d8833b0eca5ad118eb4d8ab904a1874a289e742f91108b13cc11689e6b417.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[127,356,438,491]","code_body":null,"code_caption":null,"image_caption":"[\"FIGURE 5 Injection of PP2A-Pep into the VTA of METH-addicted mice reduced drug seeking behavior as tested by CPP. (A,B) Schematic outline of the CPP procedure (A) and the CCP device (B). CPP was conducted in a device consisting of two visually distinct plexiglass compartments connected through a short gray tunnel. The CPP procedure included three phases: habituation and baseline preference (day 1), conditioning phase (days 2-6) and preference tests (days 7 and day 8). During the conditioning phase (day 2-6), mice received METH injections $(2\\\\mathrm{mg / kg},$ i. p.) in the initially non-preferred compartment and access was restricted to this compartment. After $6\\\\mathrm{h}$ mice were injected with saline in their initially preferred compartment. The control group received the same procedure, but mice were injected with two saline doses. On the test day (day 7, 24 h after the last conditioning trial), the preference test was performed. One hour after the test, mice were microinfused either with PP2A-Pep or Ctrl-Pep via the implanted cannula and tested a second time the next day (day 8). (C) METH-conditioned mice showed increased preference for the METH-paired chamber on the first test day. The data represents the mean $\\\\pm \\\\mathrm{SD}$ of 18-20 mice per condition. Statistical analysis: Two-tailed unpaired t-test. (D) Infusion of PP2A-Pep into the VTA after the first test diminished the preference for the METH-paired chamber in METH-addicted mice on the second test day. Application of the control peptide (Ctrl-Pep) had no effect. The data represents the mean $\\\\pm \\\\mathrm{SD}$ of 8-10 mice per condition. Statistical analysis: Two-way ANOVA with Tukey's multiple comparison test (ns, $p > 0.05$ ). (E) Heat maps depicting the frequencies of locations visited in the CPP chambers by mice under the different experimental conditions (representative images). \"]","image_footnote":"[]","img_path":"images/1022/3daaa7a2b9d31c12d5df1eb735863f2d5b5a986b63b8f2c4ddfd5dfbc0ae5843.jpg","list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"image"},{"bbox":"[78,862,489,919]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"with diminished $\\mathrm{GABA_B}$ receptor mediated inhibition (see e.g., Zemoura et al., 2019; Hleihil et al., 2021, 2022; Bhat et al., 2022; Balakrishnan et al., 2023). Therefore, it is reasonable to assume that the reduction in $\\mathrm{GABA_B}$ receptor immunofluorescence ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[504,87,917,693]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"staining observed in this study translates into diminished $\\mathrm{GABA_B}$ receptor mediated inhibition. The METH-induced downregulation of $\\mathrm{GABA_B}$ receptors we observed in VTA neurons is well supported by diminished $\\mathrm{GABA_B}$ receptor-mediated inhibition reported in the VTA (Arora et al., 2011; Padgett et al., 2012; Sharpe et al., 2015; Munoz et al., 2016). So far, two mechanisms have been reported to cause the reduction of $\\mathrm{GABA_B}$ receptor-mediated inhibition in the VTA. First, in GABAergic neurons, the receptors were downregulated from the cell surface involving dephosphorylation of $\\mathrm{GABA_{B2}}$ Ser-783 after a single injection of METH (Padgett et al., 2012). Second, in dopaminergic neurons, the postsynaptic $\\mathrm{GABA_B}$ receptor effectors, the GIRK channels, were downregulated instead of the receptors (Arora et al., 2011; Munoz et al., 2016). However, our data suggests a more widespread downregulation of $\\mathrm{GABA_B}$ receptors in VTA neurons than just their selective reduction in GABAergic neurons since only about $35\\%$ of neurons in the VTA are GABAergic ( $\\sim 60\\%$ are dopaminergic and $\\sim 5\\%$ glutamatergic) (Nair-Roberts et al., 2008; Ma et al., 2023). Further investigations are required to address the proportion of $\\mathrm{GABA_B}$ receptor downregulated in the different neuronal populations of the VTA. In addition, it would be interesting to analyze in detail whether solely cell surface $\\mathrm{GABA_B}$ receptors are reduced as previously reported after a single METH injection (Padgett et al., 2012) or more globally as we observed after repeated injections. However, infusion of PP2A-Pep into the VTA restored $\\mathrm{GABA_B}$ receptor expression in VTA neurons, supporting the view that the observed downregulation of $\\mathrm{GABA_B}$ receptors in METH-addicted mice is mediated by the PP2A/CaMKII pathway described above. The restoration of $\\mathrm{GABA_B}$ receptors in the VTA also reduced addictive behaviors in METH-addicted mice, suggesting that a globally restored inhibitory control by $\\mathrm{GABA_B}$ receptors in the VTA diminishes enhanced dopaminergic activity in the nucleus accumbens. Interestingly, restored $\\mathrm{GABA_B}$ receptor expression in the VTA did not normalize reduced $\\mathrm{GABA_B}$ receptor levels in the nucleus accumbens during the time frame of our analysis. Cocaine treatment most likely downregulates $\\mathrm{GABA_B}$ receptor expression in the nucleus accumbens by the same mechanism as METH treatment in the VTA (Lu et al., 2023). This suggests that simply reducing dopaminergic activity to the nucleus accumbens is not sufficient to normalize the aberrant trafficking of the receptors. It would be interesting to analyze the effect of PP2A-Pep on $\\mathrm{GABA_B}$ receptor expression in the nucleus accumbens and on addictive phenotypes of METH addicted mice. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[505,695,917,919]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Repeated psychostimulant administration in rodents leads to locomotor sensitization, a phenotype that is triggered by enhanced dopamine release within the VTA and nucleus accumbens. Reexposure to even a small quantity of the psychostimulants in the addicted subject, induces a strong activity in the VTA (Tirelli et al., 2003; Steketee and Kalivas, 2011). Accordingly, we observed a significant increase in locomotor activity upon injection of a challenge dose of the drug in METH-addicted mice. Interestingly, a single injection of PP2A-Pep into the VTA was sufficient to reverse the enhanced locomotor activity, most likely due to increased tonic inhibition mediated by the restored $\\mathrm{GABA_B}$ receptor expression in the VTA. This finding is in line with previous studies showing that the $\\mathrm{GABA_B}$ receptor agonist baclofen inhibits locomotor sensitization (Bartoletti et al., 2004; Leite-Morris et al., 2004; Cedillo and Miranda, 2013). Furthermore, selective deletion of $\\mathrm{GABA_B}$ receptors from VTA dopaminergic neurons enhanced the ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[80,38,178,48]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hleihil and Benke ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[754,38,915,48]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10.3389/fnmol.2024.1347228 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[80,948,280,958]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Molecular Neuroscience ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[490,948,505,957]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"09 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[836,948,915,959]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"8","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[77,87,487,130]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"locomotor sensitization to psychostimulants (Edwards et al., 2017), illustrating the importance of $\\mathrm{GABA_B}$ receptor mediated inhibition for controlling locomotor sensitization. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,130,489,411]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"It is believed that drug addiction is a form of Pavlovian conditioning, establishing long-term memories to stabilize drug associated behaviors. In the CPP experiment, associations are formed between a rewarding stimulus (e.g., METH) and a contextual environment (Tzschentke, 2007; McKendrick and Graziane, 2020). Re-exposure to drug-associated cues triggers memory recall, making these memories labile and sensitive to manipulations (Tronson and Taylor, 2007; Bellfy and Kwapis, 2020). In our CPP test, we aimed to interfere with METH-associated memories causing drug seeking by disturbing memory reconsolidation. This was done by infusing PP2A-Pep into the VTA $1\\mathrm{~h}$ after re-exposing the mice to the CPP test and repeating the CPP test after $24\\mathrm{~h}$ . This schedule was chosen since it is relevant in view of the development of a potential therapeutic application. Restoration of $\\mathrm{GABA_B}$ receptor expression in the VTA by PP2A-Pep after memory retrieval inhibited drug seeking behavior in the re-test session the next day. This finding suggests that METH associated memory reconsolidation has been disturbed by enhancing the expression of $\\mathrm{GABA_B}$ receptors in the VTA. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,412,489,595]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"However, our CPP protocol cannot rule out the possibility that the interfering peptide might additionally facilitate memory extinction, a process which is tightly associated with psychostimulant withdrawal (Myers and Carlezon, 2010). It is believed that memory extinction is an active learning process, where a newly learned memory is competing with the already consolidated one to weaken it. Memory consolidation and extinction have overlapping cellular and molecular mechanisms (Myers and Carlezon, 2010; Liu et al., 2019), making it difficult in our experimental set-up to discriminate among these two pathways. However, in view of the close timing of PP2A application $1\\mathrm{h}$ after memory retrieval we favor memory reconsolidation to be affected by PP2A-Pep in the first place. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,596,489,819]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We expect that PP2A-Pep is associated with negligible adverse side effects because it selectively inhibits the interaction of $\\mathrm{GABA_B}$ receptors with PP2A and does not appreciably affect $\\mathrm{GABA_B}$ receptor expression in healthy control mice. Therefore, side effects observed after global activation the receptors with the agonist baclofen such as sedation, dizziness, fatigue, insomnia, headache, paresthesia, tinnitus, and restlessness (Rolland et al., 2018) are unlikely to occur. In addition, because the peptide sequence of PP2A-Pep derived from the C-terminal domain of the $\\mathrm{GABA_B2}$ subunit is unique to $\\mathrm{GABA_B}$ receptors, we expect no off-target effects involving other receptor systems. In principle, binding of PP2A-Pep to PP2A could downregulate the expression levels of PP2A, which would affect a variety of signaling pathways involving protein dephosphorylation by PP2A. However, we previously showed that PP2A-Pep did not affect expression of PP2A (Hleihil et al., 2022). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[77,821,489,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"In conclusion, restoring $\\mathrm{GABA_B}$ receptor expression in VTA neurons of METH-addicted mice by inhibiting PP2A-mediated downregulation of the receptors inhibited locomotor sensitization and drug seeking behavior. This finding indicates that interfering with disease-relevant protein-protein interactions might be a promising alternative approach toward the development of a selective and efficient therapeutic intervention. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,87,789,105]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Data availability statement ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[505,120,917,192]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The datasets analyzed for this study have been deposited at ZENODO.org and are publicly available (doi: 10.5281/zenodo.10554456). Due to their large size, raw images and behavioral recordings can be made available upon reasonable request to the corresponding author. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,224,687,240]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ethics statement ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[505,256,917,313]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The animal study was approved by the Veterinary office of the Canton of Zurich, Waltersbachstrasse 5, CH-8006 Zurich (license ZH164/2020). The study was conducted in accordance with the local legislation and institutional requirements. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[505,344,732,361]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Author contributions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[505,376,917,463]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"MH: Conceptualization, Data curation, Formal Analysis, Writing - original draft, Writing - review & editing, Investigation, Methodology. DB: Conceptualization, Data curation, Formal Analysis, Writing - original draft, Writing - review & editing, Funding acquisition, Project administration, Resources, Supervision. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[507,494,596,513]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Funding ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[505,527,917,583]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The authors declare financial support was received for the research, authorship, and/or publication of this article. This research was funded by the Swiss National Science Foundation, grant number 31003A_182325 to DB. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[505,615,707,634]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Acknowledgments ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[529,649,904,662]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"We thank Thomas Grampp for excellent technical assistance. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[505,695,709,711]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conflict of interest ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[505,727,917,770]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[505,802,680,816]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Publisher's note ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[505,834,917,920]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"9","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher. 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(2023). Baclofen for alcohol use disorder. Cochr. Datab. Syst. Rev. 1:Cd012557. doi: 10.1002/14651858.CD012557.pub3 \",\"Arai, S., Takuma, K., Mizoguchi, H., Ibi, D., Nagai, T., Kamei, H., et al. (2009). $\\\\mathrm{GABA_B}$ receptor agonist baclofen improves methamphetamine-induced cognitive deficit in mice. Eur. J. Pharmacol. 602, 101-104. doi: 10.1016/j.ejphar.2008.10.065 \",\"Arora, D., Hearing, M., Haluk, D. M., Mirkovic, K., Fajardo-Serrano, A., Wessendorf, M. W., et al. (2011). Acute cocaine exposure weakens $\\\\mathrm{GABA_B}$ receptor-dependent G-protein-gated inwardly rectifying $\\\\mathrm{K}^+$ signaling in dopamine neurons of the ventral tegmental area. J. Neurosci. 31, 12251-12257. doi: 10.1523/JNEUROSCI.0494-11.2011 \",\"Baik, J. H. (2020). Stress and the dopaminergic reward system. Exp. Mol. Med. 52, 1879-1890. doi: 10.1038/s12276-020-00532-4 \",\"Balakrishnan, K., Hleihil, M., Bhat, M. A., Ganley, R. P., Vaas, M., Klohs, J., et al. (2023). Targeting the interaction of $\\\\mathrm{GABA_B}$ receptors with CaMKII with an interfering peptide restores receptor expression after cerebral ischemia and inhibits progressive neuronal death in mouse brain cells and slices. Brain Pathol. 33:e13099. doi: 10.1111/bpa.13099 \",\"Bardo, M. T., and Bevins, R. A. (2000). Conditioned place preference: What does it add to our preclinical understanding of drug reward? Psychopharmacology 153, 31-43. doi: 10.1007/s002130000569 \",\"Bartoletti, M., Gubellini, C., Ricci, F., and Gaiardi, M. (2004). The $\\\\mathrm{GABA_B}$ agonist baclofen blocks the expression of sensitisation to the locomotor stimulant effect of amphetamine. Behav. Pharmacol. 15, 397-401. doi: 10.1097/00008877-200409000-00014 \",\"Bartoletti, M., Gubellini, C., Ricci, F., and Giardi, M. (2005). Baclofen blocks the development of sensitization to the locomotor stimulant effect of amphetamine. Behav. Pharmacol. 16, 553-558. doi: 10.1097/01.fbp.0000179279.98029.e9 \",\"Bellfy, L., and Kwapis, J. L. (2020). Molecular mechanisms of reconsolidation-dependent memory updating. Int. J. Mol. Sci. 21, 6580. doi: 10.3390/ijms21186580 \",\"Benke, D., Michel, C., and Mohler, H. (2002). Structure of $\\\\mathrm{GABA_B}$ receptors in rat retina. J. Recept. Signal Transduc. 22, 253-266. \",\"Bhat, M. A., Esmaelli, A., Neumann, E., Balakrishnan, K., and Benke, D. (2022). Targeting the interaction of $\\\\mathrm{GABA_B}$ receptors with CHOP after an ischemic insult restores receptor expression and inhibits progressive neuronal death. Front. Pharmacol. 13:870861. doi: 10.3389/fphar.2022.870861 \",\"Bhat, M. A., Grampp, T., and Benke, D. (2023). ERK1/2-dependent phosphorylation of $\\\\mathrm{GABA_B}$ (S867/T872), controlled by CaMKII $\\\\beta$ , is required for $\\\\mathrm{GABA_B}$ receptor degradation under physiological and pathological conditions. Int. J. Mol. Sci. 24, 13436. doi: 10.3390/ijms241713436 \",\"Brebner, K., Ahn, S., and Phillips, A. G. (2005). Attenuation of d-amphetamine self-administration by baclofen in the rat: Behavioral and neurochemical correlates. Psychopharmacology 177, 409-417. doi: 10.1007/s00213-004-1968-6 \",\"Cedillo, L. N., and Miranda, F. (2013). Effects of co-administration of the $\\\\mathrm{GABA_B}$ receptor agonist baclofen and a positive allosteric modulator of the $\\\\mathrm{GABA_B}$ receptor, CGP7930, on the development and expression of amphetamine-induced locomotor sensitization in rats. Pharmacol Rep 65, 1132-1143. doi: 10.1016/s1734-1140(13) 71471-3 \",\"Cedillo, L. N., Ruiz-Garcia, R. I., Jimenez, J. C., and Miranda, F. (2019). Effect of coadministration of the $\\\\mathrm{GABA_B}$ agonist baclofen and the 5-HT2C agonist Ro60-0175 on the expression of amphetamine-induced locomotor sensitization. Exp. Brain. Res. 237, 1691-1697. doi: 10.1007/s00221-019-05540-z \",\"Chalifoux, J. R., and Carter, A. G. (2011). $\\\\mathrm{GABA_B}$ receptor modulation of synaptic function. Curr. Opin. Neurobiol. 21, 339-344. doi: 10.1016/j.conb.2011.02.004 \",\"Chen, G., and van den Pol, A. N. (1998). Presynaptic $\\\\mathrm{GABA_B}$ autoreceptor modulation of P/Q-type calcium channels and GABA release in rat suprachiasmatic nucleus neurons. J. Neurosci. 18, 1913-1922. \",\"Cheron, J., and Kerchove d'Exaerde, A. (2021). Drug addiction: From bench to bedside. Transl. Psychiatry 11, 424. doi: 10.1038/s41398-021-01542-0 \",\"Cunningham, C. L., Gremel, C. M., and Groblewski, P. A. (2006). Drug-induced conditioned place preference and aversion in mice. Nat. Protoc. 1, 1662-1670. doi: 10.1038/nprot.2006.279 \",\"de Beaurepaire, R., Sinclair, J. M. A., Heydtmann, M., Addolorato, G., Aubin, H. J., Beraha, E. M., et al. (2018). The use of baclofen as a treatment for alcohol use disorder: A clinical practice perspective. Front. Psychiatry 9:708. doi: 10.3389/fpsyt.2018.00708 \",\"Edwards, N. J., Tejeda, H. A., Pignatelli, M., Zhang, S., McDevitt, R. A., Wu, J., et al. (2017). Circuit specificity in the inhibitory architecture of the VTA regulates cocaine-induced behavior. Nat. Neurosci. 20, 438-448. doi: 10.1038/nn.4482 \",\"Fischer, K. D., Knockstedt, L. A., and Rosenberg, P. A. (2021). Glutamate homeostasis and dopamine signaling: Implications for psychostimulant addiction behavior. Neurochem Int 144, 104896. doi: 10.1016/j.neuint.2020.104896 \"]","page_idx":"10","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[509,124,917,904]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"Gahwiler, B. H., and Brown, D. A. (1985). $\\\\mathrm{GABA}_B$ -receptor-activated $\\\\mathbf{K}^{+}$ current in voltage-clamped CA3 pyramidal cells in hippocampal cultures. Proc. Natl. Acad. Sci. USA 82, 1558-1562. doi: 10.1073/pnas.82.5.1558 \",\"Garbutt, J. C. (2020). Approval of baclofen for alcohol use disorders in France: A perspective from the United States. Alcohol Alcohol 55, 48. doi: 10.1093/alcalc/agz084 \",\"Good, R. L., and Radcliffe, R. A. (2011). Methamphetamine-induced locomotor changes are dependent on age, dose and genotype. *Pharmacol. Biochem.* Behav. 98, 101-111. doi: 10.1016/j.pbb.2010.12.004 \",\"Guetg, N., Aziz, S. A., Holbro, N., Turecek, R., Rose, T., Seddik, R., et al. (2010). NMDA receptor-dependent $\\\\mathrm{GABA_B}$ receptor internalization via CaMKII phosphorylation of serine 867 in $\\\\mathrm{GABA_{B1}}$ . Proc. Natl. Acad. Sci. USA 107, 13924-13929. \",\"Harada, M., Pascoli, V., Hiver, A., Flakowski, J., and Luscher, C. (2021). Corticostriatal activity driving compulsive reward seeking. Biol. Psychiatry 90, 808-818. doi: 10.1016/j.biopsych.2021.08.018 \",\"Hearing, M., Kotecki, L., Marron Fernandez de Velasco, E., Fajardo-Serrano, A., Chung, H. J., Lujan, R., et al. (2013). Repeated cocaine weakens GABA-Girk signaling in layer 5/6 pyramidal neurons in the prelimbic cortex. Neuron 80, 159-170. doi: 10.1016/j.neuron.2013.07.019 \",\"Heinzerling, K. G., Shoptaw, S., Peck, J. A., Yang, X., Liu, J., Roll, J., et al. (2006). Randomized, placebo-controlled trial of baclofen and gabapentin for the treatment of methamphetamine dependence. *Drug Alcohol. Depend.* 85, 177-184. doi: 10.1016/j.drugalcdep.2006.03.019 \",\"Hleihil, M., Balakrishnan, K., and Benke, D. (2022). Protein phosphatase 2A regulation of $\\\\mathrm{GABA_B}$ receptors normalizes ischemia-induced aberrant receptor trafficking and provides neuroprotection. Front. Mol. Neurosci. 15:1015906. doi: 10. 3389/fnmo.2022.1015906 \",\"Hleihil, M., Vaas, M., Bhat, M. A., Balakrishnan, K., and Benke, D. (2021). Sustained baclofen-induced activation of $\\\\mathrm{GABA_B}$ receptors after cerebral ischemia restores receptor expression and function and limits progressing loss of neurons. Front. Mol. Neurosci. 14:726133. doi: 10.3389/fnml.2021.726133 \",\"Jing, L., Zhang, M., Li, J. X., Huang, P., Liu, Q., Li, Y. L., et al. (2014). Comparison of single versus repeated methamphetamine injection induced behavioral sensitization in mice. Neurosci. Lett. 560, 103-106. doi: 10.1016/j.neulet.2013.12.024 \",\"Jones, K. A., Borowsky, B., Tamm, J. A., Craig, D. A., Durkin, M. M., Dai, M., et al. (1998). $\\\\mathrm{GABA_B}$ receptor function as a heteromeric assembly of the subunits $\\\\mathrm{GABA_{BR1}}$ and $\\\\mathrm{GABA_{BR2}}$ . Nature 396, 674-679. \",\"Kaufmann, K., Malitschek, B., Schuler, V., Heid, J., Froestl, W., Beck, P., et al. (1998). $\\\\mathrm{GABA_B}$ receptor subtypes assemble into functional heteromeric complexes. Nature 396, 683-687. \",\"Kumar, P., Wu, H., McBride, J. L., Jung, K. E., Kim, M. H., Davidson, B. L., et al. (2007). Transvascular delivery of small interfering RNA to the central nervous system. Nature 448, 39-43. doi:10.1038/nature05901 \",\"Lalive, A. L., and Luscher, C. (2016). \\\"GABA $_B$ receptor functions in the mesolimbic dopamine system,\\\" in GABA $_B$ Receptor, ed. G. Colombo (Cham: Springer International Publishing), 129-154. \",\"Leggio, L., and Litten, R. Z. (2021). The $\\\\mathrm{GABA_B}$ receptor agonist baclofen helps patients with alcohol use disorder: Why these findings matter. Neuropsychopharmacology 46, 2228-2229. doi:10.1038/s41386-021-01142-y \",\"Leite-Morris, K. A., Fukudome, E. Y., Shoeb, M. H., and Kaplan, G. B. (2004). $\\\\mathrm{GABA_B}$ receptor activation in the ventral tegmental area inhibits the acquisition and expression of opiate-induced motor sensitization. J. Pharmacol. Exp. Ther. 308, 667-678. doi: 10.1124/jpet.103.058412 \",\"Li, J. Y., Yu, Y. J., Su, C. L., Shen, Y. Q., Chang, C. H., and Gean, P. W. (2023). Modulation of methamphetamine memory reconsolidation by neural projection from basolateral amygdala to nucleus accumbens. Neuropsychopharmacology 48, 478-488. doi: 10.1038/s41386-022-01417-y \",\"Li, M., Peterson, C., Xu, L., Mikosz, C. A., and Luo, F. (2023). Medical costs of substance use disorders in the US employer-sponsored insurance population. JAMA Netw Open 6, e2252378. doi:10.1001/jamanetworkopen.2022.52378 \",\"Li, S. M., Yin, L. L., Ren, Y. H., Pan, L. S., and Zheng, J. W. (2001). $\\\\mathrm{GABA_B}$ receptor agonist baclofen attenuates the development and expression of d-methamphetamine-induced place preference in rats. Life Sci. 70, 349-356. doi: 10.1016/s0024-3205(01) 01397-2 \",\"Ling, W., Shoptaw, S., and Majewska, D. (1998). Baclofen as a cocaine anti-craving medication: A preliminary clinical study. Neuropsychopharmacology 18, 403-404. doi: 10.1016/s0893-133x(97)00128-0 \",\"Liu, J. F., Tian, J., and Li, J. X. (2019). Modulating reconsolidation and extinction to regulate drug reward memory. *Eur. J. Neurosci.* 50, 2503-2512. doi: 10.1111/ejn.14072 \"]","page_idx":"10","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[80,38,179,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hleihil and Benke ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[752,38,917,49]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10.3389/fnmol.2024.1347228 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[80,948,280,958]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Frontiers in Molecular Neuroscience ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[490,948,505,957]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"11 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[836,948,915,959]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"10","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"frontiersin.org ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[80,90,487,653]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"Lu, M. F., Fu, Q., Qiu, T. Y., Yang, J. H., Peng, Q. H., and Hu, Z. Z. (2023). The CaMKII-dependent phosphorylation of $\\\\mathrm{GABA_B}$ receptors in the nucleus accumbens was involved in cocaine-induced behavioral sensitization in rats. CNS Neurosci. Ther. 29, 1345-1356. doi: 10.1111/cns.14107 \",\"Luscher, C., Jan, L. Y., Stoffel, M., Malenka, R. C., and Nicoll, R. A. (1997). G protein-coupled inwardly rectifying $\\\\mathrm{K}^+$ channels (GIRKs) mediate postsynaptic but not presynaptic transmitter actions in hippocampal neurons. Neuron 19, 687-695. \",\"Ma, S., Zhong, H., Liu, X., and Wang, L. (2023). Spatial distribution of neurons expressing single, double, and triple molecular characteristics of glutamatergic, dopaminergic, or GABAergic neurons in the mouse ventral tegmental area. J. Mol. Neurosci. 73, 345-362. doi:10.1007/s12031-023-02121-2 \",\"Maier, P. J., Marin, I., Grampp, T., Sommer, A., and Benke, D. (2010). Sustained glutamate receptor activation down-regulates $\\\\mathrm{GABA_B}$ receptors by shifting the balance from recycling to lysosomal degradation. J. Biol. Chem. 285, 35606-35614. doi: 10.1074/jbc.M110.142406 \",\"May, A. C., Upperle, R. L., and Stewart, J. L. (2020). Dark times: The role of negative reinforcement in methamphetamine addiction. Front. Psychiatry 11:114. doi: 10.3389/fpsyt.2020.00114 \",\"McKendrick, G., and Graziane, N. M. (2020). Drug-induced conditioned place preference and Its practical use in substance use disorder research. Front. Behav. Neurosci. 14:582147. doi: 10.3389/fnbeh.2020.582147 \",\"Mintz, I. M., and Bean, B. P. (1993). $\\\\mathrm{GABA_B}$ receptor inhibition of P-type $\\\\mathrm{Ca^{2+}}$ channels in central neurons. Neuron 10, 889-898. \",\"Mizoguchi, H., and Yamada, K. (2011). Pharmacologic treatment with $\\\\mathrm{GABA_B}$ receptor agonist of methamphetamine-induced cognitive impairment in mice. Curr. Neuropharmacol. 9, 109-112. doi: 10.2174/157015911795016976 \",\"Munoz, M. B., Padgett, C. L., Rifkin, R., Terunuma, M., Wickman, K., Contet, C., et al. (2016). A role for the GIRK3 subunit in methamphetamine-induced attenuation of $\\\\mathrm{GABA_B}$ receptor-activated GIRK currents in VTA dopamine neurons. J. Neurosci. 36, 3106-3114. doi: 10.1523/jneurosci.1327-15.2016 \",\"Myers, K. M., and Carlezon, W. A. Jr. (2010). Extinction of drug- and withdrawal-paired cues in animal models: Relevance to the treatment of addiction. Neurosci. Biobehav. Rev. 35, 285-302. doi: 10.1016/j.neubiorev.2010.01.011 \",\"Nair-Roberts, R. G., Chatelain-Badie, S. D., Benson, E., White-Cooper, H., Bolam, J. P., and Ungless, M. A. (2008). Stereological estimates of dopaminergic, GABAergic and glutamatergic neurons in the ventral tegmental area, substantia nigra and retrorubral field in the rat. *Neuroscience* 152, 1024-1031. doi: 10.1016/j.neuroscience.2008.01.046 \",\"Nestler, E. J. (2005). The neurobiology of cocaine addiction. Sci. Pract. Perspect. 3, 4-10. doi: 10.1151/spp05314 \",\"Ostroumov, A., and Dani, J. A. (2018). Inhibitory plasticity of mesocorticolimbic circuits in addiction and mental illness. Trends Neurosci. 41, 898-910. doi: 10.1016/j.tins.2018.07.014 \",\"Padgett, C. L., Lalive, A. L., Tan, K. R., Terunuma, M., Munoz, M. B., Pangalos, M. N., et al. (2012). Methamphetamine-evoked depression of $\\\\mathrm{GABA_B}$ receptor signaling in GABA neurons of the VTA. Neuron 73, 978-989. doi: 10.1016/j.neuron.2011.12.031 \",\"Paxinos, G., and Franklin, K. B. J. (2001). The mouse brain in steriotaxic coordinates. San Diego, CA: Academic Press. \",\"Ranaldi, R., and Poeggel, K. (2002). Baclofen decreases methamphetamine self-administration in rats. Neuroreport 13, 1107-1110. doi: 10.1097/00001756-200207020-00007 \"]","page_idx":"11","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[509,90,915,653]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"Roberts, D. C., and Brebner, K. (2000). GABA modulation of cocaine self-administration. Ann. N Y Acad. Sci. 909, 145-158. doi: 10.1111/j.1749-6632.2000.tb06680.x \",\"Rolland, B., Simon, N., and Franchitto, N. (2018). Safety challenges of using high dose baclofen for alcohol use disorder: A focused review. Front. Psychiatry 9:367. doi: 10.3389/fpsyt.2018.00367 \",\"Rolland, B., Simon, N., Franchitto, N., and Aubin, H. J. (2020). France grants an approval to baclofen for alcohol dependence. Alcohol Alcohol. 55, 44-45. doi: 10.1093/alcalc/agz082 \",\"Santos, A. E., Carvalho, C. M., Macedo, T. A., and Carvalho, A. P. (1995). Regulation of intracellular $\\\\left[\\\\mathrm{Ca}^{2+}\\\\right]$ and GABA release by presynaptic $\\\\mathrm{GABA_B}$ receptors in rat cerebrocortical synaptosomes. Neurochem. Int. 27, 397-406. \",\"Schindelin, J., Arganda-Carreras, I., Frise, E., Kaynig, V., Longair, M., Pietzsch, T., et al. (2012). Fiji: An open-source platform for biological-image analysis. Nat. Methods 9, 676-682. doi: 10.1038/nmeth.2019 \",\"Sharpe, A. L., Varela, E., Bettinger, L., and Beckstead, M. J. (2015). Methamphetamine self-administration in mice decreases GIRK channel-mediated currents in midbrain dopamine neurons. Int. J. Neuropsychopharmacol. 18, 1-10. doi: 10.1093/ijnp/pyu073 \",\"Shoptaw, S., Yang, X., Rotheram-Fuller, E. J., Hsieh, Y. C., Kintaudi, P. C., Charuvastra, V. C., et al. (2003). Randomized placebo-controlled trial of baclofen for cocaine dependence: Preliminary effects for individuals with chronic patterns of cocaine use. J. Clin. Psychiatry 64, 1440-1448. doi: 10.4088/jcp.v64 n1207 \",\"Steketee, J. D., and Kalivas, P. W. (2011). Drug wanting: Behavioral sensitization and relapse to drug-seeking behavior. Pharmacol. Rev. 63, 348-365. doi: 10.1124/pr.109.001933 \",\"Terunuma, M., Vargas, K. J., Wilkins, M. E., Ramirez, O. A., Jaureguiberry-Bravo, M., Pangalos, M. N., et al. (2010). Prolonged activation of NMDA receptors promotes dephosphorylation and alters postendocytic sorting of $\\\\mathrm{GABA_B}$ receptors. Proc. Natl. Acad. Sci. USA 107, 13918-13923. \",\"Tirelli, E., Laviola, G., and Adriani, W. (2003). Ontogenesis of behavioral sensitization and conditioned place preference induced by psychostimulants in laboratory rodents. Neurosci. Biobehav. Rev. 27, 163-178. doi: 10.1016/s0149-7634(03) 00018-6 \",\"Tronson, N. C., and Taylor, J. R. (2007). Molecular mechanisms of memory reconsolidation. Nat. Rev. Neurosci. 8, 262-275. doi: 10.1038/nrn2090 \",\"Tzschentke, T. M. (2007). Measuring reward with the conditioned place preference (CPP) paradigm: Update of the last decade. Addict. Biol. 12, 227-462. doi: 10.1111/j.1369-1600.2007.00070.x \",\"Volman, S. F., Lammel, S., Margolis, E. B., Kim, Y., Richard, J. M., Roitman, M. F., et al. (2013). New insights into the specificity and plasticity of reward and aversion encoding in the mesolimbic system. J. Neurosci. 33, 17569-17576. doi: 10. 1523/jneurosci.3250-13.2013 \",\"White, J. H., Wise, A., Main, M. J., Green, A., Fraser, N. J., Disney, G. H., et al. (1998). Heterodimerization is required for the formation of functional $\\\\mathrm{GABA_B}$ receptors. Nature 396, 679-682. \",\"Zemoura, K., Balakrishnan, K., Grampp, T., and Benke, D. (2019). $\\\\mathrm{Ca^{2+}}/$ Calmodulin-dependent protein kinase II (CaMKII) b-dependent phosphorylation of $\\\\mathrm{GABA_B}$ triggers lysosomal degradation of $\\\\mathrm{GABA_B}$ Receptors via mind bomb-2 (MIB2)-mediated Lys-63-linked ubiquitination. Mol. Neurobiol. 56, 1293-1309. doi: 10.1007/s12035-018-1142-5 \"]","page_idx":"11","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[80,38,176,48]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Hleihil and Benke ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[752,38,915,48]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"11","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"10.3389/fnmo1.2024.1347228 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This is an \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,523,302,532]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"open-access article distributed under the \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,533,315,543]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"terms of the Creative Commons Attribution \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,545,312,555]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"License (CC BY). The use, distribution or \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,556,309,565]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"reproduction in other forums is permitted, \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,567,287,576]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"provided the original author(s) and the \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,577,319,586]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"copyright owner(s) are credited and that the \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,588,319,596]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"original publication in this journal is cited, in \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,598,280,607]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"accordance with accepted academic \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,609,324,619]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"practice. No use, distribution or reproduction \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,620,302,629]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"is permitted which does not comply with \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,630,147,639]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"these terms. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[364,139,863,342]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Restoring $\\\\mathrm{GABA}_B$ receptor expression in the ventral tegmental area of methamphetamine addicted mice inhibits locomotor sensitization and drug seeking behavior \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[364,359,729,373]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Mohammad Hleihil<sup>1</sup> and Dietmar Benke<sup>1,2*</sup> \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[363,382,900,393]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$^{1}$ Institute of Pharmacology and Toxicology, University of Zurich, Zürich, Switzerland, $^{2}$ Neuroscience \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[364,394,697,404]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Center Zurich, University and ETH Zurich, Zurich, Switzerland \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[363,445,919,815]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Repeated exposure to psychostimulants such as methamphetamine (METH) induces neuronal adaptations in the mesocorticolimbic dopamine system, including the ventral tegmental area (VTA). These changes lead to persistently enhanced neuronal activity causing increased dopamine release and addictive phenotypes. A factor contributing to increased dopaminergic activity in this system appears to be reduced $\\\\mathrm{GABA_B}$ receptor-mediated neuronal inhibition in the VTA. Dephosphorylation of serine 783 (Ser783) of the $\\\\mathrm{GABA_{B2}}$ subunit by protein phosphatase 2A (PP2A) appears to trigger the downregulation $\\\\mathrm{GABA_B}$ receptors in psychostimulant-addicted rodents. Therefore, preventing the interaction of $\\\\mathrm{GABA_B}$ receptors with PP2A using an interfering peptide is a promising strategy to restore $\\\\mathrm{GABA_B}$ receptor-mediated neuronal inhibition. We have previously developed an interfering peptide (PP2A-Pep) that inhibits the $\\\\mathrm{GABA_B}$ receptors/PP2A interaction and thereby restores receptor expression under pathological conditions. Here, we tested the hypothesis that restoration of $\\\\mathrm{GABA_B}$ receptor expression in the VTA of METH addicted mice reduce addictive phenotypes. We found that the expression of $\\\\mathrm{GABA_B}$ receptors was significantly reduced in the VTA and nucleus accumbens but not in the hippocampus and somatosensory cortex of METH-addicted mice. Infusion of PP2A-Pep into the VTA of METH-addicted mice restored $\\\\mathrm{GABA_B}$ receptor expression in the VTA and inhibited METH-induced locomotor sensitization as assessed in the open field test. Moreover, administration of PP2A-Pep into the VTA also reduced drug seeking behavior in the conditioned place preference test. These observations underscore the importance of VTA $\\\\mathrm{GABA_B}$ receptors in controlling addictive phenotypes. Furthermore, this study illustrates \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,40,102,59]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"# \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[105,42,194,57]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"frontiers \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[213,43,473,57]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Molecular Neuroscience \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[734,39,863,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"TYPE Original Research \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[736,49,890,60]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PUBLISHED 07 February 2024 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[736,61,919,71]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"DOI 10.3389/fnmol.2024.1347228 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[80,948,280,958]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Molecular Neuroscience \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[490,948,505,958]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"01 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[836,948,915,959]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"0\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[364,87,917,134]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"the value of interfering peptides targeting diseases-related protein-protein interactions as an alternative approach for a potential development of selective therapeutic interventions. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[364,152,421,162]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"KEYWORDS \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[363,167,917,192]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"addiction, GABA receptor, interfering peptide, methamphetamine, protein phosphatase 2A, ventral tegmental area (VTA) \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,256,216,273]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Introduction \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,290,489,475]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Drug addiction is a complex chronic disease with biological, behavioral, and social components. Psychostimulants such as cocaine, amphetamine, and methamphetamine (METH) are highly addictive drugs, whose abuse can lead to serious health problems and significant healthcare costs (May et al., 2020; Cheron and Kerchove d'Exaerde, 2021). For instance, in 2018, 2.3 million employees receiving employer-sponsored insurance in the USA (representing $1.4\\\\%$ of the total population of employees receiving employer-sponsored insurance in the USA) were diagnosed with substance use disorders with total annual medical cost of $35.3 billion (Li M. et al., 2023). Currently, there is no effective pharmacological therapy available to cure patients from addiction. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,476,489,719]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Addiction to psychostimulants results from repeated and prolonged actions of the drug on the brain inducing long-lasting plastic neuronal adaptations (Ostroumov and Dani, 2018; Fischer et al., 2021). The effects of psychostimulants are mediated via enhanced dopamine release in the mesocorticolimbic dopamine pathway. The mesocorticolimbic circuit is involved in many cognitive processes including reward, motivation, learning, and fear (Volman et al., 2013; Baik, 2020). A key structure triggering addictive behavior is the ventral tegmental area (VTA) located in the midbrain. Acute effects of psychostimulants, such as feelings of euphoria, pleasure, and reward, are mediated via increased activity of VTA dopamine neurons projecting to the nucleus accumbens, amygdala, and medial prefrontal cortex (Nestler, 2005). Repeated and chronic consumption of psychostimulants induces plastic neuronal adaptations resulting in permanently increased dopamine release in the mesocorticolimbic dopamine pathway. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,720,490,919]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"One important factor controlling neuronal excitability and neurotransmitter release is the $\\\\mathrm{GABA_B}$ receptor, which is expressed in most neurons at axon terminals, dendritic and somatic sites (Chalifoux and Carter, 2011). $\\\\mathrm{GABA_B}$ receptors are heterodimeric assemblies composed of $\\\\mathrm{GABA_{B1}}$ and $\\\\mathrm{GABA_{B2}}$ subunits and mediate their effects by activating Gi/o proteins (Jones et al., 1998; Kaupmann et al., 1998; White et al., 1998). Their most prominent actions in the brain are the activation of G protein-coupled inwardly rectifying potassium channels (GIRK or $K_{ir}3$ ) at postsynaptic locations to reduce neuronal excitability (Gahwiler and Brown, 1985; Luscher et al., 1997) and the inhibition of voltage-gated $\\\\mathrm{Ca^{2+}}$ channels at presynaptic sites to reduce transmitter release (Mintz and Bean, 1993; Santos et al., 1995; Chen and van den Pol, 1998). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,253,919,649]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\mathrm{GABA_B}$ receptors are well expressed in the mesocorticolimbic dopamine system, where they control neuronal activity and the release of GABA, glutamate and dopamine (Lalive and Luscher, 2016). Accordingly, pharmacological activation of $\\\\mathrm{GABA_B}$ receptors by the selective agonist baclofen reduced several addictive behaviors in preclinical studies such as the development and expression of psychostimulant-induced motor sensitization (Bartoletti et al., 2004, 2005; Cedillo et al., 2019), drug seeking (Li et al., 2001), drug self-administration (Roberts and Brebner, 2000; Ranaldi and Poeggel, 2002; Brebner et al., 2005) and drug-induced cognitive impairment (Arai et al., 2009; Mizoguchi and Yamada, 2011). Early clinical studies reported some positive effects of baclofen in reducing psychostimulant consumption and drug craving (Ling et al., 1998; Shoptaw et al., 2003; Heinzerling et al., 2006). There are clinical evidences for the benefit of baclofen for the treatment of alcohol use disorders (Agabio et al., 2023), but its use is controversial (de Beaurepaire et al., 2018; Leggio and Litten, 2021). Currently, baclofen is approved for the treatment of alcohol addiction in France (Garbutt, 2020; Rolland et al., 2020). One problem of baclofen at high dosage are numerous side effects such as sedation, dizziness, fatigue, insomnia, headache, paresthesia, tinnitus, and restlessness (Rolland et al., 2018). The wide range of adverse side effects might reflect the fact that systemic application of baclofen activates all $\\\\mathrm{GABA_B}$ receptors in the body and not only in the neuronal pathways related to the addicted state. This might generally restrict the broad clinical application of baclofen and need for more specific interventions. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,650,919,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"There is considerable evidence that $\\\\mathrm{GABA_B}$ receptor-mediated inhibition is reduced in brain structures of the mesocorticolimbic system after psychostimulant treatment, diminishing inhibitory control and enhancing neuronal excitability (Arora et al., 2011; Padgett et al., 2012; Hearing et al., 2013; Sharpe et al., 2015; Munoz et al., 2016). This has been attributed to downregulation of either $\\\\mathrm{GABA_B}$ receptors and GIRK channels or both (Arora et al., 2011; Padgett et al., 2012; Hearing et al., 2013; Munoz et al., 2016; Lu et al., 2023). Aberrant downregulation of $\\\\mathrm{GABA_B}$ receptors is a response to sustained activity of glutamate receptors, which dysregulates $\\\\mathrm{GABA_B}$ receptor trafficking from the constitutive recycling pathway to lysosomal degradation (Guetg et al., 2010; Maier et al., 2010; Terunuma et al., 2010). Key steps in this aberrant regulation are the phosphorylation of Ser867 in the $\\\\mathrm{GABA_{B1}}$ subunit mediated by $\\\\mathrm{Ca^{2+}}$ -calmodulin dependent protein kinase IIβ (CaMKIIβ) (Guetg et al., 2010; Zemoura et al., 2019) and dephosphorylation of Ser783 in $\\\\mathrm{GABA_{B2}}$ by protein phosphatase 2A (PP2A) (Terunuma et al., 2010; Hleihil et al., 2022). Downregulation of cell surface and total \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,38,178,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hleihil and Benke \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[752,38,917,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10.3389/fnmol.2024.1347228 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[78,948,280,959]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Molecular Neuroscience \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[489,948,507,958]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"02 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[835,948,915,959]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"1\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[77,87,487,187]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\mathrm{GABA_B}$ receptors in the nucleus accumbens after repeated cocaine administration involved the activation of CaMKII (Lu et al., 2023), whereas selective loss of cell surface receptors was observed in the mPFC and VTA after cocaine and methamphetamine exposure, respectively, due to PP2A-mediated dephosphorylation of Ser783 in $\\\\mathrm{GABA_{B2}}$ (Padgett et al., 2012; Hearing et al., 2013). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,187,489,401]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Recently, we developed an interfering peptide (PP2A-Pep) that inhibits the interaction of $\\\\mathrm{GABA_B}$ receptors with PP2A (Hleihil et al., 2022). This peptide restored downregulated $\\\\mathrm{GABA_B}$ receptor expression after an ischemic insult, normalized neuronal overexcitation and inhibited progressive neuronal death. Since $\\\\mathrm{GABA_B}$ receptor expression and function appears to be reduced in the VTA after METH application, we tested in the present study the hypothesis whether restoring $\\\\mathrm{GABA_B}$ receptor expression in the VTA of METH-addicted mice, by blocking the interaction of $\\\\mathrm{GABA_B}$ receptors with PP2A, would reduce phenotypes associated with addiction. For this we infused PP2A-Pep into the VTA of METH-addicted mice and tested for $\\\\mathrm{GABA_B}$ receptor expression using immunohistochemistry, locomotor sensitization in the open field test (OFT) and drug seeking behavior in the conditioned place preference test (CPP). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,435,321,451]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Materials and methods \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,468,157,483]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Animals \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,499,489,612]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Experiments were performed using 8-10 weeks old adult male and female C57BL/6 mice. Mice were either purchased from Envigo (Netherlands) or bred at the Laboratory Animal Service Center of the University of Zurich (Switzerland). The mice were group-housed up to five in filter-top cages with a standard $12/12$ -h light/dark cycle and food and water available ad libitum. All experiments were approved by the veterinary office of the Canton of Zurich (license ZH164/2020). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,645,369,662]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Interfering peptide (PP2A-Pep) \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,677,489,917]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The interfering peptide (PP2A-Pep, YTIWMPENPRPGTP CDIFTNSRGKRASNGGGGRRRRRRRRFQFTQNQKKEDSKTS TSV) used in this study inhibits the interaction of PP2A with $\\\\mathrm{GABA}_B$ receptors (Hleihil et al., 2022). It contains a sequence derived from the intracellularly located C-terminal domain of $\\\\mathrm{GABA}_{B2}$ (red characters) and was rendered cell permeable by tagging it at the N-terminus with a peptide sequence derived from the Rabies virus glycoprotein followed by 9 arginine residues (gray characters). An inactive control peptide (Ctrl-Pep) contained the same amino acids but in a random sequence (except for the cell penetrating peptide sequence, YTIWMPENPRPGTPCDIFTNSRGKRASNGGGGRRRRRRRRQ KFSVNTFQEKDTKSQTS). The peptides were additionally tagged with biotin to permit their detection by immunohistochemical staining using AlexaFluor 488-conjugated streptavidin. The peptides were custom-synthesized by Pepmic Co., Ltd., Suzhou, China. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,87,880,104]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cannulation and peptide administration \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,118,917,417]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Cannula implantations were conducted using a stereotaxic frame (David Kopf Instruments and NeuroStar) under general anesthesia using continuous isoflurane inhalation via a nose mask (induction: $5\\\\%$ at $1\\\\mathrm{l / min}$ , surgery: $1.5\\\\%$ in a mixture of $\\\\mathrm{O}_2$ and air at $0.4 - 0.6\\\\mathrm{l / min}$ ). The body temperature of the mice was kept at $36.5\\\\pm 0.5^{\\\\circ}\\\\mathrm{C}$ using a feed-back controlled heating mat with a rectal temperature probe. Eyes were protected from drying using vitamin A ointment. For analgesia, mice received a subcutaneous injection of buprenorphine $(0.1\\\\mathrm{mg / kg})$ 20-30 min before surgery and an additional subcutaneous injection of lidocaine $(2\\\\mathrm{mg / kg})$ /bupivacaine $(1\\\\mathrm{mg / kg})$ at the site of incision about $5\\\\mathrm{min}$ before surgery. Before exposing the skull, the incision site was cleaned with $70\\\\%$ ethanol. A unilateral cannula implantation was performed using a 26G guide cannula (#C315GS-5/SPC, Bilaney Consultants, Germany). A hole was drilled through the skull above the VTA (anterior-posterior $-3.3$ ; medial-lateral $-0.9$ , relative to bregma). The guide cannula was placed through the hole above the VTA with a $10^{\\\\circ}$ angle (dorsal-ventral $-4.25$ ). The cannula was held in place using dental cement. All mice received post-surgical analgesia for additional 2 days $(10\\\\mathrm{mg / kg}$ carprofen, s. c. twice per day). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,417,917,531]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Peptides (0.8 μl PP2A-Pep, 100 μg/ml or 0.8 μl Ctrl-Pep, 100 μg/ml) were administered into the VTA with a rate of 100 nl/min using an automatic pump (PHD ULTRA Nanomite, Harvard Apparatus) connected by PE tubing which was attached to an injection cannula (#C315DCS-4/SPC, Bilaney Consultants, Germany) with 0.25 mm projection from the tip of the guiding cannula. Following injections, the cannula was left in place for additional 5 min to allow peptide infusion. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,560,658,576]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Behavioral tests \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,590,917,719]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Seven days prior to any experiment, the mice were habituated to the experimenter by gentle handling in the housing facility. At least $1\\\\mathrm{h}$ prior to the experiment, mice were transferred to the experimental room (50 lux dim light). All tests were performed in the light cycle. The respective experimental apparatus was cleaned with $70\\\\%$ ethanol before testing a mouse to avoid any sensory stimulation related to the previous mouse tested. In this study we used non-contingent drug administration as it ensures the expression of robust addictive phenotypes. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,733,680,747]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Open field test (OFT) \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,749,917,891]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The OFT was used to test for METH-induced locomotor sensitization (Good and Radcliffe, 2011; Jing et al., 2014). The locomotor activity was monitored in the open-field test chamber $(32\\\\mathrm{cm}\\\\times 32\\\\mathrm{cm}\\\\times 23\\\\mathrm{cm})$ with an infrared detection system under dim light conditions (50 lux). Mice were transferred from the housing facility to the experimental room $1\\\\textrm{h}$ prior to the experiment. All mice received i.p. injection (saline or METH) and immediately placed into the open field chamber $(32\\\\mathrm{cm}\\\\times 32\\\\mathrm{cm}\\\\times 23\\\\mathrm{cm})$ , where their activity was recorded for $30\\\\mathrm{min}$ . Then the mice were returned to their home cages. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,892,917,919]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"To induce methamphetamine (METH) addiction, mice were injected alternatively with either saline or methamphetamine \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,38,178,48]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hleihil and Benke \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[754,38,917,48]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10.3389/fnmol.2024.1347228 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[80,948,280,958]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Molecular Neuroscience \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[490,948,507,957]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"03 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[835,948,915,959]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"2\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[80,87,487,340]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"(METH, $2\\\\mathrm{mg / kg}$ , i. p., dissolved in saline) on every other day for 10 days (METH injections on day 1, 3, 5, 7 and 9; saline injections on day 2, 4, 6, 8, 10). Even a single METH injection has been shown to induce robust and long-lasting locomotor sensitization (Jing et al., 2014). We verified the development of locomotor sensitization in a subgroup of mice by daily open field tests. The control group received solely saline injections for 10 days. One day after the last saline injection (day 11), mice were microinfused with the peptides dissolved in saline $(0.8\\\\mu l$ PP2A-Pep, $100~\\\\mu \\\\mathrm{g / ml}$ or $0.8\\\\mu l$ Ctrl-Pep, $100~\\\\mu \\\\mathrm{g / ml})$ via the implanted cannula and then placed back to their home cage. The next day (day 12), the mice were injected with a challenging dose of METH $(1\\\\mathrm{mg / kg},$ i. p., dissolved in saline) and immediately placed into the open-field test chamber to allow free exploration for $30\\\\mathrm{min}$ . The mice were monitored by a digital camera placed above the center of the arena. From each session, $10\\\\mathrm{min}$ of the recordings were extracted for analysis with the EthoVision XT 15 software (Noldus, Wageningen, Netherlands). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,354,374,368]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conditioned place preference (CPP) \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,370,487,508]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"CCP was used to test drug seeking behavior (Cunningham et al., 2006). CPP was conducted in a two-compartment plexiglass apparatus (size of each compartment: $20\\\\mathrm{cm} \\\\times 20\\\\mathrm{cm} \\\\times 15\\\\mathrm{cm}$ ), in which the two conditioning compartments were connected through a short tunnel ( $10\\\\mathrm{cm}$ ) (Figure 5). The two conditioning chambers were visually and tactile distinct. Chamber 1 consisted of white walls with black strips and a plain white floor with gray cross lines. Chamber 2 consisted of black walls and a rough textured plexiglass floor. The behavior of the mice was recorded by a digital camera placed above the CCP device. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,511,487,651]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The CPP procedure consisted of three phases (Figure 5): habituation and baseline preference (day 1), conditioning phase (days 2-6) and preference tests (days 7 and day 8). In the habituation/preference evaluation (day 1), mice were placed in the tunnel connecting the two chambers and allowed to freely explore all compartments for $30\\\\mathrm{min}$ . This allowed to assign a preferred compartment for each mouse to conduct a biased CPP experiment: the mice received saline injections in their initially preferred chamber (saline-paired chamber) and METH injections in the non-preferred compartment (METH-paired chamber). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,652,487,777]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"During the conditioning phase (day 2-6), access was restricted to the saline-paired or METH-paired compartment. In the METH group, mice were injected with METH $(2\\\\mathrm{mg / kg}$ i.p., dissolved in saline) and immediately placed into the METH-paired chamber for $30\\\\mathrm{min}$ and then returned to the home cage. Six hours later, the mice were injected with saline and placed into the saline-paired chamber for $30\\\\mathrm{min}$ . For the control saline group, the same procedure was conducted but mice were injected with two saline doses. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,778,487,917]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On the test day (day 7), the preference test was performed $24\\\\mathrm{h}$ after the last conditioning trial by placing the mouse in the tunnel connecting the two compartments and allowing to freely explore all chambers for $10\\\\mathrm{min}$ . One hour after the test, mice were microinfused with the peptides dissolved in saline via the implanted cannula $(0.8\\\\mu l$ PP2A-Pep, $100~\\\\mu \\\\mathrm{g / ml}$ or $0.8\\\\mu l$ Ctrl-Pep, $100~\\\\mu \\\\mathrm{g / ml})$ and then placed back into their home cage. The timeline for peptide treatment was selected according to Li J. Y. et al. (2023), which predominantly affects memory reconsolidation. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,87,915,173]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"On day 8, the preference test was conducted again for $10\\\\mathrm{min}$ to test for the peptide effect. The recordings were analyzed with the EthoVision XT 15 software (Noldus, Wageningen, Netherlands) to extract the CPP data. The CPP preference score was calculated as a difference between the relative amount of time spent in the saline-paired chamber to the time spent in the METH-paired chamber. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,200,610,217]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Antibodies \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,233,915,303]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Primary antibodies: rabbit anti-GABAB2N [1:250, custom-made by GeneScript, (Benke et al., 2002)], rabbit anti-GFAP (1:500, Agilent Technologies # Z0334), guinea pig anti-NeuN (1:500, Synaptic systems # 266004), rabbit anti-Iba1 (1:500, WAKO Chemicals # 019-19741). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,304,915,374]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Secondary antibodies: donkey anti-rabbit AlexaFluor Plus 555 (1:2000, Thermo Fisher Scientific # A32794), goat anti-guinea pig AlexaFluor 647 (1:1000, Jackson ImmunoResearch # 706-605-148), and AlexaFluor 488-conjugated streptavidin (1:500, Jackson ImmunoResearch # 016-540-084). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,403,727,437]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Tissue preparation and immunohistochemistry \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,451,915,564]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"A loss of $\\\\mathrm{GABA_B}$ receptor immunofluorescence staining under ischemic conditions always correlated with a similar reduction of receptor protein signals in western blots and with diminished $\\\\mathrm{GABA_B}$ receptor mediated inhibition (see e.g., Zemoura et al., 2019; Hleihil et al., 2021, 2022; Bhat et al., 2022; Balakrishnan et al., 2023). Therefore, we assume that a reduction in $\\\\mathrm{GABA_B}$ receptor immunofluorescence staining in this study indicates diminished $\\\\mathrm{GABA_B}$ receptor mediated inhibition. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,565,915,678]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Immediately after the last behavioral test, mice were transcranially perfused with ice-cold oxygenated ACSF (125 mM NaCl, 2.5 mM KCl, 1.25 mM $\\\\mathrm{NaH_2PO_4}$ , 25 mM $\\\\mathrm{NaHCO_3}$ , 1 mM $\\\\mathrm{MgCl_2}$ , 2 mM $\\\\mathrm{CaCl_2}$ , and 25 mM glucose, pH 7.4, pH 7.4), for 2 min. Brains were then immediately dissected and cut in the medial midline. One of the hemispheres were then fixed in $4\\\\%$ PFA overnight, followed by incubation in $30\\\\%$ sucrose/PBS for $24\\\\mathrm{h}$ at $4^{\\\\circ}\\\\mathrm{C}$ for immunohistochemical staining. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[509,678,915,917]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For immunohistochemistry, free floating $40~\\\\mu \\\\mathrm{m}$ thick coronal sections were prepared using a sliding microtome (HM400; Microm) and stored in antifreeze solution (50 mM Na-phosphate buffer, pH 7.4, 1.6 M Glucose, 20 mM ethylene glycol, 6 mM sodium azide) at $-20^{\\\\circ}\\\\mathrm{C}$ for further analysis. After rinsing once in PBS, sections were incubated overnight at $4^{\\\\circ}\\\\mathrm{C}$ with primary antibody (or AlexaFluor 488-conjugated streptavidin for peptide detection) prepared in Tris-Triton solution [50 mM Tris pH 7.4, 150 mM NaCl containing $0.2\\\\%$ Triton X-100 and $5\\\\%$ normal donkey serum (NDS)]. After 3 washes for 10 min with Tris-Triton without NDS, sections were incubated with secondary antibody diluted in Tris-Triton containing $2\\\\%$ NDS for $1\\\\textrm{h}$ at room temperature. Following 3 washes for 10 min with Tris-Triton without NDS, brains were mounted onto glass slides (SuperFrost Plus, Thermo Fisher Scientific), dried and coversliped with DAKO fluorescence mounting medium supplemented with DAPI (Fluoroshield with DAPI, #F6057, Merk). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,38,176,48]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hleihil and Benke \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[754,38,915,48]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10.3389/fnmol.2024.1347228 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[80,948,278,958]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Molecular Neuroscience \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[489,948,505,957]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"04 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[835,948,915,958]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"3\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[77,87,490,244]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Images were acquired using a LSM 800 confocal microscope (Carl Zeiss). For imaging of the brain areas, a stacks of 10 optical sections at $1\\\\mu \\\\mathrm{m}$ z-spacing were taken using a plan-apochromat $10\\\\mathrm{x} / 0.45$ objective in sequential mode with a resolution of $1,024\\\\times 1,024$ pixels. The laser intensity and the detector gain were adjusted to values that avoid signal saturation and all images of one experiment was imaged with the same settings in one continuous session. For images on the cellular level, 5 optical sections with $0.7\\\\mu \\\\mathrm{m}$ z-spacing were taken using a $40\\\\mathrm{x} / 1.4$ planapochromat oil differential interference contrast objective with $0.5\\\\mathrm{x}$ digital zoom. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,244,489,330]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For quantification of $\\\\mathrm{GABA}_B$ receptor expression, all images belonging to an experiment were recorded with identical settings. Analysis was performed after merging the optical planes into a single image and an area of interest was defined. The mean fluorescence values were determined using ImageJ (Schindelin et al., 2012). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,331,489,374]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The original images recorded were too dim to be shown directly. Therefore, we enhanced the brightness of the images to make the staining more easily visible to the reader. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,406,250,422]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Statistical analysis \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,437,490,594]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The data is shown as means $\\\\pm$ SD. Immunohistochemical data was analyzed by two-way analysis of variance with type III sum of square calculations for an unbalanced data design (ANOVA, Figures 2, 3) using GraphPad Prism 8.4.3, followed by Tukey's multiple comparison test as indicated in the respective figure legends. Behavioral experiments were analyzed by two-way AVOVA with type III sum of square calculations for an unbalanced data design (Figures 4, 5D) and two-tailed unpaired $t$ -test (Figure 5C). All data sets passed tests for normal distribution and homoscedasticity. Differences were considered statistically significant when $p < 0.05$ . \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,627,159,643]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Results \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,659,426,678]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Neuron-specific uptake of PP2A-Pep \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,690,490,919]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"For testing the effect of PP2A-Pep on METH-addicted mice, the peptide was unilaterally infused into the VTA via an implanted canula (Figure 1A). Unilateral infusion had been shown to cover both sides of the VTA (Harada et al., 2021). To largely restrict uptake of PP2A-Pep (as well as the control peptide Ctrl-Pep) into neurons, it contained at the N-terminus a cell-penetrating peptide sequence derived from the Rabies virus glycoprotein (Kumar et al., 2007). We previously showed that peptides containing this sequence are selectively taken up by neurons via a receptor-mediated mechanism (Balakrishnan et al., 2023). PP2A-Pep was indeed taken up by neurons after infusion into the VTA as shown by co-immunofluorescence staining for the peptide, neurons (NeuN antibody) and nuclei (DAPI) (Figures 1B, C). The peptide was not taken up by glia cells as shown by co-staining for astrocytes with GFAP antibodies (Figure 1B) and microglia by co-staining with Iba1 antibodies (Figure 1C). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[546,117,875,266]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"A \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0769/8a4a6411f64833ff6d820b3f8fab1018340221aeb9bcb2c3aacb7434e0461078.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[546,279,653,356]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"B \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0098/f63cd539cea40f138efcdd173b27950429479dd27b1691d5c3806149faa39cbb.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[547,357,653,434]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0448/50c4cf1d1c2c65f2bd50a2531a0e097ec7f0cf36307eabedbf7717ef45f9a35a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[655,279,764,356]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0465/564c482608f863ff2c83305653b0f8c68d29626e779c5500965ee74080f0d423.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[657,357,764,434]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0711/ee62227d9609c0eced8d8c3b420eca09dd4cec345436ab9279dccb9207b43302.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[764,279,873,356]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0282/10d65e4246d9b71cfd049a8bbd180dbce782144f0bc0d4904e2be0392cd59d5a.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[766,357,873,434]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0694/74adc6aa06f3613d8a9875331e924a401f5b9594f9fd3d0cf0f60904c08a67d8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[546,451,653,527]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"C \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0494/ee62f37a3ffed7f9db4b164295f84e4e76f15b2b2f72395c3653a85ee2259ea7.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[547,529,653,605]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0038/4374b70f63c65a2bda02d3c6aa9dd8dbe92bb504a9916cd96dd8b9d8de94f567.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[655,451,764,527]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0719/d51dc9bf9da427ca0c9bb37be46978a9767e6b08d980d47654363c2f29f28a75.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[655,529,764,605]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0684/c7563a18ce8c8013d343dc1abe472a4c6a8f9f7df6622b4fe2631285ef1447cb.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[764,451,873,527]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0147/384969157f7d8112b0ab893aeb14e4658f2e397cc150b600a292cf9a29bc20db.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[766,529,873,605]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIGURE 1 \\\"]\",\n \"image_footnote\": \"[\\\"Specific uptake of PP2A-Pep into VTA neurons. (A) Infusion site and spread of PP2A-Pep. Right: Brain section stained for the infused PP2A-Pep using AlexaFluor 488-conjugated streptavidin (green) and DAPI (blue). The placement of the canula is indicated (white lines). Left: Location of section in the mouse brain atlas of Paxinos and Franklin (2001). Scale bar: $500\\\\\\\\mu m$ . (B) PP2A-Pep was selectively taken up by neurons but not by astrocytes. Brain sections containing the VTA were stained for the infused PP2A-Pep (green), Neurons (NeuN, red), astrocytes (GFAP, magenta) and nuclei (DAPI, blue). Scale bar: $20\\\\\\\\mu m$ . (C) PP2A-Pep was selectively taken up by neurons but not by microglia. Brain sections containing the VTA were stained for the infused PP2A-Pep (green), Neurons (NeuN, red), microglia (Iba1, magenta) and nuclei (DAPI, blue). Scale bar: $20\\\\\\\\mu m$ \\\"]\",\n \"img_path\": \"images/0757/e315d040a3e05407dd289f1d37fb9ebfccd727d03172e48afb9899c719286247.jpg\",\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[505,812,890,862]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"$\\\\mathrm{GABA}_B$ receptor expression was reduced in the VTA of METH-addicted mice but normalized after PP2A-Pep infusion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,877,917,921]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"First, we analyzed whether $\\\\mathrm{GABA}_B$ receptor expression in the VTA was reduced in METH-addicted mice. For this, we performed immunofluorescence staining for $\\\\mathrm{GABA}_B$ receptors and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,38,179,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hleihil and Benke \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[752,38,917,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"4\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10.3389/fnmol.2024.1347228 \",\n \"text_format\": null,\n \"text_level\": null,\n 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\"images/0547/2b9b9ea4f9fef98a3f0477e130de2aca82d62cd1a2894c03cf01df4309dfdd53.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[218,552,571,694]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"D \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0849/43d557c97f8bab274a0661d5a842bd577ea1ccd49638008422075d57882bf6ca.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[576,573,789,695]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIGURE 2 \\\"]\",\n \"image_footnote\": \"[\\\"Infusion of PP2A-Pep into the VTA of METH-addicted mice normalized $\\\\\\\\mathrm{GABA_B}$ receptor expression. Mice received multiple METH/saline (METH-addicted mice) or saline/saline (control mice) injections and were then either infused into the VTA with PP2A-Pep or the inactive Ctrl-Pep. About $24\\\\\\\\mathrm{h}$ after peptide application the mice were analyzed for $\\\\\\\\mathrm{GABA_B}$ receptor expression using $\\\\\\\\mathrm{GABA_{B2}}$ antibodies in the VTA (A), nucleus accumbens (B), hippocampus (C), and motor cortex (D) on a regional level. (A) $\\\\\\\\mathrm{GABA_B}$ receptor expression was reduced in the VTA of METH-addicted mice and restored after infusion of PP2A-Pep. Left: Representative images display peptide uptake (green) and $\\\\\\\\mathrm{GABA_B}$ receptor expression (red) in the VTA (scale bar: $100~\\\\\\\\mu \\\\\\\\mathrm{m}$ ). Right: Quantification of fluorescence intensities in the area stained for the peptide (the data was normalized to the mean immunofluorescence value of Ctrl-Pep infused into the VTA of saline treated control mice). The data represents the mean $\\\\\\\\pm$ SD of 9-10 mice per condition. Statistical analysis: Two-way ANOVA with Tukey's multiple comparison test (ns, $p > 0.05$ ). (B) $\\\\\\\\mathrm{GABA_B}$ expression was reduced in the nucleus accumbens of METH-addicted mice but unaffected by PP2A-Pep infusion into the VTA. Left: Representative images (upper row: low magnification images, scale bar $200~\\\\\\\\mu \\\\\\\\mathrm{m}$ ; lower row, high magnification images, scale bar $20~\\\\\\\\mu \\\\\\\\mathrm{m}$ ). Right: Quantification of fluorescence intensities (the data was normalized to the mean immunofluorescence value of control mice). The data represents the mean $\\\\\\\\pm$ SD of 8-10 mice per condition. Statistical analysis: Two-way ANOVA with Tukey's multiple comparison test (ns, $p > 0.05$ ). (C) $\\\\\\\\mathrm{GABA_B}$ expression remained unaffected in the hippocampus of METH-addicted mice. Left: Representative images (upper row: low magnification images, scale bar $100~\\\\\\\\mu \\\\\\\\mathrm{m}$ ; lower row, high magnification images, scale bar $20~\\\\\\\\mu \\\\\\\\mathrm{m}$ ). Right: Quantification of fluorescence intensities (the data was normalized to the mean immunofluorescence value of control mice). The data represents the mean $\\\\\\\\pm$ SD of 5-7 mice per condition. Statistical analysis: Two-way ANOVA with Tukey's multiple comparison test (ns, $p > 0.05$ ). (D) $\\\\\\\\mathrm{GABA_B}$ expression in the motor cortex remained unaffected in METH-addicted mice. Left: Representative images (upper row: low magnification images, scale bar $200~\\\\\\\\mu \\\\\\\\mathrm{m}$ ; lower row, high magnification images, scale bar $20~\\\\\\\\mu \\\\\\\\mathrm{m}$ ). Right: Quantification of fluorescence intensities (the data was normalized to the mean immunofluorescence of control mice). The data represents the mean $\\\\\\\\pm$ SD of 7-8 mice per condition. Statistical analysis: Two-way ANOVA with Tukey's multiple comparison test (ns, $p > 0.05$ ). \\\"]\",\n \"img_path\": \"images/0123/d2535f8f4c9a81d71a0f47144363ca520cb04cabf54d0231a612ae285ed4c5f3.jpg\",\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[80,38,179,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hleihil and Benke \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[752,38,915,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10.3389/fnmol.2024.1347228 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[80,948,280,958]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Molecular Neuroscience \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[489,948,507,958]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"06 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[835,948,915,959]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"5\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[119,102,606,281]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0006/c2ee56d217293bfe66fc138b7b89b5f81f4a6af9ad102bfaa0783e21bc373f7d.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[611,112,873,282]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIGURE 3 \\\"]\",\n \"image_footnote\": \"[\\\"PP2A-Pep normalized $\\\\\\\\mathrm{GABA_B}$ receptor expression in the somata of VTA neurons of METH-addicted mice. METH-addicted and control mice were infused into the VTA either with PP2A-Pep or the inactive Ctrl-Pep. About $24\\\\\\\\mathrm{h}$ after peptide application, the mice were analyzed for $\\\\\\\\mathrm{GABA_B}$ receptor expression in the soma of individual neurons that had taken up the peptide using $\\\\\\\\mathrm{GABA_{B2}}$ antibodies. Left: Representative images displaying peptide uptake (green) and $\\\\\\\\mathrm{GABA_B}$ receptor expression (orange) in VTA neurons (scale bar: $20~\\\\\\\\mu \\\\\\\\mathrm{m}$ ). Right: Quantification of fluorescence intensities (the data was normalized to the mean immunofluorescence value of Ctrl-Pep infused into the VTA of saline treated control mice). The data represents the mean $\\\\\\\\pm$ SD of 15 neurons per mouse derived from 6-8 mice per condition. The symbols in the graph display the mean values of all neurons analyzed in a single mouse. Statistical analysis: Two-way ANOVA with Tukey's multiple comparison test (ns, $p > 0.05$ ). \\\"]\",\n \"img_path\": \"images/0030/32f06784f4158fb54f3342b06408b27355364f14c582d6bc05e31c8d66ed8854.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[186,432,478,472]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"A OFT procedure \\\",\\\"C \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0823/e41cc29b183deca5394cc96f9602a3ffdf0c3ec61c0dc178fe93fade942452d0.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[512,451,643,542]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Saline-treated mice +Ctrl-Pep \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0063/ca2e86567f00e0e6f98d91b3cf88afa7de9c6353089772dd376b6d9be881676b.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[652,451,813,542]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"Saline treated mice +PP2A-Pep \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0691/6796dd4734876dcc1c1d6c35287ec020e1f6a34fbb2a19c24a6dec53f7251059.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[176,498,478,678]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"B \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0364/3aff6ab6256ec1f7e7602616f589686b067c461775b6ac1e1727ee6953a246a8.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[510,575,640,665]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"METH-treated mice +Ctrl-Pep \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/0075/d6b085d10f8f5d8379383724bd7862e8c2be4d810714399e50f4e5be532fe275.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[652,575,813,665]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"METH-treated mice +PP2A-Pep \\\"]\",\n \"image_footnote\": \"[\\\"FIGURE 4 PP2A-Pep infusion into the VTA reduced METH-induced locomotor sensitization in the OFT. (A) Schematic outline of the OFT procedure. Mice received alternating METH and saline injections for 10 days to induce METH addiction. The control group received solely saline injections. One day after the last saline injection (day 11), the interfering peptide (PP2A-Pep) or the control peptide (Ctrl-Pep) was infused into the VTA via the implanted cannula. The next day, the mice were injected with a challenge dose of METH ( $1\\\\\\\\mathrm{mg / kg}$ i.p.) and immediately placed into the open-field test chamber to allow free exploration for $30\\\\\\\\mathrm{min}$ . (B) METH-addicted mice exhibited increased locomotor activity, which was reduced to normal levels after PP2A-Pep injection into the VTA. Application of the control peptide (Ctrl-Pep) had no effect. The data represents the mean $\\\\\\\\pm$ SD of 7-11 mice per condition. Statistical analysis: Two-way ANOVA with Tukey's multiple comparison test (ns, $p > 0.05$ ). (C) Heat maps depicting the frequencies of locations visited by mice in the open field arena under the different experimental conditions (representative images). \\\"]\",\n \"img_path\": \"images/0452/e81dc07ee7219726401d625c584ad8f63a4c5be4c89016b167aae70811241b35.jpg\",\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[77,816,490,919]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"the inactive control peptide (Ctrl-Pep) infused into the VTA of saline-injected control mice and METH-addicted mice. The area of the VTA containing neurons that displayed staining for the peptide was then analyzed for $\\\\mathrm{GABA}_B$ receptor expression using antibodies directed against the $\\\\mathrm{GABA}_{B2}$ subunit. As expected, we detected a considerable reduction of $\\\\mathrm{GABA}_B$ receptor expression in the VTA in METH-addicted mice (Figure 2A). We then tested \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,816,919,919]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"whether inhibiting the interaction of $\\\\mathrm{GABA_B}$ receptors with PP2A would restore $\\\\mathrm{GABA_B}$ receptor expression. Indeed, infusion of PP2A-Pep into the VTA of METH-addicted mice normalized $\\\\mathrm{GABA_B}$ receptor expression as tested about $24\\\\mathrm{h}$ after peptide application (Figure 2A). Infusion of PP2A-Pep into the VTA of non-addicted control mice had no effect on the expression of $\\\\mathrm{GABA_B}$ receptors. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,38,179,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hleihil and Benke \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[752,38,917,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10.3389/fnmol.2024.1347228 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[78,948,280,958]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Molecular Neuroscience \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[490,948,507,958]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"07 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[835,948,915,959]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"6\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[77,87,487,187]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"One of the main projection areas of the VTA involved in addictive phenotypes is the nucleus accumbens. Like the VTA, $\\\\mathrm{GABA_B}$ receptor expression was significantly reduced in the nucleus accumbens but remained unaffected by infusion of PP2A-Pep into the VTA (Figure 2B). By contrast, $\\\\mathrm{GABA_B}$ receptor expression in the hippocampus (Figure 2C) and motor cortex (Figure 2D) was not significantly affected in METH-addicted mice. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,187,487,300]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Next, we analyzed $\\\\mathrm{GABA_B}$ receptor expression in the VTA on the cellular level. For this, we quantified immunofluorescence intensities for $\\\\mathrm{GABA_{B2}}$ in the somata of peptide positive VTA neurons. As observed on the more global, VTA wide level, $\\\\mathrm{GABA_B}$ receptor expression in the soma of individual VTA neurons was downregulated in METH-addicted mice (Figure 3). Infusion of PP2A-Pep into the VTA normalized $\\\\mathrm{GABA_B}$ receptor expression in the somata of neurons in METH-addicted mice (Figure 3). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,300,487,370]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Taken together, these results indicate that $\\\\mathrm{GABA_B}$ receptors were downregulated in the VTA and nucleus accumbens of METH-addicted mice. Infusion of PP2A-Pep into the VTA restored receptor expression to normal levels in the VTA but did not affect receptor expression in the nucleus accumbens. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,398,458,430]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PP2A-Pep infusion into the VTA reduced METH-induced locomotor sensitization \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,445,487,629]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Locomotor sensitization is a hallmark of psychostimulant-induced addiction (Tirelli et al., 2003). To test whether restoring $\\\\mathrm{GABA}_B$ receptor expression in the VTA affects locomotor sensitization, mice were injected with METH and saline for ten days followed by infusion of PP2A-Pep or Ctrl-Pep into the VTA. Locomotor sensitization was then tested the next day in the open field arena (Figure 4A). METH-addicted mice showed increased locomotor activity, which was reduced to levels of non-addicted, saline injected, control mice that were infused with the non-active control peptide (Ctrl-Pep, Figures 4B, C). Infusion of PP2A-Pep into the VTA of non-addicted control mice did not affect locomotor activity (Figures 4B, C). Thus, PP2A-Pep effectively inhibited increased locomotor in METH-addicted mice. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,657,411,707]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"PP2A-Pep infusion into the VTA of METH-addicted mice reduced drug seeking behavior \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,721,487,917]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The effect of PP2A-Pep on drug seeking behavior of METH-addicted mice was evaluated in the CPP test (Bardo and Bevins, 2000; McKendrick and Graziane, 2020). For the CPP test, we used a biased design in which the preference of the mice for one of the two compartments was determined before starting the conditioning phase (a schematic outline of the device and the experimental procedure is shown in Figures 5A, B). During the conditioning phase, the METH group received METH injections in the initially non-preferred compartment, followed by saline injections $6\\\\mathrm{h}$ later in the opposite compartment. Mice in the saline control group received injections in the preferred compartment, followed by a second saline injection $6\\\\mathrm{h}$ later in the opposite compartment. One day after the conditioning phase (day 7), mice were subjected to the CPP test and $1\\\\mathrm{h}$ thereafter the mice were microinfused \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,87,917,258]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"either with PP2A-Pep or Ctrl-Pep via the implanted cannula and returned to the home cage. A second preference test was performed the next day. As expected, METH-conditioned mice spent more time in the METH-paired compartment, indicating drug seeking behavior (Figure 5C). Infusion of PP2A-Pep into the VTA of METH-conditioned mice reduced the drug seeking behavior as the mice explored both compartments almost equally. Infusion of the inactive control peptide into the VTA of METH-conditioned mice was without effect as was infusion of PP2A-Pep into the VTA of saline control mice. These results indicate that a single administration of PP2A-Pep into the VTA of METH-addicted mice diminishes drug seeking behavior. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,290,623,306]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Discussion \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,322,917,406]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"This study indicates that repeated METH treatment in mice downregulates $\\\\mathrm{GABA_B}$ receptor expression in the VTA. Infusion of an interfering peptide that inhibits the interaction of $\\\\mathrm{GABA_B}$ receptors with PP2A into the VTA of METH-treated mice restored the expression of $\\\\mathrm{GABA_B}$ receptors. This diminished METH-induced locomotor sensitization and drug seeking behavior. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,407,917,705]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Excessive neuronal excitation triggers the downregulation of $\\\\mathrm{GABA_B}$ receptors by increasing their lysosomal degradation on the expense of recycling the receptors back to the plasma membrane (Guetg et al., 2010; Maier et al., 2010; Terunuma et al., 2010). Two key steps are involved in this pathway: CaMKII $\\\\beta$ -mediated phosphorylation of the $\\\\mathrm{GABA_B1}$ subunit at Ser867 (Guetg et al., 2010; Zemoura et al., 2019; Balakrishnan et al., 2023) and PP2A-dependent dephosphorylation of the $\\\\mathrm{GABA_B2}$ subunit at Ser783 (Terunuma et al., 2010; Hleihil et al., 2022). Elevated intracellular $\\\\mathrm{Ca^{2+}}$ levels due to enhanced neuronal activity activates CaMKII $\\\\beta$ to induce phosphorylation of Ser867 in $\\\\mathrm{GABA_B1}$ (Maier et al., 2010). Recent evidence suggests that CaMKII $\\\\beta$ not directly phosphorylate $\\\\mathrm{GABA_B1}$ but activates ERK1/2 which in turn phosphorylate Ser867 and the adjacent Thr872 (Bhat et al., 2023). Phosphorylation of $\\\\mathrm{GABA_B1}$ is required to enable K63-linked ubiquitination of the receptor by the E3-ubiquitin ligase MIB-2, which serves as a signal for their lysosomal degradation (Zemoura et al., 2019). In addition to CaMKII $\\\\beta$ , activation of PP2A under these conditions dephosphorylates Ser783 in $\\\\mathrm{GABA_B2}$ . This inhibits fast recycling of the receptors and directs them into the degradation pathway controlled by CaMKII $\\\\beta$ (Terunuma et al., 2010; Hleihil et al., 2022). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,707,917,876]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In the present study we found that in most neurons of the VTA analyzed, $\\\\mathrm{GABA_B}$ receptors were downregulated after repetitive METH treatment in mice. Likewise, we found a similar reduction of $\\\\mathrm{GABA_B}$ receptors in the nucleus accumbent, a main projection area of the VTA neurons associated with drug addiction. This finding is well supported by a recent study indicating reduced $\\\\mathrm{GABA_B}$ receptor expression in the nucleus accumbens of cocaine-addicted rats involving CaMKII dependent phosphorylation of the receptors (Lu et al., 2023). By contrast, $\\\\mathrm{GABA_B}$ receptor expression in the hippocampus and motor cortex remained unaffected, reassuring that the changes in immunostaining we detected were not caused by factors unrelated to the treatment. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,877,917,919]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In all our previous studies, a loss of $\\\\mathrm{GABA_B}$ receptor immunofluorescence staining always correlated with a similar reduction of receptor protein signals in western blots and \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,38,178,48]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hleihil and Benke \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[752,38,915,48]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10.3389/fnmo1.2024.1347228 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[78,948,280,958]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Molecular Neuroscience \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[489,948,505,957]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"08 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[835,948,915,959]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"7\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"frontiersin.org \",\n \"text_format\": 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null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[127,356,438,491]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": \"[\\\"FIGURE 5 Injection of PP2A-Pep into the VTA of METH-addicted mice reduced drug seeking behavior as tested by CPP. (A,B) Schematic outline of the CPP procedure (A) and the CCP device (B). CPP was conducted in a device consisting of two visually distinct plexiglass compartments connected through a short gray tunnel. The CPP procedure included three phases: habituation and baseline preference (day 1), conditioning phase (days 2-6) and preference tests (days 7 and day 8). During the conditioning phase (day 2-6), mice received METH injections $(2\\\\\\\\mathrm{mg / kg},$ i. p.) in the initially non-preferred compartment and access was restricted to this compartment. After $6\\\\\\\\mathrm{h}$ mice were injected with saline in their initially preferred compartment. The control group received the same procedure, but mice were injected with two saline doses. On the test day (day 7, 24 h after the last conditioning trial), the preference test was performed. One hour after the test, mice were microinfused either with PP2A-Pep or Ctrl-Pep via the implanted cannula and tested a second time the next day (day 8). (C) METH-conditioned mice showed increased preference for the METH-paired chamber on the first test day. The data represents the mean $\\\\\\\\pm \\\\\\\\mathrm{SD}$ of 18-20 mice per condition. Statistical analysis: Two-tailed unpaired t-test. (D) Infusion of PP2A-Pep into the VTA after the first test diminished the preference for the METH-paired chamber in METH-addicted mice on the second test day. Application of the control peptide (Ctrl-Pep) had no effect. The data represents the mean $\\\\\\\\pm \\\\\\\\mathrm{SD}$ of 8-10 mice per condition. Statistical analysis: Two-way ANOVA with Tukey's multiple comparison test (ns, $p > 0.05$ ). (E) Heat maps depicting the frequencies of locations visited in the CPP chambers by mice under the different experimental conditions (representative images). \\\"]\",\n \"image_footnote\": \"[]\",\n \"img_path\": \"images/1022/3daaa7a2b9d31c12d5df1eb735863f2d5b5a986b63b8f2c4ddfd5dfbc0ae5843.jpg\",\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"image\"\n },\n {\n \"bbox\": \"[78,862,489,919]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"with diminished $\\\\mathrm{GABA_B}$ receptor mediated inhibition (see e.g., Zemoura et al., 2019; Hleihil et al., 2021, 2022; Bhat et al., 2022; Balakrishnan et al., 2023). Therefore, it is reasonable to assume that the reduction in $\\\\mathrm{GABA_B}$ receptor immunofluorescence \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[504,87,917,693]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"staining observed in this study translates into diminished $\\\\mathrm{GABA_B}$ receptor mediated inhibition. The METH-induced downregulation of $\\\\mathrm{GABA_B}$ receptors we observed in VTA neurons is well supported by diminished $\\\\mathrm{GABA_B}$ receptor-mediated inhibition reported in the VTA (Arora et al., 2011; Padgett et al., 2012; Sharpe et al., 2015; Munoz et al., 2016). So far, two mechanisms have been reported to cause the reduction of $\\\\mathrm{GABA_B}$ receptor-mediated inhibition in the VTA. First, in GABAergic neurons, the receptors were downregulated from the cell surface involving dephosphorylation of $\\\\mathrm{GABA_{B2}}$ Ser-783 after a single injection of METH (Padgett et al., 2012). Second, in dopaminergic neurons, the postsynaptic $\\\\mathrm{GABA_B}$ receptor effectors, the GIRK channels, were downregulated instead of the receptors (Arora et al., 2011; Munoz et al., 2016). However, our data suggests a more widespread downregulation of $\\\\mathrm{GABA_B}$ receptors in VTA neurons than just their selective reduction in GABAergic neurons since only about $35\\\\%$ of neurons in the VTA are GABAergic ( $\\\\sim 60\\\\%$ are dopaminergic and $\\\\sim 5\\\\%$ glutamatergic) (Nair-Roberts et al., 2008; Ma et al., 2023). Further investigations are required to address the proportion of $\\\\mathrm{GABA_B}$ receptor downregulated in the different neuronal populations of the VTA. In addition, it would be interesting to analyze in detail whether solely cell surface $\\\\mathrm{GABA_B}$ receptors are reduced as previously reported after a single METH injection (Padgett et al., 2012) or more globally as we observed after repeated injections. However, infusion of PP2A-Pep into the VTA restored $\\\\mathrm{GABA_B}$ receptor expression in VTA neurons, supporting the view that the observed downregulation of $\\\\mathrm{GABA_B}$ receptors in METH-addicted mice is mediated by the PP2A/CaMKII pathway described above. The restoration of $\\\\mathrm{GABA_B}$ receptors in the VTA also reduced addictive behaviors in METH-addicted mice, suggesting that a globally restored inhibitory control by $\\\\mathrm{GABA_B}$ receptors in the VTA diminishes enhanced dopaminergic activity in the nucleus accumbens. Interestingly, restored $\\\\mathrm{GABA_B}$ receptor expression in the VTA did not normalize reduced $\\\\mathrm{GABA_B}$ receptor levels in the nucleus accumbens during the time frame of our analysis. Cocaine treatment most likely downregulates $\\\\mathrm{GABA_B}$ receptor expression in the nucleus accumbens by the same mechanism as METH treatment in the VTA (Lu et al., 2023). This suggests that simply reducing dopaminergic activity to the nucleus accumbens is not sufficient to normalize the aberrant trafficking of the receptors. It would be interesting to analyze the effect of PP2A-Pep on $\\\\mathrm{GABA_B}$ receptor expression in the nucleus accumbens and on addictive phenotypes of METH addicted mice. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,695,917,919]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Repeated psychostimulant administration in rodents leads to locomotor sensitization, a phenotype that is triggered by enhanced dopamine release within the VTA and nucleus accumbens. Reexposure to even a small quantity of the psychostimulants in the addicted subject, induces a strong activity in the VTA (Tirelli et al., 2003; Steketee and Kalivas, 2011). Accordingly, we observed a significant increase in locomotor activity upon injection of a challenge dose of the drug in METH-addicted mice. Interestingly, a single injection of PP2A-Pep into the VTA was sufficient to reverse the enhanced locomotor activity, most likely due to increased tonic inhibition mediated by the restored $\\\\mathrm{GABA_B}$ receptor expression in the VTA. This finding is in line with previous studies showing that the $\\\\mathrm{GABA_B}$ receptor agonist baclofen inhibits locomotor sensitization (Bartoletti et al., 2004; Leite-Morris et al., 2004; Cedillo and Miranda, 2013). Furthermore, selective deletion of $\\\\mathrm{GABA_B}$ receptors from VTA dopaminergic neurons enhanced the \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,38,178,48]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hleihil and Benke \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[754,38,915,48]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10.3389/fnmol.2024.1347228 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[80,948,280,958]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Molecular Neuroscience \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[490,948,505,957]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"09 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[836,948,915,959]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"8\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[77,87,487,130]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"locomotor sensitization to psychostimulants (Edwards et al., 2017), illustrating the importance of $\\\\mathrm{GABA_B}$ receptor mediated inhibition for controlling locomotor sensitization. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,130,489,411]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"It is believed that drug addiction is a form of Pavlovian conditioning, establishing long-term memories to stabilize drug associated behaviors. In the CPP experiment, associations are formed between a rewarding stimulus (e.g., METH) and a contextual environment (Tzschentke, 2007; McKendrick and Graziane, 2020). Re-exposure to drug-associated cues triggers memory recall, making these memories labile and sensitive to manipulations (Tronson and Taylor, 2007; Bellfy and Kwapis, 2020). In our CPP test, we aimed to interfere with METH-associated memories causing drug seeking by disturbing memory reconsolidation. This was done by infusing PP2A-Pep into the VTA $1\\\\mathrm{~h}$ after re-exposing the mice to the CPP test and repeating the CPP test after $24\\\\mathrm{~h}$ . This schedule was chosen since it is relevant in view of the development of a potential therapeutic application. Restoration of $\\\\mathrm{GABA_B}$ receptor expression in the VTA by PP2A-Pep after memory retrieval inhibited drug seeking behavior in the re-test session the next day. This finding suggests that METH associated memory reconsolidation has been disturbed by enhancing the expression of $\\\\mathrm{GABA_B}$ receptors in the VTA. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,412,489,595]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"However, our CPP protocol cannot rule out the possibility that the interfering peptide might additionally facilitate memory extinction, a process which is tightly associated with psychostimulant withdrawal (Myers and Carlezon, 2010). It is believed that memory extinction is an active learning process, where a newly learned memory is competing with the already consolidated one to weaken it. Memory consolidation and extinction have overlapping cellular and molecular mechanisms (Myers and Carlezon, 2010; Liu et al., 2019), making it difficult in our experimental set-up to discriminate among these two pathways. However, in view of the close timing of PP2A application $1\\\\mathrm{h}$ after memory retrieval we favor memory reconsolidation to be affected by PP2A-Pep in the first place. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,596,489,819]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We expect that PP2A-Pep is associated with negligible adverse side effects because it selectively inhibits the interaction of $\\\\mathrm{GABA_B}$ receptors with PP2A and does not appreciably affect $\\\\mathrm{GABA_B}$ receptor expression in healthy control mice. Therefore, side effects observed after global activation the receptors with the agonist baclofen such as sedation, dizziness, fatigue, insomnia, headache, paresthesia, tinnitus, and restlessness (Rolland et al., 2018) are unlikely to occur. In addition, because the peptide sequence of PP2A-Pep derived from the C-terminal domain of the $\\\\mathrm{GABA_B2}$ subunit is unique to $\\\\mathrm{GABA_B}$ receptors, we expect no off-target effects involving other receptor systems. In principle, binding of PP2A-Pep to PP2A could downregulate the expression levels of PP2A, which would affect a variety of signaling pathways involving protein dephosphorylation by PP2A. However, we previously showed that PP2A-Pep did not affect expression of PP2A (Hleihil et al., 2022). \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[77,821,489,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"In conclusion, restoring $\\\\mathrm{GABA_B}$ receptor expression in VTA neurons of METH-addicted mice by inhibiting PP2A-mediated downregulation of the receptors inhibited locomotor sensitization and drug seeking behavior. This finding indicates that interfering with disease-relevant protein-protein interactions might be a promising alternative approach toward the development of a selective and efficient therapeutic intervention. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,87,789,105]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Data availability statement \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,120,917,192]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The datasets analyzed for this study have been deposited at ZENODO.org and are publicly available (doi: 10.5281/zenodo.10554456). Due to their large size, raw images and behavioral recordings can be made available upon reasonable request to the corresponding author. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,224,687,240]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Ethics statement \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,256,917,313]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The animal study was approved by the Veterinary office of the Canton of Zurich, Waltersbachstrasse 5, CH-8006 Zurich (license ZH164/2020). The study was conducted in accordance with the local legislation and institutional requirements. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,344,732,361]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Author contributions \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,376,917,463]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"MH: Conceptualization, Data curation, Formal Analysis, Writing - original draft, Writing - review & editing, Investigation, Methodology. DB: Conceptualization, Data curation, Formal Analysis, Writing - original draft, Writing - review & editing, Funding acquisition, Project administration, Resources, Supervision. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[507,494,596,513]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Funding \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,527,917,583]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The authors declare financial support was received for the research, authorship, and/or publication of this article. This research was funded by the Swiss National Science Foundation, grant number 31003A_182325 to DB. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,615,707,634]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Acknowledgments \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[529,649,904,662]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"We thank Thomas Grampp for excellent technical assistance. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,695,709,711]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Conflict of interest \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,727,917,770]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,802,680,816]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Publisher's note \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[505,834,917,920]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher. \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[78,38,178,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Hleihil and Benke \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[752,38,917,49]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10.3389/fnmol.2024.1347228 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"header\"\n },\n {\n \"bbox\": \"[78,948,280,959]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"Frontiers in Molecular Neuroscience \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[490,948,505,958]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"10 \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"page_number\"\n },\n {\n \"bbox\": \"[835,948,915,959]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"9\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"frontiersin.org \",\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"footer\"\n },\n {\n \"bbox\": \"[80,87,200,104]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": null,\n \"page_idx\": \"10\",\n \"sub_type\": null,\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": \"References \",\n \"text_format\": null,\n \"text_level\": \"1\",\n \"type\": \"text\"\n },\n {\n \"bbox\": \"[80,124,489,906]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"Agabio, R., Saulle, R., Rosner, S., and Minozzi, S. (2023). Baclofen for alcohol use disorder. Cochr. Datab. Syst. Rev. 1:Cd012557. doi: 10.1002/14651858.CD012557.pub3 \\\",\\\"Arai, S., Takuma, K., Mizoguchi, H., Ibi, D., Nagai, T., Kamei, H., et al. (2009). $\\\\\\\\mathrm{GABA_B}$ receptor agonist baclofen improves methamphetamine-induced cognitive deficit in mice. Eur. J. Pharmacol. 602, 101-104. doi: 10.1016/j.ejphar.2008.10.065 \\\",\\\"Arora, D., Hearing, M., Haluk, D. M., Mirkovic, K., Fajardo-Serrano, A., Wessendorf, M. W., et al. (2011). Acute cocaine exposure weakens $\\\\\\\\mathrm{GABA_B}$ receptor-dependent G-protein-gated inwardly rectifying $\\\\\\\\mathrm{K}^+$ signaling in dopamine neurons of the ventral tegmental area. J. Neurosci. 31, 12251-12257. doi: 10.1523/JNEUROSCI.0494-11.2011 \\\",\\\"Baik, J. H. (2020). Stress and the dopaminergic reward system. Exp. Mol. Med. 52, 1879-1890. doi: 10.1038/s12276-020-00532-4 \\\",\\\"Balakrishnan, K., Hleihil, M., Bhat, M. A., Ganley, R. P., Vaas, M., Klohs, J., et al. (2023). Targeting the interaction of $\\\\\\\\mathrm{GABA_B}$ receptors with CaMKII with an interfering peptide restores receptor expression after cerebral ischemia and inhibits progressive neuronal death in mouse brain cells and slices. Brain Pathol. 33:e13099. doi: 10.1111/bpa.13099 \\\",\\\"Bardo, M. T., and Bevins, R. A. (2000). Conditioned place preference: What does it add to our preclinical understanding of drug reward? Psychopharmacology 153, 31-43. doi: 10.1007/s002130000569 \\\",\\\"Bartoletti, M., Gubellini, C., Ricci, F., and Gaiardi, M. (2004). The $\\\\\\\\mathrm{GABA_B}$ agonist baclofen blocks the expression of sensitisation to the locomotor stimulant effect of amphetamine. Behav. Pharmacol. 15, 397-401. doi: 10.1097/00008877-200409000-00014 \\\",\\\"Bartoletti, M., Gubellini, C., Ricci, F., and Giardi, M. (2005). Baclofen blocks the development of sensitization to the locomotor stimulant effect of amphetamine. Behav. Pharmacol. 16, 553-558. doi: 10.1097/01.fbp.0000179279.98029.e9 \\\",\\\"Bellfy, L., and Kwapis, J. L. (2020). Molecular mechanisms of reconsolidation-dependent memory updating. Int. J. Mol. Sci. 21, 6580. doi: 10.3390/ijms21186580 \\\",\\\"Benke, D., Michel, C., and Mohler, H. (2002). Structure of $\\\\\\\\mathrm{GABA_B}$ receptors in rat retina. J. Recept. Signal Transduc. 22, 253-266. \\\",\\\"Bhat, M. A., Esmaelli, A., Neumann, E., Balakrishnan, K., and Benke, D. (2022). Targeting the interaction of $\\\\\\\\mathrm{GABA_B}$ receptors with CHOP after an ischemic insult restores receptor expression and inhibits progressive neuronal death. Front. Pharmacol. 13:870861. doi: 10.3389/fphar.2022.870861 \\\",\\\"Bhat, M. A., Grampp, T., and Benke, D. (2023). ERK1/2-dependent phosphorylation of $\\\\\\\\mathrm{GABA_B}$ (S867/T872), controlled by CaMKII $\\\\\\\\beta$ , is required for $\\\\\\\\mathrm{GABA_B}$ receptor degradation under physiological and pathological conditions. Int. J. Mol. Sci. 24, 13436. doi: 10.3390/ijms241713436 \\\",\\\"Brebner, K., Ahn, S., and Phillips, A. G. (2005). Attenuation of d-amphetamine self-administration by baclofen in the rat: Behavioral and neurochemical correlates. Psychopharmacology 177, 409-417. doi: 10.1007/s00213-004-1968-6 \\\",\\\"Cedillo, L. N., and Miranda, F. (2013). Effects of co-administration of the $\\\\\\\\mathrm{GABA_B}$ receptor agonist baclofen and a positive allosteric modulator of the $\\\\\\\\mathrm{GABA_B}$ receptor, CGP7930, on the development and expression of amphetamine-induced locomotor sensitization in rats. Pharmacol Rep 65, 1132-1143. doi: 10.1016/s1734-1140(13) 71471-3 \\\",\\\"Cedillo, L. N., Ruiz-Garcia, R. I., Jimenez, J. C., and Miranda, F. (2019). Effect of coadministration of the $\\\\\\\\mathrm{GABA_B}$ agonist baclofen and the 5-HT2C agonist Ro60-0175 on the expression of amphetamine-induced locomotor sensitization. Exp. Brain. Res. 237, 1691-1697. doi: 10.1007/s00221-019-05540-z \\\",\\\"Chalifoux, J. R., and Carter, A. G. (2011). $\\\\\\\\mathrm{GABA_B}$ receptor modulation of synaptic function. Curr. Opin. Neurobiol. 21, 339-344. doi: 10.1016/j.conb.2011.02.004 \\\",\\\"Chen, G., and van den Pol, A. N. (1998). Presynaptic $\\\\\\\\mathrm{GABA_B}$ autoreceptor modulation of P/Q-type calcium channels and GABA release in rat suprachiasmatic nucleus neurons. J. Neurosci. 18, 1913-1922. \\\",\\\"Cheron, J., and Kerchove d'Exaerde, A. (2021). Drug addiction: From bench to bedside. Transl. Psychiatry 11, 424. doi: 10.1038/s41398-021-01542-0 \\\",\\\"Cunningham, C. L., Gremel, C. M., and Groblewski, P. A. (2006). Drug-induced conditioned place preference and aversion in mice. Nat. Protoc. 1, 1662-1670. doi: 10.1038/nprot.2006.279 \\\",\\\"de Beaurepaire, R., Sinclair, J. M. A., Heydtmann, M., Addolorato, G., Aubin, H. J., Beraha, E. M., et al. (2018). The use of baclofen as a treatment for alcohol use disorder: A clinical practice perspective. Front. Psychiatry 9:708. doi: 10.3389/fpsyt.2018.00708 \\\",\\\"Edwards, N. J., Tejeda, H. A., Pignatelli, M., Zhang, S., McDevitt, R. A., Wu, J., et al. (2017). Circuit specificity in the inhibitory architecture of the VTA regulates cocaine-induced behavior. Nat. Neurosci. 20, 438-448. doi: 10.1038/nn.4482 \\\",\\\"Fischer, K. D., Knockstedt, L. A., and Rosenberg, P. A. (2021). Glutamate homeostasis and dopamine signaling: Implications for psychostimulant addiction behavior. Neurochem Int 144, 104896. doi: 10.1016/j.neuint.2020.104896 \\\"]\",\n \"page_idx\": \"10\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[509,124,917,904]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"Gahwiler, B. H., and Brown, D. A. (1985). $\\\\\\\\mathrm{GABA}_B$ -receptor-activated $\\\\\\\\mathbf{K}^{+}$ current in voltage-clamped CA3 pyramidal cells in hippocampal cultures. Proc. Natl. Acad. Sci. USA 82, 1558-1562. doi: 10.1073/pnas.82.5.1558 \\\",\\\"Garbutt, J. C. (2020). Approval of baclofen for alcohol use disorders in France: A perspective from the United States. Alcohol Alcohol 55, 48. doi: 10.1093/alcalc/agz084 \\\",\\\"Good, R. L., and Radcliffe, R. A. (2011). Methamphetamine-induced locomotor changes are dependent on age, dose and genotype. *Pharmacol. Biochem.* Behav. 98, 101-111. doi: 10.1016/j.pbb.2010.12.004 \\\",\\\"Guetg, N., Aziz, S. A., Holbro, N., Turecek, R., Rose, T., Seddik, R., et al. (2010). NMDA receptor-dependent $\\\\\\\\mathrm{GABA_B}$ receptor internalization via CaMKII phosphorylation of serine 867 in $\\\\\\\\mathrm{GABA_{B1}}$ . Proc. Natl. Acad. Sci. USA 107, 13924-13929. \\\",\\\"Harada, M., Pascoli, V., Hiver, A., Flakowski, J., and Luscher, C. (2021). Corticostriatal activity driving compulsive reward seeking. Biol. Psychiatry 90, 808-818. doi: 10.1016/j.biopsych.2021.08.018 \\\",\\\"Hearing, M., Kotecki, L., Marron Fernandez de Velasco, E., Fajardo-Serrano, A., Chung, H. J., Lujan, R., et al. (2013). Repeated cocaine weakens GABA-Girk signaling in layer 5/6 pyramidal neurons in the prelimbic cortex. Neuron 80, 159-170. doi: 10.1016/j.neuron.2013.07.019 \\\",\\\"Heinzerling, K. G., Shoptaw, S., Peck, J. A., Yang, X., Liu, J., Roll, J., et al. (2006). Randomized, placebo-controlled trial of baclofen and gabapentin for the treatment of methamphetamine dependence. *Drug Alcohol. Depend.* 85, 177-184. doi: 10.1016/j.drugalcdep.2006.03.019 \\\",\\\"Hleihil, M., Balakrishnan, K., and Benke, D. (2022). Protein phosphatase 2A regulation of $\\\\\\\\mathrm{GABA_B}$ receptors normalizes ischemia-induced aberrant receptor trafficking and provides neuroprotection. Front. Mol. Neurosci. 15:1015906. doi: 10. 3389/fnmo.2022.1015906 \\\",\\\"Hleihil, M., Vaas, M., Bhat, M. A., Balakrishnan, K., and Benke, D. (2021). Sustained baclofen-induced activation of $\\\\\\\\mathrm{GABA_B}$ receptors after cerebral ischemia restores receptor expression and function and limits progressing loss of neurons. Front. Mol. Neurosci. 14:726133. doi: 10.3389/fnml.2021.726133 \\\",\\\"Jing, L., Zhang, M., Li, J. X., Huang, P., Liu, Q., Li, Y. L., et al. (2014). Comparison of single versus repeated methamphetamine injection induced behavioral sensitization in mice. Neurosci. Lett. 560, 103-106. doi: 10.1016/j.neulet.2013.12.024 \\\",\\\"Jones, K. A., Borowsky, B., Tamm, J. A., Craig, D. A., Durkin, M. M., Dai, M., et al. (1998). $\\\\\\\\mathrm{GABA_B}$ receptor function as a heteromeric assembly of the subunits $\\\\\\\\mathrm{GABA_{BR1}}$ and $\\\\\\\\mathrm{GABA_{BR2}}$ . Nature 396, 674-679. \\\",\\\"Kaufmann, K., Malitschek, B., Schuler, V., Heid, J., Froestl, W., Beck, P., et al. (1998). $\\\\\\\\mathrm{GABA_B}$ receptor subtypes assemble into functional heteromeric complexes. Nature 396, 683-687. \\\",\\\"Kumar, P., Wu, H., McBride, J. L., Jung, K. E., Kim, M. H., Davidson, B. L., et al. (2007). Transvascular delivery of small interfering RNA to the central nervous system. Nature 448, 39-43. doi:10.1038/nature05901 \\\",\\\"Lalive, A. L., and Luscher, C. (2016). \\\\\\\"GABA $_B$ receptor functions in the mesolimbic dopamine system,\\\\\\\" in GABA $_B$ Receptor, ed. G. Colombo (Cham: Springer International Publishing), 129-154. \\\",\\\"Leggio, L., and Litten, R. Z. (2021). The $\\\\\\\\mathrm{GABA_B}$ receptor agonist baclofen helps patients with alcohol use disorder: Why these findings matter. Neuropsychopharmacology 46, 2228-2229. doi:10.1038/s41386-021-01142-y \\\",\\\"Leite-Morris, K. A., Fukudome, E. Y., Shoeb, M. H., and Kaplan, G. B. (2004). $\\\\\\\\mathrm{GABA_B}$ receptor activation in the ventral tegmental area inhibits the acquisition and expression of opiate-induced motor sensitization. J. Pharmacol. Exp. Ther. 308, 667-678. doi: 10.1124/jpet.103.058412 \\\",\\\"Li, J. Y., Yu, Y. J., Su, C. L., Shen, Y. Q., Chang, C. H., and Gean, P. W. (2023). Modulation of methamphetamine memory reconsolidation by neural projection from basolateral amygdala to nucleus accumbens. Neuropsychopharmacology 48, 478-488. doi: 10.1038/s41386-022-01417-y \\\",\\\"Li, M., Peterson, C., Xu, L., Mikosz, C. A., and Luo, F. (2023). Medical costs of substance use disorders in the US employer-sponsored insurance population. JAMA Netw Open 6, e2252378. doi:10.1001/jamanetworkopen.2022.52378 \\\",\\\"Li, S. M., Yin, L. L., Ren, Y. H., Pan, L. S., and Zheng, J. W. (2001). $\\\\\\\\mathrm{GABA_B}$ receptor agonist baclofen attenuates the development and expression of d-methamphetamine-induced place preference in rats. Life Sci. 70, 349-356. doi: 10.1016/s0024-3205(01) 01397-2 \\\",\\\"Ling, W., Shoptaw, S., and Majewska, D. (1998). Baclofen as a cocaine anti-craving medication: A preliminary clinical study. Neuropsychopharmacology 18, 403-404. doi: 10.1016/s0893-133x(97)00128-0 \\\",\\\"Liu, J. F., Tian, J., and Li, J. X. (2019). Modulating reconsolidation and extinction to regulate drug reward memory. *Eur. J. 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F., Fu, Q., Qiu, T. Y., Yang, J. H., Peng, Q. H., and Hu, Z. Z. (2023). The CaMKII-dependent phosphorylation of $\\\\\\\\mathrm{GABA_B}$ receptors in the nucleus accumbens was involved in cocaine-induced behavioral sensitization in rats. CNS Neurosci. Ther. 29, 1345-1356. doi: 10.1111/cns.14107 \\\",\\\"Luscher, C., Jan, L. Y., Stoffel, M., Malenka, R. C., and Nicoll, R. A. (1997). G protein-coupled inwardly rectifying $\\\\\\\\mathrm{K}^+$ channels (GIRKs) mediate postsynaptic but not presynaptic transmitter actions in hippocampal neurons. Neuron 19, 687-695. \\\",\\\"Ma, S., Zhong, H., Liu, X., and Wang, L. (2023). Spatial distribution of neurons expressing single, double, and triple molecular characteristics of glutamatergic, dopaminergic, or GABAergic neurons in the mouse ventral tegmental area. J. Mol. Neurosci. 73, 345-362. doi:10.1007/s12031-023-02121-2 \\\",\\\"Maier, P. J., Marin, I., Grampp, T., Sommer, A., and Benke, D. (2010). Sustained glutamate receptor activation down-regulates $\\\\\\\\mathrm{GABA_B}$ receptors by shifting the balance from recycling to lysosomal degradation. J. Biol. Chem. 285, 35606-35614. doi: 10.1074/jbc.M110.142406 \\\",\\\"May, A. C., Upperle, R. L., and Stewart, J. L. (2020). Dark times: The role of negative reinforcement in methamphetamine addiction. Front. Psychiatry 11:114. doi: 10.3389/fpsyt.2020.00114 \\\",\\\"McKendrick, G., and Graziane, N. M. (2020). Drug-induced conditioned place preference and Its practical use in substance use disorder research. Front. Behav. Neurosci. 14:582147. doi: 10.3389/fnbeh.2020.582147 \\\",\\\"Mintz, I. M., and Bean, B. P. (1993). $\\\\\\\\mathrm{GABA_B}$ receptor inhibition of P-type $\\\\\\\\mathrm{Ca^{2+}}$ channels in central neurons. Neuron 10, 889-898. \\\",\\\"Mizoguchi, H., and Yamada, K. (2011). Pharmacologic treatment with $\\\\\\\\mathrm{GABA_B}$ receptor agonist of methamphetamine-induced cognitive impairment in mice. Curr. Neuropharmacol. 9, 109-112. doi: 10.2174/157015911795016976 \\\",\\\"Munoz, M. B., Padgett, C. L., Rifkin, R., Terunuma, M., Wickman, K., Contet, C., et al. (2016). A role for the GIRK3 subunit in methamphetamine-induced attenuation of $\\\\\\\\mathrm{GABA_B}$ receptor-activated GIRK currents in VTA dopamine neurons. J. Neurosci. 36, 3106-3114. doi: 10.1523/jneurosci.1327-15.2016 \\\",\\\"Myers, K. M., and Carlezon, W. A. Jr. (2010). Extinction of drug- and withdrawal-paired cues in animal models: Relevance to the treatment of addiction. Neurosci. Biobehav. Rev. 35, 285-302. doi: 10.1016/j.neubiorev.2010.01.011 \\\",\\\"Nair-Roberts, R. G., Chatelain-Badie, S. D., Benson, E., White-Cooper, H., Bolam, J. P., and Ungless, M. A. (2008). Stereological estimates of dopaminergic, GABAergic and glutamatergic neurons in the ventral tegmental area, substantia nigra and retrorubral field in the rat. *Neuroscience* 152, 1024-1031. doi: 10.1016/j.neuroscience.2008.01.046 \\\",\\\"Nestler, E. J. (2005). The neurobiology of cocaine addiction. Sci. Pract. Perspect. 3, 4-10. doi: 10.1151/spp05314 \\\",\\\"Ostroumov, A., and Dani, J. A. (2018). Inhibitory plasticity of mesocorticolimbic circuits in addiction and mental illness. Trends Neurosci. 41, 898-910. doi: 10.1016/j.tins.2018.07.014 \\\",\\\"Padgett, C. L., Lalive, A. L., Tan, K. R., Terunuma, M., Munoz, M. B., Pangalos, M. N., et al. (2012). Methamphetamine-evoked depression of $\\\\\\\\mathrm{GABA_B}$ receptor signaling in GABA neurons of the VTA. Neuron 73, 978-989. doi: 10.1016/j.neuron.2011.12.031 \\\",\\\"Paxinos, G., and Franklin, K. B. J. (2001). The mouse brain in steriotaxic coordinates. San Diego, CA: Academic Press. \\\",\\\"Ranaldi, R., and Poeggel, K. (2002). Baclofen decreases methamphetamine self-administration in rats. Neuroreport 13, 1107-1110. doi: 10.1097/00001756-200207020-00007 \\\"]\",\n \"page_idx\": \"11\",\n \"sub_type\": \"ref_text\",\n \"table_body\": null,\n \"table_caption\": null,\n \"table_footnote\": null,\n \"text\": null,\n \"text_format\": null,\n \"text_level\": null,\n \"type\": \"list\"\n },\n {\n \"bbox\": \"[509,90,915,653]\",\n \"code_body\": null,\n \"code_caption\": null,\n \"image_caption\": null,\n \"image_footnote\": null,\n \"img_path\": null,\n \"list_items\": \"[\\\"Roberts, D. C., and Brebner, K. (2000). GABA modulation of cocaine self-administration. Ann. N Y Acad. Sci. 909, 145-158. doi: 10.1111/j.1749-6632.2000.tb06680.x \\\",\\\"Rolland, B., Simon, N., and Franchitto, N. (2018). Safety challenges of using high dose baclofen for alcohol use disorder: A focused review. Front. Psychiatry 9:367. doi: 10.3389/fpsyt.2018.00367 \\\",\\\"Rolland, B., Simon, N., Franchitto, N., and Aubin, H. J. (2020). France grants an approval to baclofen for alcohol dependence. Alcohol Alcohol. 55, 44-45. doi: 10.1093/alcalc/agz082 \\\",\\\"Santos, A. E., Carvalho, C. M., Macedo, T. A., and Carvalho, A. P. (1995). Regulation of intracellular $\\\\\\\\left[\\\\\\\\mathrm{Ca}^{2+}\\\\\\\\right]$ and GABA release by presynaptic $\\\\\\\\mathrm{GABA_B}$ receptors in rat cerebrocortical synaptosomes. Neurochem. Int. 27, 397-406. \\\",\\\"Schindelin, J., Arganda-Carreras, I., Frise, E., Kaynig, V., Longair, M., Pietzsch, T., et al. (2012). Fiji: An open-source platform for biological-image analysis. Nat. Methods 9, 676-682. doi: 10.1038/nmeth.2019 \\\",\\\"Sharpe, A. L., Varela, E., Bettinger, L., and Beckstead, M. J. (2015). Methamphetamine self-administration in mice decreases GIRK channel-mediated currents in midbrain dopamine neurons. Int. J. Neuropsychopharmacol. 18, 1-10. doi: 10.1093/ijnp/pyu073 \\\",\\\"Shoptaw, S., Yang, X., Rotheram-Fuller, E. J., Hsieh, Y. C., Kintaudi, P. C., Charuvastra, V. C., et al. (2003). Randomized placebo-controlled trial of baclofen for cocaine dependence: Preliminary effects for individuals with chronic patterns of cocaine use. J. Clin. Psychiatry 64, 1440-1448. doi: 10.4088/jcp.v64 n1207 \\\",\\\"Steketee, J. D., and Kalivas, P. W. (2011). Drug wanting: Behavioral sensitization and relapse to drug-seeking behavior. Pharmacol. Rev. 63, 348-365. doi: 10.1124/pr.109.001933 \\\",\\\"Terunuma, M., Vargas, K. J., Wilkins, M. E., Ramirez, O. A., Jaureguiberry-Bravo, M., Pangalos, M. N., et al. (2010). Prolonged activation of NMDA receptors promotes dephosphorylation and alters postendocytic sorting of $\\\\\\\\mathrm{GABA_B}$ receptors. Proc. Natl. Acad. Sci. USA 107, 13918-13923. \\\",\\\"Tirelli, E., Laviola, G., and Adriani, W. (2003). Ontogenesis of behavioral sensitization and conditioned place preference induced by psychostimulants in laboratory rodents. Neurosci. Biobehav. Rev. 27, 163-178. doi: 10.1016/s0149-7634(03) 00018-6 \\\",\\\"Tronson, N. C., and Taylor, J. R. (2007). Molecular mechanisms of memory reconsolidation. Nat. Rev. Neurosci. 8, 262-275. doi: 10.1038/nrn2090 \\\",\\\"Tzschentke, T. M. (2007). Measuring reward with the conditioned place preference (CPP) paradigm: Update of the last decade. Addict. Biol. 12, 227-462. doi: 10.1111/j.1369-1600.2007.00070.x \\\",\\\"Volman, S. F., Lammel, S., Margolis, E. B., Kim, Y., Richard, J. M., Roitman, M. F., et al. (2013). New insights into the specificity and plasticity of reward and aversion encoding in the mesolimbic system. J. Neurosci. 33, 17569-17576. doi: 10. 1523/jneurosci.3250-13.2013 \\\",\\\"White, J. H., Wise, A., Main, M. J., Green, A., Fraser, N. J., Disney, G. H., et al. (1998). Heterodimerization is required for the formation of functional $\\\\\\\\mathrm{GABA_B}$ receptors. Nature 396, 679-682. \\\",\\\"Zemoura, K., Balakrishnan, K., Grampp, T., and Benke, D. (2019). $\\\\\\\\mathrm{Ca^{2+}}/$ Calmodulin-dependent protein kinase II (CaMKII) b-dependent phosphorylation of $\\\\\\\\mathrm{GABA_B}$ triggers lysosomal degradation of $\\\\\\\\mathrm{GABA_B}$ Receptors via mind bomb-2 (MIB2)-mediated Lys-63-linked ubiquitination. Mol. 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behavior"}}},{"rowIdx":74,"cells":{"abstract":{"kind":"string","value":""},"author":{"kind":"string","value":"Bo Xiu|Rui Zhang|Aijia Shang|Fuyun Liu"},"content_list":{"kind":"list like","value":[{"bbox":"[91,129,903,178]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Chinese expert consensus on diagnosis and management of split cord malformation ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[91,198,457,217]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bo Xiu $^{1}$ (区), Fuyun Liu $^{2}$ , Aijia Shang $^{3}$ , Rui Zhang $^{4}$ ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[89,236,885,324]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"$^{1}$ Department of Neurosurgery, The 7th Medical Center of PLA General Hospital, Beijing 100700, China \",\"$^{2}$ Department of Pediatric Orthopaedics, Zhengzhou University The 3rd Hospital, Zhengzhou 450052, Henan, China \",\"$^{3}$ Department of Neurosurgery, The 1st Medical Center of PLA General Hospital, Beijing 100853, China \",\"$^{4}$ Department of Neurosurgery, Shandong Provincial Hospital Affiliated to Shandong First Medical University, Ji'an 250021, Shandong, China \"]","page_idx":"0","sub_type":"text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[93,341,227,358]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ARTICLEINFO ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[96,366,274,382]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Received: 26 April 2020 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[98,384,251,400]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Revised: 3 June 2020 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[98,402,264,418]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Accepted: 24 June 2020 ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[96,422,294,490]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"© The authors 2020. This is an open access article under the CC BY-NC License (http://creativecommons.org/licenses/by-nc/4.0/) ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[96,500,205,514]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"KEYWORDS ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[96,519,274,605]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"split cord malformation; \ntethered cord syndrome; \nneural tube defect; \ndiagnosis; surgery; \nconsensus ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[344,342,459,358]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ABSTRACT ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[341,368,906,561]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Split cord malformation (SCM) is a neural tube defect that the spinal cord is longitudinally separated into two hemicords with individual functions, which causes severe spinal cord impairment and sensorimotor deficit. As a kind of myelodysplasia and a special type of tethered cord syndrome, SCM is not widely understood, and common issues in the diagnosis and treatment of SCM should be clarified. In this paper, the Chinese Split Cord Malformation Working Group made a consensus for SCM on embryopathogenesis and types, clinical presentations, neuroimaging assessment, indications and principle of the surgery, surgical techniques and nuances, prognosis and follow up. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[93,643,241,659]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"1 Introduction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[89,676,485,870]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Split cord malformation (SCM) is a neural tube defect that the spinal cord is longitudinally separated into two hemicords with individual functions, which causes severe spinal cord impairment and sensorimotor deficit. French pathologist Ollivier d'Angers initiated \"diastematomyelia\" from the Greek \"diastema\" and \"myelos\", meaning \"cleft\" and \"spinal cord\" to denote the malformation in 1837 [1, 2]. Some authors named the malformation as \"split notochord ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[508,637,903,792]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"syndrome\" [3] for its ontogenetic mechanism. Inconsistent definitions of the malformation increased the vagueness and confusions. Therefore, the Chinese Split Cord Malformation Working Group adopted the term \"split cord malformation\", which was named by Dachling Pang in 1992, for standardization, and reached a global consensus [4]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[508,793,903,871]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Split cord malformation is a kind of myelodysplasia and a special type of tethered cord syndrome, accounting for $4\\% - 9\\%$ of all congenital spinal cord malformations. The deficit ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,45,323,74]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Journal of Neurorestoratology DOI 10.26599/JNR.2020.9040010 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[781,45,903,74]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2020,8(2):83-92 ISSN 2324-2426 ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[93,77,207,94]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"REVIEW ARTICLE ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[91,894,464,910]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Corresponding author: Bo Xiu, E-mail: boxiu@scsurgery.com ","text_format":null,"text_level":null,"type":"page_footnote"},{"bbox":"[93,936,218,956]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"0","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"TSINGHUA UNIVERSITY PRESS ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[89,94,486,170]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"occurs more commonly in lower thoracic and lumbar segments than in upper thoracic segments, while rarely in cervical and lumbosacral spine [5-7]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[89,173,486,370]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"A considerable number of neurosurgeons and spinal surgeons lack sufficient understanding of the occurrence, clinical characteristics and treatment of the disease, which leads to misdiagnosis and mistreatment. After collective discussion, the Chinese Split Cord Malformation Working Group has reached the following consensus on the diagnosis and treatment of SCM. The participants are listed in the end of this consensus (Appendix 1). ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,393,478,413]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2 Embryopathogenesis and classification ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[91,434,312,452]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.1 Embryopathogenesis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[89,461,486,875]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The pathogenesis of SCM remains controversial. Unified theory of embryogenesis [4] firstly proposed by Pang et al. contends that all SCMs originate from one basic ontogenetic error occurring around the time when the primitive neurenteric canal closes, during the $3^{\\mathrm{rd}}$ to the $4^{\\mathrm{th}}$ week of embryogenesis. Maternal risk factor for SCMs may include: folate deficiency, in utero exposure to certain viruses or chemical compounds, maternal diabetes mellitus and genetic disorders, etc. The aforementioned risk factors may contribute to an abnormal mid-line adhesion or fistula between the ectoderm and endoderm, which splits the notochord and neural plate. The consequential notochord dehiscence induces neuroschisis and mesenchyme condensation in the vicinity of the fistula. The aggregated mesenchymal cells differentiate into osseo-cartilaginous, fibrous tissues and vascular structures, then bisect the spinal canal and spinal cord. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,877,486,915]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Three pathophysiological elements lead to clinical presentations: 1. Asymmetrical develop ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[508,94,903,309]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"ments of the hemicords, which often result in a tenuous and dysplastic hemicord on one side; 2. Shearing and tethering effect on the spinal cord from the bisecting elements. Interior and posterior fibrous bands in the split and aberrant nerves originating from the interior aspect of the hemicord may contribute to the tethering effect as well [8, 9]; 3. Traction injury of the terminal conus medullaris due to mild-to-severe low-lying conus medullaris and thickened filum terminale. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,316,669,332]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"2.2 Classification ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[508,344,906,540]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Postulated by Pang et al. [1], SCMs can be categorized into two types. Type I SCM accounts for $40\\%$ of all SCMs and is a condition in which two hemicords, each with individual central canal and pia mater, inhabit their own dural sacs and are transfixed by a dural-sheathed rigid, bony median septum. Type II SCM accouts for $60\\%$ of all SCMs and indicates that two hemicords share a single dural sac despite separated by a nonrigid fibrous septum. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[508,542,905,639]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Pang Classification has catered for clinical requirements and became a commonly recognized method. There are reports that cannot be rigorously validated by Pang classification, albeit it is rare [10-12]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,664,743,683]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3 Clinical presentations ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[508,701,903,798]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Split cord malformation, occurring more frequently in females than in males [13], usually has an insidious onset with slow progression. Initial hospital visit is often made before adolescent age. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,806,707,821]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.1 Cutaneous lesions ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[508,834,903,910]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Split cord malformation has many related cutaneous lesions including hypertrichosis, hyperpigmentation, dimples, pilonidal sinus, hemangiomas and subcutaneous lipoma [14, 15]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[94,45,117,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"84 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[697,44,903,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Journal of Neurorestoratology ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[700,936,903,952]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"1","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Journal of Neurorestoratology ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[91,89,485,147]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Cutaneous lesions are essential for an early diagnosis and assessment of SCM as primary reasons for SCM patients' hospital visits. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,155,285,170]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.2 Lower extremities ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[89,181,485,412]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Leg problems are the second earliest and most frequent symptoms associated with SCM following dermatological lesions. A pivotal element pertinent to the symptom is the very existence of the asymmetrical development of the hemicords [8]. A thinner maldeveloped hemicord compared to the better developed side often promises ipsilateral neural deficit such as atrophy of the leg and foot, ankle deformity, pain and numbness. While in the case of two equally divided hemicords, leg symptoms can be slight and unnoticeable. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,420,210,435]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.3 Scoliosis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[89,446,485,700]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Scoliosis can be detected in $60\\%$ of all SCM patients, especially among those who have asymmetrically developed hemicords [16-19], and even be the only symptom in some cases [8, 20]. Scoliosis can be compensation to the tilted pelvis attributed to atrophy of the ipsilateral leg controlled by a maldeveloped hemicord, or a result of myodynia imbalance surrounding the column, which is secondary to tethering-derived neural deficit. Vertebral body deformity, often seen in SCM patients, is another possible factor contributing to the development of scoliosis. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,711,339,728]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.4 Lower back and leg pain ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[89,737,485,912]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"For SCM patients with symmetric hemicords, lower back pain is usually the main complaint without obvious lower extremity change or scoliosis. Pain can be either unilateral or bilateral, although the latter is more common. Traction from the septum damages the proximal cord and causes syringomyelia regardless of the symmetry of two hemicords. Syringomyelia can exist anywhere in the cord, from site above SCM ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[508,90,901,127]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"to either side of hemicords, which causes unilateral or bilateral lower back and leg pain. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,134,741,152]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"3.5 Sphincter dysfunction ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[506,162,903,397]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Bladder and bowel dysfunction is a less common symptom among SCM patients. It is usually not severe and primarily affects older children and adults. Sphincter dysfunction includes constipation or inactive bowel movement, frequent and urgent micturition, and enuresis, etc. The location of the septum is pivotal to the sphincter dysfunction. A higher located septum means relatively less significant low-lying conus, which promises less possibility of sphincter dysfunction with a later onset and slighter severity. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[508,400,905,704]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Each aforementioned symptom can happen alone or simultaneously in one patient. Children are more likely to have deformity, leg weakness and sphincter dysfunction and adults tend to present with lower back and leg pain. But some SCM Type II patients could be asymptomatic with only radiographic evidence [21]. Misdiagnosis and missed diagnosis are what we saw quite often in the triage of patients who were referred to orthopedic care. Unnecessary orthopedic procedures were performed based on a misdiagnosis of talipes equinovarus or idiopathic scoliosis. Delayed medical consultation would result in more neurological impairments and more orthopedic deformities. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,728,782,747]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4 Neuroimaging assessment ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,769,792,787]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4.1 Magnetic resonance imaging ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[506,795,903,912]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Besides reflecting associated vertebra anomalies noted ahead, sagittal images can help to locate the septum accurately. Microseptum and fibrous band at split point, which are always subtle and undistinguishable, should be recognized carefully in these images as tethering factors. The ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,45,297,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Journal of Neurorestoratology ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[880,45,901,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"85 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[93,939,218,959]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"2","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"TSINGHUA UNIVERSITY PRESS ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[89,90,486,247]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"axial images covering whole lesions can directly show the hemicords, two dural sacs, septum-pointed direction and abnormal fila terminale. Coronal images can reveal the relationship between septum and the convergence point of two hemicords. A full spine magnetic resonance imaging (MRI) is the best to present all coexisting lesions [12, 14, 16, 22]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,254,331,272]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4.2 Computed tomography ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[89,280,486,477]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Computed tomography (CT) is valuable for studying bone structure, and the high-resolution CT can even catch a minuscule bony spur [15]. Three-dimension (3D) reconstruction CT plays a significant role in presenting spinal deformities including laminospinous defect, abnormal bifurcation of spinous process, hemivertebra and bifida vertebra. Especially, it provides 3D images of bony septum for clinicians to plan a surgery. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,485,184,502]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4.3 X-ray ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[89,512,486,689]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"X-ray scan can assist to show enostosis and spinal deformities such as scoliosis, kyphosis, hemivertebra, butterfly vertebra, Klippel-Feil syndrome and spina bifida. However, it lacks high resolution on bony ridge at the mid-line of spinal canal. With high sensibility in detecting and accuracy in typing, associated neuroimaging information is vital for clinical diagnosis and surgical plans. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,696,294,714]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"4.4 Ultrasound testing ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[89,724,486,905]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Ultrasound testing is the most favorable method in prenatal screening for its convenience and non-invasive nature. Detected by ultrasound testing [23], the lesion lying in the midline of two partially widened vertebral posterior ossification centers is a special characteristic of SCM, which may be further confirmed by fetal MRI. Ultrasound testing can also help to assess tethering severity and monitor post-operative ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[508,90,903,128]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"remission among the children under 12 months old by the pulsation of conus medullaris. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[508,129,906,209]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Electrophysiological tests, urogenital sonography, post-void residual volume (PVR) and urodynamic testing are optional and can serve in assessing surgical outcomes. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,233,901,253]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5 Indications and principle of the surgery ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[508,269,903,326]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Surgery is the only effective way to remove abnormal anatomic structures tethering spinal cord in SCM. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,335,650,351]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5.1 Indications ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,362,660,379]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5.1.1 Type I SCM ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[508,389,906,650]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Surgical timing and indications of Type I SCM are similar to the ones of spina bifida occulta with tethered cord [7, 24]. Once diagnosed, infants and children should accept immediate surgical intervention [25, 26]. For adults, if there is no symptom or the condition is stable in recent years, the operation is unnecessary, and the condition can be observed continuously; however, if there is lumbago and leg pain or the condition tends to worsen, the operation should be performed to relieve the pain, prevent or delay the further aggravation of nerve damage caused by tethered cord. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,660,666,678]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5.1.2 Type II SCM ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[508,686,906,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The surgical indications of Type II SCM are controversial. Some authors preferred an aggressive intervention regardless of symptoms and fibrous bands [13, 26], while others suggested a conservative monitoring for progression [21]. Lan et al. argued that the patients of Type II SCM may benefit few from surgery for lacking of symptoms or fibrous bands [9]. Shang et al. proposed that congenitally asymmetric hemicords cause most neural deficits in Type II SCM patients, whose growth hardly deteriorate ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[94,45,117,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"86 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[697,44,903,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Journal of Neurorestoratology ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[702,936,903,952]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"3","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Journal of Neurorestoratology ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[89,90,486,248]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"conditions unless cord tethering is involved, and surgery should only be considered with a progressive disease [27]. On the contrary, Pang et al. recommended Type II SCM patients to accept surgeries, because tethering factors surely existing in many cases will weaken neural functions constantly and surgery may protect them against deterioration [28]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[89,250,486,491]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Chinese Split Cord Malformation Working Group has reached a consensus that close monitoring on disease progression should be the initial option for children, while surgery is necessary if there exists: (1) symptoms resulted from tethered cord; or (2) evidence of fibrous bands, syringomyelia or low-lying conus medullaris on MRI [29]. In our practice, we performed surgeries on Type II SCM patients based on the same indications with Type I and identified fibrous bands-tethered cord in almost every procedure. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,500,334,517]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"5.2 Principle of the Surgery ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[89,525,486,623]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Remove any anatomic abnormality tethering spinal cord including osteocartilaginous tissues, fibrous septum and fixed connective tissues. Pay more attention to integrity of intervertebral facets for higher stability and less complications. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[93,647,423,665]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6 Surgical techniques and nuances ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[91,688,246,705]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6.1 Bony septum ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[89,715,486,914]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Precise localization of bony septum is crucial in SCM treatment, while it could be laborious on asymmetrical Type I patients, especially with scoliosis. For less bleeding and accidental injuries, it is beneficial for surgeons to review neuroimages preoperatively and correlate imaging manifestations to anatomical landmarks intraoperatively. Elder age, sturdy septa base and narrow spinal canal challenge neurosurgeons. Unossified septa will turn to be ossified ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[508,90,906,914]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"after toddler hood, which leads to a difficult extirpation. Impeded by ventrally originating blood supply, any manipulation without clear identification of spinal cord may bring about a catastrophic, massive epidural bleeding within a narrow surgical field. What we recommend, to preserve the spinal cord, is that an early duotomy of the sacs flanking the septum should be performed after rongeuring the septum surface. Then, resect the residuals meticulously. Next, cut the vessels, aberrant nerves and fibrous bands between the interior cord and septum by electrocoagulation and gently pull the patty-covered cord outwards to avoid subarachnoid hemorrhage. Blood irrigation to the septum and adhesion will be cut off once a clear demarcation is dissected off. Also, once identified, supplying vessels of septum should be cut off. Commonly, the septa grow from the anteroinferior to the posterosuperior with a sturdy base, which require a resection following this longitudinal deviation, especially for the tightly squeezed cord. Microsurgical techniques and delicate instruments such as mini Friedman bone rongeurs, $3\\mathrm{mm}$ high speed burr and ultrasonic bone scalpel will facilitate the extirpation of the septa. To circumvent further injury, little residual of the septa base is acceptable if the spinal cord is released from the oppression of septa. Covering rough bony bed with bone wax can achieve hemostasis and avoid air embolism and cord-shearing injury. Suture is unnecessary because ventral dura is so clung to the vertebra that CSF hardly leaks out. However, the dorsal dura needs a water-tight closure. Notably, insufficient resection of connective tissues (such as meninges in the split and fibrous bands), or unopened dura, can result in an incomplete detethering of the cord [7], which we should avoid. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,45,297,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Journal of Neurorestoratology ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[880,45,901,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"87 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[93,939,218,959]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"4","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"TSINGHUA UNIVERSITY PRESS ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[94,87,241,105]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6.2 Type II SCM ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[94,113,483,587]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Type II SCM tethered by taut fibrous bands or septa requires a surgical performance. Many septa in Type II SCMs are too tenuous to appreciate in MRI and sometimes a surgical exploration could be inconclusive with unexperienced surgeons. However, a septum splitting and tethering the cord is identifiable in all Type I SCMs and most Type II SCMs in the lower end of the split, where the two hemicords rejoin. Fibrous bands from the posteroinferior to the anterosuperior terminate at the inferior pole of the split where surgery incision should be pinpointed. Dimples in the skin is the cutaneous end of the fibrous band indicating the deep end and an excellent mark to make incision. For patients without skin dimples, surgeons should carefully look for the defects in spine processes or laminae at the inferior end of the split where fibrous bands underneath bone defects can be easily detected in the epidural space. CT and MRI images are indispensable to localize the penetrating point of fibrous band on dura. Pang et al. reported that ventral tethering could occur in about $21\\%$ of Type II SCMs [28]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[94,596,269,611]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6.3 Aberrant nerves ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[94,622,483,758]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Aberrant nerves originate from inner sides of hemicords and terminate at the dura on the surface of osseous septum or dorsal dura. Once aberrant nerves, serving as adhesive and tethering factors [8], are confirmed nonfunctional by neuro-electrophysiology monitoring, rhizotomy would be rational and necessary. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[94,767,269,783]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6.4 Filum terminale ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[94,794,483,909]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Transection of the fila terminale is unnecessary without sphincter dysfunction or low-lying conus medullaris. However, severe low-lying conus medullaris, especially combined with thick fila terminale and lipoma, is a solid indicator of fila terminale transection regardless ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[513,90,901,482]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"of the symptoms. Some authors suggested fila terminale transection on all SCM pediatric patients [30, 31] instead of a delayed surgery and close monitoring on asymptomatic or stable adult patients, while the surgery should be performed if condition deteriorates. Extirpation of the septa should be performed prior to filum terminale transaction [32]. Retraction of the spinal cord after filum terminale transection can cover the whole view of septum and make the resection more challenging. Filum terminale transection and spectrum transection should be performed simultaneously considering only the former may exacerbate the symptom. In a rare case, two hemicords will not rejoin and a condition called \"duplicate fila terminale\" occurs. Although no splitting spetum is between two fila terminale, the transection of both fila terminale is required to fully release the tethering of spinal cord. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[513,491,861,509]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6.5 Accompanying spinal cord pathology ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[513,518,901,693]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"If the accompanying pathologies coexist in the symptom progression or are potentially deleterious in the future, they can be removed at one stage with detethering procedures, especially within the same operation field with the septa. Most common comorbidities to SCM are congenital masses such as neuroenteric cyst, spinal bifida and other types of tethered cord syndrome [33]. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[513,702,627,717]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6.6 Scoliosis ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[513,728,901,905]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Scoliosis should be corrected after SCM surgery. Scoliosis correction before Type I SCM surgery could lead to spinal cord retraction in pediatric patients [16]. Some spine surgeons proposed a single stage surgery of both SCM (especially Type II) and scoliosis [34-36]. However, we recommend a second stage scoliosis correction following Type I SCM surgery to reduce the surgical risks. SCM extirpation may arrest the ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[94,45,114,56]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"88 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[699,45,903,59]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Journal of Neurorestoratology ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[702,936,903,951]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"5","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Journal of Neurorestoratology ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[89,90,486,247]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"progression of preexisting scoliosis in non Chiari II pediatric patients with a Cobb angle $< 30^{\\circ}$ . Scoliosis correction is needless if the scoliosis stabilizes or improves after SCM extirpation. A single-staged surgery can be performed at surgeons' discretion on Type II SCM patients with only filum terminale tethering but no fibrous bands. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[89,254,485,289]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"6.7 Intraoperative neuroelectrophysiology monitoring ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[89,297,488,540]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Intraoperative neuro-electrophysiology monitoring (IONM) can ensure the safety of surgical practice and is highly recommended. Constant EMG monitoring can detect nerve irritation in real time, and intraoperative electrostimulation is suggested to distinguish the functional nerves from tethered bands and non-functional nerves. Alteration in latency and amplitude of somatosensory evoked potential (SEP) and motor evoked potential (MEP) are essential real-time parameters on examining sensorimotor pathway integrity. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[91,564,348,582]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"7 Prognosis and follow up ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[89,595,488,917]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Recommended early surgical intervention promises a satisfactory recovery, especially for patients presenting with mild symptoms or recent onset of symptom. However, those with severe long-term symptoms improve indistinctively after SCM extirpation [37]. Pain can be remarkably relieved, and other symptoms including leg weakness, sphincter dysfunction, scoliosis and leg deformity may be partially diminished or just remain unchanged. Importantly, surgery can only contribute to detethering the cord and preventing further injury, but cannot reverse the pre-existing malformation like leg atrophy due to asymmetrical hemicord. But delayed progression after surgery can also benefit young patients. Regular postoperative follow-up should include ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[508,90,906,492]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"MRI scan. The first-time postoperative follow-up needs to include the assessment on the extent of the septum extirpation, patency of the subarachnoid space and the extent of cord detethering, etc. Routine follow-ups should assess potential risks of retethering, the relapsing of syringomyelia or congenital tumors, which rely on not only MRI but also the combination of neuroimaging, history, symptoms and IONM [38]. Of note, leg deformities may be more prominent as children grow up. It can be the result of retethering or asymmetrical hemicord development, which can be distinguished by routine EMG and SEP test. Symptoms in retethering patients will deteriorate while it remained unchanged in patients with asymmetrical hemicords. A whole spine X-ray is required for SCM patients with scoliosis to assess the progression and necessity for scoliotic surgical intervention. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[509,519,694,535]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Conflict of interests ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[508,556,905,594]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"The authors declare that they have no conflict of interests. ","text_format":null,"text_level":null,"type":"text"},{"bbox":"[511,623,614,638]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"References ","text_format":null,"text_level":"1","type":"text"},{"bbox":"[509,655,905,910]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"[1] Saker E, Loukas M, Fisahn C, et al. Historical perspective of split cord malformations: A tale of two cords. Pediatr Neurosurg. 2017, 52(1): 1-5. \",\"[2] Babu R, Reynolds R, Moreno JR, et al. Concurrent split cord malformation and teratoma: Dysembryology, presentation, and treatment. J Clin Neurosci. 2014, 21(2): 212-6. \",\"[3] Bentley JF, Smith JR. Developmental posterior enteric remnants and spinal malformations: the split notochord syndrome. Arch Dis Child. 1960, 35: 76-86. \",\"[4] Pang D, Dias MS, Ahab-Barmada M. Split cord malformation: Part I: a unified theory of embryogenesis for double spinal cord malformations. Neurosurgery. 1992, 31(3): 451-480. \"]","page_idx":"6","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[91,44,297,61]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Journal of Neurorestoratology ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[880,45,901,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"89 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[93,939,120,959]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"# ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[122,940,217,950]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"TSINGHUA ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[125,950,218,957]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"6","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"UNIVERSITY PRESS ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[93,129,906,634]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":"images/0829/dfad4ec8df190b59afddabdf40a3e79dd6a60c479ec07b319873c3b97cd7155b.jpg","list_items":null,"page_idx":"7","sub_type":null,"table_body":"<table><tr><td>1</td><td>Nan Bo</td><td>Shanghai Children's Medical Center</td></tr><tr><td>2</td><td>Qian Chen</td><td>Shenzhen Children's Hospital</td></tr><tr><td>3</td><td>Ruoping Chen</td><td>Xinhua Hospital Affiliated to School of Medicine, Shanghai Jiaotong University</td></tr><tr><td>4</td><td>Shuo Gu</td><td>Hainan Women and Children's Medical Center</td></tr><tr><td>5</td><td>Xiaosheng He</td><td>Xijing Hospital of Air-force Military Medical University</td></tr><tr><td>6</td><td>Wen Jin</td><td>Shanxi Children's Hospital</td></tr><tr><td>7</td><td>Cuicui Li</td><td>The 7th Medical Center of PLA General Hospital</td></tr><tr><td>8</td><td>Fangcheng Li</td><td>Guangzhou Women and Children's Medical Center</td></tr><tr><td>9</td><td>Hao Li</td><td>Fudan University Children's Hospital</td></tr><tr><td>10</td><td>Fuyun Liu</td><td>Zhengzhou University the Third Hospital</td></tr><tr><td>11</td><td>Wei Liu</td><td>Beijing Tiantan Hospital, Capital Medical University</td></tr><tr><td>12</td><td>Lin Qi</td><td>Children's Hospital Affiliated to Zhengzhou University</td></tr><tr><td>13</td><td>Xiang Qi</td><td>Beijing Children's Hospital, Capital Medical University</td></tr><tr><td>14</td><td>Aijia Shang</td><td>The First Medical Center of PLA General Hospital</td></tr><tr><td>15</td><td>Zhipeng Shen</td><td>The Children's Hospital , Zhejiang University</td></tr><tr><td>16</td><td>Hangyu Shi</td><td>Xi'an Children's Hospital</td></tr><tr><td>17</td><td>Benzhang Tao</td><td>The First Medical Center of PLA General Hospital</td></tr><tr><td>18</td><td>Feng Wan</td><td>Tongji Hospital Affiliated to Huazhong University of Science & Technology</td></tr><tr><td>19</td><td>Guangyu Wang</td><td>Qilu Children's Hospital of Shandong University</td></tr><tr><td>20</td><td>Jvlei Wang</td><td>Tangdu Hospital of Air-force Military Medical University</td></tr><tr><td>21</td><td>Linlin Wang</td><td>Institute of Reproductive and Child Health, Peking University Health Science Center</td></tr><tr><td>22</td><td>Shuihua Wu</td><td>Hunan Children's Hospital</td></tr><tr><td>23</td><td>Bo Xiu</td><td>The 7th Medical Center of PLA General Hospital</td></tr><tr><td>24</td><td>Zhi Yuan</td><td>The Second Hospital of Lanzhou University</td></tr><tr><td>25</td><td>Xuan Zhai</td><td>The Children's Hospital, Chongqing Medical University</td></tr><tr><td>26</td><td>Rui Zhang</td><td>Shandong Provincial Hospital Affiliated to Shandong First Medical University</td></tr></table>","table_caption":"[\"Appendix 1 Participants of consensus on the diagnosis and treatment of SCM from Chinese Split Cord Malformation Working Group (In alphabetical order of expert's family name). \"]","table_footnote":"[]","text":null,"text_format":null,"text_level":null,"type":"table"},{"bbox":"[91,644,485,908]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"[5] Luo YG, Li W, Dong CQ, et al. The clinical characteristics and surgical treatment of split cord malformation in children (in Chinese). J Clin Ped Sur. 2013, 12(1): 31-35. \",\"[6] Sinha S, Agarwal D, Mahapatra AK. Split cord malformations: an experience of 203 cases. Childs Nerv Syst. 2006, 22(1): 3-7. \",\"[7] Feng F, Tan HN, Li XY, et al. Radiographic characteristics in congenital scoliosis associated with split cord malformation: a retrospective study of 266 surgical cases. BMC Musculoskelet Disord. 2017, 18(1): 420. \",\"[8] Xiu B. Common problems and countermeasures in the diagnoses and treatment of split cord malformations (in Chinese). J Clin Pediatr Surg. \"]","page_idx":"7","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[511,642,903,912]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"2019, 18(2): 84-87. \",\"[9] Lan BS, Wang KZ, Liu AQ, et al. Pathological factors and treatment of nerve injury of diastenatomyelia (in Chinese). J Clin Orth. 2002, 5(2): 94-97. \",\"[10] Meena RK, Doddamani RS, Gurjar HK, et al. Type 1.5 split cord malformations: An uncommon entity. World Neurosurg. 2020, 133: 142-149. \",\"[11] Salunke P, Kovai P, Malik V, et al. Mixed split cord malformation: are we missing something? Clin Neurol Neurosurg. 2011, 113(9): 774-778. \",\"[12] Mahapatra AK. Split cord malformation - A study of 300 cases at AIIMS 1990-2006. J Pediatr Neurosci. 2011, 6(Suppl 1): S41-S45. \",\"[13] Hu WM, Liu P, Liu FY, et al. Clinical analysis of 225 cases of split cord malformation in children (in \"]","page_idx":"7","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[94,45,117,58]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"90 ","text_format":null,"text_level":null,"type":"page_number"},{"bbox":"[697,45,903,60]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Journal of Neurorestoratology ","text_format":null,"text_level":null,"type":"header"},{"bbox":"[700,936,903,951]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":null,"page_idx":"7","sub_type":null,"table_body":null,"table_caption":null,"table_footnote":null,"text":"Journal of Neurorestoratology ","text_format":null,"text_level":null,"type":"footer"},{"bbox":"[93,89,486,913]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"Chinese). Chin J Pediatr Surg. 2018, 39(3): 204-208. \",\"[14] Xiu B. A practical classification of spinal neural tube defects and its clinical significance (in Chinese). Chin J Neurosurg Dis Res. 2017, 16(5): 393-396. \",\"[15] Pang D. Split cord malformation: Part II: Clinical syndrome. Neurosurgery. 1992, 31(3): 481-500. \",\"[16] Erşahin Y. Split cord malformation types I and II: a personal series of 131 patients. Childs Nerv Syst. 2013, 29(9): 1515-1526. \",\"[17] Rokos J. Pathogenesis of diastematomyelia and spina bifida. J Pathol. 1975, 117(3): 155-161. \",\"[18] Guthkelch AN. Diastematomyelia with median septum. Brain. 1974, 97: 729-742. \",\"[19] Feng F, Shen JX, Zhang JG, et al. Characteristics and clinical relevance of the osseous spur in patients with congenital scoliosis and split spinal cord malformation. J Bone Jo Surg. 2016, 98(24): 2096-2102. \",\"[20] Heemskerk JL, Kruyt MC, Colo D, et al. Prevalence and risk factors for neural axis anomalies in idiopathic scoliosis: a systematic review. Spine J. 2018, 18(7): 1261-1271. \",\"[21] Cheng B, Li Y, Wang KZ, et al. Single tubular type of dia stematomyelia a retrospective study of 23 cases (in Chinese). Chin J Orthop. 2003, 23(5): 275-278. \",\"[22] Liu FY, Chen XL. Diagnosis and treatment of split spinal cord malformations combined with congenital spinal deformity (in Chinese). Chin J Appl Clin Pediatr. 2016, 31(11): 805-809. \",\"[23] Kline-Fath BM, Calvo-Garcia MA, O'Hara SM, et al. Water imaging (hydrography) in the fetus: The value of a heavily T2-weighted sequence. Pediatr Radiol. 2007, 37(2): 133-140. \",\"[24] Xiu B. Attention should be paid to standardizing the classification of spinal canal malformation (in Chinese). Natl Med J China. 2017, 97(48): 3761-3762. \",\"[25] Liu FY, Luo XF, Xia B, et al. Clinical study of congenital scoliosis with intraspinal deformity in children (in Chinese). Orthopedic J Chin. 2012, 20(9): 789-791. \",\"[26] Jin HM, Sun LP, Bao N, et al. Diagnosis and treatment of diastematomyelain children (in Chinese). Chin J Pediatr Surg. 2003, 24(5): 430-432. \",\"[27] Shang AJ, Zhang YZ, Qiao GY, et al. Surgical treatment of split cord malformation (in Chinese). $J$ \"]","page_idx":"8","sub_type":"ref_text","table_body":null,"table_caption":null,"table_footnote":null,"text":null,"text_format":null,"text_level":null,"type":"list"},{"bbox":"[513,90,903,914]","code_body":null,"code_caption":null,"image_caption":null,"image_footnote":null,"img_path":null,"list_items":"[\"Chin PLA Postgrad Med Sch. 2010, 31(4): 307-308, 311. \",\"[28] Pang D. Ventral tethering in split cord malformation. Neurosurg Focus. 2001, 10(1): e6. \",\"[29] Liu FY, Xia B, Si PC, et al. Management of split cord malformations associated with other deformities in children (in Chinese). Orthopedic J China. 2006. 14(11): 808-811. \",\"[30] Barutcuoglu M, Selcuki M, Selcuki D, et al. Cutting filum terminale is very important in split cord malformation cases to achieve total release. Childs Nerv Syst. 2015, 31(3): 425-432. \",\"[31] Barutcuoglu M, Umur AS, Özdemir S, et al. Double split cord malformations in a child: Types I and II at 2 different levels in the same route: Report of a rare case. Neurosurg Q. 2015, 25(4): 495-498. \",\"[32] Lee GYF, Paradiso G, Tator CH, et al. Surgical management of tethered cord syndrome in adults: Indications, techniques, and long-term outcomes in 60 patients. J Neurosurg Spine. 2006, 4: 123-131. \",\"[33] Ozturk E, Sonmez G, Mutlu H, et al. Split-cord malformation and accompanying anomalies. J De Neuroradiol. 2008, 35(3): 150-156. \",\"[34] Shen JX, Zhang JG, Feng F, et al. Corrective surgery for congenital scoliosis associated with split cord malformation: It may be safe to leave diastematomyelia untreated in patients with intact or stable neurological status. J Bone Joint Surg Am. 2016, 98(11): 926-936. \",\"[35] Hui H, Tao HR, Jiang XF, et al. Safety and efficacy of 1-stage surgical treatment of congenital spinal deformity associated with split spinal cord malformation. Spine. 2012, 37(25): 2104-2113. \",\"[36] Ayvaz M, Akalan N, Yazici M, et al. Is it necessary to operate all split cord malformations before corrective surgery for patients with congenital spinal deformities? Spine. 2009, 34(22): 2413-2418. \",\"[37] Cui ZQ, Xiu B, Sun ZX, et al. Microsurgical treatment of type I diastematomyelia in 75 children (in Chinese). Chin J Neurosurg. 2012, 28(7): 710-713. \",\"